HIGH ACCURACY WIRELESS RANGING FOR PHASE ALIGNMENT IN DISTRIBUTED MICROWAVE BEAMFORMING ARRAYS By Sean Michael Ellison A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of Electrical Engineering – Doctor of Philosophy 2020 ABSTRACT HIGH ACCURACY WIRELESS RANGING FOR PHASE ALIGNMENT IN DISTRIBUTED MICROWAVE BEAMFORMING ARRAYS By Sean Michael Ellison In recent years, there has been an increasing interest in distributed antenna arrays due to their poten- tial to provide improvements in performance, scalability, robustness, and cost over classic phased antenna arrays. Distributed arrays synthesize a large aperture using small, low-cost, and low-power devices, supporting improvements that would otherwise be too costly or bulky to achieve in a sin- gle system. Such arrays also have the flexibility to be scaled by adding or removing elements from the array, depending on the application at hand. Within distributed antenna arrays, there are gener- ally two classes that are considered: incoherent distributed arrays, where little or no coordination is performed between nodes in the array, yielding a collection of individual wireless systems; and coherent distributed arrays, where element coordination is performed at the level of the radio fre- quency phase. While incoherent arrays are easier to implement, their improvements, such as gain and signal-to-noise ratio, generally scale only as the square-root of the number of elements, yield- ing diminishing returns. Coherent arrays achieve sensitivity improvements directly proportional to the number of elements in the array, yielding significant improvements as the array scales. How- ever, distributed coherence requires significantly more coordination between nodes. The electrical states that need to be aligned to enable coherent beamforming include: each device’s internal clock frequencies; relative timing of information symbols; and alignment of the beamforming phase. In general, there are two methods to achieve alignment: closed-loop and open-loop. Closed loop is only feasible to applications that have reliable feedback from the receive location, such as com- munications systems. Open-loop requires the nodes to coordinate without feedback, but opens the application space to instances where there is no feedback from the destination such as radar and remote sensing. In this work, I focus on the alignment of the phase of the beamforming signals in open- loop coherent distributed antenna arrays. I present a distributed antenna array supporting open- loop distributed beamforming at 1.5 GHz. Based on a scalable, high-accuracy internode ranging technique, I demonstrate open-loop beamforming experiments using three transmitting nodes. To support distributed beamforming without feedback from the destination, the relative positions of the nodes in the distributed array must be known with accuracies below λ 15 of the beamforming carrier frequency to ensure that the array maintains at least 90% coherent beamforming gain at the receive location. For operations at microwave frequencies, this leads to range estimation ac- curacies of centimeters or less. I present scalable, high-accuracy waveforms and new approaches to refine range measurements to significantly improve the estimation accuracy. Using one of the designed waveforms with a three-node array, I demonstrate high-accuracy ranging simultaneously between multiple nodes, from which phase corrections on two secondary nodes are implemented to maintain beamforming with the primary node, thereby supporting open-loop distributed beam- forming. Upon movement of the nodes, the range estimation is used to dynamically update the phase correction, maintaining beamforming as the nodes move. I show the first open-loop dis- tributed beamforming at 1.5 GHz with two-node and three-node arrays, demonstrating the ability to implement and maintain phase-based beamforming without feedback from the destination. TABLE OF CONTENTS LIST OF TABLES . . . LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii CHAPTER 1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 CHAPTER 2 RANGING WAVEFORM CLASSIFICATION METRICS AND CLAS- 2.1 Derivation of Waveform Estimation Characteristics SIC RANGING WAVEFORMS . . . . . . . . . . . . . . . . . . . . . . . . 16 . . . . . . . . . . . . . . . . . 16 2.1.1 Ambiguity Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.1.2 Cramer_Rao Lower Bound (CRLB) . . . . . . . . . . . . . . . . . . . . . 17 2.1.3 Relative Positional and Oscillator Phase Accuracy Requirements to Co- 2.1.4 herently Beamform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Signal-to-Noise Ratio (SNR) . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.2 Radar Ranging Waveforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.2.1 Linear Frequency Modulated Waveform (LFMW) . . . . . . . . . . . . . . 24 . . . . . . . . . . . . . . . . . . . . 27 2.2.2 2.2.3 . . . . . . . . . . . . . . . . . . . . 34 Stepped-Frequency Waveform (SFW) Pulsed Two-Tone Waveform (PTTW) CHAPTER 3 WAVEFORMS DESIGNED FOR HIGH-ACCURACY RANGING FOR DISTRIBUTED BEAMFORMING ARRAYS . . . . . . . . . . . . . . . . 40 3.1 Time Domain Duplex: Joint Range-Doppler Estimation Using Prolate Spheroidal . Wave Functions (PSWF) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.1.1 Theory of Prolate Spheroidal Wave Functions . . . . . . . . . . . . . . . . 41 3.1.2 Numerical Approximation of the PSWFs . . . . . . . . . . . . . . . . . . . 42 3.1.3 Range and Doppler Estimation Bounds . . . . . . . . . . . . . . . . . . . 44 3.1.4 Remote Sensing Waveform Design Based on PSWF of the Zeroth Order . . 45 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.1.4.1 3.1.4.2 Measurements 3.2 Time Domain Duplex: Joint Phase Alignment and Frequency Transfer for Wide- . . . . . . band Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.2.1 Waveform Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.2.1.1 Ambiguity Function . . . . . . . . . . . . . . . . . . . . . . . . 60 3.2.1.2 Range Estimation Lower Bounds . . . . . . . . . . . . . . . . . 74 Frequency Estimation Lower Bound . . . . . . . . . . . . . . . . 76 3.2.1.3 3.2.2 Experimental Setup and Signal Processing . . . . . . . . . . . . . . . . . . 77 3.2.2.1 Waveform Processing . . . . . . . . . . . . . . . . . . . . . . . 78 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . 79 3.2.2.2 . 84 . . . . . . . . . . . . . . . . . . . . . 84 IEEE 802.11 Standards and Legacy Preamble Format . . . . . . . . . . . . 85 3.3 Frequency Division: Phase Alignment for Networked Systems 3.3.1 Orthogonal Frequency Division Multiplexing (OFDM) 3.3.2 iv 3.3.3 Theoretical ranging Accuracy of the Legacy Preamble and the Stepped- Frequency Waveform . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4 Experimental Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 . 92 3.4 Pulse Encoding: Two-Tone Stepped Frequency Waveform (TTSFW) . . . . . . . . 96 Frequency Domain Multiplexing (FDM) Approach . . . . . . . . . . . . . 97 . 98 . 106 . 106 . . . . . . . . . . . . . . . . . 109 3.5 Comparison to Traditional Wideband Ranging Methods . . . . . . . . . . . . . . . 115 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3.1 Waveform Properties . . . . . . . . . . . . . . . . . . . . . . . 3.4.3.2 Wireless Ranging Measurements 3.4.1 3.4.2 Waveform Design . 3.4.3 Measurements . . . CHAPTER 4 DEMONSTRATION OF WIRELESS PHASE ALIGNMENT IN DIS- TRIBUTED BEAMFORMING . . . . . . . . . . . . . . . . . . . . . . . . 117 4.1 Range Estimation Waveform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 4.2 Ranging Requirements for Open-Loop Distributed Beamforming . . . . . . . . . . 118 4.3 Range Estimation Refinement for Node Motion . . . . . . . . . . . . . . . . . . . 120 4.4 Distributed Antenna Array and Open-Loop Distributed Beamforming Experiments 122 . 4.5 Two-Node Experiment . 124 . 127 4.6 Three-Node Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CHAPTER 5 APPLICATION OF PHASE ALIGNED COHERENT DISTRIBUTED . . . ARRAYS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 5.1 Antenna Array Dynamics and Impacts on Wireless Communication . . . . . . . . 130 5.1.1 Estimation of Bit-Error-Rate of Communications Signals . . . . . . . . . . 132 5.1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 5.2 Dynamic Distributed Antenna Array Design and implementation . . . . . . . . . . 135 5.2.1 Ranging Waveform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 5.2.2 Estimation Ability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 5.3 Experimental Validation of Secure Communication . . . . . . . . . . . . . . . . . 138 Simulation . . . . . CHAPTER 6 CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 APPENDIX A MATLAB FORMULATION OF BASEBAND PSWF . . . . . . . . . 147 APPENDIX B MATLAB FORMULATION OF DUAL PULSE PSWF . . . . . . . . 151 APPENDIX C MATLAB PSWF SIMULATION . . . . . . . . . . . . . . . . . . . . 157 APPENDIX D MATLAB PSWF MEASUREMENT PROCESSING . . . . . . . . . 164 APPENDIX E MATLAB NETWORK SYSTEM SIMULATION . . . . . . . . . . . 172 APPENDIX F MATLAB NETWORK SYSTEM MEASUREMENT PROCESSING . 175 APPENDIX G MATLAB TTSFW SIMULATION . . . . . . . . . . . . . . . . . . . 180 APPENDIX H MATLAB DISTRIBUTED BEAMFORMING MEASUREMENT PROCESSING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 APPENDIX I MATLAB SECURE TRANSMISSION SIMULATION FOR THE LINEAR CASE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 APPENDIX J MATLAB SECURE TRANSMISSION MEASUREMENT PROCESS- ING FOR THE LINEAR CASE . . . . . . . . . . . . . . . . . . . . 191 APPENDIX K MATLAB SECURE TRANSMISSION SIMULATION FOR THE SINUSOIDAL CASE . . . . . . . . . . . . . . . . . . . . . . . . . . 194 APPENDIX L MATLAB SECURE TRANSMISSION MEASUREMENT PROCESS- ING FOR THE SINUSOIDAL CASE . . . . . . . . . . . . . . . . . 198 BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 vi LIST OF TABLES Table 3.1: Processing Gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 Table 3.2: Comparison of discussed waveform attributes . . . . . . . . . . . . . . . . . . . 106 Table 3.3: Comparison of Waveform Performance . . . . . . . . . . . . . . . . . . . . . . 116 vii LIST OF FIGURES Figure 1.1: Relative gains that can be obtained through incoherent and coherent beam- forming methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Figure 1.2: Transmission from independent wireless systems, signal 1 and signal 2, adding destructively due to phase mismatch. This example is a worst case scenario of where signals cancel each other out. For in-phase operations, the individ- ual signal amplitudes sum, boosting the overall signal. . . . . . . . . . . . . . . Figure 1.3: Transmission from independent wireless systems, signal 1 and signal 2, with two separate reference frequencies. Here the combination of the two sig- nals creates an baseband modulation equivalent to the frequency difference. For operations at the same underlying frequency, a constant amplitude signal would be apparent. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 1.4: Transmission from independent wireless systems, signal 1 and signal 2, with time misalignment. The phase mismatches due to phase encoded data cre- ate an amplitude modulation to the signal and, at worst case, can result in destructive interference essentially nulling out the signal. For a time aligned signal, all information pulses should add constructively, essentially summing the individual signals. This is much like the phase alignment aspect but on a pulse by pulse basis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 1.5: Transmission from independent wireless systems, signal 1 and signal 2, pro- viding perfect ideal coherence by applying phase alignment, frequency syn- chronization, and time alignment. . . . . . . . . . . . . . . . . . . . . . . . . . Figure 1.6: Receiver-coordinated distributed transmit beamforming allows a network of transmitters to achieve longer communication ranges or higher data rates [1]. . Figure 1.7: Phase synchronization using receiver feedback based on the received SNR [2]. . . Figure 1.8: M-source distributed beamforming system model using round-rip synchro- nization [3]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 1.9: System model with an N-node transmit cluster and a single target node. Node 0 corresponding to the target and node 1 corresponding to the primary node are assumed to have a direct link to the transmit nodes [4]. . . . . . . . . . Figure 1.10: Communication model for a sensor network with minimal feedback from the target [5]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii 3 3 4 5 6 6 7 8 9 Figure 1.11: Coherent operation of radar sensor network through GPS synchronization [6]. . 11 Figure 1.12: Model for a sensor network with feedbackless coordination where θ1, θ2, and θ3 are the corresponding phase shifts provided from internode distances [7]. . . 11 Figure 1.13: Range estimation based phase alignment of a hierarchical primary-secondary topology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Figure 2.1: Degradation of coherent gain with relative oscillator phase errors between node internal oscillators [8]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Figure 2.2: Degradation of coherent gain with relative internode positional errors [8]. . . . . 22 Figure 2.3: (a) LFMW waveform in the time domain. (b) LFMW in the frequency domain. . 24 Figure 2.4: (a) Ambiguity function of the LFMW waveform. (b) Intensity plot of the ambiguity function. (c) LFMW matched filter (zero Doppler cut). (d) LFMW Doppler response (zero time cut). . . . . . . . . . . . . . . . . . . . . . . . . . 26 Figure 2.5: (a) Representation of the SFW with 50% duty cycle time domain. (b) SFW time-frequency plot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Figure 2.6: (a) Ambiguity function of the SFW. (b) Intensity plot of the ambiguity func- tion. (c) Matched filter of the SFW (zero Doppler cut). (d) Doppler response of SFW (zero time cut). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Figure 2.7: (a) The baseband PTTW in the time domain. (b) PTTW in the frequency domain. 35 Figure 2.8: (a) Ambiguity function of the PTTW. (b) Intensity plot of the ambiguity function. (c) Matched filter of the PTTW (zero Doppler cut). (d) Doppler response of the PTTW (zero time cut). . . . . . . . . . . . . . . . . . . . . . . 37 Figure 3.1: CRLB for Doppler (2.11) and range (2.12) of the baseband PSWFs for the cases of c = π (a), c = 4π (b), c = 7π (c), and c = 10π (d) on the right and left respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 Figure 3.2: Time domain representation of the temporally bifurcated waveform formed from concatenating two n = 0 PSWFs. This example was taken at c = 3π. . . . 48 Figure 3.3: Time and frequency bifurcated waveform with 100% (left) and 10% (right) time modulation and 100% (a), 50% (b), and 10% (c) modulation in frequency. . 49 ix Figure 3.4: (a) Comparison of the optimal CRLB to the CRLB of the spatial uncertainty of the dual pulse waveform with varying spectral bandwidth where the per- centage refers to 2W ∆f . (b) Comparison of the optimal CRLB to the CRLB of the frequency uncertainty of the dual pulse waveform with varying temporal instantaneous bandwidth where the percentage refers to 2T(cid:48) T . . . . . . . . . . . 50 Figure 3.5: (a) Time representation of the waveform with 10% modulation. (b) Spectral representation with 10% modulation. (c) Ambiguity function of the dual pulse PSWF. (d) Intensity plot of the ambiguity function showing ambiguous nature in both time and frequency. (e) Matched filter of the dual pulse PSWF (zero Doppler cut). (f) Doppler response of the dual pulse PSWF (zero time cut). 52 Figure 3.6: Pseudo-random white noise for disambiguation for 10% occupation of both time and frequency, relative to the measurement waveform (left), and 25% occupancy in time and frequency (right). (a) Pseudo-random white noise in the time domain. (b) Time-frequency plot of the pseudo-random white noise. (c) Matched filter response for the measurement waveform (blue) and disambiguation (red). (d) Doppler response of the measurement waveform (blue) and disambiguation (red). (e) Combined ambiguity function of mea- surement waveform and disambiguation demonstrating suppression of side- lobes in both time and Doppler. . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Figure 3.7: LFMW for disambiguation for 10% occupation of both time and frequency, relative to the measurement waveform (left), and 25% occupancy in time and frequency (right). (a) LFMW in the time domain. (b) Time-frequency plot of the LFMW. (c) Matched filter response for the measurement waveform (blue) and disambiguation (red). (d) Doppler response of the measurement waveform (blue) and disambiguation (red). (e) Combined ambiguity function of measurement waveform and disambiguation demonstrating suppression of sidelobes in both time and Doppler. . . . . . . . . . . . . . . . . . . . . . . . . 54 Figure 3.8: (a) Block diagram of the wireless experimental setup. (b) Image of experi- mental setup in the semi-enclosed arch range. . . . . . . . . . . . . . . . . . . 57 Figure 3.9: (a) Time-frequency plot of a measured waveform with T = 10 µs and W = 6 MHz. (b) Comparison of measured results, simulation, and CRLB for the dual pulse PSWF as a spatial estimator. (c) Comparison of measured results, simulation, and CRLB for the dual pulse PSWF as a frequency estimator. . . . . 58 x Figure 3.10: Ranging is based on widely-spaced two-tone signal with 100 MHz of band- width, with the lower tone modulated by a 10 MHz frequency reference. (a) Spectrum of the joint ranging and frequency transfer signal using single- sideband modulation. (b) Spectrum of the joint ranging and frequency trans- fer signal using double-sideband modulation. (c) Spectrum of the joint rang- ing and frequency transfer signal using amplitude modulation. . . . . . . . . . . 61 Figure 3.11: (a) Ambiguity function of the PTTW with single sideband modulation. (b) Intensity plot of the ambiguity function. (c) Matched filter of single sideband modulation (zero Doppler cut). (d) Doppler response of the single sideband modulation (zero time cut). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 Figure 3.12: (a) Ambiguity function of the PTTW with double sideband modulation. (b) Intensity plot of the ambiguity function. (c) Matched filter of double side- band modulation (zero Doppler cut). (d) Doppler response of the double . . . . . . . . . . . . . . . . . . . . . . . 69 sideband modulation (zero time cut). Figure 3.13: (a) Ambiguity function of the PTTW with amplitude modulation. (b) Inten- sity plot of the ambiguity function. (c) Matched filter of amplitude modula- tion (zero Doppler cut). (d) Doppler response of the amplitude modulation (zero time cut). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 3.14: (a) Image of the joint ranging and frequency transfer system experimental setup in an indoor wireless test range. The distance between the transceiver and the corner reflector was 4.57 m.(b) Block diagram of measurement sys- tem used in the experiments. The transmitter, represented by the Arbitrary Waveform Generator (AWG) generates the multi-tone signal, and the oscil- loscope samples the received signals. . . . . . . . . . . . . . . . . . . . . . . . 74 . 78 Figure 3.15: Measured delay estimation accuracy versus SNR with the frequency refer- ence included using (a) single-sideband, (b) double-sideband, and (c) ampli- tude modulation. The separation of the ranging tones is given in the legend, with the CRB for the 200 MHz tone separation from (3.46), (3.48), and (3.50) for (a), (b), and (c) respectively plotted as the solid line. . . . . . . . . . . . . . 81 Figure 3.16: Comparison of all three modulation formats showing consistent performance regardless of the modulation format. . . . . . . . . . . . . . . . . . . . . . . . 82 Figure 3.17: Variance on the estimate of the 10 MHz frequency reference versus SNR using (a) single-sideband modulation, (b) double-sideband modulation, and (c) amplitude modulation. The separation of the ranging tones is given in the legend. The red dashed line indicates the 18° requirement to ensure P (Gc ≥ 0.9) ≈ 1 for large arrays. The CRB given by (3.52) is shown in the solide line. . 83 xi Figure 3.18: Channel splitting using an FDM method where the sinc(·) representation comes from the Fourier of a rectangular time domain envelope. The spac- ing of each channel is equivalent to 1/T to ensure that the peak is at a null of all of the other sinc(·) functions. . . . . . . . . . . . . . . . . . . . . . . . . . 85 Figure 3.19: Visualization of Legacy preamble. . . . . . . . . . . . . . . . . . . . . . . . . 86 Figure 3.20: (a) Ambiguity function of the Legacy preamble. (b) Intensity plot of the ambiguity function. (c) Matched filter of the Legacy preamble (zero Doppler cut). (d) Doppler response of the Legacy preamble (zero time cut). . . . . . . . 89 Figure 3.21: Time-frequency plot of the Legacy preamble training field and SFW with a 50% duty cycle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 Figure 3.22: (a) Simulated IEEE 802.11n waveform with 20 MHz bandwidth with pay- load (blue) and L-STF preamble (red). (b) SFW with Tr = 8 µs with a 50% duty cycle and containing the 12 data sub-carriers present in the L-STF. (c) Spectrum of both the 802.11n L-STF and the SFW. (d) Normalized zero time delay matched filter output for both L-STF and SFW demonstrating the similarities in the temporal waveform shapes. . . . . . . . . . . . . . . . . . . 93 Figure 3.23: (a) Measured IEEE 802.11ac waveform with 160 MHz bandwidth. (b) Mea- sured spectrum of the waveform. (c) Measured matched filter output. . . . . . . 95 Figure 3.24: (a) Block diagram of the ranging experiment (AWG = arbitrary waveform generator, LNA = low-noise amplifier). The corner reflector was placed a distance of 1.5 m from the ranging system. Delay estimation processing was implemented offline in MATLAB. (b) Image of experimental setup up in the semi-enclosed arch range. (c) Measurement variances along with simulated data for 1000 Monte Carlo iterations over various SNR values. The simulated performance of the SFW and Legacy preamble waveforms yielded nearly identical delay estimation, verifying the theoretical similarities of the lower bounds described earlier. The measured performance of the Legacy preamble waveform was close to the simulated performance and comparable to the lower bound (CRLB). The Legacy preamble waveform achieved a lowest ranging error of 1.9mm with a preprocessing SNR of 19 dB combined with 31 dB of processing gain for a total SNR of 50 dB. . . . . . . . . . . . . . . . 96 Figure 3.25: Channel splitting using an FDM method for a two-tone system where half of the bandwidth is allocated to the first tone, − ∆f 2 , of the N channels and the second half is a allocated to the upper tone, ∆f 2 . . . . . . . . . . . . . . . . . . 98 Figure 3.26: (a) Two-tone stepped-frequency waveform in the time domain. (a) TTSFW in the time-frequency domain . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 xii Figure 3.27: (a) Ambiguity function of the TTSFW. (b) Intensity plot of the ambiguity function. (c) Matched filter of the TTSFW (zero Doppler cut). (d) Doppler response of TTSFW (zero time cut). . . . . . . . . . . . . . . . . . . . . . . . 103 Figure 3.28: (a) The bounds and simulation of TTSFW vs. number of pulse with 4 MHz of bandwidth total bandwidth and 30 dB preprocessing SNR. (b) The bounds and simulation of TTSFW vs. SNR with the pulse number fixed to N = 4. . . . 106 Figure 3.29: (a) Measured two pulse TTSFW in time domain. (b) Two pulse TTSFW in the frequency domain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 Figure 3.30: Measured matched filter output of the PTTW and TTSFW waveforms pro- duced on the X310 using two pulses . . . . . . . . . . . . . . . . . . . . . . . 107 Figure 3.31: Comparison of variance of PTTW and TTSFW with equal ∆f to the CRLB of the TTSFW. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 Figure 3.32: (a) Measured four pulse TTSFW in time domain. (b) Four pulse TTSFW in the frequency domain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 Figure 3.33: Measured matched filter output of the PTTW and TTSFW waveforms pro- duced on the X310 using four pulses. . . . . . . . . . . . . . . . . . . . . . . . 110 Figure 3.34: (a) Schematic of single SDR in the arch range. (b) Image of experimental setup. (c) Range Measurement results. (d) Variance of measurement. . . . . . . 111 Figure 3.35: (a) Schematic of two SDRs in the arch range. (b) Image of experimental setup. (c) Range Measurement results. (d) Variance of measurement. . . . . . . 113 Figure 3.36: (a) Schematic of three SDRs in the arch range. (b) Image of experimental setup. (c) Range Measurement results. (d) Variance of measurement. . . . . . . 114 Figure 4.1: Measured waveform supporting ranging between two nodes: (a) time do- main; (b) frequency domain. . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 Figure 4.2: Measured waveform supporting ranging between five nodes: (a) time do- main; (b) frequency domain. . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 Figure 4.3: Block diagram of the Kalman filter used for range estimation refinement with moving nodes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 Figure 4.4: Block diagrams of the nodes. . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Figure 4.5: RF chain internal to each of the X310 SDRs. . . . . . . . . . . . . . . . . . . . 124 xiii Figure 4.6: (a) Block diagram of the distributed two-node experiment. (b) Image of the experimental setup in the semi-enclosed arch range. (c) Measured results of coherent gain with and without performing range-based phase correction. . . . . 126 Figure 4.7: (a) Block diagram of the distributed three-node experiment. (b) Image of the experimental setup in the semi-enclosed arch range. (c) Measured results of coherent gain with and without performing range-based phase correction. . . . . 128 Figure 5.1: Flight paths are taken over a 1 m extent at a 1.5 GHz beamforming frequency assuming a dipole transmitter and broadside beamforming direction. (a) Lin- ear flight path (left) and resulting array radiation pattern over time (right). (b) Sinusoidal flight path (left) and resulting array radiation pattern over time (right).131 Figure 5.2: Comparison of BER performance of a static case of two dipoles at 1 m sep- aration, and the estimated performance of the two designated flight paths at 1.5 GHz and 12 dB of SNR. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 (a) Block diagram of the simulation. (b) Simulated BER over −90° to 90° domain compared to calculated estimation for the linear case. The snapshots of data show the 100 points of demodulated data (blue) and the location of the constellation points (red) at angles −20° and 0°. . . . . . . . . . . . . . . . 135 Figure 5.3: Figure 5.4: Method of performing alignment of electrical states of time, through high amplitude preamble, and phase, range estimation between primary and sec- ondary elements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 . . Figure 5.5: (a) Ranging waveform in the time domain. frequency domain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 (b) Ranging waveform in the Figure 5.6: Block diagram of the experimental setup . . . . . . . . . . . . . . . . . . . . . 139 Figure 5.7: (b) (a) Image of the experimental setup in the semi-enclosed arch range. Image of the antenna setup for ranging and beamforming on each element. (c) Farthest separation of the elements mounted on the linear actuator. (d) Closest separation of the elements mounted on the linear actuator. . . . . . . . . 140 Figure 5.8: (a) Comparison of simulated and actual flight path for the linear case. (b) Comparison of simulated and actual flight path for the sinusoidal case. . . . . . 141 Figure 5.9: (a) Comparison of simulated, measured, and calculated estimation of the BER for the linear case. (b) Comparison of simulated, measured, and cal- culated estimation of the BER for the sinusoidal case. . . . . . . . . . . . . . . 143 xiv CHAPTER 1 INTRODUCTION Advancements in the capabilities of wireless systems strongly depend on the achievable gain, power, and resolution of the system, and the ability to scale these metrics. With traditional single- platform systems, improving such aspects for radar, remote sensing, and communications requires modifications of the devices in the system, the system efficiency, or the antenna aperture size, all of which can represent significant cost drivers. To overcome the challenge of continually upgrading single-system capabilities, there is growing interest in distributed antenna array systems, where collections of small, low power, and relatively inexpensive wireless systems are coordinated to mimic the performance of a single, large system, thereby increasing the capabilities of the overall array system through array gain and/or larger array aperture area to achieve performance other- wise unattainable with a single platform [8–10]. To address this challenge, recent research has focused on the development of distributed wireless technologies implemented in Multiple Input Multiple Output (MIMO) [11–14] or distributed beamforming [5, 15–17] applications. Such dis- tributed wireless systems enable direct performance scalability by adding or removing inexpensive nodes from the array [18,19]. Applications of such disaggregated arrays include small satellites or hybrid satellite-terrestrial cooperative systems [20], cubesat swarm operations for remote sensing, drone constellations for soil moisture measurements, and distributed communications for greater throughput and higher-reliability connections [21], among others. Distributed arrays that operate in an incoherent sense have the benefit of requiring min- imal coordination to align the electrical states of the elements [22–25]. In essence, each element acts as its own entity with minimal cooperation within the system as a whole. Coherent distributed arrays are a specific subset of distributed arrays in which platforms are coordinated at the level of radio frequency (RF) phase to enable phased-array beamforming [5, 8, 26–28]. Here the elements act as a single system rather than independent entities seen in the incoherent method. While re- quiring additional overhead to perform coherence operations, the array gain scales proportionally 1 Figure 1.1: Relative gains that can be obtained through incoherent and coherent beamforming methods. to the number of elements within the array, much like classic single platform systems, whereas the gain of the incoherent methodology scales proportionally to the square-root of the number of elements (see Fig. 1.1). This is due to the phase of coherent signals adding constructively while incoherent signals add with a random phase at the risk of destructive interference. Therefore, co- herent operations allow for significantly improved signal strength enabling long range connections to be made and higher obtainable signal-to-noise ratios (SNR) thereby improving signal reliability. To support distributed coherent operations: • The wireless systems on each platform must be phase-aligned at the level of the RF beam- forming frequency phase to ensure that signals add constructively [29, 30]. A system in which the independent signals are not phase aligned can cause significant degradation in the coherent gain (see Fig. 1.2). 2 Figure 1.2: Transmission from independent wireless systems, signal 1 and signal 2, adding de- structively due to phase mismatch. This example is a worst case scenario of where signals cancel each other out. For in-phase operations, the individual signal amplitudes sum, boosting the overall signal. • Frequency synchronization must be implemented to insure all elements are operating at the same reference frequency [31–34]. A shift in the reference frequency between elements re- sults in a frequency shift of the derived signals. This is important since the internal oscillators on separate devices in the distributed array will drift over time if left unsynchronized. This drift results in significant errors in information demodulation for communication systems and estimation errors for remote sensing and radar applications (see Fig. 1.3). Figure 1.3: Transmission from independent wireless systems, signal 1 and signal 2, with two separate reference frequencies. Here the combination of the two signals creates an baseband mod- ulation equivalent to the frequency difference. For operations at the same underlying frequency, a constant amplitude signal would be apparent. 3 • Time alignment between nodes must be implemented so that symbol information has suffi- cient overlap at the target destination [35–41]. This aspect of coordination is required for ap- plications transmitting pulsed information such as symbols in communications waveforms. Mismatched time can result in phase interference between pulses making demodulation of information increasingly difficult (see Fig. 1.4). Figure 1.4: Transmission from independent wireless systems, signal 1 and signal 2, with time misalignment. The phase mismatches due to phase encoded data create an amplitude modulation to the signal and, at worst case, can result in destructive interference essentially nulling out the signal. For a time aligned signal, all information pulses should add constructively, essentially summing the individual signals. This is much like the phase alignment aspect but on a pulse by pulse basis. Only after these aspects of coordination are addressed can coherent beamforming be accomplished. Note that time alignment is only applicable to pulsed information systems and can be neglected for continuous wave transmission signals. An image of ideal coherent beamforming completing all three tasks of alignment can be seen in Fig. 1.5. Of these three coordination aspects, phase alignment is the more challenging task due to the high-precision required to align the carrier phases. Previous works have explored a closed- loop methodology where the task of phase alignment is performed using feedback from the target location. In this formulation, information is relayed back to the array, indicating what phase cor- rections need to be made to enable coherence [1–4, 42–48]. Among these closed-loop approaches are: 4 Figure 1.5: Transmission from independent wireless systems, signal 1 and signal 2, providing per- fect ideal coherence by applying phase alignment, frequency synchronization, and time alignment. 1. Receiver-coordinated explicit-feedback in which carrier phase and frequency synchroniza- tion is performed with unmodulated beacon signals on each node (see Fig. 1.6). A full duplexed method of exploring separate frequency bands for beacons and beamforming car- rier [43] proved to be effective for time-invariant and time-varying single-path channel en- vironments but degrades significantly in the presence of multipath. A similar method was investigated using a half duplex system where a single frequency is used for beacons and the beamforming carrier [1]. Here a time-slotted round-trip carrier synchronization proto- col is implemented which has the benefit of channel reciprocity in a multipath environment and proved to be effective in time-invariant and time-variant channels as well as multipath environments. 5 Figure 1.6: Receiver-coordinated distributed transmit beamforming allows a network of transmit- ters to achieve longer communication ranges or higher data rates [1]. Figure 1.7: Phase synchronization using receiver feedback based on the received SNR [2]. 2. A phase alignment method where the phase of each node is randomly updated and the receiver responds with a single bit indicating the impact to the received SNR (see Fig. 1.7) [2,44]. If the phase perturbation has a negative impact, the phase is reverted to the state prior to the adjustment; if the SNR is improved or unaffected then the new phase correction is kept. This algorithm for phase alignment is scalable and converges over time. It proves to be effective for static systems with linear channels. Improvements on this method using 3-bit feedback in each timeslot [45] conveys the direction of the motion of the receiver relative to the transmitter in addition to the received signal strength extending this methodology to time-variant systems. 3. A method of round-trip synchronization (see Fig. 1.8) [3, 46] where a time slotted approach for system beacons and beamforming carrier signal utilize the same frequency. In the first time slot of this approach the base station transmits a carrier beacon to all of the elements in 6 Figure 1.8: M-source distributed beamforming system model using round-rip synchronization [3]. the array where each element independently calculates the appropriate carrier phase offset as well as a reference frequency shift from this beacon signal. The subsequent time slots consist of elements within the array transmitting their relative beacons in which the rest of elements calculate local frequency and phase offsets creating a fully connected array. The final time slot is beamforming back to the base station. This method presents a solution applicable to time-invariant, time-variant, and multipath environments as well as arrays con- sisting of mobile nodes, but suffers from serious scalability constraints due to the increasing amount of overhead with array size. Some of the overhead constraints could potentially be alleviated using beacons on multiple frequency bands [43] but estimations will suffer due to the inapplicability of channel reciprocity. 4. A time slotted approach for a retrodirective array that corrects for time-varying phase and frequency offset. A broadcast from the primary node is sent to all the secondary nodes in the array with a message that contains the best estimate of the complex channel state based on previous estimations [4, 47, 48]. The secondary nodes then respond individually with updated predictions of future channel estimations provided from an extended Kalman filter 7 Figure 1.9: System model with an N-node transmit cluster and a single target node. Node 0 corresponding to the target and node 1 corresponding to the primary node are assumed to have a direct link to the transmit nodes [4]. (EKF) [4, 48]. This method uses minimal communication with a base station where the ma- jority of the coordination is done by the array elements themselves. However, this approach to a retrodirective array presents limited scalability due to the increased update wait time needed from the primary node to receive a response from each individual secondary node and required upwards of 1 s to correct errors due to EKF estimation presenting limitations of extended beamforming operations. 5. A method of using an element hierarchical primary-secondary structure to perform synchro- nization requiring minimal feedback from the target destination (see Fig. 1.10) [5, 49]. This method uses a timeslotted approach in which signal transmissions are interleaved with bea- con broadcasts, consisting of the beamforming carrier, sent from the primary element to all the subsequent secondary elements where the beamforming center frequency is shared by the beacon to ensure channel reciprocity. The secondary nodes, after individual communication with the primary node, can then perform a channel estimation based on a preamble response that precedes the beacon message and the reciprocal can be applied to the baseband signal to ensure a coherent output. This method presents issues with scalability, especially for mobile 8 Figure 1.10: Communication model for a sensor network with minimal feedback from the target [5]. systems, due to the increasing amount of overhead with array size due to the time duplexed nature of this process. This results in a high probability of element channels changing before the transmission is sent stemming from either the physical motion of the platforms or refer- ence clock drift over a period of time leading the channel estimation to no longer become accurate. Although these feedback methods are effective, they are restricted to applications where reliable feedback can be provided by the receiver. Numerous situations arise where such feedback is not available, particularly in cases where individual nodes in the array do not have sufficient sensitivity to close a link to a base station on their own. Furthermore, closed-loop architectures are inherently unable to support wireless applications beyond communications, such as remote sensing, imaging, and radar where coherent feedback is generally not present. The work presented here focuses on an open-loop array where no feedback from the des- tination is assumed [5–8, 49–51]. Due to this lack of feedback, the elements of the array must self align their electrical states [8,52]. Methodologies utilizing closed-loop and retrodirective architec- tures do not inherently provide node localization due to the cooperative nature of these systems. For arbitrary beamforming it is necessary to obtain locations of nodes as the interference of the 9 beamforming phase will depend on spatial position. Although open-loop arrays are a considerably more daunting task to develop than closed-loop approaches, there are huge versatility benefits due to the ability to arbitrarily steer beams. Using open-loop concepts can extend the application space to scattering problems as well as communications but, due to the challenging nature of this prob- lem, this method is less investigated. The current methodologies of enabling open-loop coherent distributed arrays (CDAs) include: 1. Leveraging the global positioning systems (GPS) for estimations in position and frequency reference (see Fig. 1.11) [6, 50, 51]. This method obtains position to typical trilateration operations with accuracies on the order of 10 cm and a frequency reference from the pulse- per-second (PPS) signal that has timing accuracies of approximately 10 ns. However, the information accuracy provided by GPS is far too coarse to enable coherent beamforming in the microwave region and also only provides a solution when reliable GPS signals can be obtained. 2. An alternate method of using a primary-secondary methodology where the primary node transmits a beacon signal consisting of the carrier and timing reference. The secondary nodes, with knowledge of internode distance to the primary node, can then account for the delays between said nodes with the appropriate timing and phase shift [7, 8]. The primary node can then send another trigger signal to initiate secondary nodes to begin transmis- sion with the appropriate baseband modulation to enable coherent beamforming. While [7] presents an in-depth analysis of this operation, the method at which the internode distances are obtain is not presented and is validated through simulation alone. In [8] proof of concept of this method is experimentally validated using a single primary and secondary node but the issue of scalability based on this technique is not expressed. From these works it can be seen that the two most fundamental aspects to enable coherent open-loop CDAs are the alignment of the phase of the beamforming signal and a stable frequency reference to which all of the elements can align to. Of these two, phase alignment presents the more 10 Figure 1.11: Coherent operation of radar sensor network through GPS synchronization [6]. Figure 1.12: Model for a sensor network with feedbackless coordination where θ1, θ2, and θ3 are the corresponding phase shifts provided from internode distances [7]. 11 challenging problem due to the accuracies needed to enable beamforming in the millimeter wave and microwave regions. To achieve phase alignment and implement a phase-based beamsteering operation, the relative positions of the individual nodes must be known to within a fraction of the wavelength of the beamforming frequency. Previous works have shown that these internode rang- ing measurements must have accuracies of less than λ/15 to have no more than 0.5 dB reduction in coherent gain with a probability of 90% [8]. This calls for accuracies on a sub-centimeter level for millimeter wave and microwave operations. Furthermore, this accuracy must be obtained before nodes move out of coherence due to array dynamics. This has been achieved in the past using optical systems providing estimates of element locations [53, 54]. However, such systems are not easily scalable and require accurate tracking and pointing for each node connection. This dissertation focuses on a method to perform phase alignment that is more con- ducive to scalability, system implementation, and cost than optical tracking by using microwave radar responses to estimate internode localization. For airborne CDAs, a microwave system also presents a solution to environmental challenges, such as cloud cover or fog, that optical systems cannot overcome due to the significant attenuation at optical frequencies. Classic microwave tech- niques of obtaining range estimations, through time of flight of a return signal, are often opti- mized through transmit waveform design. The most common radar ranging waveforms used in practice include linear frequency modulated waveforms (LFMW) [55–58] and stepped frequency waveforms (SFW) [59–62]. Waveform characteristics and estimation abilities for the LFMW and SFW are discussed in-depth in Sections 2.2.1 and 2.2.2 respectively. These waveforms are com- monly used due to their ability to unambiguously estimate range, but are not designed for optimal range estimation. Previous works have shown that spectrally-sparse waveforms can achieve near- optimal ranging accuracy but come at the cost of estimation ambiguity [8]. Therefore, waveform designs optimized for range estimation, based on spectrally-sparse signals, that address the scala- bility needs for open-loop CDAs are derived and discussed in Chapter 3 where methods to address the challenge of measurement ambiguity are discussed as well. The overall system structure used in this work follows the hierarchical primary-secondary 12 Figure 1.13: Range estimation based phase alignment of a hierarchical primary-secondary topol- ogy. scheme seen in [5, 7, 49] due to the potential for scalability. Beamforming from the primary node sets a reference point in space to which the subsequent secondary node(s) can estimate their rel- ative distance. With this knowledge, the secondary nodes can adjust the phase of their respective beamforming signals to account for the proportional phase shift from primary to secondary node (see Fig. 1.13) where the beamforming frequency and angle to the target destination are assumed to be known a priori. This requires all of the secondary nodes to perform their estimation sig- nal processing independently and therefore work on a pairwise basis with the single primary node improving the robustness and scalability. This method is more conducive to the addition or subtrac- tion of secondary nodes due to desired design constraints or node failure. To avoid the bottleneck to scalability using this approach seen in previous works, different frequency bands are utilized for the range estimation and beamforming where beamforming is typically the lower of the two due to limitations on measurement accuracies that can be achieved by practical systems. By using differ- ent frequency bands, timeslotting can be avoided, resulting in a full duplex system. A full duplexed system allows for constant range measurements to be taken, and therefore continual updates to the phase of the beamforming signal, enabling greater scalability and faster array dynamics of mobile platform by reducing the rate at which nodes move out of coherent range before the next update is given. This becomes particularity important for platforms that undergo quick relative motion, due to designated flight paths, obstacle avoidance, system vibration, or turbulence. Mobile arrays that are able to quickly change element positions will aid in applications such as synthetic aperture radar (SAR) imaging [63–66] and interferometric imaging [67, 68] where a variety of electrical baselines are needed to accommodate image resolution. 13 To demonstrate the feasibility and scalability of this concept, wireless experimental vali- dation of a phase coherent distributed array with more than two nodes is shown in Chapter 4 using an estimation waveform derived in Chapter 3. Nodes undergo relative motion thereby demon- strating the ability of the system to quickly implement phase corrections for mobile systems, en- abling applications such as airborne and spaceborne radar, remote sensing, and imaging. Using this method, beamforming in the L-band region is shown to be obtainable with no more than 12.5 MHz of bandwidth allocated to the range estimation waveform, making applications such as L-band SAR for vegetation and forestry measurements obtainable using off-the-shelf equipment. To expand upon these applications, a method for using array motion to increase wireless security is derived and experimentally validated in Chapter 5. This is made possible by the change of array electrical baselines resulting from array dynamics, thus changing the locations in space of con- structive interference while maintaining coherence at the desired target destination. This method can be used in remote sensing applications as well, but is demonstrated through a communication signal because the performance of signal coherence over various angles can easily be observed from the bit-error-ratio (BER). The principal contributions of this work include the following: • New derivations and experimental evaluations of novel ranging waveforms that can easily be scaled to accommodate a large collection of nodes. The scalability methods that are investi- gated include time domain duxplexing, frequency division multiplexing, and pulse encoding. The waveforms that are discussed include: a waveform obtaining near-optimal range and ve- locity estimation using an optimally approximate time- and band-limited baseband envelope; a waveform to perform joint ranging and frequency transfer based on a spectrally sparse sig- nal model; a method of using a communications preamble that is common to many IEEE standards to perform ranging and communication within the same frequency band without interfering with the information throughput; and a highly scalable pulse encoded spectrally sparse waveform with inherent disambiguation potential. 14 • Experimental demonstration of phase alignment supporting up to 120 nodes with two and three transmitting nodes in a dynamic open-loop distributed beamforming array with greater than 90% coherent gain. This is performed using the primary-secondary hierarchical topol- ogy where phase compensation comes from a range estimations provided by the waveforms derived in this work. This method is directly scalable to larger arrays. • An experimental evaluation of a novel approach to secure wireless operations using a two- element dynamic open-loop CDA with designated flight paths utilizing the beamforming approach described in this work. Methods in the literature, such as parasitic arrays [69–71], periodically switched arrays [72–74], and directional modulation [75–78], classically trade transmission power for the added benefit of security. This method of using CDAs can provide optimal power transfer as well as operational security. A wireless beamforming experiment is proven to have approximately 80% maximum power transfer, which can be improved upon by allocating more bandwidth to the estimation process or alternate flight paths, while providing high levels of confusion at off angle directions. 15 CHAPTER 2 RANGING WAVEFORM CLASSIFICATION METRICS AND CLASSIC RANGING WAVEFORMS In this chapter I discuss common metrics for evaluating the performance of radar signals, namely the ambiguity function which shows a received signal response to time delay and a Doppler fre- quency shift and the Cramer-Rao lower bound which defines the ideal variance that can be obtained by a signal for an unknown variable. These metrics are evaluated for common waveforms used in practice, and are then used to compare the performance of different waveforms in the following chapter for the purpose of enabling phase alignment of distributed beamforming. 2.1 Derivation of Waveform Estimation Characteristics 2.1.1 Ambiguity Function The response to a waveform in both time and frequency is given by the ambiguity function [79, 80]. The ambiguity function represented by the cross-correlation of the transmitted signal with a Doppler shifted version. The ambiguity function is given by (cid:90) ∞ −∞ s∗(τ − t)s(τ )ej2πfDτ dτ AF (t, fD) = (2.1) where s(t) is the time domain representation of the waveform and fD is the Doppler shift. The matched filter, equivalent to the autocorrelation or zero Doppler cut of the ambiguity function, is used to measure the response in time. The matched filter is an optimal linear filter that maxi- mizes the SNR of the measurement by correlating the transmitted and received signals providing an additional processing gain equivalent to the time-bandwidth product, T BWr, where T is the signal time duration and BWr is the receiver bandwidth representing a maximum measurement processing gain. The zero time cut, t = 0, shows the waveforms response to Doppler. The ideal variance of the main peaks of both the matched filter and Doppler cuts of the ambiguity function are equivalent to the ideal Cramer-Rao Lower Bound (CRLB). 16 2.1.2 Cramer_Rao Lower Bound (CRLB) Often in remote sensing applications it is desirable to estimate both time or frequency of a measure- ment as these metrics provide information about the behavior of a target. A time delay measure- ment gives range information where a frequency shift provides radial velocity through Doppler. The theoretical accuracy of these measurements as an estimator depends on the signal variance which is inversely proportional to the Fisher Information [81]. The Fisher Information is a mea- sure of the amount of information that can be extracted from a signal. This performance limitation is given by the CRLB. The CRLB, which is defined by the noise terms, gives a theoretical lim- itation on the amount of information that can be extracted from a signal but does not provide a method in which to design a waveform to achieve this theoretical bound. The general form of a radar signal can be expressed as Sr(x) = αs(x; u) + w(x) (2.2) where w(x) is assumed to be additive Gaussian white noise (AWGN), α is the complex coefficient containing amplitude and phase information, and u is the parameter to be estimated which for this work is either velocity, given through Doppler, or position, given through the proportional time delay. For an unknown parameter u and a deterministic α, the CRLB can be derived as [82] s(x; u)dx (2.3) (cid:32)(cid:90) (cid:12)(cid:12)(cid:12)(cid:12)∂s(x; u) ∂u (cid:12)(cid:12)(cid:12)(cid:12)2 σ2(ˆu − u) ≥ N0 2|α|2 (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ∂s(x; u)∗ ∂u dx − 1 Es (cid:12)(cid:12)(cid:12)(cid:12)2(cid:33)−1 where Es is the signal energy and N0 2 is the noise power spectral density. For an estimation of a return signal time delay, which gives information about the radial distance, can be expressed in its general form from (2.2) as sr(t) = αg(t − τ ) + w(t) (2.4) where τ is the delay and g(t) is the transmitted signal. The variance can then be solved from (2.3) with the model (2.4) where s(x; u) = g(t − τ ). The first integral in (2.3) can be solved using Parseval’s theorem while the second integral can be solved using Plancherel’s theorem, also known as Rayleigh’s theory, which states the squared modulus of a function is equal to the integral of the 17 squared modulus of it’s spectrum. This corresponds the Parseval’s theorem for conservation of energy of a Fourier series. The first integral of (2.3) can be solved for by (cid:90) (cid:12)(cid:12)(cid:12)(cid:12)∂s(x; u) ∂u (cid:12)(cid:12)(cid:12)(cid:12)2 (cid:12)(cid:12)(cid:12)(cid:12)2 (cid:90) (cid:12)(cid:12)(cid:12)(cid:12) ∂ (cid:90) (cid:12)(cid:12)(cid:12)(cid:12) ∂ (cid:90) ∂τ g(t − τ ) dt G(f )e−j2πf τ ∂τ (2πf )2 |G(f )|2 df dx = = = = ζ2 f (cid:12)(cid:12)(cid:12)(cid:12)2 df (2.5) where ζ2 f is the mean-squared bandwidth which is equivalent to the second moment of the fre- quency spectrum. A similar formation can be made for the second integral in (2.3) (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) (cid:18) ∂s(x; u) ∂u (cid:19)∗ s(x; u)dx g(t − τ ) g(t − τ )dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) (cid:18) ∂ (cid:18)(cid:90) (cid:12)(cid:12)(cid:12)(cid:12)2 = = ∂τ 2πf |G(f )|2 df (cid:19)∗ (cid:19)2 (cid:12)(cid:12)(cid:12)(cid:12)2 (2.6) where µ2 f is the mean frequency equivalent to the first moment of the frequency spectrum. This = µ2 f results in an uncertainty of a radar time delay estimator of 2|α|2(cid:16) τ ≥ σ2 N0 f − 1 ζ2 Es µ2 f (cid:17) (2.7) An estimation of a return signal radial velocity can be expressed in its general with inbound velocity v (cid:28) c where c is the speed of light from (2.2) as sr(t) = αg(t)ej2πfDt + w(t) (2.8) where fD = 2 fcv c and fc is the radio frequency (RF) center frequency. The variance can then be solved from (2.3) with the model (2.8) where s(x; u) = g(t)ej2πfDt. The first integral of (2.3) can 18 be solved for by (cid:90) (cid:12)(cid:12)(cid:12)(cid:12)∂s(x; u) ∂u (cid:12)(cid:12)(cid:12)(cid:12)2 dx = = = (cid:90) (cid:12)(cid:12)(cid:12)(cid:12) ∂ (cid:90) (cid:12)(cid:12)(cid:12)(cid:12) ∂ (cid:90) ∂fD g(t)ej2πfDt ∂fD (2πt)2 |g(t)|2 dt (cid:12)(cid:12)(cid:12)(cid:12)2 (cid:12)(cid:12)(cid:12)(cid:12)2 G(f − fD) df dt (2.9) = ζ2 t where ζ2 t is the mean-squared time duration equivalent to the second moment of the time spectrum. The second integral of (2.3) can be solved by (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) (cid:18) ∂s(x; u) ∂u (cid:19)∗ s(x; u)dx (cid:12)(cid:12)(cid:12)(cid:12)2 = = (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) (cid:18) ∂ (cid:18)(cid:90) (cid:19)∗ (cid:19)2 G(f − fD) ∂fD 2πt|g(t)|2dt (cid:12)(cid:12)(cid:12)(cid:12)2 G(f − fD)df (2.10) = µ2 t where µ2 t is the mean time duration corollary to the first moment of the time domain. The uncer- tainty of a radar waveform as a frequency estimator is then 2|α|2(cid:16) ≥ σ2 fD N0 t − 1 ζ2 Es µ2 t (cid:17) (2.11) Therefore, the measurement ability of a radar waveform depends on the first and second moments of the Fourier of the domain of interest. From the temporal and spectral uncertainties the estimation of radial distance and velocity can be derived. The spatial estimation is given from the uncertainty of the time delay (2.7) as x ≥ c2 σ2 4 σ2 τ (2.12) where σ2 x is the variance over space and the factor of four comes from the two way propagation seen by typical radar measurements. The estimation of velocity can be expressed from the spectral uncertainty (2.11) as v ≥ c2 σ2 4f 2 c σ2 fD (2.13) where σ2 v is the variance in radial velocity. 19 2.1.3 Relative Positional and Oscillator Phase Accuracy Requirements to Coherently Beam- form In a coherent distributed transmit operation, the set of nodes in the array transmits a waveform with the appropriate relative phasing and timing such that the transmitted waveforms arrive at the target in-phase and time aligned, ensuring that the waveforms add constructively at the destination. While there are many drivers of coordination error in open-loop distributed arrays, here I focus on two of the most prominent, which are errors in the measurement of the distances between each node pair and locking the oscillators to the same frequency. Both ensure that the phase errors are sufficiently low to enable high-gain distributed beamforming. For a distributed array of N arbitrarily-placed transmitting nodes emitting continuous-wave signals to a given direction, the signal incident at a point in the far-field of each individual node can be given by [8] N(cid:88) sr(t) = hnαn (t) ej(2πf t+φn+φs,n) (2.14) n=1 where N is the number of elements in the array, hn is the channel response between the target and node n, αn is the amplitude, and φn is the phase error, given by φn = 2π λ (dn + δdn) cos(θn) + δφc (2.15) where δdn is the error in the antenna separation measurement, θn is the beamsteering angle relative to the platform orientation, and δφc is the relative phase error of the oscillator. With perfect alignment and position measurements, the ideal received signal is N(cid:88) (cid:16) (cid:17) si(t) = C 2πf t+ 2π λ dncosθn j e . (2.16) To evaluate the effect of the antenna separation and oscillator phase error terms, the received signal n=1 power of (2.14) relative to (2.16), given by Gc = |srs∗ r| (cid:12)(cid:12)sis∗ (cid:12)(cid:12) , i (2.17) is evaluated. The standard deviations of the error terms (phase error σφ and internode range error σd) are varied, and the probability that the signal power exceeds a given threshold P (Gc ≥ X), 20 Figure 2.1: Degradation of coherent gain with relative oscillator phase errors between node internal oscillators [8]. where 0 ≤ X ≤ 1 is the fraction of the ideal coherent signal power gain, is determined through Monte Carlo simulation. In this work I evaluate the probability that the received signal exceed 0.9 of the ideal gain (X = 0.9), corresponding to a coherent gain degradation of 0.5 dB. In Fig. 2.1 the probability of the received signal power exceeding 0.9 is shown for errors in the oscillator phase σφ, corollary to the square-root of the CRLB of frequency in (2.11), for coherent distributed arrays consisting of N = 2, 3, 5, 10, 20, 100, and 1000 nodes. As the number of nodes in the system increases, the transition area where P (Gc ≥ 0.9) = 1 becomes sharper, converging to a point below which P (Gc ≥ 0.9) is approximately 1, and above which is zero. The area where P (Gc ≥ 0.9) = 1 also increases as the number of platforms increases due to the decreasing variance of the received signal power as the number of elements in the network increases. As N → ∞, this cutoff error below which P (Gc ≥ 0.9) ≈ 1 approaches 18°. The oscillator phase error represents an aggregate error which may be due to a number of factors, but 21 0.9)00.10.20.30.40.50.60.70.80.91N = 2N = 3N = 5N = 10N = 20N = 100N = 1000 Figure 2.2: Degradation of coherent gain with relative internode positional errors [8]. the most prominent is phase migration due to frequency drift between the platform oscillators. If the oscillators are locked continuously, phase migration is minimal since the phase error output of a phase-locked loop (PLL) is theoretically zero; in this case the PLL noise may contribute as a factor in the error. In such cases the relative frequency drift contributes as the largest error driver. In Fig. 2.2 the probability of the received signal power exceeding 0.9 is shown for errors in the inter-node range measurement σd, corollary to the square-root of the positional CRLB from (2.12). A similar trend with increasing array size is seen. As N → ∞, this cutoff error below which P (Gc ≥ 0.9) ≈ 1 approaches λ 15. Note that this does not match the relative phase error of 18°, which would correlate to a phase error of λ 20, because the pointing angle θn is also a random variable, and at some angles the relative range error does not produce significant phase errors in (2.15). For example, if θn = 0°, the relative range error can be arbitrary, without increasing the total phase error while at other angles, such as end-fire θn = 90°, a more stringent approximation of λ 20 is required due to direct impact that position has on the total phase error. 22 00.020.040.060.080.10.120.140.160.180.2 0.9)N = 2N = 3N = 5N = 10N = 20N = 100N = 1000 2.1.4 Signal-to-Noise Ratio (SNR) Determining the parametric performance of the system required an estimate of the received SNR, which is estimated in this work using an eigenvalue decomposition approach [83]. The benefit of this approach is that it can directly estimate the signal power and the noise power without needing to distinguish the individual frequencies within a signal. SNR is equivalent to 2|α|2Es N0 equations in (2.7) and (2.11). The received signal can be represented by the sampled matrix is the CRLB  X = χ1,1 χ1,2 . . . χ1,L χ2,1 χ2,2 ... ... . . . χ2,L ... ... χN,1 χN,2 . . . χN,L  (2.18) (2.20) (2.21) (2.22) where N is the total number of samples per capture and L is then total number of signal observa- tions. From this the covariance matrix can be computed as Rx = 1 N XXH (2.19) where XH is the Hermitian of the matrix X. Once the covariance matrix is calculated, the eigenvalues λl are calculated using singular- value decomposition, and the resulting eigenvalues are rank ordered from largest to smallest. Be- cause the various tones in the modulated signal are generated by the same system, they are corre- lated, yielding a single eigenvalue λ1 which is the largest eigenvalue for non-negative SNR. The remaining λ2 − λL eigenvalues represent the noise, and therefore the average noise power level can be estimated by L(cid:88) l=2 λl γ2 = 1 L − 1 The signal power level then is calculated using Ps = λ1 − γ2 L from which the SNR can be obtained using (2.20) and (2.21) by (cid:18) Ps (cid:19) γ2 SNRdB = 10 log10 23 (a) (b) Figure 2.3: (a) LFMW waveform in the time domain. (b) LFMW in the frequency domain. 2.2 Radar Ranging Waveforms 2.2.1 Linear Frequency Modulated Waveform (LFMW) One of the most widely used ranging waveforms is the LFMWW [55–58] due to its ability to achieve high range resolutions while providing an unambiguous estimation of the time delay. This waveform is used as a comparison to quantify the quality of the designed waveforms in the fol- lowing sections and chapters. A the general form of a LFMWW utilizes a single frequency that sweeps linearly over a given time and bandwidth, given by (cid:18) t (cid:19) s(t) = 1√ T rect T ejπkt2 (2.23) where T is the time duration and k is modulation rate given by k = BW T , where BW is the sweep bandwidth. An image of the time and frequency domain representations of an LFMW can be seen in Fig. 2.3. The range and Doppler response of the LFMW can be characterized through the ambi- guity function. Inputting the time domain representation of the LFMW (2.23) in the ambiguity function (2.1) AF (t, fD) = (cid:90) ∞ −∞ rect 1 T (cid:18) τ − t (cid:19) T (cid:16) τ (cid:17) T e−jπk(τ−t)2 rect 24 ejπkτ 2 ej2πfDτ dτ (2.24) Truncation of the integral can be performed due to rect(·) being a time limited function with the bounds [− T 2 ]. After combining exponential terms 2 , T AF (t, fD) = 1 T − T 2 e−jπkt2 e−jπk(t2−2τ t)ej2πfDτ dτ (cid:16) (cid:34) j2π(kt+fD) e t+ T 2 (cid:17) − e−jπ(kt+fD)T (cid:35) (2.25) = j2T π (kt + fD) (cid:90) t+ T 2 Extracting a factor of ejπ(kt+fD)t from the function so that function in the brackets contains a complex exponential subtracted by its conjugate AF (t, fD) = e−jπkt2 j2T π (kt + fD) ejπ(kt+fD)t (cid:104) ejπ(kt+fD)(t+T ) − e−jπ(kt+fD)(t+T )(cid:105) A sine function can now be formed from the complex exponentials where sin(θ) = 1 j2 AF (t, fD) = e−jπkt2 ejπ(kt+fD)t T π (kt + fD) sin π (kt + fD) (t + T ) A sinc(·) function can now be formed from the sin(·) function by multiplying by t+T AF (t, fD) = e−jπkt2 ejπ(kt+fD)t + 1 sinc π (kt + fD) (t + T ) This derivation shown above represents the case of t < 0 case but is symmetric for the case of t > 0. The general case of the derivation can be represented by AF (t, fD) = e−jπkt2 ejπ(kt+fD)t sinc π (kt + fD) (T − |t|) The magnitude of the ambiguity function can can now be taken and like terms can be combined (cid:19) 1 − |t| (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) (cid:16) (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) |AF (t, fD)| = ejπ(kt+fD)t(cid:12)(cid:12)(cid:12) = 1. An image of the ambiguity function of the LFMW can be seen (2.30) sinc T π (kt + fD) (T − |t|) resulting in [80] (cid:12)(cid:12)(cid:12)e−jπkt2 where in Fig. 2.4. 25 (cid:16) (cid:16) (cid:16) (cid:19) (cid:18) t T (cid:18) 1 − |t| (cid:19) T (cid:16) (2.26) ejθ − e−jθ(cid:17) (2.27) (2.28) (cid:17) t+T (cid:17) (cid:17) (2.29) (a) (c) (b) (d) Figure 2.4: (a) Ambiguity function of the LFMW waveform. (b) Intensity plot of the ambiguity function. (c) LFMW matched filter (zero Doppler cut). (d) LFMW Doppler response (zero time cut). The measurement ability of an LFMW as a positional estimator, by (2.12), or a velocity estimator through Doppler, (2.13), is characterized by the first and second moment of the fre- quency and time spectrums respectively. This waveform inherently has a zero first moment in both domains, µf = µt = 0, so therefore the estimation ability relies solely on the second moment. The second moment of the frequency spectrum can be calculated by approximating the LFMW bandwidth as a rectangular function. The normalized second moment of the frequency spectrum, 26 where the normalization factor is the signal energy, is given by Applying the limits of integration as [− BW 2 , BW BW df BW df (cid:17)(cid:12)(cid:12)(cid:12)2 (cid:82) (2πf )2|S(f )|2df (cid:82) |S(f )|2df (cid:82) (2πf )2(cid:12)(cid:12)(cid:12)rect (cid:16) f (cid:82)(cid:12)(cid:12)(cid:12)rect (cid:17)(cid:12)(cid:12)(cid:12)2 (cid:16) f 2 ] bounded by the rect(·) function (cid:82) BW/2 (cid:82) BW/2 (cid:20)(cid:16) BW (cid:104) BW 3 (cid:17)3 −(cid:16)− BW (cid:105) −BW/2(2πf )2df BW 2 + BW 2 (cid:17)3(cid:21) −BW/2 df 2 2 4π2 3 4π2 3 4 BW π2BW 2 3 ζ2 f = = ζ2 f = = = = (2.31) (2.32) (2.33) This result can be seen in [80]. This gives a positional uncertainty of 4(cid:0)π2BW 2(cid:1) SNR 3c2 x ≥ σ2 A similar argument can be made for the second moment of the time domain due to rectangular envelope used to modulate this waveform providing an uncertainty in velocity of (cid:0)π2T 2(cid:1) SNR 3c2 v ≥ σ2 4f 2 c (2.34) 2.2.2 Stepped-Frequency Waveform (SFW) The stepped-frequency waveform (SFW) was designed as a pulse compression method that achieves effective wideband measurements through several consecutive pulses with narrow instantaneous bandwidths [59–62]. This waveform can be thought of a discrete LFMW and is also capable of 27 (a) (b) Figure 2.5: (a) Representation of the SFW with 50% duty cycle time domain. (b) SFW time- frequency plot. achieving high range resolution. This is performed by having each consecutive narrowband pulse increase its frequency by δf and can be modeled as N−1 2(cid:88) (cid:18)t − nTr (cid:19) T ej2πnδf t (2.35) s(t) = 1√ N rect n=− N−1 2 where δf is the discrete frequency step between each pulse, N is the number of pulses, T is the pulse duration, and nTr is the center time of each pulse. An image of the waveform model, with a 50% temporal duty cycle, in both the time domain and the time-frequency domain can be seen in Fig. 2.5. There are two main advantages to using a SFW: the range resolution is increased while maintaining narrow instantaneous bandwidth [84], and, while not completely unambiguous like the LFMW, the next highest sidelobe of the matched filter is shifted by t = 1 δf . The ambiguity function can be solved for by inputting the time domain representation of the SFW from (2.35) in the ambiguity function equation (2.1) AF (t, fD) = 1 N (cid:90) ∞ −∞ N−1 2(cid:88) rect n=− N−1 2 × rect (cid:18) τ − t − nTr (cid:19) (cid:18) τ − nTr T T 28 (cid:19) e−j2πnδf (τ−t) (2.36) ej2πnδf τ ej2πfDτ dτ The summation and integral are linear processes therefore the integration can be brought inside the summation AF (t, fD) = 1 N N−1 2(cid:88) n=− N−1 2 (cid:90) ∞ −∞ rect (cid:18) τ − t − nTr (cid:19) T (cid:18) τ − nTr (cid:19) T rect ej2πnδf t ej2πfDτ dτ A substitution can be made such that ˆτ = τ − nTr AF (t, fD) = 1 N ej2πnδf t (cid:90) ∞ −∞ rect (cid:18) ˆτ − t T (cid:19) rect (cid:18) ˆτ T (cid:19) ej2πfD(ˆτ +nTr)dˆτ (2.37) (2.38) Due to the time limited nature of the rect(·) function, the integral is bounded by [− T 2 , T 2 ] AF (t, fD) = 1 N = 1 N ej2πn(δf t+fDTr) ej2πn(δf t+fDTr) ej2πfD ˆτ dˆτ (cid:32) (cid:17) (cid:16) t+ T 2 j2πfD e − e −j2πfD T 2 (cid:33)(cid:35) (2.39) N−1 2(cid:88) n=− N−1 2 N−1 2(cid:88) 2(cid:88) 2 N−1 n=− N−1 n=− N−1 2 2 (cid:90) t+ T (cid:34) − T 2 1 j2πfD Extracting a factor of ejπfDt so the function in the parenthesis contains a complex exponential subtracted by its conjugate N−1 2(cid:88) AF (t, fD) = 1 N n=− N−1 2 (cid:34) ejπfDt j2πfD ej2πn(δf t+fDTr) ejπfD(t+T ) − e−jπfD(t+T )(cid:17)(cid:35) (cid:16) (cid:16) (2.40) ejθ − e−jθ(cid:17) (cid:34) (cid:16) (cid:17)(cid:35) A sine function can now be formed from the complex exponentials where sin(θ) = 1 j2 N−1 2(cid:88) n=− N−1 2 AF (t, fD) = 1 N ej2πn(δf t+fDTr) ejπfDt πfD sin πfD(t + T ) By multiplying by a factor of t+T N−1 2(cid:88) t+T , a sinc(·) function can now be formed (cid:16) ej2πn(δf t+fDTr)(cid:104) ejπfDt (t + T ) sinc (cid:17)(cid:105) πfD(t + T ) AF (t, fD) = 1 N n=− N−1 2 29 (2.41) (2.42) This derivation shown above represents the case of t < 0 case but is symmetric for the case of (cid:104) 1 N t > 0. The general case of the derivation can be represented by (cid:16) (cid:17)(cid:105) N−1 2(cid:88) AF (t, fD) = ejπfDt (T − |t|) sinc πfD (T − |t|) n=− N−1 2 The sum in (3.59) can be evaluated by first expanding the summation Ssum = ej2πn(δf t+fDTr) N−1 2(cid:88) n=− N−1 −j2π N−1 2 = e 2 (δf t+fDTr)... + 1... + ej2π N−1 2 (δf t+fDTr) A secondary summation can be formed by multiplying (2.44) with ej2π(δf t+fDTr) Ssumej2π(δf t+fDTr) = e −j2π N−3 2 (δf t+fDTr)... + 1... + ej2π N +1 2 (δf t+fDTr) ej2πn(δf t+fDTr) (2.43) (2.44) (2.45) Subtracting the terms in (2.44) from (2.45) Ssumej2π(δf t+fDTr) − Ssum = ejπ(N +1)(δf t+fDTr) − e−jπ(N−1)(δf t+fDTr) (2.46) Ssum can now be solved for Ssum = ejπ(N +1)(δf t+fDTr) − e−jπ(N−1)(δf t+fDTr) ej2π(δf t+fDTr) − 1 (2.47) A factor of ejπ(δf t+fDTr) can be extracted from both the numerator and denominator. Sine terms can the be formed on the numerator and denominator where sin(θ) = 1 j2 Ssum = ejπ(δf t+fDTr) ejπ(δf t+fDTr) ejπN(δf t+fDTr) − e−jπN(δf t+fDTr) ejπ(δf t+fDTr) − e−jπ(δf t+fDTr) (cid:16) (cid:16) πN (δf t + fDTr) sin sin π (δf t + fDTr) Ssum = Plugging (3.60) back into (3.59) AF (t, fD) = ejπfDt (T − |t|) sinc 1 N (cid:16) πfD (T − |t|) 30 πN (δf t + fDTr) π (δf t + fDTr) (cid:17) (cid:17) (cid:16) (cid:17)sin (cid:16) sin (cid:16) (2.48) ejθ − e−jθ(cid:17) (2.49) (2.50) (cid:17) (cid:17) The magnitude can now be taken where |ejπfDt|= 1 resulting in [85] |AF (t, fD)| = (T − |t|) sinc πfD (T − |t|) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) 1 N (cid:16) (cid:17)sin (cid:16) (cid:16) πN (δf t + fDTr) sin π (δf t + fDTr) (cid:17) (cid:17) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (2.51) A plot of the SFW ambiguity function can be see in Fig. 2.6. When looking at the fd = 0 cut, this function is maximized at every t = n at t = 1 δf and therefore the next highest sidelobe is N δf , it can now easily be seen that the lobe beamwidth is inversely proportional to the number of pulses N. The δf . Considering the nulls of the zero Doppler cut, which occurs at t = ± 1 disambiguation properties of this waveform can clearly be seen to be inversely proportional to the frequency step δf. Therefore large frequency steps result in unambiguous measurements but this requires a large amount of bandwidth. The estimation ability of this waveform is again derived from the temporal and spectral uncertainty in (2.7) and (2.11). The first moment of this waveform, like the LFMW, is funda- mentally zero, µt = µf = 0, and therefore can be neglected in both calculations resulting in the dictation of the CRLB calculation stemming from the second moment of the Fourier of the domain of interest. For the calculation of ζ2 f an approximation is made such that each pulse consists of a delta function in the frequency domain at each of the underlying pulse modulation frequencies such that ζ2 f = = (cid:82) (2πf )2|S(f )|2df (cid:82) |S(f )|2df 2(cid:80) N−1 n=− N−1 2 N−1 2(cid:80) n=− N−1 2 (cid:82) (2πf )2 |δ (f − nδf )|2 df (cid:82) |δ (f − nδf )|2 df (2.52) Due the fundamental property of delta functions being zero outside of the specified shifted value, 31 (a) (c) (b) (d) Figure 2.6: (a) Ambiguity function of the SFW. (b) Intensity plot of the ambiguity function. (c) Matched filter of the SFW (zero Doppler cut). (d) Doppler response of SFW (zero time cut). the inner product resulting from the squared terms, can be neglected reducing this calculation to 4π2 (nδf )2 n=− N−1  N−1  2(cid:80) 12N(cid:0)N 2 − 1(cid:1)(cid:105) 4π2δf 2(cid:104) 1 π2δf 2(cid:0)N 2 − 1(cid:1) 2 N N ζ2 f = = = 4(cid:2)π2δf 2(cid:0)N 2 − 1(cid:1)(cid:3) SNR 3c2 x ≥ σ2 32 3 Resulting in a spatial uncertainty from (2.12) of (2.53) (2.54) which provides the same accuracy of the LFMW for δf √ N 2 − 1 = BW where BW is the LFMW sweep bandwidth. The spectral uncertainty can be calculated from the second moment of the rectangular pulse train used as a baseband modulation for this waveform such that Applying the limits [nTr − T 2 , nTr + T Resulting in an uncertainty in the velocity from (2.13) of (cid:82) (2πt)2|S(t)|2dt (cid:82) |S(t)|2dt 2(cid:80) N−1 n=− N−1 2 N−1 2(cid:80) ζ2 t = = 2 T T dt dt dt (2πt)2dt 2 nTr− T 2 2 nTr− T 2 (cid:82) (2πt)2(cid:12)(cid:12)(cid:12)rect (cid:17)(cid:12)(cid:12)(cid:12)2 (cid:16) t−nTr (cid:17)(cid:12)(cid:12)(cid:12)2 (cid:82)(cid:12)(cid:12)(cid:12)rect (cid:16) t−nTr n=− N−1 2 ] due to rect(·) being a time limited function (cid:82) nTr+ T (cid:82) nTr+ T (cid:20)(cid:16) (cid:104)(cid:16) 4π2(cid:104) 12N(cid:0)N 2 − 1(cid:1)(cid:105) (cid:104) 1 + T 2(cid:105) (cid:17) c π2(cid:2)T 2 (cid:0)N 2 − 1(cid:1) + T 2(cid:3) SNR (cid:17)3 −(cid:16) (cid:17) −(cid:16) (cid:105) (cid:17)3(cid:21) (cid:17)(cid:105) r T + T 3 nTr − T nTr − T + N T 3 4 nTr + T 2 nTr + T 2 N 2 − 1 3n2T 2 (cid:111) N T N T c2 2 2 4 N−1 2(cid:80) n=− N−1 2 N−1 2(cid:80) n=− N−1 N−1 2 2(cid:80) 4π2 n=− N−1 2 N−1 2(cid:80) ζ2 t = = n=− N−1 N−1 2 n=− N−1 2 = 2(cid:80) 4π2(cid:110) = π2(cid:104) = T 2 r 3T 2 r T (cid:16) v ≥ σ2 4f 2 r 33 (2.55) (2.56) (2.57) which decreases the uncertainty from a single rectangular pulse of duration T by spreading the temporal energy nonuniformly over the the time domain if the duty cycle is less that 100%. 2.2.3 Pulsed Two-Tone Waveform (PTTW) The optimal performance of a time delay estimator maximizes the second moment of the frequency spectrum given by a waveform consisting of two frequencies at the opposite edges of the band of interest. In this case continuous wave waveform is needed due to the representation in the Fourier domain which results a delta function at the designated frequencies, yielding a perfect two-tone signal. In practice, it is necessary that the signal is time-limited due to system constraints. To assess the effects of this time limitation, a two-tone waveform modulated by a square temporal pulse is analyzed here. In the frequency domain, this creates sinc(·) functions at the designated two frequencies whose bandwidth is inversely proportional to the pulse duration. This results in a decrease of the mean squared bandwidth and in turn raises the minimum obtainable variance. As long as the pulse bandwidth is small compared to the frequency separation of the two tones, this reduction in accuracy has little effect on the variance. The time domain representation of a pulsed two-tone waveform (PTTW) which can be (cid:18) t (cid:19)(cid:16) T e−jπ∆f t + ejπ∆f t(cid:17) expressed as 1 2 rect s(t) = (2.58) where rect(·) is the rectangular pulse function, T is the pulse duration, and ∆f is the equivalent to the spectral separation between the two tones (i.e. f2 − f1) where f2 and f1 are the higher and lower frequency tones respectively. An image of an example of the PTTW can be seen in the time and frequency domains can be seen in Fig. 2.7. Here a baseband representation of the PTTW is represented therefore only the beat frequency between the two frequencies are present. If this waveform were to be upconverted to a carrier, a modulated sinusoid would be created. The range and Doppler response of the PTTW can again be observed through inputting 34 (a) (b) Figure 2.7: (a) The baseband PTTW in the time domain. (b) PTTW in the frequency domain. the time domain representation of the PTTW from (2.58) into the ambiguity function (2.1) AF (t, fD) = 1 4 (cid:90) ∞ −∞ rect × rect (cid:18) τ − t (cid:19)(cid:16) ejπ∆f (τ−t) + e−jπ∆f (τ−t)(cid:17) e−jπ∆f τ + ejπ∆f τ(cid:17) (cid:17)(cid:16) (cid:16) τ ej2πfDτ dτ T T (2.59) Truncation of the integral can be performed due to rect(·) being a time limited function with the bounds [− T 2 , T 2 ] (cid:90) t+ T (cid:16) e−jπ∆f t + ejπ∆f t + ejπ∆f (2τ−t) + e−jπ∆f (2τ−t)(cid:17) ej2πfDτ dτ AF (t, fD) = = 2 1 4 e−jπ∆f t + ejπ∆f t − T 2 (cid:34) (cid:16) (cid:17) j2πfD e t+ T 2 j8πfD (cid:35) − e−jπfDT (cid:34) (cid:34) (cid:16) (cid:16) j2π(fD−∆f) e j2π(∆f +fD) e t+ T 2 t+ T 2 e−jπ∆f t j8π (∆f + fD) ejπ∆f t j8π (fD − ∆f ) + + (cid:17) (cid:17) − e−jπ(∆f +fD)T − e−jπ(fD−∆f)T (cid:35) (cid:35) (2.60) Extracting a factor from each function so that every parenthesis contains a complex exponential 35 subtracted by its conjugate AF (t, fD) = + j8πfD e−jπ∆f tejπ(∆f +fD)t j8π (∆f + fD) ejπ∆f tejπ(fD−∆f)t j8π (fD − ∆f ) + (cid:16) ejπfDt e−jπ∆f t + ejπ∆f t(cid:17) (cid:104) ejπfD(t+T ) − e−jπfD(t+T )(cid:105) ejπ(∆f +fD)(t+T ) − e−jπ(∆f +fD)(t+T )(cid:105) (cid:104) ejπ(fD−∆f)(t+T ) − e−jπ(fD−∆f)(t+T )(cid:105) (cid:104) (cid:16) (cid:16) e−jπ∆f t + ejπ∆f t(cid:17) ejθ − e−jθ(cid:17) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) π (∆f + fD) (t + T ) π (fD − ∆f ) (t + T ) ejπfDt (t + T ) sinc πfD(t + T ) (t + T ) sinc (t + T ) sinc (cid:16) AF (t, fD) = 4 + + ejπfDt 4 ejπfDt 4 ejπ∆f t cancels in the second and third terms and sinc functions can now be formed by multiplying each function by t+T t+T such that sinc(θ) = 1 j2θ (2.61) (2.62) The terms ejπfDt represents the case of t < 0 case but is symmetric for the case of t > 0. The general case of the and (t + T ) can then be extracted from each term. This derivation shown above 4 derivation can be represented by AF (t, fD) = ejπfDt (T − |t|) 4 |AF (t, fD)| = (cid:12)(cid:12)(cid:12)(cid:12)(T − |t|) (cid:12)(cid:12)(cid:12)ejπfDt(cid:12)(cid:12)(cid:12) = 1 and 4 where the (cid:16) sinc + sinc + sinc (2.63) (cid:104)(cid:16) e−jπ∆f t + ejπ∆f t(cid:17) (cid:17) (cid:16) (cid:17) πfD (T − |t|) (cid:16) (cid:17)(cid:105) π (∆f + fD) (T − |t|) π (fD − ∆f ) (T − |t|) (cid:16) π (∆f + fD) (T − |t|) π (fD − ∆f ) (T − |t|) (cid:104) (cid:12)(cid:12)(cid:12)e−jπ∆f t + ejπ∆f t(cid:12)(cid:12)(cid:12) = 2. An image of the ambiguity function for (cid:17) (cid:17)(cid:105)(cid:12)(cid:12)(cid:12)(cid:12) πfD (T − |t|) (2.64) + sinc + sinc 2 sinc (cid:16) (cid:17) (cid:16) The magnitude of the ambiguity function can can now be taken and like terms can be combined a PTTW can be seen in Fig. 2.8. From this image, it can be seen that there is a repeated lobing pattern in the time domain. This arises from the beat frequency between − ∆f 2 and ∆f 2 . In the 36 (a) (c) (b) (d) Figure 2.8: (a) Ambiguity function of the PTTW. (b) Intensity plot of the ambiguity function. (c) Matched filter of the PTTW (zero Doppler cut). (d) Doppler response of the PTTW (zero time cut). frequency domain the response is a sinc(·) function which stems from the overlaying rectangular pulse modulation in the time domain. While having the benefit of increased measurement accuracy, a PTTW waveform has a drawback which is that the measurement is highly ambiguous due to the large number of lobes in the temporal response. It is typically challenging to track the correct peak lobe especially in the presence of noise. Previous works have used an interweaved wideband pulse with bandwidth equal to ∆f 4 to have effective peak tracking [86]. The uncertainty of the time and Doppler estimation are again measured through the tem- 37 poral and spectral uncertainty of the matched filter and Doppler cuts of the ambiguity function giving a measure of the stability of this waveform on a measurement by measurement basis. The calculation on of the CRLB, resulting from the first and second moment of the Fourier of the do- main of interest, can be formulated solely from the second moment due to the PTTW inherently having a zero first moment µt = µf = 0. An approximation can be made for calculating the second moment of the of the frequency domain by using delta functions at each frequency − ∆f 2 and ∆f 2 where the assumption is made that the individual pulse bandwidth 1 T is much less than the overall bandwidth ∆f. This insures that the modulation bandwidth stemming from the time limited aspect of the waveform has negligible degradation to the overall performance of the optimal continuous wave case. Calculation of the normalized mean squared bandwidth can be made using (2.5) (cid:82) (2πf )2|S(f )|2df (cid:82) |S(f )|2df (cid:82) (2πf )2(cid:12)(cid:12)(cid:12)δ (cid:16) (cid:82)(cid:12)(cid:12)(cid:12)δ (cid:17) (cid:16) (cid:20)(cid:16) ∆f (cid:16) ∆f (cid:17)2 f + ∆f 2 f + ∆f 2 4π2 2 + 2 (cid:17) (cid:16) (cid:17)2(cid:21) + δ (1 + 1) ζ2 f = = = (cid:16) f − ∆f 2 (cid:17)(cid:12)(cid:12)(cid:12)2 + δ f − ∆f 2 (cid:17)(cid:12)(cid:12)(cid:12)2 df df (2.65) = π2∆f 2 The mean-squared bandwidth is maximized when the bandwidth, ∆f, is also maximized and thus providing the minimum variance. This is the driving motivation behind spectrally sparse wave- forms as a method for ranging. This results in a spatial uncertainty from (2.12) of [86] 4(cid:0)π2∆f 2(cid:1) SNR c2 x ≥ σ2 (2.66) providing an improvement of a factor of three from what can be achieved from the what can be seen by a waveform occupying the same bandwidth with equal spectral power at each frequency (i.e. LFMW with sweep bandwidth ∆f). The uncertainty of velocity (2.13) can be calculated from the spectral uncertainty (2.9) and (2.11) where the response is again linked to the rectangular modulation in the time domain. 38 Following the previous derivations (cid:0)π2T 2(cid:1) SNR 3c2 v ≥ σ2 4f 2 c (2.67) which has the same Doppler uncertainty as the LFMW case for the same time duration T . 39 CHAPTER 3 WAVEFORMS DESIGNED FOR HIGH-ACCURACY RANGING FOR DISTRIBUTED BEAMFORMING ARRAYS In this chapter I derive waveforms for range estimation, using the metrics discussed in the last chap- ter, to enable phase alignment of distributed beamforming. These waveforms are validated through experimentation. The waveforms in the previous chapter are used as a quality comparison for the new waveform designs. The specific waveforms I discuss in this chapter are: a waveform for near optimal estimation of position and velocity; a spectrally sparse waveform enabling joint frequency transfer and phase alignment; a method for using a standard communications waveform preamble to enable networked systems to perform position and velocity estimation without effecting the in- formation throughput; and a pulse encoded dual-tone waveform for high-accuracy scalable range estimation. These waveform designs demonstrate different methods to address scalability for dis- tributed systems that consist of more than two nodes, where multiple inter-node communication links are required. The advantages and disadvantages of time domain duplexing (TDD), frequency domain multiplexing (FDM), and pulse encoding for the purpose of scalability are discussed in this chapter. 3.1 Time Domain Duplex: Joint Range-Doppler Estimation Using Prolate Spheroidal Wave Functions (PSWF) In this section a waveform design to provide near optimal performance as both a position and velocity estimator is presented. Polate spheroidal wave functions (PSWF) are employed due to their unique ability to maximally concentrate signal energy into a finite time and bandwidth. Waveforms with finite durations reduce the spread of energy, in either temporal or spectral do- mains, resulting in an increase of the second moment of that respective domain and thereby an improvement of the theoretical estimation ability by reducing the CRLB. The PSWF presents a solution that is approximately finite in both domains and, therefore, is an optimal baseband wave- 40 form representation. Here an optimized mean squared duration and bandwidth are designed by concatenating two PSWFs to create a time envelope that is modulated by a two-tone signal. Due to this waveform’s near maximum time and bandwidth occupancy, a time duplex method for scala- bility would need to be implemented where each primary to secondary connection of a distributed array would have individual time slots to perform their estimations. 3.1.1 Theory of Prolate Spheroidal Wave Functions Since the early 1930 PSWFs have been known as the eigenfunctions to the Sturm-Liouville oper- ators Lc given by Lc(ψ) = (cid:16) 1 − x2(cid:17) d2ψ dx2 − 2x − c2x2φ dψ dx (3.1) for any c > 0. This function arises from solving the Helmholtz equation ∆Ψ + k2Ψ = 0 over a prolate spheroid. Later in the 1960s Slepian, Pollak, and Landau arrived at the same function from the perspective of nonclassical versions of uncertainty principles [87–91]. Here they discovered the PSWFs to be the eigenfunctions of a self-adjoint integral operator defined by (cid:16) sin (cid:90) T 2 − T 2 (cid:17) 2πW (t − s) π(t − s) λnψn(t) = ψn(s)ds n = 0, 1, 2, . . . (3.2) where λn are the eigenvalues, ψn are the PSWFs, W is bandwidth, and T is the time duration. The values of W and T are bound by the value c = πT W known as the Slepian frequency. For any given value such that T > 0 and W > 0 the PSWFs in (3.2) represent a countably infinite set of real functions which have the following properties: 1. ψn(t) represent an orthonormal set of band-limited functions which for unit energy implies (cid:90) ∞ −∞ ψn(t)ψm(t)dt = 0, n (cid:54)= m 1, n = m n, m = 0, 1, 2, 3 . . . 41 2. In the time interval [−T /2, T /2] the values of ψn(t) are also orthogonal yet do not contain the entire concentration of energy such that λn ≤ 1. ψn(t)ψm(t)dt = λn, n = m n (cid:54)= m n, m = 0, 1, 2, 3 . . . (cid:90) T 2 − T 2 0, 3. ψn(t) is even or odd with even and odd values of n respectively. 4. There is exactly n zeros in the interval [−T /2, T /2]. The functions ψn(t) represent eigenfunctions to the equation in (3.2) where the values of λn represent the corresponding eigenvalues. Properties of the eigenvalues include the following: 1. They rank from largest to smallest values with the value of n such that 1 ≥ λ0 ≥ λ1 ≥ λ2 ≥ . . . 2. They are the kernel of the sinc(·) operator 3. They approach 0 as the value of n approaches infinity: lim n→∞ λn = 0 Slepian discovered that values of the eigenvalues are nearly 1 until 2c π when at which they superex- ponentially approach 0 at a rate of approximately log(c) [92]. The transition point at which these values approach zero plays an important role by providing a truncation value to approximate the PSWFs. 3.1.2 Numerical Approximation of the PSWFs There are many different methods to approximate these functions through numerical models, among them the quadrature method which uses the Gaussian quadrature formula [93–95] and the Whittaker-Shannon sampling theorem method [96, 97], but the classic method is the Bouwkamp 42 algorithm which uses the Legendre-Galerkin method [98, 99] and is the approach used in this dis- sertation. The Bouwkamp algorithm is based off of representing the PSWF as a set of normalized Legendre polynomials ψn(t) = ∞(cid:88) βnk ¯Pk(t) (3.3) where ¯Pk(t) are the normalized Legendre polynomial and βnk are the expansion coefficients. The Legendre polynomials are a set of orthogonal polynomials which satisfies the three-term recursion k=0 relation Pk+1(t) = 2k + 1 k + 1 tPk(t) − k k + 1 Pk−1(t) k ≥ 1 with the initial conditions P0(t) = 1 P1(t) = t (3.4) (3.5) These Legendre polynomials are normalized to create an orthonormal basis set using the normal- ization factor Pk(t) = k + 1 2 Pk(t) (3.6) The expansion coefficients βnk present in (3.3) are calculated through an infinite symmetric peta- diagonal matrix A [95, 100–102] known as the Galerkin matrix (cid:114) (3.7) (3.8) (3.9)  A = a0,0 0 a0,2 0 0 a1,1 0 a1,3 a2,0 ... 0 0 ... 0 a2,2 ... 0 0 ... 0 0 0 a2,4 ... 0 0 0 0 0 0 ··· ··· ··· 0 ... ... ··· ak,k−2 0 ak,k 0 ... 0 0 ...  where the nonzero terms are defined as ak,k = k(k + 1) + c2 2k(k + 1) − 1 (2k + 3)(2k − 1) (2k + 3)(cid:112)(2k + 1)(2k + 5) (k + 1)(k + 2) c2 ak,k+2 = ak+2,k = The eigenvalues and eigenvectors to this matrix A can be solved by the eigen-problem (A − χnI) βn = 0 43 where χn are the eigenvalues and βn = (βn0, βn1, βn2, . . . ) is the eigenvector. From the orthogo- nality of Parseval’s identity, ∞(cid:88) |βnk|2 = 1 ∀n ≥ 0 (3.10) k=0 with the parity βnk = 0 if n + k is odd. In essence this states that if n is even only the even values of βnk are used and similarly if n is odd only the odd values of βnk are used. To compute the series representation in (3.3) the values of k have to extend to infinity which is not feasible in any practical system. Instead an approximation is made by truncating the infinite sum such that ψn(t) ≈ M(cid:88) βnk ¯Pk(t) (3.11) k=0 The truncation value is taken such that that M > 2c π which is the limit after which the eigenvalues λn superexponentially approach zero [103]. It is also to be mentioned that this method only works for the bounds [−1, 1] outside of which the result will experience extremely large errors. Values of |t|≥ 1 can be approximated by a series of Bessel functions [97] but for this dissertation I am only interested behavior of the function inside the bounds of [−1, 1]. 3.1.3 Range and Doppler Estimation Bounds The bounds on spatial and spectral parameter estimation of the PSWFs can be calculated by the second moment of the Fourier of the parameter of interest due to the PSWF inherently having a zero first moment in both time and frequency. The joint measurement is fundamentally limited by the radar uncertainty principle σtσf = 1 πSNR (3.12) Due to nonexistence of a closed form solution to PSWFs, the bounds in (2.11) and (2.12) is solved numerically. Therefore, these bounds need to calculated explicitly for the values of T and W on a case by case basis. A nominal value of T = 10 µs, which is indicative of the length of waveforms used in remote sensing, is chosen in this work. This results in bandwidth values of W = 100 kHz, 44 W = 400 kHz, W = 700 kHz, and W = 1 MHz for the values of c = π, c = 4π, c = 7π, and c = 10π respectively. For other values of T the estimation ability of the optimal, LFMW, and rectangular pulse cases scale proportionality providing the same conclusion but at a shifted value. Images of the calculated CRLBs for the baseband PSWFs as a time and frequency estimator can be seen in Fig. 3.1. For the case of c = π the bounds converge after n = 1 and provide no improvement with increasing order. This is due to the majority of the energy lying outside of the bounds in both time and frequency domains and thus poorly representing the signal, providing invalid estimation performance. As the value of c increases the concentration of energy within the given frequency window increases to become nearly maximum for all values of n and therefore improves the range accuracy. As for the frequency estimation, the performance slightly degrades as the value of c increases. This is due to the concentration of energy condensing to zero in the time domain. When comparing the performance of the baseband PSWFs to the optimal infinitesimally thin two tone and pulse waveforms and a classic LFMW that occupy the same time and bandwidth, it can be seen that the range estimation performance degrades as the value of c increases. This is again due the larger percentage of the energy concentration condensing to zero in both time and frequency leaving unused time and bandwidth at the edges of the windowed domains resulting in a reduction of the second moment in both time and frequency thereby reducing the obtainable estimation variance. 3.1.4 Remote Sensing Waveform Design Based on PSWF of the Zeroth Order From the analysis in Section 2.2.3, the second moment of the frequency spectrum is maximized when the spectral energy is concentrated at two bifurcated frequencies at opposite ends of band of interest. By maximizing the second moment, the variance is minimized thus providing the most accurate estimation of the time delay. Similarly, from the mean squared duration (2.9) by separating the signal energy into two bifurcated temporal pulses at the ends of a time window, the second moment of the time domain is maximized and thus providing the most accurate estimation of frequency [86, 104]. In the time-frequency domain the optimal joint range-Doppler waveform 45 (a) (b) (c) (d) Figure 3.1: CRLB for Doppler (2.11) and range (2.12) of the baseband PSWFs for the cases of c = π (a), c = 4π (b), c = 7π (c), and c = 10π (d) on the right and left respectively. 46 would be represented by four delta functions at the four corners of the time-frequency window of interest. Here each domain will be approached independently. In [104] the effects of temporally bifurcated rectangular pulses with varying bandwidth was explored. It was found from this study that pulses with higher instantaneous bandwidths have improved Doppler estimation performance over ones with low instantaneous bandwidth due to the shorter temporal duration and therefore the ability to achieve higher temporal second moments. Here I use a similar approach but instead of rectangular pulses, the PSWF of the 0th order is used due to the maximal concentration of energy in both domains and, therefore, represents a nearly-optimal design approaching a delta function in both time and frequency. A depiction of the concatenated n = 0 PSWFs with various pulse bandwidths can be seen in Fig. 3.2 where the percentage in the legend refers to the ratio of the overall waveform bandwidth to the pulse instantaneous bandwidth such that 2T(cid:48) T where T(cid:48) is the duration of the individual pulses. This waveform represents a baseband signal which can be used as an envelope for a frequency bifurcated signal of which is inversely proportional the bandwidth σ2 dual pulse PSFW can be represented by (cid:16) ψ (t + t0) + ψ (t − t0) M(cid:88) (cid:16) Pk (t − t0) + Pk (t + t0) β0k s(t) = = (cid:16) i.e. f (t) = ej2πf1t + ej2πf2t(cid:17) the performance t ∼ 1/(f2 − f1)2 as seen in Section 2.2.3. This (cid:17)(cid:16) e−jπ∆f t + ejπ∆f t(cid:17) (cid:17)(cid:16) e−jπ∆f t + ejπ∆f t(cid:17) (3.13) k=0 where ψ(t) from (3.11) is shifted in time by t0 and ∆f is f2 − f1. The initial conditions to the shifted Legendre polynomials become Applying these initial conditions to (3.13) (cid:114) M(cid:88) k=0 s(t) = 2 β0k k + P0(t ± t0) = 1 P1(t ± t0) = t ± t0 (cid:18) 2k + 1 k + 1 1 2 Pk(t) + t0 47 (cid:19)(cid:16) e−jπ∆f t + ejπ∆f t(cid:17) (3.14) (3.15) Figure 3.2: Time domain representation of the temporally bifurcated waveform formed from con- catenating two n = 0 PSWFs. This example was taken at c = 3π. The time-frequency plots of several examples of this time and frequency bifurcated waveform with various modulation percentages can be seen in Fig. 3.3. Time- and band-limited modulation of an optimal signal (i.e. four delta functions in the corners of the time-frequency window) results in a reduction of the mean-squared bandwidth/- duration and degrades the performance of the parameter estimation ability. The effect of this degradation is dependent on the total bandwidth ∆f and total duration T and therefore if the mod- ulation represents a small portion of the overall domain the effect to the estimation performance (cid:16) i.e. 2T(cid:48) T (cid:17) (cid:16) (cid:17) is negligible. Here the effects of modulation on the estimation ability in both time and frequency i.e. 2W ∆f the results of which can be seen in Fig. 3.4 (a) and (b) for the spatial and spectral uncertainties respectively. From these plots it can be seen that the effects of the modula- tion decrease with percent coverage. For the case of the spatial bound, when 2W = ∆f errors two orders of magnitude over the optimal can be expected. When the spectral bandwidth is reduced to 2W = 0.1∆f the estimation of spatial accuracy is nearly equivalent to the optimal. For the case of 48 (a) (b) (c) Figure 3.3: Time and frequency bifurcated waveform with 100% (left) and 10% (right) time mod- ulation and 100% (a), 50% (b), and 10% (c) modulation in frequency. 49 (a) (b) Figure 3.4: (a) Comparison of the optimal CRLB to the CRLB of the spatial uncertainty of the dual pulse waveform with varying spectral bandwidth where the percentage refers to 2W (b) ∆f . Comparison of the optimal CRLB to the CRLB of the frequency uncertainty of the dual pulse waveform with varying temporal instantaneous bandwidth where the percentage refers to 2T(cid:48) T . the spectral bound, when 2T(cid:48) = T errors of roughly half an order of magnitude above the optimal can be expected. When the temporal bandwidth is reduced to 2T(cid:48) = 0.1T the estimation ability is within 90% of the optimal error. It is to be noted that these bounds can be further improved to more closely represent the optimal result by further reducing the modulation percentage in the 50 respective domain and therefore, more closely representing a delta functions. The ambiguity function for the dual PSWF can be calculated by inputting the time do- AF (t, fD) = β0q q + main representation of the waveform (3.15) into the the ambiguity function equation (2.1), yielding (cid:114) (cid:114) (cid:90) ∞ −∞ 2 ×2 Q(cid:88) M(cid:88) q=0 k=0 (cid:18) Pq(t − τ ) + t0 (cid:18) 2q + 1 q + 1 (cid:19)(cid:16) ejπ∆f (t−τ ) + e−jπ∆f (t−τ )(cid:17) (cid:19)(cid:16) e−jπ∆f τ + ejπ∆f τ(cid:17) ej2πfDτ dτ 1 2 1 2 β0k k + Pk(τ ) + t0 2k + 1 k + 1 (3.16) Due to the orthogonality principle of Legendre polynomials k (cid:54)= q = 0 reduces (3.16) to AF (t, fD) = 4 (cid:90) ∞ −∞ M(cid:88) k=0 (cid:18) (cid:18) × β2 0k k + 1 2 (cid:19)(cid:18) Pk(t − τ ) + t0 (cid:19)(cid:16) ejπ∆f (t−τ ) + e−jπ∆f (t−τ )(cid:17) (cid:19)(cid:16) 2k + 1 k + 1 e−jπ∆f τ + ejπ∆f τ(cid:17) ej2πfDτ dτ (3.17) Pk(τ ) + t0 2k + 1 k + 1 From here the calculation of the ambiguity function is less tractable to be analyzed analytically due to the recursion properties of the Legendre polynomials. At this point the rest of the calculation is carried out numerically and can be seen in Fig. 3.5. Here we can see the ambiguous nature of this waveform. From the zoomed in portion of the center of the intensity plot in Fig 3.5(d) the spectral and temporal ambiguities of the underlying beat pattern in time, resulting from the two frequencies, and Doppler, from the two temporal pulses, can be seen. As for the zoomed in portion on the far left, ambiguities in time are present as only one temporal pulse has been correlated at this point. To disambiguate this waveform two approaches can be made: the use of a waveform representing a square domain in time and frequency given by pseudo-random white noise; and a waveform representing a sloped domain in time and frequency given by an LFMW. The amount of time and bandwidth allocated to the disambiguation waveform dictate the performance. Examples of using pseudo-random white noise and an LFMW to disambiguate can be seen in Fig. 3.6 and Fig. 3.7 respectively. This method of disambiguation represents a time-duplexed scheme where the measurement waveform and disambiguation waveform are sent in separate time slots as to not 51 (a) (c) (e) (b) (d) (f) Figure 3.5: (a) Time representation of the waveform with 10% modulation. (b) Spectral represen- tation with 10% modulation. (c) Ambiguity function of the dual pulse PSWF. (d) Intensity plot of the ambiguity function showing ambiguous nature in both time and frequency. (e) Matched filter of the dual pulse PSWF (zero Doppler cut). (f) Doppler response of the dual pulse PSWF (zero time cut). 52 (a) (b) (c) (d) (e) Figure 3.6: Pseudo-random white noise for disambiguation for 10% occupation of both time and frequency, relative to the measurement waveform (left), and 25% occupancy in time and frequency (right). (a) Pseudo-random white noise in the time domain. (b) Time-frequency plot of the pseudo- random white noise. (c) Matched filter response for the measurement waveform (blue) and disam- biguation (red). (d) Doppler response of the measurement waveform (blue) and disambiguation (red). (e) Combined ambiguity function of measurement waveform and disambiguation demon- strating suppression of sidelobes in both time and Doppler. 53 (a) (b) (c) (d) (e) Figure 3.7: LFMW for disambiguation for 10% occupation of both time and frequency, relative to the measurement waveform (left), and 25% occupancy in time and frequency (right). (a) LFMW in the time domain. (b) Time-frequency plot of the LFMW. (c) Matched filter response for the measurement waveform (blue) and disambiguation (red). (d) Doppler response of the measurement waveform (blue) and disambiguation (red). (e) Combined ambiguity function of measurement waveform and disambiguation demonstrating suppression of sidelobes in both time and Doppler. 54 interfere with the measurement ability of the derived dual PSWF. An alternative method could be used, due to the dual PSWF sparsity in both time and frequency, where the disambiguation pulse is embedded between the two PSWF pulses. This method would save time, as only one time slot is required rather than two, and reduce computational complexity as the calculation of a single matched filter and Doppler response would be required. However, embedding the disambiguation method would reduce that second moment in both time and frequency domains and therefore re- duce the obtainable measurement accuracy where the impact in both disambiguation ability and degradation of the accuracy depends on the energy and occupational time and bandwidth allocated to the disambiguation pulse. 3.1.4.1 Simulation To evaluate the estimation ability of this waveform in comparison to the derived bounds, a simula- tion is conducted in MATLAB. The waveform parameters that are considered consist of a temporal bifurcation of 2T(cid:48) ∆f = 10% where W = 6 MHz. An image of the waveform for the time and frequency domain can be seen in T = 10% where T = 10 µs and a frequency bifurcation of 2W Fig. 3.5(a). This waveform is then corrupted with AWGN so that the estimation performance at multiple SNR levels can be evaluated. The SNR of the signal is evaluated using an eigenvalue decomposition method from Section 2.1.4. The positional uncertainty, again solved using (2.12), is calculated from the time of flight temporal uncertainty. An estimation of time of flight of the received signal can be evaluated from the peak value of a matched filtering process. An image of the matched filter output can be seen in Fig. 3.5(e). The peak value was then interpolated using a built-in MATLAB spline interpolator by 1000 points to improve the measurement estimation and avoid binning around the peak. The waveform variance is then calculated from these 1000 peak values. The matched filter process also maximizes the SNR of the signal by providing an additional processing gain equivalent to the time-bandwidth product where the time component is the time duration T and the bandwidth is the receiver bandwidth which is linked to the sample rate and filtering of the signal. 55 The estimation of the frequency was performed by evaluating the one-sided frequency spectrum of the matched filter autocorrelation function and taking the peak value. An image of the Doppler response of the dual pulse waveform can be seen in Fig. 3.5(f) where the lobing structure appears from the interference of the overlapping instantaneous bandwidths of the temporal pulses and is concurrent with the findings in [104]. The peak frequency of the spectrum was interpolated by 1000 points using MATLAB’s build-in spline function and the variance was taken over the 1000 Monte Carlo iterations. 3.1.4.2 Measurements Wireless measurements are performed in a semi-enclosed arch range using a two standard gain 2 − 12 GHz wideband horn antennas oriented in a quasi monostatic fashion at the edge of the range. A corner reflector is placed at the center of the range approximately 1.25 m from the transmitter and receiver. The signal is generated at the RF level with a center frequency of 3 GHz using a M8190A 12 GSa/s Keysight arbitrary waveform generator (AWG). Variable attenuation is applied to the output of the AWG so that multiple SNR levels could be evaluated. The receiving antenna is connected to a MSO-X 92004A Keysight Infiniium High-Performance Oscilloscope. The scope captures are triggered using a threshold detection on a rectangular pulse provided by the AWG. Both the AWG and the scope’s local oscillators are locked using an external 10 MHz source provided by an Agilent MXG Analog Signal Generator. 1000 observations are recorded on the scope and processed offline in MATLAB where digital down conversion to baseband was performed and a low pass filter with a cutoff frequency of 100 MHz is applied. The estimation process is performed in the same way as the simulation which was conducted for 1000 Monte Carlo iterations to provide statistical relevance comparatively to the measured data. A block diagram of the experimental setup along with an image of the setup in the arch range can be seen in Fig. 4.6(a) and (b) respectively. An image of a measured waveform in the time-frequency domain where T = 10 µs and W = 6 MHz can be seen in Fig. 3.9(a). The results of both spatial and frequency estimation 56 (a) (b) Figure 3.8: (a) Block diagram of the wireless experimental setup. (b) Image of experimental setup in the semi-enclosed arch range. for both simulation and measurements compared to the calculated bounds can be seen in Fig. 3.9(b) and (c) respectively. From this plot it can be seen that measurements closely match the simulation and the calculated bounds providing a near-optimal estimation in both temporal and spectral domains. At the highest performing point, measured accuracies of σm = 3.1 mm and σf = 216 Hz are obtained at 13 dB preprocessing SNR with 33 dB of processing gain. This enables a beamforming frequency of 6.45 GHz with an accuracy of λ 15. 57 (a) (b) (c) Figure 3.9: (a) Time-frequency plot of a measured waveform with T = 10 µs and W = 6 MHz. (b) Comparison of measured results, simulation, and CRLB for the dual pulse PSWF as a spatial estimator. (c) Comparison of measured results, simulation, and CRLB for the dual pulse PSWF as a frequency estimator. 58 3.2 Time Domain Duplex: Joint Phase Alignment and Frequency Transfer for Wideband Systems In this section I address a waveform used for inter-node coordination to jointly enable the measurement of the range between platforms, while simultaneously transferring a frequency reference in a stable manner such that the frequency can be used to discipline a local oscillator. Here the waveforms used to enable estimation of delay (which yields an estimate of the range) as well as estimation of the frequency reference, which would be input to a PLL to align the oscillator of the receiving node to that of the transmitting node are described. These waveforms are based on the spectrally sparse model of the PTTW which uses a large portion of the system bandwidth. For this reason, like the dual PSFW, this method represents a time duplex scheme for scalability where each node has a specific time slot to perform its measurement to acquire both a range estimation and the frequency reference. 3.2.1 Waveform Design In the prior chapter, the use of the two-tone continuous wave (TTCW) signal for optimal ranging accuracy was described, but is generally not obtainable due to system constrains, leading to the PTTW from Section 2.2.3. Due to the high-accuracy capability of the PTTW signal, it is used as the base for the joint ranging and frequency transfer waveform in this work. A frequency reference, which is chosen to be 10 MHz, clock signal is modulated onto the lower of the two tones, and the phase error is measured to assess the stability of a frequency alignment process. The normalized PTTW signal is given by s1(t) = 1 2 rect (cid:18) t (cid:19)(cid:16) T e−jπ∆f t + ejπ∆f t(cid:17) (3.18) where ∆f is the frequency difference between the two frequencies of the two-tone waveform. Three different types of modulation schemes are implemented for the frequency reference fm: single sideband, double sideband, and amplitude modulation, shown respectively in the following: (cid:18) t (cid:19)(cid:16) H[cos(2πfmt)]e−jπ∆f t + ejπ∆f t(cid:17) s2(t) = 1 2 rect T (3.19) 59 (cid:18) t (cid:19)(cid:16) cos(2πfmt)e−jπ∆f t + ejπ∆f t(cid:17) (cid:19)(cid:110)(cid:104) e−jπ∆f t + ejπ∆f t(cid:111) (cid:16) (cid:17)(cid:105) 1 + sin 2πfmt T s3(t) = 1 2 s4(t) = 1 3 rect rect (cid:18) t T (3.20) (3.21) where H is the Hilbert transform. Fig. 3.10 shows theses waveforms simulated in MATLAB and measured using an M8190A 12 GSa/s AWG. Here the baseline PTTW waveform uses a 150 MHz tone separation, the lower of the two tones is then modulated with a 10 MHz frequency reference signal. The above expressions can be evaluated to determine the theoretical performance of each waveform in terms of delay estimation and frequency estimation. The error in estimating the delay can be determined by evaluating each of the three waveforms using (2.5) and (2.12). To make the analysis more general, the signals si above can be rewritten in their baseband forms as (cid:18) t (cid:18) t (cid:18) t T (cid:19)(cid:16) (cid:19)(cid:16) (cid:19)(cid:16) ejπ(−∆f +2fm)t + ejπ∆f t(cid:17) e−j2π(∆f +2fm)t + ejπ(−∆f +2fm)t + ejπ∆f t(cid:17) e−jπ(∆f +2fm)t + e−jπ∆f t + ejπ(−∆f +2fm)t + ejπ∆f t(cid:17) s2(t) = s3(t) = s4(t) = 1 2 1 3 1 4 rect rect rect T T (3.22) (3.23) (3.24) The above formulation takes a more general approach to the location of the tones in terms of their relative separations. 3.2.1.1 Ambiguity Function The response of the PTTW using single sideband modulation to estimate time delay and Doppler is given by the ambiguity function (2.1) with the model of the time domain waveform from (3.22) AF2(t, fD) = 1 4 (cid:90) ∞ −∞ rect × rect (cid:18) τ − t (cid:19)(cid:16) ejπ(∆f−2fm)(τ−t) + e−jπ∆f (τ−t)(cid:17) e−jπ(∆f−2fm)τ + ejπ∆f τ(cid:17) (cid:17)(cid:16) (cid:16) τ ej2πfDτ dτ T T (3.25) 60 (a) (b) (c) Figure 3.10: Ranging is based on widely-spaced two-tone signal with 100 MHz of bandwidth, with the lower tone modulated by a 10 MHz frequency reference. (a) Spectrum of the joint ranging and frequency transfer signal using single-sideband modulation. (b) Spectrum of the joint ranging and frequency transfer signal using double-sideband modulation. (c) Spectrum of the joint ranging and frequency transfer signal using amplitude modulation. 61 Truncation of the integral can be performed due to time domain rectangular envelope, rect(·), being a time limited function with the bounds [− T 2 , T 2 ], yielding e−jπ(∆f−2fm)t + ejπ∆f t + ejπ[(∆f−2fm)(τ−t)+∆f τ ] (cid:90) t+ T 2 (cid:16) − T 2 AF2(t, fD) = 1 4 AF2(t, fD) = e−jπ(∆f−2fm)t + ejπ∆f t j8πfD e−jπ(∆f−2fm)t j8π (∆f − fm + fD) ejπ∆f t j8π (fD − ∆f + fm) + + ej2πfDτ dτ j2πfD e (cid:16) (cid:35) t+ T 2 + e−jπ[(2∆f−2fm)τ−∆f t](cid:17) (cid:17) (cid:16) j2π(∆f−fm+fD) (cid:16) j2π(fD−∆f +fm) − e−jπfDT (cid:17) (cid:17) t+ T 2 t+ T 2 (cid:34) (cid:34) (cid:34) e e − e−jπ(∆f−fm+fD)T − e−jπ(fD−∆f +fm)T (cid:35) (cid:35) (3.26) Extracting a factor from each function so that every parenthesis contains a complex exponential subtracted by its conjugate (cid:16) e−jπ(∆f−2fm)t + ejπ∆f t(cid:17) ejπfDt AF2(t, fD) = (cid:104) ejπfD(t+T ) − e−jπfD(t+T )(cid:105) e−jπ(∆f−2fm)tejπ(∆f−fm+fD)t j8π (∆f − fm + fD) + + (cid:104) (cid:104) j8πfD ejπ∆f tejπ(fD−∆f +fm)t j8π (fD − ∆f + fm) ejπ(∆f−fm+fD)(t+T ) − e−jπ(∆f−fm+fD)(t+T )(cid:105) ejπ(fD−∆f +fm)(t+T ) − e−jπ(fD−∆f +fm)(t+T )(cid:105) ejθ − e−jθ(cid:17) (cid:16) e−jπ(∆f−2fm)t + ejπ∆f t(cid:17) (cid:16) (cid:16) (cid:17) (cid:17) πfD(t + T ) π (∆f − fm + fD) (t + T ) (cid:17) π (fD − ∆f + fm) (t + T ) ejπfDt (t + T ) sinc ejπ(fm+fD)t ejπ(fm+fD)t (t + T ) sinc (t + T ) sinc (3.27) (3.28) (cid:16) (cid:16) 4 + + 4 4 each function by t+T t+T such that sinc(θ) = 1 j2θ AF2(t, fD) = ejπ∆f t cancels in the second and third terms and sinc functions can now be formed by multiplying The terms ejπfDt represents the case of t < 0 case but is symmetric for the case of t > 0. The general case of the and (t + T ) can then be extracted from each term. This derivation shown above 4 62 derivation can be represented by AF2(t, fD) = ejπfDt (T − |t|) 4 (cid:104)(cid:16) e−jπ(∆f−2fm)t + ejπ∆f t(cid:17) (cid:16) (cid:17) (cid:16) (cid:17) πfD (T − |t|) (cid:16) (cid:17)(cid:105) π (∆f − fm + fD) (T − |t|) π (fD − ∆f + fm) (T − |t|) sinc +ejπfmt sinc +ejπfmt sinc (3.29) The magnitude of the ambiguity function can can now be taken (cid:12)(cid:12)(cid:12)(cid:12)(T − |t|) 4 (cid:104) (cid:16) |AF2(t, fD)| = 2 sinc πfD (T − |t|) (cid:17) + sinc (cid:16) π (∆f − fm + fD) (T − |t|) (cid:16) π (fD − ∆f + fm) (T − |t|) (cid:17) (cid:17)(cid:105)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)e−jπ(∆f−2fm)t + ejπ∆f t(cid:12)(cid:12)(cid:12) = 2. An image of the (3.30) + sinc (cid:12)(cid:12)(cid:12)ejπfDt(cid:12)(cid:12)(cid:12) = 1, (cid:12)(cid:12)(cid:12)ejπfmt(cid:12)(cid:12)(cid:12) = 1, and where the ambiguity function of single sideband modulation with ∆f = 50 MHz and fm = 10 MHz along with the matched filter and Doppler response can be seen in Fig. 3.11. This figure resembles the ambiguity function of the PTTW due to single sideband modulation containing two frequencies. of the time domain waveform from (3.23) The ambiguity function of the PTTW using double sideband modulation with the model (cid:90) ∞ −∞ rect (cid:18) τ − t (cid:19)(cid:16) ejπ(∆f−2fm)(τ−t) + ejπ(∆f +2fm)(τ−t) + e−jπ∆f (τ−t)(cid:17) (cid:17)(cid:16) (cid:16) τ e−jπ(∆f−2fm)τ + e−jπ(∆f +2fm)τ + ejπ∆f τ(cid:17) ej2πfDτ dτ T × rect T AF3(t, fD) = 1 9 (3.31) AF3(t, fD) = Truncation of the integral due to rect(·) envelope 2 1 9 e−jπ(∆f−2fm)t + ejπ[4fmτ−(∆f +2fm)t] + e−jπ[(2∆f−2fm)τ−∆f t] (cid:16) +e−jπ(∆f +2fm)t + e−jπ[4fmτ +(∆f−2fm)t] + e−jπ[(2∆f +2fm)τ−∆f t] (cid:90) t+ T +ejπ∆f t + ejπ[(∆f−2fm)(τ−t)+∆f τ ] + ejπ[(∆f +2fm)(τ−t)+∆f τ ](cid:17) − T 2 ej2πfDτ dτ (3.32) 63 (a) (c) (b) (d) Figure 3.11: (a) Ambiguity function of the PTTW with single sideband modulation. (b) Intensity plot of the ambiguity function. (c) Matched filter of single sideband modulation (zero Doppler cut). (d) Doppler response of the single sideband modulation (zero time cut). 64 e−jπ(∆f−2fm)t + e−jπ(∆f +2fm)t + ejπ∆f t j18πfD e−jπ(∆f +2fm)t j18π (2fm + fD) + ejπ∆f t j18π (fD − ∆f + fm) e−jπ(∆f−2fm)t j18π (fD − 2fm) + ejπ∆f t j18π (fD − ∆f − fm) e−jπ(∆f−2fm)t j18π (fD + ∆f − fm) e−jπ(∆f +2fm)t j18π (fD + ∆f + fm) (cid:34) (cid:34) (cid:34) j2π(2fm+fD) e (cid:16) j2π(fD−∆f +fm) e t+ T 2 (cid:34) (cid:34) (cid:34) j2π(fD−2fm) (cid:16) e j2π(fD−∆f−fm) (cid:16) e j2π(fD+∆f−fm) (cid:16) e j2π(fD+∆f +fm) e t+ T 2 t+ T 2 (cid:17) (cid:17) (cid:17) (cid:17) (cid:34) (cid:16) (cid:17) (cid:16) t+ T 2 − e−jπfDT − e−jπ(2fm+fD)T (cid:17) j2πfD e t+ T 2 t+ T 2 (cid:16) − e−jπ(fD−∆f +fm)T (cid:17) − e−jπ(fD−2fm)T t+ T 2 − e−jπ(fD−∆f−fm)T − e−jπ(fD+∆f−fm)T − e−jπ(fD+∆f +fm)T (cid:35) (cid:35) (cid:35) (cid:35) (cid:35) (cid:35) (cid:35) AF3(t, fD) = + + + + Creating a complex exponential subtracted by its conjugate can be made by extracting a factor in (3.33) 65 each parenthesis AF3(t, fD) = ejπfDt (cid:16) e−jπ(∆f−2fm)t + e−jπ(∆f +2fm)t + ejπ∆f t(cid:17) ×(cid:104) j18πfD e−jπ(∆f +2fm)tejπ(2fm+fD)t + j18π (2fm + fD) ejπ∆f tejπ(fD−∆f +fm)t j18π (fD − ∆f + fm) e−jπ(∆f−2fm)tejπ(fD−2fm)t ejπfD(t+T ) − e−jπfD(t+T )(cid:105) ejπ(2fm+fD)(t+T ) − e−jπ(2fm+fD)(t+T )(cid:105) (cid:104) (cid:104) ejπ(fD−∆f +fm)(t+T ) − e−jπ(fD−∆f +fm)(t+T )(cid:105) (cid:104) ejπ(fD−2fm)(t+T ) − e−jπ(fD−2fm)(t+T )(cid:105) (cid:104) ejπ(fD−∆f−fm)(t+T ) − e−jπ(fD−∆f−fm)(t+T )(cid:105) j18π (fD − 2fm) + ejπ∆f tejπ(fD−∆f−fm)t ejπ(fD+∆f−fm)(t+T ) − e−jπ(fD+∆f−fm)(t+T )(cid:105) (cid:104) j18π (fD − ∆f − fm) e−jπ(∆f−2fm)tejπ(fD+∆f−fm)t (cid:104) ejπ(fD+∆f +fm)(t+T ) − e−jπ(fD+∆f +fm)(t+T )(cid:105) e−jπ(∆f +2fm)tejπ(fD+∆f +fm)t + + + + j18π (fD + ∆f − fm) j18π (fD + ∆f + fm) ejπ∆f t cancels in the third, fifth, sixth, and seventh terms and the ej2πfmt cancel in the second and fourth terms. Sinc functions can now be formed by multiplying each function by t+T t+T such (3.34) 66 that sinc(θ) = 1 j2θ ejθ − e−jθ(cid:17) (cid:16) e−jπ(∆f−2fm)t + e−jπ(∆f +2fm)t + ejπ∆f t(cid:17) (cid:16) ejπfDt 9 ejπ(fD−∆f)t (t + T ) sinc (t + T ) sinc 9 ejπ(fD−∆f)t (t + T ) sinc (t + T ) sinc (t + T ) sinc (t + T ) sinc AF3(t, fD) = + + + + + 9 ejπ(fD+fm)t + 9 ejπ(fD−fm)t 9 ejπ(fD+fm)t ejπ(fD−fm)t 9 9 (t + T ) (cid:16) (cid:16) × sinc πfD (t + T ) (cid:16) π (2fm + fD) (t + T ) π (fD − ∆f + fm) (t + T ) (cid:16) π (fD − 2fm) (t + T ) (cid:16) π (fD − ∆f − fm) (t + T ) (cid:16) π (fD + ∆f − fm) (t + T ) (cid:16) π (fD + ∆f + fm) (t + T ) (cid:17) (cid:17) (cid:17) (cid:17) (cid:17) (cid:17) (cid:17) (3.35) The terms ejπfDt represents the case of t < 0 case but is symmetric for the case of t > 0. The general case is given and (t + T ) can then be extracted from each term. This derivation shown above 9 represented by AF3(t, fD) = 9 ejπfDt (T − |t|) (cid:104)(cid:16) e−jπ(∆f−2fm)t + e−jπ(∆f +2fm)t + ejπ∆f t(cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) πfD (T − |t|) (cid:16) (cid:17) (cid:17) π (fD − ∆f + fm) (T − |t|) π (2fm + fD) (T − |t|) + ejπfmt sinc (cid:16) (cid:17) (cid:17)(cid:105) + e−jπfmt sinc π (fD − ∆f − fm) (T − |t|) π (fD − 2fm) (T − |t|) + e−jπfmt sinc π (fD + ∆f + fm) (T − |t|) π (fD + ∆f − fm) (T − |t|) × sinc (cid:16) +e−jπ∆f t sinc (cid:16) +e−jπ∆f t sinc +ejπfmt sinc (3.36) 67 (cid:104) 3 sinc + sinc + sinc (cid:17) (cid:17) |AF3(t, fD)| = (cid:16) The magnitude of the ambiguity function can can now be taken and like terms can be combined (cid:12)(cid:12)(cid:12)(cid:12)(T − |t|) (cid:16) π (fD − ∆f − fm) (T − |t|) (cid:16) (cid:16) (cid:16) πfD (T − |t|) π (2fm + fD) (T − |t|) 9 (cid:17) (cid:16) π (fD − ∆f + fm) (T − |t|) π (fD − 2fm) (T − |t|) (cid:16) π (fD + ∆f − fm) (T − |t|) π (fD + ∆f + fm) (T − |t|) (cid:12)(cid:12)(cid:12)ejπfDt(cid:12)(cid:12)(cid:12) = 1, (cid:17) (cid:17) (cid:17) (cid:17)(cid:105)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)e−jπ∆f t(cid:12)(cid:12)(cid:12) = 1, and (cid:12)(cid:12)(cid:12)e−jπ(∆f−2fm)t + e−jπ(∆f +2fm)t +ejπ∆f t(cid:12)(cid:12)(cid:12) = 3. An image of the ambiguity function of double (cid:12)(cid:12)(cid:12)e±jπfmt(cid:12)(cid:12)(cid:12) = 1, where the magnitude of exponential terms (3.37) + sinc + sinc + sinc + sinc sideband modulation with ∆f = 50 MHz and fm = 10 MHz along with the matched filter and Doppler response can be seen in Fig. 3.12. of the time domain waveform from (3.24) Finally, the ambiguity function of the PTTW using amplitude modultion with the model 1 16 (cid:19)(cid:16) (cid:18) τ − t (cid:90) ∞ −∞ rect (cid:17)(cid:16) (cid:16) τ e−jπ(∆f−2fm)τ + e−jπ∆f τ + e−jπ(∆f +2fm)τ + ejπ∆f τ(cid:17) T ejπ(∆f−2fm)(τ−t) + ejπ∆f (τ−t) + ejπ(∆f +2fm)(τ−t) +e−jπ∆f (τ−t)(cid:17) AF4(t, fD) = ej2πfDτ dτ (3.38) × rect T (cid:90) t+ T (cid:16) The integral can be truncated due to rect(·) envelope such that 2 AF4(t, fD) = 1 16 e−jπ(∆f−2fm)t + e−jπ[(∆f−2fm)t+2fmτ ] + e−jπ[(∆f−2fm)t+4fmτ ] − T 2 +ejπ[(∆f−2fm)(τ−t)+∆f τ ] + ejπ[2fmτ−∆f t] + e−jπ∆f t + e−jπ[∆f t+2fmτ ] +ejπ∆f (2τ−t) + ejπ[4fmτ−(∆f +2fm)t] + ejπ[2fmτ−(∆f +2fm)t] + e−jπ(∆f +2fm)t +ejπ[(∆f +2fm)(τ−t)+∆f τ ] + ejπ[∆f t−(2∆f−2fm)τ ] + e−jπ∆f (2τ−t) + ejπ[∆f t−(2∆f +2fm)τ ] +ejπ∆f t(cid:17) ej2πfDτ dτ (3.39) 68 (a) (c) (b) (d) Figure 3.12: (a) Ambiguity function of the PTTW with double sideband modulation. (b) Intensity plot of the ambiguity function. (c) Matched filter of double sideband modulation (zero Doppler cut). (d) Doppler response of the double sideband modulation (zero time cut). 69 AF4(t, fD) = j32πfD e−jπ(∆f−2fm)t + e−jπ(∆f +2fm)t + ejπ∆f t + e−jπ∆f t (cid:16) (cid:17) (cid:17) e−jπ(∆f−2fm)t + e−jπ∆f t j2πfD e t+ T 2 (cid:34) (cid:34) (cid:34) × + e (cid:16) j2π(fD−fm) (cid:16) j2π(fD−2fm) (cid:16) e j32π (fD − fm) (cid:34) e−jπ(∆f−2fm)t j32π (fD − 2fm) + j2π(fD+∆f−fm) t+ T 2 e e−jπ(∆f−2fm)t j32π (fD + ∆f − fm) e−jπ∆f t + e−jπ(∆f +2fm)t + j2π(fD+fm) e t+ T 2 (cid:34) (cid:34) (cid:17) t+ T 2 − e−jπfDT − e−jπ(fD−fm)T t+ T 2 − e−jπ(fD−2fm)T (cid:17) (cid:16) (cid:16) (cid:16) t+ T 2 − e−jπ(fD+fm)T − e−jπ(fD+∆f−fm)T (cid:17) (cid:17) (cid:17) − e−jπ(fD+∆f)T − e−jπ(fD+2fm)T − e−jπ(fD+∆f +fm)T − e−jπ(fD−∆f +fm)T (cid:17) − e−jπ(fD−∆f)T t+ T 2 − e−jπ(fD−∆f−fm)T (cid:35) (cid:35) (cid:35) (cid:35) (cid:35) (cid:35) (cid:35) (cid:35) (cid:35) (cid:35) (cid:35) + + + j2π(fD+∆f) e t+ T 2 j32π (fD + fm) e−jπ∆f t (cid:34) + j32π (fD + ∆f ) e−jπ(∆f +2fm)t j32π (fD + 2fm) + e−jπ(∆f +2fm)t j32π (fD + ∆f + fm) ejπ∆f t j32π (fD − ∆f + fm) ejπ∆f t + j32π (fD − ∆f ) + ejπ∆f t j32π (fD − ∆f − fm) (cid:34) (cid:34) (cid:34) e e e t+ T 2 j2π(fD+2fm) e j2π(fD+∆f +fm) (cid:16) (cid:16) j2π(fD−∆f +fm) (cid:17) (cid:17) (cid:16) j2π(fD−∆f) (cid:17) (cid:16) e j2π(fD−∆f−fm) t+ T 2 (cid:34) t+ T 2 Extracting a factor from each function so that every parenthesis contains a complex exponential (3.40) 70 subtracted by its conjugate AF4(t, fD) = ×(cid:104) + + + + + + e−jπ(∆f−2fm)t + e−jπ(∆f +2fm)t + ejπ∆f t + e−jπ∆f t j32π (fD + ∆f − fm) j32π (fD − fm) e−jπ(∆f−2fm)tejπ(fD−2fm)t (cid:16) e−jπ(∆f−2fm)t + e−jπ∆f t(cid:17) ejπfD(t+T ) − e−jπfD(t+T )(cid:105) j32πfDe−jπfDt (cid:104) ejπ(fD−fm)(t+T ) − e−jπ(fD−fm)(t+T )(cid:105) ejπ(fD−fm)t (cid:104) ejπ(fD−2fm)(t+T ) − e−jπ(fD−2fm)(t+T )(cid:105) (cid:104) ejπ(fD+∆f−fm)(t+T ) − e−jπ(fD+∆f−fm)(t+T )(cid:105) j32π (fD − 2fm) e−jπ(∆f−2fm)tejπ(fD+∆f−fm)t e−jπ∆f t + e−jπ(∆f +2fm)t(cid:17) (cid:16) (cid:104) ejπ(fD+fm)(t+T ) − e−jπ(fD+fm)(t+T )(cid:105) ejπ(fD+∆f)(t+T ) − e−jπ(fD+∆f)(t+T )(cid:105) (cid:104) (cid:104) ejπ(fD+2fm)(t+T ) − e−jπ(fD+2fm)(t+T )(cid:105) (cid:104) ejπ(fD+∆f +fm)(t+T ) − e−jπ(fD+∆f +fm)(t+T )(cid:105) (cid:104) ejπ(fD−∆f +fm)(t+T ) − e−jπ(fD−∆f +fm)(t+T )(cid:105) ejπ(fD−∆f)(t+T ) − e−jπ(fD−∆f)(t+T )(cid:105) (cid:104) ejπ(fD−∆f−fm)(t+T ) − e−jπ(fD−∆f−fm)(t+T )(cid:105) j32π (fD + 2fm) e−jπ(∆f +2fm)tejπ(fD+∆f +fm)t j32π (fD + ∆f + fm) ejπ∆f tejπ(fD−∆f +fm)t j32π (fD − ∆f + fm) + j32π (fD + ∆f ) e−jπ(∆f +2fm)tejπ(fD+2fm)t ejπ∆f tejπ(fD−∆f)t j32π (fD − ∆f ) + ejπ(fD+fm)t j32π (fD + fm) e−jπ∆f tejπ(fD+∆f)t + ejπ∆f tejπ(fD−∆f−fm)t j32π (fD − ∆f − fm) + (cid:104) Sinc functions can now be formed by multiplying each function by t+T t+T such that (3.41) 71 (cid:16) ejθ − e−jθ(cid:17) yielding sinc(θ) = 1 j2θ e−jπ(∆f−2fm)t + e−jπ(∆f +2fm)t + ejπ∆f t + e−jπ∆f t AF4(t, fD) = 16e−jπfDt e−jπ(∆f−2fm)t + e−jπ∆f t(cid:17) ejπ(fD−fm)t (cid:16) (cid:16) × (t + T ) sinc πfD (t + T ) (cid:16) π (fD − fm) (t + T ) (cid:16) π (fD − 2fm) (t + T ) π (fD + ∆f − fm) (t + T ) (cid:16) (cid:16) (cid:16) π (fD + ∆f ) (t + T ) π (fD + fm) (t + T ) π (fD + 2fm) (t + T ) (cid:16) (cid:16) π (fD + ∆f + fm) (t + T ) (cid:16) π (fD − ∆f + fm) (t + T ) π (fD − ∆f ) (t + T ) (cid:16) π (fD − ∆f − fm) (t + T ) (cid:17) (cid:17) (cid:17) (cid:17) (cid:17) (cid:17) (cid:17) (cid:17) (cid:17) (cid:17) (cid:17) (cid:16) + (cid:16) + 16 ejπ(fD−∆f)t + 16 ejπ(fD+fm)t (t + T ) sinc (t + T ) sinc e−jπ∆f t + e−jπ(∆f +2fm)t(cid:17) + (t + T ) sinc 16 ejπ(fD+fm)t 16 + + + + ejπfDt 16 ejπ(fD−∆f)t (t + T ) sinc (t + T ) sinc (t + T ) sinc + 16 ejπ(fD−fm)t 16 ejπ(fD+fm)t 16 + ejπ(fD−fm)t 16 (t + T ) sinc (t + T ) sinc ejπfDt 16 (t + T ) sinc (t + T ) sinc The terms ejπfDt represents the case of t < 0 case but is symmetric for the case of t > 0. The general case of the and (t + T ) can then be extracted from each term. This derivation shown above 16 (3.42) 72 derivation can be represented by AF4(t, fD) = (cid:16) (cid:16) (cid:16) (T − |t|) 16e−jπfDt (cid:16) e−jπ(∆f−2fm)t + e−jπ∆f t(cid:17) + e−jπ∆f t + e−jπ(∆f +2fm)t(cid:17) (cid:16) π (fD − 2fm) (T − |t|) (cid:17) (cid:16) (cid:16) (cid:17) π (fD + ∆f ) (T − |t|) + sinc (cid:16) (cid:17) π (fD + ∆f + fm) (T − |t|) + ejπfmt sinc + e−jπfmt sinc π (fD − ∆f ) (T − |t|) + ejπfmt sinc (cid:16) e−jπfmt sinc e−jπ(∆f−2fm)t + e−jπ(∆f +2fm)t + ejπ∆f t + e−jπ∆f t(cid:17) (cid:17) (cid:17) (cid:17) (cid:17) (cid:17) (cid:17) (cid:17) (cid:16) × sinc πfD (T − |t|) π (fD − fm) (T − |t|) (cid:16) π (fD + ∆f − fm) (T − |t|) (cid:16) π (fD + fm) (T − |t|) ejπfmt sinc (cid:16) + e−jπ∆f t sinc π (fD + 2fm) (T − |t|) (cid:16) π (fD − ∆f + fm) (T − |t|) π (fD − ∆f − fm) (T − |t|) (cid:17) + + sinc +e−jπ∆f t sinc +e−jπfmt sinc The magnitude of the ambiguity function can can now be taken and like terms can be combined |AF4(t, fD)| = (cid:16) 4 sinc (cid:104) (cid:16) (cid:12)(cid:12)(cid:12)(cid:12)(T − |t|) (cid:17) (cid:16) πfD (T − |t|) 16 π (fD − 2fm) (T − |t|) (cid:17) (cid:16) π (fD + fm) (T − |t|) +2 sinc (cid:16) π (fD + 2fm) (T − |t|) π (fD − ∆f + fm) (T − |t|) + sinc + sinc + sinc + sinc + sinc + 2 sinc (cid:17) (cid:17) (cid:17) (cid:16) (cid:16) (cid:16) π (fD − fm) (T − |t|) (cid:16) π (fD + ∆f − fm) (T − |t|) (cid:16) π (fD + ∆f ) (T − |t|) + sinc (cid:16) π (fD + ∆f + fm) (T − |t|) π (fD − ∆f ) (T − |t|) + sinc π (fD − ∆f − fm) (T − |t|) (cid:17) (cid:17) (cid:17) (cid:17) (cid:17) (cid:17)(cid:105)(cid:12)(cid:12)(cid:12)(cid:12) + sinc (3.43) (3.44) An image of the ambiguity function of amplitude modulation format with ∆f = 50 MHz and fm = 10 MHz along with the matched filter and Doppler response can be seen in Fig. 3.13. 73 (a) (c) (b) (d) Figure 3.13: (a) Ambiguity function of the PTTW with amplitude modulation. (b) Intensity plot of the ambiguity function. (c) Matched filter of amplitude modulation (zero Doppler cut). (d) Doppler response of the amplitude modulation (zero time cut). 3.2.1.2 Range Estimation Lower Bounds The spatial estimation ability from (2.12) for the PTTW signal with the frequency reference fm modulated on the lower of the two tones can be approximated by delta functions in the frequency domain. While the first moment for these waveforms is not entirely negligible, the modulated tone fm is assumed to be small compared to the overall total waveform bandwidth ∆f. Therefore, the spatial estimation can be approximated by the second moment of the frequency spectrum (2.5). 74 The second moment using single sideband modulation (3.22) is given by ζ2 f,2 = = = = f + f + (cid:82) (2πf )2 |S2(f )|2 df (cid:82) |S2(f )|2 df (cid:82) (2πf )2(cid:12)(cid:12)(cid:12)δ (cid:17)(cid:105) (cid:16) ∆f (cid:104) (cid:82)(cid:12)(cid:12)(cid:12)δ (cid:16) ∆f (cid:104) (cid:17)(cid:105) 2 − fm (cid:17)2(cid:21) (cid:20)(cid:16) ∆f (cid:16) ∆f (cid:17)2 2 − fm 2 − fm (cid:20) (cid:21) 2 − fm∆f + f 2 ∆f 2 (cid:20)∆f 2 (cid:21) 1 + 1 − fm∆f + f 2 1 + 1 4π2 4π2 + m 2 m + δ = 2π2 2 (cid:104) f − ∆f (cid:104) + δ f − ∆f 2 2 (cid:105)(cid:12)(cid:12)(cid:12)2 (cid:105)(cid:12)(cid:12)(cid:12)2 df df (3.45) (3.46) (cid:3) SNR This gives a positional uncertainty form of x,2 ≥ σ2 4π2(cid:2)∆f 2 − 2fm∆f + 2f 2 c2 m Similarly, for modulation using double sideband modulation (3.23) the positional uncer- tainty can be derived by the second moment (cid:105)(cid:17) 2 + fm + δ (cid:104) (cid:17)(cid:105) (cid:16) ∆f (cid:17)2 2 − fm + δ f + 2 + fm 1 + 1 + 1 (cid:17)(cid:105) + δ (cid:104) f + (cid:16) ∆f (cid:16) ∆f (cid:17)(cid:105) 2 − fm (cid:17)2(cid:21) (cid:16) ∆f 2 − fm + 2 ζ3 f,3 = = = = (cid:82) (2πf )2 |S3(f )|2 df (cid:82) |S3(f )|2 df (cid:82) (2πf )2(cid:12)(cid:12)(cid:12)δ (cid:16) ∆f (cid:104) (cid:82)(cid:12)(cid:12)(cid:12)δ (cid:16) ∆f (cid:104) (cid:20)(cid:16) ∆f (cid:17)2 (cid:21) (cid:20) 2 + fm 4π2 f + f + + 4π2 (cid:20) 3∆f 2 4 + 2f 2 m 3 ∆f 2 + (cid:21) 8f 2 m 3 = π2 Resulting in a positional uncertainty of x,3 ≥ σ2 (cid:20) 4π2 c2 ∆f 2 + 8f 2 m 3 (cid:21) SNR 75 (cid:104) f − ∆f (cid:104) + δ f − ∆f 2 2 (cid:105)(cid:12)(cid:12)(cid:12)2 (cid:105)(cid:12)(cid:12)(cid:12)2 df df (3.47) (3.48) Finally, adding the frequency reference to the PTTW signal using amplitude modulation (3.24) is similar to double-sideband modulation, except that the carrier tone at the lower frequency is not suppressed, resulting in four frequency tones. The positional uncertainty can be derived from the second moment ζ3 f,4 = = (cid:82) (2πf )2 |S4(f )|2 df (cid:82) |S4(f )|2 df (cid:82) (2πf )2(cid:12)(cid:12)(cid:12)δ (cid:16) ∆f (cid:104) (cid:82)(cid:12)(cid:12)(cid:12)δ (cid:16) ∆f (cid:104) (cid:20)(cid:16) ∆f (cid:17)2 (cid:105) = π2(cid:104) ∆f 2 + 2f 2 m 2 + fm 4π2 f + f + + = 2 + fm 2 + fm (cid:17)(cid:105) (cid:17)(cid:105) (cid:17)2 (cid:16) ∆f + δ + 2 + δ f + ∆f 2 (cid:104) (cid:104) (cid:105) (cid:16) ∆f f + ∆f 2 2 − fm (cid:105) (cid:17)2 + δ f + (cid:16) ∆f (cid:104) (cid:16) ∆f (cid:17)(cid:105) 2 − fm (cid:17)2(cid:21) (cid:16) ∆f 2 − fm + δ (cid:104) f + + 2 (cid:17)(cid:105) + δ 1 + 1 + 1 + 1 (cid:104) f − ∆f (cid:104) + δ f − ∆f 2 2 (cid:105)(cid:12)(cid:12)(cid:12)2 (cid:105)(cid:12)(cid:12)(cid:12)2 df df (3.49) (3.50) Resulting in a positional uncertainty of x,4 ≥ σ2 4π2(cid:2)∆f 2 + 2f 2 c2 m (cid:3) SNR From the equations formed for the three modulation formats in (3.46), (3.48), and (3.50) it can be noticed that the bounds are are only a function of PTTW tone separation and modulation bandwidth. 3.2.1.3 Frequency Estimation Lower Bound Estimation of the frequency of a signal is dependent on the duration over which the signal is observed. In particular, the accuracy in estimating the frequency of a signal is inversely dependent on the mean-square duration of the signal (2.9). Calculating this error for the three waveforms analyzed here is generally complicated due to the fact that the signals vary in amplitude as a function of time. However, the objective is not to estimate the frequency of the entire signal, but only of the frequency reference signal modulated onto the waveform. To accomplish this, the signal is first demodulated from the PTTW signal, recovering only the frequency reference fm. For such a waveform with a rectangular envelope of duration T , the mean-square duration is given 76 by [105] yielding a frequency estimation error from (2.11) of ζ2 t = (πT )2 3 f ≥ σ2 3 (πT )2 SNR (3.51) (3.52) which is the same for all three modulated waveforms and provides the same performance as the LFMW and PTTW. 3.2.2 Experimental Setup and Signal Processing An experimental ranging system is designed to test the accuracy of estimating the range and fre- quency using the above waveforms. The transmit antenna is connected to a Keysight M8191 AWG which is used to generate the various waveforms. The waveforms are transmitted through a stan- dard gain 3.95 − 5.85 GHz horn antenna and reflected off of a corner reflector that was placed at the center of a wireless test range 4.57 m from the transmitter (see Fig. 3.14(a)). The received sig- nal is then captured using a MSO-X 92004A Keysight Infiniium High-Performance Oscilloscope. The trigger signal, which consisted of the pulse envelope of the transmitted waveform, is sent from the AWG to the oscilloscope, where an internal threshold detector is used to trigger the waveform capture. The Oscilloscope and AWG reference clocks are locked to a master 10 MHz reference produced by a Agilent MXG Analog Signal Generator. The received waveform amplitude levels are changed using different combinations of attenuators equaling 6, 9, 15, 20, and 30 dB reductions from the maximum signal strength so that the performance could be evaluated over different SNR levels. A 20 dB gain low noise amplifier (LNA) is used preceding the attenuators. The attenuators and amplifier are placed on the receive side to effect total signal amplitude including noise from the transmitter and from the receiver hardware. A block diagram of the measurement system can be seen in Fig. 3.14(b). 77 (a) (b) Figure 3.14: (a) Image of the joint ranging and frequency transfer system experimental setup in an indoor wireless test range. The distance between the transceiver and the corner reflector was 4.57 m.(b) Block diagram of measurement system used in the experiments. The transmitter, represented by the Arbitrary Waveform Generator (AWG) generates the multi-tone signal, and the oscilloscope samples the received signals. 3.2.2.1 Waveform Processing The delay, and thus the range, is measured by processing the received signal with a matched filter using an analytic copy of the transmitted signal and estimating the time of the peak in the filter output. Each captured measurement is a triggered 5 µs segment which contains a 2 µs pulse of the modulated signal surrounded by noise. Each capture is digitally filtered 10 MHz above and below the frequency band of interest. 500 of these captures are collected per waveform, per tone sepa- ration, and per SNR. A manual disambiguation process is performed by choosing the maximum of the first matched filter output then, for all following matched filter outputs, the nearest peak to 78 the first matched filter maximum. This approach allows appropriate estimation of the accuracy without requiring consideration of ambiguities. Spline interpolation is performed around the 10 points nearest to the estimated peak to alleviate quantization issues and improve the estimator per- formance. The processing gain resulting form the matched filter where BWr is the filtered receiver bandwidth yields the gains given in Table 3.1. Table 3.1: Processing Gain Tone Separation Processing Gain 50 MHz 100 MHz 22.04 dB 24.15 dB 150 MHz 25.6 dB 200 MHz 26.6 dB Frequency estimation is achieved by demodulating the 10 MHz reference signal from the PTTW signal. The received waveforms are downconverted using the frequency of lower tone of the PTTW. Once the signals are downconverted digital low-pass filtering with a 20 MHz cutoff frequency is applied to filter out higher signals and harmonics. Amplitude modulation requires an additional high-pass filter with a 5 MHz cutoff frequency to filter out additional lower frequency signals resulting from the four tone modulation format. Each capture is processed only around the 2 µs waveform duration. The frequency estimator is implemented by measuring the zero-crossings of the 10 MHz signal which is done by rectifying the signal and calculating the temporal differences in the minima of the rectified signal. The standard deviation of the zero crossings are taken for each capture, from which the overall mean was calculated. The exact modulated frequency is detected and the standard deviation is converted to an angular variation versus this frequency, yielding the total phase error due to frequency mismatch. 3.2.2.2 Experimental Results The delay estimation error for the three modulation formats is shown in Fig. 3.15, along with the theoretical lower bounds for 200 MHz tone separation calculated from (3.46), (3.48), and (3.50).The variance is expected to decrease with tone separation, since increasing the tone separa- tion serves to increase the mean-square bandwidth. It is clear that the delay estimation performance 79 tracks precisely as predicted versus SNR and tone separation, despite the presence of the 10 MHz frequency reference modulated onto the lower tone. A comparison of the performance of all three modulation formats is shown in Fig. 3.16. Effectively, the mean-square bandwidth of the signal is largely unchanged in the presence of the frequency reference, leading to consistent delay estimates. In each case, the 200 MHz tone separation achieves a delay accuracy on the order of 10−21 s2, or a range accuracy of 4.7 mm, supporting coherent operations in arrays operating at frequencies up (cid:16) (cid:17) to 4.2 GHz with degradation of less than 0.5 dB i.e. λ 15 . (cid:16) (cid:17) The measured error in the frequency estimation of each of the modulation types, along with the theoretical lower bound, is shown in Fig. 3.17. The performance is dependent on SNR as predicted, but is largely independent of the tone separation. The frequency of the higher tone is arbitrary as long as it is above the filter cutoff, however any remaining residual signal can degrade the estimation performance. In order to ensure that the coherent beam degrades by less than 0.5 dB compared to ideal coherent gain, frequency estimation of less than 18° i.e. λ 20 is required, which is represented by the dashed red line. The results indicate that this metric is achievable for all modulation formats with reasonable SNR. 80 (a) (b) (c) Figure 3.15: Measured delay estimation accuracy versus SNR with the frequency reference in- cluded using (a) single-sideband, (b) double-sideband, and (c) amplitude modulation. The sepa- ration of the ranging tones is given in the legend, with the CRB for the 200 MHz tone separation from (3.46), (3.48), and (3.50) for (a), (b), and (c) respectively plotted as the solid line. 81 Figure 3.16: Comparison of all three modulation formats showing consistent performance regard- less of the modulation format. 82 (a) (b) (c) Figure 3.17: Variance on the estimate of the 10 MHz frequency reference versus SNR using (a) single-sideband modulation, (b) double-sideband modulation, and (c) amplitude modulation. The separation of the ranging tones is given in the legend. The red dashed line indicates the 18° re- quirement to ensure P (Gc ≥ 0.9) ≈ 1 for large arrays. The CRB given by (3.52) is shown in the solide line. 83 3.3 Frequency Division: Phase Alignment for Networked Systems A common application of interest for distributed systems is communication in which in- formation is sent from point A to point B. This section discusses the use of distributed systems that are acting as a communication system in a dual radar-comm approach to the range estima- tion. Many current communication systems utilize an orthogonal frequency division multiplexing (OFDM) methodology for information transfer [84,106–109] where a deterministic preamble pro- ceeds the modulated information for synchronization purposes. These preambles present a unique set of orthogonal frequency bands for each user in the system and present a solution to simultane- ous internode measurements without the time delays needed for a time duplex scheme. This is a method to address distributed scalability without interfering with information transfer. 3.3.1 Orthogonal Frequency Division Multiplexing (OFDM) Coherent distributed arrays consisting of large numbers of nodes generally necessitate some mul- tiplexing approach to enable internode ranging between multiple node pairs. While this may be accomplished using time-domain multiplexing, where the range between each node is done in se- quence, with mobile nodes there is inherently some time limitations after which the motion of the nodes, either from intentional movement or inherent platform vibration, causes the measurement to no longer be sufficiently accurate. It is thus preferable to begin with a orthogonal frequency division multiplexing (OFDM) approach where multiple measurements can be accomplished si- multaneously, after which additional time-multiplexing may be included. OFDM contains some inherent time-domain information, since the bandwidth of the channels used for each node pair dictates limits on the waveform length. If a half-duplex system is available then there is a limitation on the minimum detectable time delay that can be measured due to bandwidth constraints such that Tmin = 2 BW where BW is the total available bandwidth. With this minimum available time the minimum detectable range can be calculated as Rmin = Tminc where c is the speed of light. This Tmin is now the lower limit of the pulse length for a half duplex 84 Figure 3.18: Channel splitting using an FDM method where the sinc(·) representation comes from the Fourier of a rectangular time domain envelope. The spacing of each channel is equivalent to 1/T to ensure that the peak is at a null of all of the other sinc(·) functions. system. For a full duplex system there is no minimum range limitation due to the ability to transmit and receive simultaneously. The dwell time for each transmission measurement is at least 2T where T is the pulse length which is fixed to the max dimension of the array such that T = Rmax c so there is sufficient time for the signal to travel the full extent of the array. Therefore the minimum waveform length is defined by the type of system that is available and by the dimensions of the array. It is to be noted that these are minimum temporal limits and in practice waveforms are much longer than this mostly due to hardware limitations. 3.3.2 IEEE 802.11 Standards and Legacy Preamble Format Here IEEE 802.11a/g/n/ac/ax standards are considered due to their general ubiquity in wireless systems. These standards utilize an OFDM architecture to modulate information onto carriers at frequencies of 2.4 GHz or 5 GHz, with bandwidths of 20 MHz, 40 MHz, 80 MHz, or 160 MHz, depending on the standard [110]. As with any WiFi format, each packet of data is preceded by a preamble to help synchronize the receiver, reduce channel noise, and reduce errors. All of the 85 Figure 3.19: Visualization of Legacy preamble. standards have the same format type known as Legacy. A visualization of the packet format using a Legacy preamble can be seen in Fig. 3.19. Synchronization of the carrier frequencies, which is performed in both non-high throughput and high throughput cases, is accomplished by the trans- mitter simultaneously bursting a symbol consisting of the set of sub-carriers that are used in the following message. This is known as the training field which can either be short or long lasting 8 µs or 16 µs and is denoted by Legacy short training field (L-STF) and Legacy long training field (L-LTF) respectively. The Legacy preamble training fields can be represented generally by ej2πnδf t (3.53) sLegacy(t) = 1√ N N−1 2(cid:88) n=− N−1 2 (cid:18) t (cid:19) rect T where N is the number of sub-carriers, T is the total waveform duration, and δf is the separation between the carrier tones. These training fields are present in each packet sent from the transmitter and therefore are a reliable source for measurement repeatability within a short time frame, assum- ing a consistent stream of information is being sent. Therefore, the training field will be used for the basis to obtain a range measurement. The time and Doppler response of using the legacy preamble for radar measurements can be observed through the ambiguity function (2.1) by 86 (cid:90) ∞ −∞ 1 N N−1 2(cid:88) n= N−1 2 (cid:18) τ − t (cid:19) T rect e−j2πnδf (τ−t) rect (cid:16) τ (cid:17) T AF (t, fD) = ej2πnδf τ ej2πfDτ dτ (3.54) The summation and integral are linear processes therefore the integration can be brought inside the summation AF (t, fD) = 1 N (cid:90) ∞ −∞ rect (cid:18) τ − t (cid:19) T (cid:16) τ (cid:17) T rect ej2πnδf t N−1 2(cid:88) n=− N−1 2 ej2πfDτ dτ (3.55) The integral, due to the rectangular time domain envelope, can be truncated due to the time limited nature of the rect(·) function with bounds [− T 2 ], yielding AF (t, fD) = 1 N = 1 N 2 2 , T (cid:90) t+ T (cid:34) − T 2 1 j2πfD ej2πnδf t ej2πnδf t ej2πfDτ dτ (cid:32) (cid:17) (cid:16) t+ T 2 j2πfD − e−jπfDT e (cid:33)(cid:35) (3.56) N−1 2(cid:88) 2(cid:88) 2 N−1 n=− N−1 n=− N−1 2 N−1 2(cid:88) n=− N−1 2 (cid:104) Extracting a factor of ejπfDt so the function in the parenthesis contains a complex exponential subtracted by its conjugate AF (t, fD) = 1 N (cid:34) (cid:16) ejπfDt j2πfD ej2πnδf t ejπfD(t+T ) − e−jπfD(t+T )(cid:17)(cid:35) (cid:16) t+T such that sinc θ = 1 j2θ (3.57) ejθ − e−jθ(cid:17) (cid:16) (cid:17)(cid:105) ejπfDt (t + T ) sinc πfD(t + T ) (3.58) A sinc(·) function can now be formed by multiplying by t+T AF (t, fD) = 1 N N−1 2(cid:88) ej2πnδf t(cid:104) n=− N−1 2 This derivation shown above represents the case of t < 0 case but is symmetric for the case of t > 0. The general case of the derivation can be represented by AF (t, fD) = 1 N ejπfDt (T − |t|) sinc πfD (T − |t|) ej2πnδf t (3.59) (cid:17)(cid:105) N−1 2(cid:88) n=− N−1 2 (cid:16) 87 The summation of the complex exponentials containing the index of summation follow the same approach as the SFW in Section 2.2.2 Ssum = Plugging (3.60) back into (3.59) AF (t, fD) = 1 N ejπfDt (T − |t|) sinc The magnitude can now be taken where (cid:16) (cid:16) (cid:17) (cid:17) sin ej2πnδf t = πN δf t sin πδf t N−1 2(cid:88) n=− N−1 2 (cid:16) πfD (T − |t|) (cid:17)sin (cid:12)(cid:12)(cid:12)ejπfDt(cid:12)(cid:12)(cid:12) = 1 resulting in [85] (cid:16) (cid:17)sin (cid:16) πfD (T − |t|) (cid:16) (cid:16) (cid:16) πN δf t sin πδf t (cid:17) (cid:17) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:17) (cid:17) πN δf t sin πδf t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) 1 N |AF (t, fD)| = (T − |t|) sinc (3.60) (3.61) (3.62) An image of an example of the legacy preamble ambiguity function along with the matched filter and Doppler responses can be seen in Fig. 3.20. 3.3.3 Theoretical ranging Accuracy of the Legacy Preamble and the Stepped-Frequency Waveform The SFW, from Section 2.2.2, is a commonly-used radar waveform.The SFW is again represented by (cid:18)t − nTr (cid:19) T rect N−1 2(cid:88) n=− N−1 2 s(t) = 1√ N ej2πnδf t (3.63) where Tr is the period between pulses, N is the number of pulses, and δf is the frequency step for every consecutive pulse. This is very similar to the model of the L-STF and L-LTF but in this case the sub-carriers are transmitted all at once rather than at different times. A time frequency plot comparing the formulation of the training field to the SFW can be seen in Fig. 3.21. The positional and velocity estimation accuracy of this waveform is given by the CRLBs in (2.12) and (2.13) respectively where the estimation ability is again solely dependent on the 88 (a) (c) (b) (d) Figure 3.20: (a) Ambiguity function of the Legacy preamble. (b) Intensity plot of the ambiguity function. (c) Matched filter of the Legacy preamble (zero Doppler cut). (d) Doppler response of the Legacy preamble (zero time cut). second moments due to preamble format being zero mean time and frequency, µf = µt = 0. The second moment of the frequency spectrum (2.5) can be approximated by delta functions in the the 89 Figure 3.21: Time-frequency plot of the Legacy preamble training field and SFW with a 50% duty cycle. frequency domain by ζ2 f = = = = = Resulting in a positional uncertainty of x ≥ σ2 (cid:82) (2πf )2|S(f )|2df (cid:82) |S(f )|2df 2(cid:80) N−1 n=− N−1 2 4π2 n=− N−1 2 N−1 (cid:82) (2πf )2 |δ (f − nδf )|2 df (cid:82) |δ (f − nδf )|2 df 2(cid:80)  N−1  2(cid:80) 4π2δf 2(cid:104) 1 12 N(cid:0)N 2 − 1(cid:1)(cid:105) π2δf 2(cid:0)N 2 − 1(cid:1) n=− N−1 2 N (nδf )2 N 3 4(cid:2)π2δf 2(cid:0)N 2 − 1(cid:1)(cid:3) SNR 3c2 90 (3.64) (3.65) The spectrum of the SFW, given the appropriate T , δf, and N, is thus identical to the spectrum of the Legacy preamble, and therefore the mean-squared bandwidth of the Legacy pream- ble is equal to that of the SFW, ζ2 f (cid:12)(cid:12)(cid:12)Legacy (cid:12)(cid:12)(cid:12)SFW = ζ2 f . (3.66) Thus the performance of a Legacy preamble as a delay (and therefore ranging) estimator is equal to that of the SFW for an equivalent SNR. The uncertainty in velocity can solved for by (2.13) and (2.9) (cid:1)(cid:12)(cid:12)2 (cid:82) (2πt)2|s(t)|2dt (cid:82) |s(t)|2dt (cid:82) (2πt)2(cid:12)(cid:12)rect(cid:0) t (cid:1)(cid:12)(cid:12)2 (cid:82)(cid:12)(cid:12)rect(cid:0) t (cid:82) T /2−T /2(2πt)2dt (cid:82) T /2−T /2 dt (cid:20)(cid:16) T (cid:17)3 −(cid:16)− T T T dt 2 2 4π2 3 dt (cid:17)3(cid:21) T 2 + T 2 π2T 2 3 ζ2 t = = = = = (3.67) (3.68) Giving a uncertainty in velocity of (cid:0)π2T 2(cid:1) SNR 3c2 v ≥ σ2 4f 2 c This has the same performance as the LFMW and PTTW for the same duration T . Comparing (3.67) to (2.56) the Legacy preamble has reduced performance in velocity estimation relative to the SFW due to the uniform distribution of energy over the time domain. To demonstrate the similarities of the ranging performance of the Legacy preamble and the SFW, a simulation of these two waveforms are shown in Fig. 3.22 where in Fig. 3.22(a) there is an example of an 802.11n waveform with 20 MHz of bandwidth and data on sub-carriers ±4, ±8, ±12, ±16, ±20, and ±24. This is then modeled as a SFW in Fig. 3.22(b) where N = 12, T = 8 µs, and δf = 1.25 MHz (4 ∗ 312.5 kHz) where 312.5 kHz is the standard 802.11n channel 91 bandwidth and the center frequency (DC at baseband after demodulation) is neglected. Both the corresponding spectrum of the L-STF and the SFW are given in Fig. 3.22(c). The normalized matched filter output can be seen in Fig. 3.22(d) for both waveforms. The matched filter produces the same output shape for both the L-STF and the SFW which, therefore, reinforces the conclusion that they provide the same accuracy as a time delay estimator. Using the Legacy preamble over the SFW as an estimator entails a processing gain re- duction by a factor of N due to the reduction in time-bandwidth product. However, it should be noted that temporal pulse compression of multitone signals is not classically done in radar sys- tems. Phase-modulated waveforms are prevalent in radar, since such signals can be implemented with constant amplitude, and transmitters can then be operated at a maximum power level at all times (typically near the 1− dB compression point where efficiency is high), yielding optimal sen- sitivity. Multitone signals, however, are amplitude modulated, which necessitates operating trans- mitters below the compression point. Communications systems commonly operate in this region due to the use of signals such as quadrature amplitude modulation (QAM), thus pulse compression of Legacy preambles is feasible when implemented in communications systems. 3.3.4 Experimental Validation To measure the variance of the Legacy preamble as a range estimator, a wireless experiment is conducted in a semi-enclosed antenna range. The 802.11 waveform is produced using MATLAB’s WLAN toolbox on a host computer connected to a Keysight M8191 AWG and is received by a Keysight MSO-X 92004A Oscilloscope. Taking into consideration the memory depth of the scope, the 802.11 waveform with the shortest time duration is chosen for this ranging experiment. This correlates to the standard that supports the largest bandwidth of 160 MHz and therefore 802.11ac is chosen and given a simple payload of [1 0 0 1]. For other formats of 802.11 with the Legacy preamble, the results scale proportionally with the supported bandwidth as the method of informa- tion modulation does not interfere with the range estimation process. An image of the time and frequency domain representations of the waveform used for the ranging experiment can be seen in 92 (a) (b) (c) (d) Figure 3.22: (a) Simulated IEEE 802.11n waveform with 20 MHz bandwidth with payload (blue) and L-STF preamble (red). (b) SFW with Tr = 8 µs with a 50% duty cycle and containing the 12 data sub-carriers present in the L-STF. (c) Spectrum of both the 802.11n L-STF and the SFW. (d) Normalized zero time delay matched filter output for both L-STF and SFW demonstrating the similarities in the temporal waveform shapes. 93 Figs. 3.23(a) and 3.23(b) respectively along with the resulting matched filter output in Fig. 3.23(c). There are only two channels in the 802.11ac format that support 160 MHz bandwidth operation, channels 50 and 114 centered at 5.25 GHz and 5.57 GHz respectively. Channel 50 is chosen for this experiment. The baseband signal is upconverted to a RF carrier using a double balanced mixer with 10 dB insertion loss. The LO signal was generated by a Agilent MXG Analog signal gen- erator. To counteract the loss of the mixer, the RF signal is amplified on both of the transmit and receive ends by a 22 dB power amplifier and a 17 dB LNA respectively. The internal clock fre- quencies of the AWG, scope, and signal generator are all connected via SMA cables to ensure that all operations are derived from the same reference signal. The transmit and receive antennas are two standard gain 3.95− 5.85 GHz horn antennas that are placed close together in a quasi-monostatic fashion at the edge of the semi-enclosed arch range. A corner reflector was placed in the center of the range at approximately 1.5 m distance from the ranging system. An image of the block diagram of the setup along with an image of the experi- mental setup in the semi-enclosed arch range can be seen in Figs. 3.24(a) and 3.24(b) respectively. The SNR of the received signal is changed using attenuators on the input to the scope. The scope capture is triggered by an internal threshold detection from a rectangular pulse sent by AWG and the received waveform is downloaded to MATLAB where the matched filter and range estimation processing is implemented offline. The peak of each matched filter output is upsampled by 1000 points using spline interpolation. At each attenuation level the variance of 100 estimated values are calculated, and the SNR is estimated using the eigenvalue decomposition process from Section 2.1.4. The resulting measured variances, simulations of the performance of the 802.11ac waveform along with the corresponding SFW, and the theoretical bound can be seen in Fig. 3.24(c). At all SNR values the SFW and Legacy preamble achieved similar ranging performance in simulation, supporting the use of the Legacy preamble as a ranging waveform. The measured Legacy preamble ranging performance closely matches the simulation, and is comparable to the lower bound. At 50 dB postprocessing SNR (19 dB preprocessing SNR and 31 dB of processing gain determined from the time-bandwidth product) the Legacy preamble was able to obtaina two-way ranging error 94 (a) (b) (c) Figure 3.23: (a) Measured IEEE 802.11ac waveform with 160 MHz bandwidth. (b) Measured spectrum of the waveform. (c) Measured matched filter output. of σx = 1.9 mm providing a maximum beamforming frequency of 10.5 GHz with the accuracy of λ 15 of the coherent transmitted signal. 95 (a) (b) (c) Figure 3.24: (a) Block diagram of the ranging experiment (AWG = arbitrary waveform generator, LNA = low-noise amplifier). The corner reflector was placed a distance of 1.5 m from the ranging system. Delay estimation processing was implemented offline in MATLAB. (b) Image of experi- mental setup up in the semi-enclosed arch range. (c) Measurement variances along with simulated data for 1000 Monte Carlo iterations over various SNR values. The simulated performance of the SFW and Legacy preamble waveforms yielded nearly identical delay estimation, verifying the theoretical similarities of the lower bounds described earlier. The measured performance of the Legacy preamble waveform was close to the simulated performance and comparable to the lower bound (CRLB). The Legacy preamble waveform achieved a lowest ranging error of 1.9mm with a preprocessing SNR of 19 dB combined with 31 dB of processing gain for a total SNR of 50 dB. 3.4 Pulse Encoding: Two-Tone Stepped Frequency Waveform (TTSFW) This section I analyze another method of using frequency division to provide simulta- neous high accuracy range measurements between multiple nodes for a distributed array. Here 96 a pulse encoded waveform, based off of the spectrally sparse PTTW from Section 2.2.3 and the disambiguation properties of the SFW from Section 2.2.2, is implemented in a one, two, and three node system. It is shown that this waveform has scalability potential of N ! where N is the num- ber of pulses providing each unique primary to secondary node connection with a unique pulse structure to look for on return utilizing the same pulse structure and supporting simultaneous range measurements with equivalent accuracy to multiple node pairs without the use of time scheduling. 3.4.1 Frequency Domain Multiplexing (FDM) Approach Similar to the networked system, this method uses a frequency division method but here the domain is split in two to accommodate the structure of the PTTW. Once the limit on temporal length, T , of the waveform is determined, the bandwidth that each pulse occupies can be estimated as fch = 1 T . Using frequency domain multiplexing (FDM) the number of simultaneous connections can be estimated as m = BW fch where BW is the total waveform bandwidth [111]. It can now be assumed that m connections can be made in 2T accounting for two way propagation. Given metrics relative to the coherence time of the channel between the elements, such as the vibration profile of the platforms, a desired update rate ∆t can be chosen. The total number of connectable nodes can then be derived from the number of m connections that can be made in ∆t. An estimation of this number can be expressed as N = BW ∆t 2 (3.69) Once the domain is divided into bands, two bands are designated for a given node pair to form a channel. Channels are given two bands to support the spectral sparsity of the two-tone structure of this waveform, as described in this section. To prevent measurement bias, the channels are sequentially organized such that the tone separation of each channel remains constant (see Fig. 3.25). 97 Figure 3.25: Channel splitting using an FDM method for a two-tone system where half of the bandwidth is allocated to the first tone, − ∆f 2 , of the N channels and the second half is a allocated to the upper tone, ∆f 2 . 3.4.2 Waveform Design The multi-node high-accuracy ranging waveform is based on a spectrally-sparse, two-tone wave- form. In this section, the multi-node stepped-frequency waveform is derived and discussed. To combine the advantage of both the resolution of the PTTW and the disambiguation ability of the SFW, a two-tone stepped-frequency waveform (TTSFW) is developed. This is done by first choos- ing the individual pulse bandwidth, ∆f, then monotonically increasing these frequencies by δf keeping the bandwidth of every pulse the same. This waveform can be scaled similarly to that of the PTTW but instead of having only one pulse, here the waveform includes an arbitrary N pulses, essentially stitching N different PTTW pulses into one waveform. Since more resources are being used per node connection it is intuitive that the scalability would decrease, however this is not the case. In fact the same waveform with N pulses can be used to service N ! connections. This is done by shifting the order of pulses (frequency steps) so that there are N ! unique waveforms utilizing the same pulses, each 98 (a) (b) Figure 3.26: (a) Two-tone stepped-frequency waveform in the time domain. (a) TTSFW in the time-frequency domain in a different order. The baseband TTSFW signal can be modeled as s(t) = √ 1 2 N N−1 2(cid:88) n=− N−1 2 (cid:18) t − nTr (cid:19)(cid:16) e−jπ∆f t + ejπ∆f t(cid:17) rect T ej2πnδf t (3.70) Using the scalability approach described above, the frequency step and pulse bandwidth can be expressed in terms of the full system bandwidth as δf = BW 2N−1 . An image of the time frequency spectrum along with the waveform in the time domain can be seen in 2N−1 and ∆f = N δf = N BW Fig. 3.26. The disambiguation properties of the waveform, along with its response to both time delay and Doppler, can be observed from the ambiguity function of the TTSFW. Inputting the time domain representation of the TTSFW in (3.70) into the ambiguity function in (2.1) T 99 AF (t, fD) = 1 4N (cid:90) ∞ −∞ N−1 2(cid:88) rect n=− N−1 2 (cid:18) τ − t − nTr (cid:19)(cid:16) (cid:18) τ − nTr T (cid:19)(cid:16) ejπ∆f (τ−t) + e−jπ∆f (τ−t)(cid:17) e−jπ∆f τ + ejπ∆f τ(cid:17) × rect ×e−j2πnδf (τ−t) ej2πnδf τ ej2πfDτ dτ (3.71) The summation and integral are linear processes therefore the integration can be brought inside the summation AF (t, fD) = 1 4N N−1 2(cid:88) n=− N−1 2 ej2πnδf t (cid:90) ∞ −∞ rect × rect (cid:18) τ − t − nTr (cid:19)(cid:16) ejπ∆f (τ−t) + e−jπ∆f (τ−t)(cid:17) (cid:18) τ − nTr (cid:19)(cid:16) e−jπ∆f τ + ejπ∆f τ(cid:17) ej2πfDτ dτ T T (3.72) A substitution can be made such that ˆτ = τ − nTr AF (t, fD) = 1 4N N−1 2(cid:88) n=− N−1 2 (cid:90) ∞ −∞ rect (cid:18) ˆτ (cid:19)(cid:16) (cid:19)(cid:16) ejπ∆f (ˆτ +nTr−t) + e−jπ∆f (ˆτ +nTr−t)(cid:17) (cid:18) ˆτ − t e−jπ∆f (ˆτ +nTr) + ejπ∆f (ˆτ +nTr)(cid:17) ej2πfD(ˆτ +nTr)dˆτ T ej2πnδf t × rect T Truncation of the integral around [− T envelope, yielding 2 , T (3.73) 2 ] can be performed time limited nature of the rect(·) AF (t, fD) = 1 4N = 1 4N N−1 2(cid:88) n=− N−1 2 N−1 2(cid:88) n=− N−1 2 (cid:34) (cid:90) t+ T 2 − T 2 ej2πn(δf t+fDTr) e−jπ∆f t + ejπ∆f t + ejπ∆f [2(ˆτ +nTr)−t] (cid:34) ej2πn(δf t+fDTr) ejπ∆f (2nTr−t) j2π (∆f + fD) ejπ∆f (t−2nTr) j2π (fD − ∆f ) + + + e−jπ∆f [2(ˆτ +nTr)−t] (cid:17) (cid:16) j2πfD e t+ T 2 j2πfD (cid:32) e−jπ∆f t + ejπ∆f t (cid:16) (cid:16) j2π(fD−∆f) e j2π(∆f +fD) e (cid:32) (cid:32) (cid:17) (cid:17) t+ T 2 − e−jπ(∆f +fD)T t+ T 2 − e−jπ(fD−∆f)T (cid:35) ej2πfD ˆτ dˆτ (cid:33) − e−jπfDT (cid:33) (cid:33)(cid:35) (3.74) 100 Extracting a factor from each function so that every parenthesis contains a complex exponential subtracted by its conjugate N−1 AF (t, fD) = 1 4N 2(cid:88) n=− N−1 2 ej2πn(δf t+fDTr) ejπ∆f (2nTr−t)ejπ(∆f +fD)t + ejπ∆f (t−2nTr)ejπ(fD−∆f)t j2π (∆f + fD) (cid:16) + j2π (fD − ∆f ) (cid:34)(cid:16) e−jπ∆f t + ejπ∆f t(cid:17) ejπfDt j2πfD ×(cid:16) ejπfD(t+T ) − e−jπfD(t+T )(cid:17) ejπ(∆f +fD)(t+T ) − e−jπ(∆f +fD)(t+T )(cid:17) (cid:16) ejπ(fD−∆f)(t+T ) − e−jπ(fD−∆f)(t+T )(cid:17)(cid:35) ejπ∆f t cancels in the second and third terms and sinc functions can now be formed by multiplying each function by t+T t+T such that sinc(θ) = 1 j2θ AF (t, fD) = 1 4N ej2πn(δf t+fDTr) N−1 2(cid:88) n=− N−1 2 (cid:16) ejθ − e−jθ(cid:17) (cid:34)(cid:16) e−jπ∆f t + ejπ∆f t(cid:17) (cid:16) ×ejπfDt (t + T ) sinc +ejπ(fDt+2nTr) (t + T ) sinc +ejπ(fDt−2nTr) (t + T ) sinc (cid:16) The terms ejπfDt and (t + T ) can then be extracted from each term. AF (t, fD) = ejπfDt (t + T ) 4N ej2πn(δf t+fDTr) N−1 2(cid:88) n=− N−1 2 (3.75) (3.76) (cid:17) (cid:17) (cid:17)(cid:35) (3.77) (cid:17) (cid:17) (cid:17)(cid:35) πfD(t + T ) (cid:16) π (∆f + fD) (t + T ) π (fD − ∆f ) (t + T ) (cid:34)(cid:16) e−jπ∆f t + ejπ∆f t(cid:17) (cid:16) × sinc πfD(t + T ) (cid:16) π (∆f + fD) (t + T ) π (fD − ∆f ) (t + T ) +ej2πnTr sinc +e−j2πnTr sinc (cid:16) 101 The summations of the complex exponentials containing the index of summation follow the same approach as the SFW in Section 2.2.2 where ej2πn(δf t+fDTr) = πN (δf t + fDTr) π (δf t + fDTr) (cid:17) (cid:17) (3.78) sin sin πTr πN Tr (cid:16) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:17) (cid:40)(cid:16) (cid:17) sin sin e±j2πnTr = (cid:16) (cid:16) (cid:16) N−1 2(cid:88) 2(cid:88) 2 N−1 n=− N−1 n=− N−1 2 Ssum = Ssum = 4N (cid:16) (cid:16) (cid:17) (cid:17) (cid:104) sin + πN Tr sin πTr Plugging these values of the summations back into (3.77) AF (t, fD) = ejπfDt (t + T ) sin πN (δf t + fDTr) sin π (δf t + fDTr) e−jπ∆f t + ejπ∆f t(cid:17) (cid:16) × sinc (cid:16) (cid:17) (cid:17)(cid:105)(cid:41) πfD(t + T ) sinc π (∆f + fD) (t + T ) + sinc π (fD − ∆f ) (t + T ) (3.79) + 4N sinc (cid:16) (cid:17) sin sin 2 sinc sin πTr sin πN Tr This derivation shown above represents the case of t < 0 case but is symmetric for the case of (cid:16) (cid:16) (cid:17) (cid:17) (cid:40) (cid:17) π (δf t + fDTr) πN (δf t + fDTr) π (∆f + fD) (T − |t|) t > 0. The general case of the magnitude derivation can be represented by |AF (t, fD)| = (cid:16) (cid:16) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(T − |t|) (cid:17) (cid:17)(cid:105)(cid:41)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:17) (cid:104) (cid:16) (cid:12)(cid:12)(cid:12)e−jπ∆f + ejπ∆f(cid:12)(cid:12)(cid:12) = 2. A plot of this function can be seen in Fig. 3.27. (cid:12)(cid:12)(cid:12)ejπfDt(cid:12)(cid:12)(cid:12) = 1 and πfD (T − |t|) (cid:16) where Looking at the fD = 0 cut of the ambiguity function, this function is maximized at every t = nN ∆f . Since the lobing pattern of the PTTW occurs at every t = n ∆f the disambiguation properties of this waveform are now apparent: for an N pulse system, there is N − 1 consecutive lobes notched out of the matched filter output of the PTTW with the same ∆f. It is important to note, however, π (fD − ∆f ) (T − |t|) (3.80) + sinc 102 (a) (c) (b) (d) Figure 3.27: (a) Ambiguity function of the TTSFW. (b) Intensity plot of the ambiguity function. (c) Matched filter of the TTSFW (zero Doppler cut). (d) Doppler response of TTSFW (zero time cut). that the spectrally sparse nature of the TTSFW waveform, along with the ability to simultaneously transmit multiple waveforms simultaneously, makes the TTSFW more beneficial to implement than a traditional LFMW. The uncertainty of the estimation ability of the TTSFW can be derived using (2.12) and (2.13), where again this waveform is fundamentally zero mean frequency and zero mean time duration, µt = µf = 0, making the CRLB solely dependent on the second moment. The mean- squared bandwidth, for the uncertainty in position, can be solved for by making the approximation that the spectrum of each pulse of the TTSFW consists of two delta functions at − ∆f 2 + nδf and 103 (cid:17)(cid:105) + δ + δ f −(cid:16) ∆f (cid:104) (cid:104) f −(cid:16) ∆f (cid:17)2 2 + nδf (cid:17)(cid:105)(cid:12)(cid:12)(cid:12) df (cid:17)(cid:105)(cid:12)(cid:12)(cid:12) df 2 + nδf 2 + nδf ∆f 2 + nδf such that n=0 f + (cid:82) (2πf )2|S(f )|2df (cid:82) |S(f )|2df (cid:82) (2πf )2(cid:12)(cid:12)(cid:12)δ (cid:104) (cid:16) ∆f N−1(cid:80) 2 − nδf (cid:82)(cid:12)(cid:12)(cid:12)δ (cid:104) (cid:16) ∆f (cid:17)(cid:105) N−1(cid:80) 2 − nδf (cid:17)2 (cid:16) ∆f (cid:16) ∆f N−1(cid:80) 2 − nδf N−1(cid:80) (cid:16) ∆f (cid:17)2 N−1(cid:80) (1 + 1) 4π2 f + n=0 n=0 n=0 + 8π2 n=0 ζ2 f = = = = 2 + (nδf )2 2N (2πδf )2 (cid:20) 1 (cid:16) (πδf )2(cid:0)N 2 − 1(cid:1) 12 N N (cid:17)(cid:21) N 2 − 1 = π2∆f 2 + = π2∆f 2 + (3.81) (3.82) (3.83) The definitions for ∆f = N δf = N BW 2N−1 = BW 2− 1 N these into the mean squared bandwidth and δf = BW 2N−1 can now be applied. Plugging 3 (cid:32) (cid:33)2 + (πBW )2(cid:0)N 2 − 1(cid:1) 3(cid:0)4N 2 + 4N + 1(cid:1) 2 N 2−1 4N 2+4N +1 (πBW )2(cid:16) (cid:16) (cid:17) (cid:17) c2 3 SNR ζ2 f = π2 BW 2 − 1 N (cid:32) π2 x ≥ σ2 4 (cid:33)2 BW 2− 1 N + This gives a positional uncertainty of from (2.12) as Looking at the limits of the mean-squared bandwidth, if N = 1 then the second term becomes zero and the first term becomes π2BW 2. This is exactly the result found in (2.65) in the two-tone case which is to be expected since when N = 1 the TTSFW reduces to a two-tone waveform. Looking at the opposite bound, when N = ∞, the the first term approaches (πBW )2 4 (3.84) (cid:32) π2 lim N→∞ (cid:33)2 = BW 2 − 1 N 104 and the second term (cid:34) 3(cid:0)4N 2 + 4N + 1(cid:1)(cid:35) (cid:0)N 2 − 1(cid:1) = (πBW )2 3 (πBW )2 lim N→∞ (3.85) The term in (3.84) is the two tone case when the individual pulse bandwidth is BW 2 . This follows the scalability approach described previously for a large number N (see Fig. 3.25). The term in (3.85) is exactly the bound that is obtained by using a LFMW signal with equal amplitude across the frequency band BW [105]. This is again to be expected due to the equal distribution of power over the spectrum emulating the approximation used to derive the mean-squared bandwidth of (cid:104)− BW (cid:105) the LFMW (i.e. a rectangular function bounded by 2 , BW 2 ). It can be deduced that the first term in (3.82) is a measure of the mean squared bandwidth of the individual pulses and the second term is a measure of the mean squared bandwidth of the overall waveform. A plot of the TTSFW’s accuracy vs. number of pulses used can be seen in Fig. 3.28(a). From this figure, the best obtainable bound is for when N = 1 which is the two-tone case. The value of the mean-squared bandwidth then increases in a logarithmic fashion until the upper bound of 7(πBW )2 approximately N = 20. To test the robustness of the waveform as the number of pulses increases, is reached at 12 1000 Monte Carlo simulations are performed with 4 MHz of total bandwidth at approximately 30 dB preprocessing SNR and duration of 500 µs sampled at 25 MHz. This is to emulate what is achievable by the measurements in the following section. The simulated results as a function of the number of pulses, N, can also be seen on the Fig. 3.28(a) while a plot of the simulation with a fixed N and variable SNR can be seen in Fig. 3.28(b). The uncertainty in velocity follows the same derivation in Section 2.2.2 where the time characteristics of a rectangular pulse train used as an envelope are the same as the SFW and results in v ≥ σ2 c π2(cid:2)T 2 r 4f 2 (cid:0)N 2 − 1(cid:1) + T 2(cid:3) SNR c2 (3.86) A comparison of the beneficial attributes of the TTSFW in comparison to the waveforms derived in Chapter 2 can be seen in Table 3.2. 105 (a) (b) Figure 3.28: (a) The bounds and simulation of TTSFW vs. number of pulse with 4 MHz of band- width total bandwidth and 30 dB preprocessing SNR. (b) The bounds and simulation of TTSFW vs. SNR with the pulse number fixed to N = 4. Table 3.2: Comparison of discussed waveform attributes Rank based on CRLB (lowest to highest) Waveform Scalable Unambiguous 1 2 3 4 Full Bandwidth Two Tone TTSFW PTTW Full Bandwidth LFMW No Yes Yes No No Yes No Yes 3.4.3 Measurements 3.4.3.1 Waveform Properties A comparison test for the accuracy of the TTSFW and the PTTW was performed using an Ettus X310 software defined radio (SDR) which has an operational bandwidth of 10 MHz – 6 GHz, and an instantaneous bandwidth of 160 MHz. The SDR interfaces with a host computer though a 10 GHz Ethernet cable and is processed using LabView. Due to processing rate limitations the sampling rate of the SDR is confined to 25 MHz. To ensure that the resulting matched filter digitization is too coarse, both waveforms are taken to have equivalent pulse bandwidth ∆f to be 106 (a) (b) Figure 3.29: (a) Measured two pulse TTSFW in time domain. frequency domain. (b) Two pulse TTSFW in the Figure 3.30: Measured matched filter output of the PTTW and TTSFW waveforms produced on the X310 using two pulses 4 MHz and a pulse duration of 1 ms with duty cycle of 50%. To confirm that both waveforms have the same integration time of 0.5 ms, the time is split into two equal 0.25 ms pulses of the TTSFW. The corresponding bandwidth ∆f for the TTSFW is 2 MHz which means that every other lobe of the the PTTW is notched out. An image of the TTSFW in the time domain and in the frequency domain can be seen in Fig. 3.29 and an image of the comparison of the matched filter outputs of the TTSFW and PTTW can be seen in Fig. 3.30. 107 Figure 3.31: Comparison of variance of PTTW and TTSFW with equal ∆f to the CRLB of the TTSFW. Measurements are taken using a loopback architecture, where the transmit channel on the SDR is connected directly to the receive channel via a SMA cable. The received waveform is passed through a matched filter and then interpolated eight times using a built-in LabView spline function to improve the accuracy of the measurement. The interpolation was limited to 8 due to processing limitations and the desire to keep the processing running real-time. Increased laten- cies resulted in dropped packets, leads to incorrect measurements. The peak of 100 matched filter outputs are recorded and averaged together. The received signal SNR is estimated using the eigen- value decomposition approach for Section 2.1.4. The processing gain resulting from using the matched filter for the case of T = 0.5 ms and a receiver bandwidth of BWr = 12.5 MHz is 38 dB. A comparison of the variance of the two pulse TTSFW and a PTTW with equivalent ∆f vs the derived CRLB for the TTSFW in (3.83) can be seen in Fig. 3.31. The variances are comparable indicating that the accuracy obtained by the TTSFW is equivalent to that of the PTTW with equal ∆f. 108 (a) (b) Figure 3.32: (a) Measured four pulse TTSFW in time domain. frequency domain. (b) Four pulse TTSFW in the 3.4.3.2 Wireless Ranging Measurements Wireless ranging measurements are taken on a 24 ft semi-enclosed arch range with a corner reflec- tor or an SDR-based active repeater placed in the middle of the range. Horn antennas are placed on the edge of the arch using a wideband (2 − 12 GHz) standard gain horn antenna as the transmitter and a narrowband (3.5 − 5.5 GHz) standard gain horn antenna as the receiver. An Ettus X310 SDR is used to create the TTSFW. The horn antenna have greater directivity at higher frequencies which helps with multipath error from the lab environment. For this reason the center frequency is chosen to be 5.25 GHz due to the limiting bandwidth of the narrowband horns. The ranging waveform consists of a four-pulse TTSFW with each pulse bandwidth equal to ∆f = 4 MHz and increasing δf = 1 MHz with every consecutive pulse. The sampling fre- quency is chosen to be 25 MHz which is the highest obtainable stable sample rate with the equip- ment setup. The total waveform duration is 1 ms, in which each pulse has a duration of 250 µs and a duty cycle of 50%. These waveforms have a preprocessing SNR of approximately 30 dB. An image of the waveform in the time domain and frequency domain can be seen in Fig. 4.1. The disambiguation properties of the four pulse waveform can be seen in Fig. 3.33. 109 Figure 3.33: Measured matched filter output of the PTTW and TTSFW waveforms produced on the X310 using four pulses. The first measurement is taken using a single SDR transmitting to a corner reflector. This experiment is representative of a single secondary node where a secondary node determines its relative range to a single point in the array. The corner reflector is placed at the far end of the range (15 ft from the transmitter) and moved 10 ft towards the transmitter in 10 in increments. The return waveform is then passed through the matched filter and interpolated 8 times to improve the accuracy. The peak of the matched filter is then selected by using a simple peak finding operation since the properties of the TTSFW take care of the disambiguation. At each distance 100 matched filter peaks are averaged together. After the measurement is taken and the position values are computed, a simple post- processing calibration procedure is implemented to account for any static delays that are inherent to the system. The calibration procedure is performed by taking the average of the differences between the expected value and the measured value at each point and subtracting this average from all points. An image of the performance of a single SDR can be seen in Fig. 3.34. The second measurement is to test the performance of multiple SDRs ranging to a corner reflector simultaneously. This experiment is representative of two secondary nodes performing simultaneous ranging. To do this measurement, all of the experiment parameters from the single 110 (a) (b) (c) (d) Figure 3.34: (a) Schematic of single SDR in the arch range. (b) Image of experimental setup. (c) Range Measurement results. (d) Variance of measurement. 111 SDR case remained the same, however two separate SDRs and sets of horn antennas are used. The reference oscillators of the two SDRs are locked via cable; in future work this will be implemented with a wireless link. The two sets of antennas are placed in the same area at different elevations. The waveforms used by two SDRs contains the same set of frequencies but where the first SDR steps up in frequency, the second steps down in frequency, implementing the different step cycles mentioned in the previous section. This results in no interference between the waveforms, and in turn allows for no needed coordination of the waveform start times between the SDRs. An image of the performance of both SDRs can be seen in Fig. 3.35. The third measurement is conducted to demonstrate the scalability of the waveform to three separate SDRs. The corner reflector is removed from the setup and replaced by an SDR acting This helps to increase the SNR of the received signals by changing the signal decay from 1 as an active repeater that capturs the incoming signals and retransmits them with increased gain. r4 to 1 r2 due to the retransmit gain, and further reduce the effects of multipath. This experiment is represen- tative of two secondary nodes ranging to a primary node which here is given by the repeater. In a fully distributed array this primary node would be sending out the reference beamforming signal in which the secondary nodes, now with knowledge of the distance to the primary, phase adjust their output creating a coherent system. The center frequencies of transmit and receive are separated to also help with multipath issues. The two SDRs on the side of the range are transmitting at 5 GHz and the repeater responds at 5.25 GHz. The performance at each distance can be seen in Fig. 3.36. Clearly, the ranging waveform produces accurate range measurements, below 1 in2 variance for multiple nodes simultaneously. The achieved ranging accuracy of σx = 8.03 mm (or 0.1 in2) enables coherent transmission up to 2.49 GHz in a two-way ranging measurement, following from the requirement that the error is within λ 15 of the coherent transmission signal. 112 (a) (b) (c) (d) Figure 3.35: (a) Schematic of two SDRs in the arch range. (b) Image of experimental setup. (c) Range Measurement results. (d) Variance of measurement. 113 (a) (b) (c) (d) Figure 3.36: (a) Schematic of three SDRs in the arch range. (b) Image of experimental setup. (c) Range Measurement results. (d) Variance of measurement. 114 3.5 Comparison to Traditional Wideband Ranging Methods Traditional hardware-based approaches to high-accuracy ranging entail the use of wide- band waveforms such as frequency modulated continuous wave (FMCW) waveforms or pulsed LFMW. The accuracy of the estimate of the delay (range) does scale inversely with increasing bandwidth, as indicated by (2.12) and (2.11), however the waveforms typically used for millimeter- scale ranging use bandwidths of hundreds of MHz or even GHz, which generally increase cost and can lead to non-idealities such as passband ripple or increased insertion loss from wideband impedance matching. In contrast, spectrally-sparse waveforms like those demonstrated herein can be implemented with individually narrow-band hardware channels, easing cost and reducing some of the undesirable hardware imperfections of wideband systems. The receiver hardware can also be simplified by leveraging the spectrally sparse nature of the waveforms; the ability to subsample spectrally sparse signals with a low-rate digitizer and digitally reconstruct the waveform to achieve sub-mm ranging accuracy has been previously demonstrated [52]. High-accuracy ranging is typically approached by using wideband waveforms which achieve both good resolution and good accuracy. While resolution is dependent on overall band- width, the accuracy is dependent on the mean-square bandwidth as shown earlier, and as such the accuracy can be improved with spectrally-sparse waveforms [86]. Commonly used waveforms include FMCW and ultra-wideband (UWB) waveforms. A one-dimensional high-accuracy rang- ing system was developed using FMCW waveform in [112] with a ranging standard deviation of 0.1 mm. The waveform had 2 GHz bandwidth with 2 ms sweep time at a center frequency of 75 GHz, with range estimates averaged over 2000 samples, increasing the SNR by 16.5 dB. This system was tested for a distance up to 1.43 m in an anechoic chamber. Three-dimensional ranging was implemented in [113] using UWB waveforms with a frequency range of 5.4 to 10.6 GHz, demonstrating dynamic tracking with a root mean square error (RMSE) of 5.24 mm, and static positioning RMSE of 1.98 mm using averaging over 106 samples. An accuracy of 0.1 mm was also reported for other short-range localization radars [114,115]. Other radars using smaller band- widths yielded less accurate measurements, e.g. in [116] a waveform with 500 MHz bandwidth 115 was used on a center frequency of 23.75 GHz to achieve a range resolution of 30 cm. Compared to traditional ranging methods, the use of spectrally-sparse waveforms has benefits not only in terms of accuracy, but also hardware implementation; as described later, the challenge of implementing instantaneously wideband signals in hardware is alleviated with the use of only a few frequency tones. In particular, the basic two-tone ranging approach enables direct scalability, by changing the separation of the two signal tones. Previously, the ability to estimate range with 0.3 mm accuracy using only two frequency tones separated by up to 4 GHz was demonstrated [117]. A comparison of the results of my approaches with the literature using a figure of merit (FOM) aimed at including the benefits of spectral sparsity and ranging accuracy. The FOM is defined as the accuracy per unit occupied bandwidth, or the product of the waveform bandwidth, spectral occupancy (the fraction of the total bandwidth occupied by the waveform), and ranging accuracy. The FOM and other parameters are compared in Table 3.3. The spectral occupancy is calculated by the fraction of the bandwidth of the waveform tones relative to the total (cid:16) (cid:17) bandwidth i.e. 1 T . Table 3.3: Comparison of Waveform Performance Ref. [112] [113] [114] [115] [116] Waveform FMCW UWB FMCW Spread Spectrum FMCW Range-Doppler S. M. Ellison S. M. Ellison Joint Range-Frequency S. M. Ellison S. M. Ellison Rad-Comm TTSFW Bandwidth (MHz) Spectral Occupancy (%) Range Accuracy (mm) FOMa (mm/MHz) 2000 5200 2000 3120 500 120 200 160 6 100 100 100 100 100 0.83 0.75 18.75 0.27 0.1 5.24 0.1 0.1 300 3.1 4.7 1.9 8.03 200 27,248 200 312 150,000 3 7 57 0.13 aFOM = bandwidth × spectral occupancy × range accuracy 116 CHAPTER 4 DEMONSTRATION OF WIRELESS PHASE ALIGNMENT IN DISTRIBUTED BEAMFORMING In this chapter I demonstrate a two- and three-node open-loop coherent distributed beamforming system consisting of one primary node and one to two secondary nodes. To estimate the internode ranging (i.e. the term d in (2.15)) in the Section 2.2.3, I investigated the use of spectrally-sparse waveforms for high-accuracy ranging, showing that a two-tone waveform obtains near-optimal ranging accuracy. Addressing the ambiguity and scalability challenges that arise with a simple two-tone waveform, I developed a TTSFW in Section 3.4. This chapter expands on the prior derived waveform, TTSFW, to enable arbitrary beamforming in an distributed open-loop sense by implementing the range estimations to maintain phase alignment for a dynamic array operating with a continuous wave beamforming signal at 1.5 GHz. 4.1 Range Estimation Waveform Due to the need for scalability the TTSFW is used. In the case of a two-node array, consisting of a primary node and a single secondary node, only a single pulse N = 1 is required since only a single unique connection is made. Fig. 4.1 shows an example implementation of this waveform in the time and frequency domains. For this case the baseband waveform has a duration of 1 ms with a 50% duty cycle and frequencies f1 = 500 kHz and f2 = 11.5 MHz, which matches the bandwidth that has been demonstrated to obtain high ranging accuracy from Section 3.4. In the case of a three-node array, consisting of a primary node and two secondary nodes, there are two unique connections and therefore a multipulse waveform is required such that N ≥ 2. I demonstrate here a waveform supporting more than three nodes, with N = 5 to illustrate the scalability. An image of the waveform in the time domain and frequency domain can be seen in Fig. 4.2. The time duration of the baseband waveform was 200 µs per pulse with a 50% duty cycle, frequencies f1 = 500 kHz and f2 = 5.5 MHz, and a step size of δf = 1 MHz. To ensure that 117 (a) (b) Figure 4.1: Measured waveform supporting ranging between two nodes: (a) time domain; (b) frequency domain. each connection has a unique pulse signature, one secondary node begins its waveform with the pulse containing the lowest frequency pair of tones and increases the frequency by the frequency step, while the second secondary node begins with the pulse with the highest frequency pair and decreases the frequency by the frequency step; this approach to unique pulse-to-pulse signatures per node is easily extendable. The pulse labels in Fig. 4.2(b) given by Pulse a − b indicate the pulse order relative to the secondary such that the index a is associated with secondary node one while b is associated with secondary node two. 4.2 Ranging Requirements for Open-Loop Distributed Beamforming Inherent uncertainly is present in any system that attempts to estimate a parameter due to the presence of random noise. A measure of the variance or stability of an estimate of a random 118 (a) (b) Figure 4.2: Measured waveform supporting ranging between five nodes: (a) time domain; (b) frequency domain. variable is given by the CRLB, which for position estimation is given by (2.12). The mean-square bandwidth for the two waveforms, which will dictate the CRLB due to this waveform inherently having a zero first moment in both domains, with the aforementioned parameters yields (cid:12)(cid:12)(cid:12)N =1 (cid:12)(cid:12)(cid:12)N =5 ζ2 f,2 node ζ2 f,3 node = π2BW 2 = 1.942 × 1015 = 507π2BW 2 1000 = 6.0547 × 1014 The lower bound on delay estimation for the two waveforms can now be evaluated con- sidering an SNR of 30 dB; this closely matches that can be obtained in typical cooperative ranging techniques. Note that, unlike a traditional radar ranging measurement which undergoes propaga- tion losses in both directions, the cooperative ranging only suffers losses in one direction, with the 119 primary node repeating the signal with added gain. Thus, relatively high SNR values are feasible. The processing gain resulting from the matched filter process is equivalent to the time-bandwidth product of N T BWr, where T is the non-zero time duration of the pulse, N is the number of pulses, and BWr is the receiver noise bandwidth which for this work, since no additional filtering outside the analog bandwidth of the system is used, is equal to the sampling bandwidth of 12.5 MHz. For the two node experiment, T = 250 µs, thus the processing gain is 35 dB resulting in a post processing SNR of 65 dB. Using (2.12) the bound on the accuracy of a two-way measurement is σx = 2.44 mm. The distance uncertainty sets the maximum operation frequency that can be achieved by f ≤ c 20σx (4.1) where σx is the standard deviation of the two way distance measurement and the factor of 20 derives from the coherent gain statistical analysis for the end-fire array configuration for Section 2.1.3. For this case the resulting maximum frequency is limited to f2 node ≤ 6.14 GHz. For the three node experiment, N T resulting from the summation of all the pulses was equal to 500 µs, yielding a processing gain of 38 dB resulting in a post processing SNR of 68 dB. The distance estimation accuracy can then be calculated using (2.12) and is equal to σx = 2.42 mm. The maximum operational frequency for the three node is thus derived to be f3 node ≤ 6.18 GHz. 4.3 Range Estimation Refinement for Node Motion Phase alignment of the beamforming signals to produce a phase coherent system at the target destination is performed by estimating the range between the primary node and the cor- responding secondary node and applying the appropriate phase shift. For this experiment, the captures the signals and retransmits. Thereby, the propagation losses are proportional to 1 ranging signal is transmitted from the secondary node(s) to the primary node where a repeater r2 rather r4 which is seen by typical radar measurements. The center frequencies of the transmit and receive of the repeater are separated by 1 GHz. This ensures that the desired signal dominate any than 1 120 Figure 4.3: Block diagram of the Kalman filter used for range estimation refinement with moving nodes. multipath and any crosstalk can be neglected. After the signal is received, the secondary node estimates the time of flight by matched filtering the return signal. The peak of the matched filter is spline interpolated in real-time in LabView with 1000 points to avoid discretization errors. The matched filter output peak value is tracked using a 1-D Kalman filter. A Kalman filter gives the optimal state estimation for linear systems in the presence of Gaussian noise [118]. The model of the filter, shown in Fig. 4.3, is ˆxn = ˆxn−1 + Kn(zn − ˆxn−1) (4.2) where ˆxn is the prediction of the current state, ˆxn−1 is the prediction of the previous state, zn is the measurement at the current state, and Kn is the current state Kalman gain given by Kn = σ2 n−1 n−1 + σ2 σ2 M (4.3) where σ2 M = 3 × 10−5 is determined by measuring the variance of 1000 peak values. The current state uncertainty is then n−1 is the previous state uncertainty and the measurement variance σ2 updated by n = (1 − K)σ2 σ2 n−1 + σ2 c (4.4) c = 5 × 10−6 is added to model the array dynamics. This added uncertainty is used to model the internode range of the system as a constant value where an additional constant uncertainty of σ2 with a small, random perturbation to account for both positive and negative radial motion. The 121 experiments in the following section have induced motion that is proportional to the coherent frequency wavelength of the 1.5 GHz (20 cm), which is roughly 2% of the sampling interval of the matched filter at 25 MHz (12 m). Since this induced motion is very small compared to the discretization of the matched filter, this motion acts like a fluctuation on an otherwise constant value and therefore a 1-D Kalman filter is sufficient. However, if the motion is much larger than a fraction of the matched filter sampling rate, resulting in discontinuities in the matched filter output, the Kalman filter will diverge. If this is the case, other techniques such as a higher dimensional Kalman filter, such as an EKF [119–121] to linearize the discontinuities or an unscented Kalman filter (UKF) [122–124], may be needed. The time delay estimate found from the output of the Kalman filter is then converted to a phase shift of the operational frequency. This phase shift is then applied to the beamforming carrier signal on the secondary node. Thus, as the primary and secondary change their relative positions, the outputs remain phase locked at the target location. 4.4 Distributed Antenna Array and Open-Loop Distributed Beamforming Experiments The architecture investigated in this work is based on a single primary node and mul- tiple secondary nodes. As the objective is to demonstrate the ability to simultaneously mea- sure internode range between multiple nodes with sufficient accuracy to support beamforming, a continuous-wave transmitted signals is used, and the reference oscillators are locked via cable. Wireless frequency alignment can be implemented in various ways for a fully wireless system including: an adjunct self-mixing circuit [34, 125]; two-way time transfer [126, 127]; reference broadcast synchronization [128]; timing synch protocol for sensor networks [129]; flooding time synch protocol [130]; and ultra wideband pulse time of arrival [131–133]. An image of the block diagram of the primary and secondary nodes can be seen in Fig. 4.4 (a) and (b) respectively. Each node consisted of two Ettus X310 SDRs, each of which are connected to one host computer run- ning Windows 7 with 32 GB of RAM via 10 GB Ethernet cables. The X310s utilize two UBX 160 daughterboards which have operational bandwidths from DC to 6 GHz with an instantaneous 122 (a) (b) Figure 4.4: Block diagrams of the nodes. bandwidth of up to 160 MHz. These daughterboards support complex up- and down-conversion for in-phase and quadrature mixing as well as internal amplification equivalent to 30 dB and 33.5 dB for the transmit and receive sides, respectively. A block diagram of the X310 RF chain can be seen in Fig. 4.5. The SDRs interface with the host computer using using LabVIEW 2018 where a maximum sampling rate of 25 MHz is possible and is limited by the data throughput between Lab- VIEW and the SDRs, restricting the maximum achievable instantaneous bandwidth to 12.5 MHz. One SDR on each node transmits the beamforming signal while the second SDR is used to either implement ranging to the primary node, or to capture and retransmit any incoming ranging signals from the secondary nodes. Each secondary node transmits a version of the ranging wave- form with a distinct stepped-frequency pattern. The primary node repeats any incoming signals in a continuous manner (i.e. no time scheduling was required). Each secondary node then processes the received signal via matched filter followed by the Kalman filter refinement step. Both of the two 123 Figure 4.5: RF chain internal to each of the X310 SDRs. and three node experiments described below yields SNR values of approximately 30 dB, which are determined using an eigenvalue decomposition approach from Section 2.1.4. The secondary nodes then calculate the range, from which the relative phase of the beamforming carrier signal is updated based on the measured range estimation. The beamforming signals are transmitted from each node at a carrier frequency of 1.5 GHz using 1.35 − 9.5 GHz ultra wideband log periodic antennas. Transmission of the ranging signals from the secondary nodes are implemented at a carrier frequency of 4.25 GHz and, after reception at the primary node, are retransmitted at a carrier frequency of 5.25 GHz, providing frequency diversity to mitigate crosstalk and multipath. The beamformed signals are captured on a Keysight MSO-X 92004A oscilloscope. The power levels of the individual signals are also recorded at each location by selectively turning on individual transmitters. 4.5 Two-Node Experiment Two experiments are conducted, one with two transmitting nodes and one with three transmitting nodes. In both cases the arrays are steering to the end-fire direction. The primary node is moved in both experiments, inducing relative phase errors between all nodes that is corrected via range estimation. An initial calibration procedure is implemented by adjusting the phases of 124 the transmitted signals until maximum gain is obtained; after this point no adjustment is performed aside from that implemented by the ranging system. The calibration procedure effectively amounts to calibration of the phase delays present in each node, and can reasonably be implemented without monitoring the received power by using, e.g., power couplers at the transmitter outputs. The two-node distributed array consists of a primary node and a single secondary node separated initially by a distance of 1.5 m. The receiving antenna at the destination is located at a distance of 1.5 m from the secondary node, and 3 m from the initial position of the primary node. The diagram of the setup is shown in Fig. 4.6(a), where the red colored signals indicate the ranging and the blue signals represent beamforming. Fig. 4.6(b) shows the experimental system in a semi- enclosed arch range in the laboratory. The beamforming and ranging antennas are set on foam blocks that are easily repositioned by manually sliding to different positions. Absorber is used to mitigate reflections from nearby objects that are not fully shielded by the mounted absorber in the range. The primary node is moved by hand in approximately 2 cm increments towards the secondary to simulate motion in the array for a total of 20 cm, equal to the wavelength of the beamforming frequency. At each distance 100 snapshots containing 15 cycles of the 1.5 GHz signal are captured on the scope. The resulting 1,500 peak values are averaged to give the measured amplitude at each distance. Fig. 4.6(c) shows the measured results of the beamforming experiment. The red dashed lines show the individual amplitudes of the two transmitters, as well as the ideal summation of the two signal, denoted as max amplitude, which represents the maximum possible beamformed amplitude that can be achieved with error-free phase correction. The black dashed line indicates the 90% coherent gain threshold. Two beamforming measurements are represented in the plot. The first is an uncorrected beamforming experiment where the ranging system is not utilized to update the transmission phase of the secondary node, represented by the orange curve. The signal begins initially at a high coherent amplitude value, before decreasing to a null and finally increasing again to a high amplitude value, clearly showing the constructive and destructive interference expected when no phase correction is implemented. The blue line shows the result of performing the same 125 (a) (b) (c) Figure 4.6: (a) Block diagram of the distributed two-node experiment. (b) Image of the experimen- tal setup in the semi-enclosed arch range. (c) Measured results of coherent gain with and without performing range-based phase correction. 126 measurement with the ranging system automatically updating the phase of the beamforming signal. The phase corrected beamforming signal amplitude is close to the ideal level for the majority of the test, indicating successful beamforming when the nodes undergo relative motion. 4.6 Three-Node Experiment The two-node system is expanded upon to create a three-node open-loop beamforming system. The array is oriented in an end-fire configuration, again to demonstrate the most chal- lenging beamforming case. A diagram of the setup is shown in Fig. 4.7(a), where the red colored signals indicate ranging and the blue signals represent beamforming. Fig. 4.7(b) shows the ex- perimental system in a semi-enclosed arch range in the laboratory. The two secondary nodes are separated by 0.6 m, with first secondary node a distance of 1 m from the receiving antenna. The primary node starts at a distance of 1 m from the second secondary node. The two secondary nodes remain stationary, and range to the primary node, which captures and retransmits both signals. No time scheduling is used for individual measurements; the TTSFW supports simultaneous multin- ode operation thus no scheduling is required. The primary node is again moved in 2 cm increments for a total of 20 cm. In the same way as the two-node experiment, at each distance 100 snapshots containing fifteen cycles of the 1.5 GHz signal are captured. The resulting 1,500 peak values are again averaged to give the measured amplitude at each distance. The measured results of the three-node experiment are shown in Fig. 4.7(c). Here the two secondary node individual powers are again constant resulting from their stationary positions. The uncorrected beamforming measurement showed a clear null, dropping to 7% of the available total power when the primary node location causes destructive interference. When the ranging system is utilized, the system maintains a high-gain beamforming signal, achieving a coherent power level consistently above the 90% threshold. The amplitude of the beamforming signal closely matches the ideal level, indicating successful beamforming of a three node system when undergoing relative node motion. 127 (a) (b) (c) Figure 4.7: (a) Block diagram of the distributed three-node experiment. (b) Image of the exper- imental setup in the semi-enclosed arch range. (c) Measured results of coherent gain with and without performing range-based phase correction. 128 CHAPTER 5 APPLICATION OF PHASE ALIGNED COHERENT DISTRIBUTED ARRAYS Phases arrays have advantageous properties when it comes to communication, namely they max- imize power transfer in a desired direction. However, traditional phased array architectures also transmit power, and therefore information, to undesired directions through sidelobes. This infor- mation can be recovered by an eavesdropper with a sufficiently sensitive receiver. The array side- lobes are inherent to the system and created through the array design. Past works have addressed the mitigation of sidelobe energy with unique array configurations designed through numerical optimization. However, this technique has limited functionality due to the inability to fully elim- inate sidelobe energy, and therefore the information can still be recovered by eavesdroppers. To combat the drawbacks faced by traditional phased arrays, time modulated arrays are designed to change array characteristics over time, which provides another degree of freedom. These arrays present a method for improved communication security as well as the ability to provide multiple independent data streams to different locations enabling signals utilizing on-off keying [134], fre- quency shift keying [135], pulse-position modulation [136], or amplitude- and phase-based mod- ulations [69, 70]. Types of time modulated arrays include: parasitic arrays where driven elements are loaded by switching parasitic elements which modulate the amplitude and phase of signals to create a distinct symbol constellation in the direction of the desired receiver and different char- acteristics in undesired locations [69–71]; periodically switched arrays in which elements of the array are switched on or off to synthetically change the electrical baselines of elements in the ar- ray causing the radiation pattern to change and thus change the locations of sidelobes [72–74]; and directional modulation where the driven elements are given additional baseband phase shifts creating unique modulation constellations which can be demodulated at the intended receiver but creates distortion at undesired directions making information unrecoverable [75–78]. A significant drawback to all these methods is a reduction in mainbeam power, and therefore impose a trade-off for secure communications: in switching arrays, elements are selectively turned off, effectively 129 reducing the total possible output power; parasitic arrays trade radiated power to power parasitic elements and require extensive trial-and-error analysis to obtain the desired effects; and directional modulation imparts the change of element weights to alter the sidelobes, which also results in a reduced mainbeam gain. CDAs present a viable solution to both secure communications and maximum power transfer. Distributed arrays are unique in the fact that they operate similar to phased arrays but consist of several elements physically distributed in space, which enables spatial position to be used as another degree of freedom. When elements of an array move physical position relative to each other, the electrical baselines of the array are also changed and thus so are the locations of sidelobes. This is similar to the effects of the periodic switching arrays but, whereas periodic switching arrays suffer from reduced power due to unused elements, CDAs do not reduce power levels, as the mainbeam power can remain unaltered as the elements physically move. 5.1 Antenna Array Dynamics and Impacts on Wireless Communication The energy radiated from multiple antennas that work together in a cooperative array add constructively and destructively at different points in space. The radiation pattern of a static antenna array can be calculated by the array factor [137] given by In(θ, φ)ejk(rn·ˆr) AF (θ, φ) = N(cid:88) (5.1) n=1 where In(θ, φ) is the element pattern of the nth element, rn is the location of the nth element, ˆr is the unit vector in the direction of rn, and k = 2π λ is the wave number where λ is the wave- length of the beamforming frequency. The position of the elements, given a constant beamforming frequency, dictate aspects of the radiation pattern such as mainbeam width as well as location and number of sidelobes. In a dynamic array, elements change their physical position, and there- fore the radiation pattern characteristics are dynamic with time. However, as long as coherence is maintained, the power delivered in the mainbeam remains unchanged. This can be demonstrated through a simple two element array where two different flight paths was well as their subsequent radiation patterns over time can be seen in Fig. 5.1. From this image it can seen that as the spatial 130 (a) (b) Figure 5.1: Flight paths are taken over a 1 m extent at a 1.5 GHz beamforming frequency assuming a dipole transmitter and broadside beamforming direction. (a) Linear flight path (left) and result- ing array radiation pattern over time (right). (b) Sinusoidal flight path (left) and resulting array radiation pattern over time (right). location changes with time, so does the location and intensity of sidelobes. This is the key to this method of masking the information sent or filtering received information to/from a single location other than the desired one. The ability to change the location and intensity of radiated sidelobes aid in the security of wireless systems for both transmit and receive operations. In a transmit system, the relocation of sidelobe energy results in signal amplitude and phase fluctuations at off angle directions making it increasingly difficulty to accurately demodulate a received signal and thus challenging for an eavesdropper to recover transmitted information over an extended period of time. In a receive 131 system, the change in location and intensity of sidelobes energy makes it difficult to jam as the effectiveness of the addition of uncorrelated signals fluctuate over an extend period of time. In addition to this, if a location of a jammer or eavesdropper can be estimated, a flight path can be designed such that a null is steered to that specific direction while maintaining the variation in sidelobe characteristics in all other directions. 5.1.1 Estimation of Bit-Error-Rate of Communications Signals Many applications can benefit from increased transmission security including radar, remote sens- ing, and imaging; but one of the most prominent impacts is in the area of communication. Here I demonstrate the degradation of reliable signal information at angles outside of the mainbeam through determination of the bit-error-ratio (BER) of the communication signal. The probability of bit error in a phase-based symbol modulation is calculated through the complimentary Gaussian error function [75, 138] and is defined as Pb(error) = 1 2 erfc (cid:32)(cid:115) (cid:33) Eb N0 (5.2) where Eb is the bit energy and N0 2 is the noise power spectral density. For a binary phase shift key (BPSK) signal Eb = Es where Es is the symbol energy. The symbol energy is calculated through the power of the radiation pattern over a set of angles. To account for element motion of dynamic arrays, rn with time, I define the average radiation pattern as the weighted integration of the array factor over the total flight path in all space. Here the integral weight is the spatial probability density function (PDF) of the path. Without loss of generality, the position is calculated in Cartesian coordinates (ˆx, ˆy, ˆz) and is expressed as AF (θ, φ) = p(rn,ˆx,ˆy,ˆz)e jk(rn,ˆx,ˆy,ˆz·ˆr) dx dy dy (5.3) N(cid:88) n=1 (cid:90)(cid:90)(cid:90) rn,∈{ˆx,ˆy,ˆz},max In rn,∈{ˆx,ˆy,ˆz},min where p(rn,ˆx,ˆy,ˆz) is the spatial PDF of the flight trajectory. From this it can seen that to estimate the BER the exact flight path does not need to be known, only the distribution over space is required. 132 Therefore, if a vibration profile of a mobile platform or deviation from a preset flight path is known, the BER can be estimated through the PDF alone. Here I consider the example of two-element linear array lying on the x axis, from which (5.3) reduces to (cid:90) rˆx,max rˆx,min AF (θ) = 1 + p(rˆx)ejkrˆxsinθdrˆx (5.4) (5.5) (5.6) The PDF of the linear flight path, shown in Fig. 5.1, can be solved for by p(rˆx) = 1 rˆx,max − rˆx,min as it has a uniform distribution over space. The sinusoidal path can be solved for by (cid:115) p(rˆx) = π 1 − 1 (cid:18) 2x−rˆx,max−rˆx,min rˆx,max−rˆx,min (cid:19) and is defined over the extent [0, π] where symmetry exists for [π, 2π] assuming a uniform sampling rate, or data rate, over the entire path. In addition to this, if nonuniform sampling is performed such that the data is pulsed at different locations along the flight path, unique radiation patterns can be designed depending on the desired goal of the user. The BER of a nonuniform data rate can be estimated using the above process as well, assuming that the spatial probably is known. It is to be noted here that (5.3) is a calculation of the average radiation pattern, and provides an estimation of the BER of a dynamic CDA rather than a lower bound. This is due to time instances where off angle directions have improved or reduced performance relative to the mean depending on the current state of the array. A demonstration of the improved BER performance that CDAs can provide can be seen in Fig. 5.2 where the two designated flight paths from Fig. 5.1 are compared a static two-element array with 1 m of separation. This example was taken at a beamforming frequency of 1.5 GHz and a SNR, given by Eb N0/2, of 12 dB. From this figure it can seen that the reliability of information at broadside is maintained while reducing the performance at off angle directions up to a two orders of magnitude, therefore improving the transmission security and reducing the chance of recoverable information for an eavesdropper. 133 Figure 5.2: Comparison of BER performance of a static case of two dipoles at 1 m separation, and the estimated performance of the two designated flight paths at 1.5 GHz and 12 dB of SNR. 5.1.2 Simulation The effectiveness of this method is verified through simulation by calculating the array pattern (5.4) over a 90° to 90° domain for a time T . The array pattern is then used to modulate the amplitude of a pseudo-random bit sequence (PRBS) of 300 kbits. AWGN is then applied to the bit sequence to create an SNR of 12 dB in the mainbeam, oriented at broadside, and a constant noise power- spectral density is maintained at all angles. The demodulation process is performed by mapping the bit plus noise by its Euclidean distance to the nearest constellation point. The demodulated data is then compared to the original PRBS and the BER is calculated at each angle. A block diagram of this process can be seen in Fig. 5.3(a) along with an example of the the simulation results in comparison to the calculated estimation in Fig. 5.3(b). Here the disruption of the demodulated data (blue) in comparison to the correct constellation points (red) in off angle directions can be seen. 134 (a) (b) Figure 5.3: (a) Block diagram of the simulation. (b) Simulated BER over −90° to 90° domain compared to calculated estimation for the linear case. The snapshots of data show the 100 points of demodulated data (blue) and the location of the constellation points (red) at angles −20° and 0°. 5.2 Dynamic Distributed Antenna Array Design and implementation In this chapter an open-loop CDA, where information is sent from two different elements, is implemented. The fundamental electrical states that must be coordinated are phase and, since pulsed information is to be sent, time. Time alignment is performed by a preamble consisting of a high-amplitude pulse that is sent initially for calibration; time alignment is relatively stationary after this initial calibration, but long-term systems would require periodic recalibration using the same approach. Phase alignment must be performed continually due to the motion of the platforms, 135 Figure 5.4: Method of performing alignment of electrical states of time, through high amplitude preamble, and phase, range estimation between primary and secondary elements. and is implemented through a range estimation between elements. Static phase offsets which do not change during the course of the operation, such as static phase delays through the electronic systems, and are calibrated prior to operation. The structure of the dynamic distributed array follows the structure described in Chapter 4 and is achieved using a hierarchical centralized architecture consisting of a single primary ele- ment and one subsequent secondary element. An image depicting this method of time alignment, through a high amplitude preamble, and phase alignment, through a range estimation, can be seen in Fig. 5.4. 5.2.1 Ranging Waveform The ranging waveform used for this work is the TTSFW from Section 3.4. This waveform is generated on an Ettus X310 SDR which is connected to a host computer via 10 GB Ethernet cable and interfaced with using LabView 2018. The achievable sampling rate using this setup is 20 MHz, where the limiting factor is LabView processing speed, providing an instantaneous bandwidth of 10 MHz. For this experiment, which consists of two elements, only one unique connection is made and, therefore, only a single pulse of the TTSFW is required. The remaining parameters are chosen to be ∆f = 6 MHz and Tr = 500 µs with a 50% duty cycle. An image of the TTSFW used in this experiment for both time and frequency can be seen in Fig. 5.5. 136 (a) (b) Figure 5.5: (a) Ranging waveform in the time domain. (b) Ranging waveform in the frequency domain. 5.2.2 Estimation Ability The estimation ability of the TTSFW, given by (3.83), is again dictated by the mean-squared band- width of the signal which is given by (3.82) which for the aforementioned waveform parameters is derived as ζ2 f (cid:12)(cid:12)(cid:12)N =1 = π2BW 2 = 3.5531 × 1014 (5.7) The CRLB for the TTSFW, (3.83), can be evaluated at an SNR of 24 dB which represents the worst case scenario seen from the measured data and is estimated through a eigenvalue decomposition method described in Section 2.1.4. The processing gain from the matched filter process, given by T BWr, where T = 250 µs and BWr is the reciever noise bandwidth which, since no filter is being used, is 10 MHz. The processing gain is equivalent to 34 dB giving a total post processing SNR of 58 dB. The maximum theoretical positional accuracy, given these parameters, can be found to be σx = 1 cm. The relative ranging accuracy for broadside beamforming must be known within λ 15, unlike end-fire that requires λ 20, to have a high probability of achieving 90% f ≤ c 15σx 137 (5.8) with provides an obtainable beamforming frequency of f ≤ 1.9964 GHz. To stay close to this limit, the beamforming frequency for the experiment is chosen to be 1.5 GHz. To ensure that the range estimation provides an estimate of the beamforming phase be- fore the elements move out of coherent range with a degradation of 10%, the maximum velocity of the element relative motion can be calculated as v = λ 15Tupdate (5.9) where v is the maximum velocity, λ is the wavelength of the beamforming frequency, and Tupdate is the update rate. The ranging process on the host computer of the secondary element is measured to have an average processing time of 60 ms. This leads to a maximum tolerable velocity of 0.2 m/s to ensure a coherent gain of approximately 80% where 10% degradation comes from the uncertainty in the ranging process and 10% from the motion of the elements moving out of relative coherence before a new phase update is available. 5.3 Experimental Validation of Secure Communication Wireless measurements are conducted in a semi-enclosed arch range where the receiver consists of a 0.5−6 GHz horn antenna mounded along the outer rail of the range and a two element open-loop CDA is mounted on a linear actuator in the middle of the range 3.05 m away from the receiver. The receiving antenna is rotated around the outer rail of the range on a movable platform in 10° increments to evaluate the performance of the array at angles off of broadside. The received signals are captured on an Ettus X310 SDR where signals are downcoverted to baseband using the internal local oscillator (LO), which is frequency locked to the elements within the array to emulate worst case scenario of an eavesdropper having perfect clocking information. The baseband waveform is then off-boarded to MATLAB for processing. The primary and secondary elements of the CDA are each equipped with a 0.698 − 2.69 GHz dipole antenna for beamforming and two 1.3 − 9.5 GHz ultra wideband log periodic antenna for ranging. Beamforming operation tooks place at 1.5 GHz while the ranging is performed at 4 GHz and 5.5 GHz for transmit and receive respectively. The separation of ranging center frequencies is to avoid cross talk, and chosen such 138 Figure 5.6: Block diagram of the experimental setup that they lie far outside the instantaneous bandwidths achievable on each radio. Due to the linear actuator only having a single moving platform, the primary element is mounted stationary on the end while the secondary element is mounted to the movable platform, therefore, all of the array’s motion is done solely by the secondary element. A block diagram of the experimental setup can be seen in Fig. 5.6 where the red arrowed lines represent the internode ranging while the blue arrowed lines represent beamforming. An image of the the experimental setup in the range can be seen in Fig. 5.7(a) along with the antenna configuration on each element can in Fig. 5.7(b). The full extent that the linear actuator can achieve is 1 m but due to the antenna mounting structure the motion of the array is limited 0.744 m. The farthest separation of the elements is limited to 0.934 m and the closest range the elements achieve is 0.196 m to avoid collision of the log periodic antennas. The element positions for these two limiting factors can be seeing in Fig. 5.7(c) and (d) for the cases of farthest and closest extents respectively. The simulation was run with these range limits and a comparison to the true flight path, taken from the linear actuator data, can be seen in Fig. 5.8. The RMSE of the deviation of the actual path from the simulated path is 4.7 mm and 1.16 cm for the linear and sinusoidal cases respectively. It should be noted 139 (a) (c) (b) (d) Figure 5.7: (a) Image of the experimental setup in the semi-enclosed arch range. (b) Image of the antenna setup for ranging and beamforming on each element. (c) Farthest separation of the elements mounted on the linear actuator. (d) Closest separation of the elements mounted on the linear actuator. here that the sinusoidal case is only captured from [0, π] due to memory depth constraints of the saving process. This limited path will have no effect the resulting BER, as the probability in (5.4) is symmetric around π. The maximum velocity of the relative motion between the elements is measured to be 0.1923 m/s which is below the limit of 0.2 m/s required for 10% degradation of the coherent signal. The coherent gain for these two flight paths is measured by comparing the coherent sum of the two elements to the sum of the two elements radiating independently and is found to be 79.9% and 78.78% (∼ 1 dB degradation) for the linear and sinusoidal case respectively. Higher levels of coherence can be achieved by one of the following ways: lowering the beamforming frequency; increasing the ranging waveform bandwidth to lower the positional uncertainty; reducing the processing overhead to provide a quicker update to the beamforming phase; or reducing the velocity of the relative motion between the elements. 140 (a) (b) Figure 5.8: (a) Comparison of simulated and actual flight path for the linear case. (b) Comparison of simulated and actual flight path for the sinusoidal case. Calibration of the data is performed prior to operation by placing the array in a static state at the farthest extent of the linear actuator and the receiving horn, oriented at the desired beamforming direction, is connected to a MSO-X 92004A Keysight Infiniium High-Performance Oscilloscope. The timing on the output of the secondary element is altered until sufficient over- lap with the primary element the high amplitude time alignment pulse is achieved. Once time alignment is complete, phase alignment is accomplished by alternating the output phase of the sec- ondary element until a maximum amplitude of the beamforming signal can be seen on the scope. After calibration is performed the receiving antenna is then connected to a X310 SDR for capture where data is collected at a bit rate of 100 kbits/s. The PRBS is generated using a built-in Lab- View maximum length sequence (MLS) block for the time durations depicted in Fig. 5.8. The linear actuator motion is then started and the array is set in constant relative motion. Once the data is collected and off-boarded to MATLAB, processing is performed by first stripping off the time alignment pulse. The data is demodulated by averaging the samples of each bit plus noise and map- ping this mean value to a constellation point by the Euclidean distance. Measurements are taken at 12.85 dB and 12.05 dB SNR determined by an eigenvalue decomposition method, from Section 2.1.4, for the linear and sinusoidal paths respectively. The measured results of a −60° to 60° extent measured in 10° increments can be seen in Fig. 5.9(a) and (b) for the linear and sinusoidal cases 141 respectively and are compared to simulation and estimation value of the BER. It can be noticed from these results that the positive angles have improved performance compared to the negative angles where the result is expected to be symmetric around 0°. The analysis for the estimation, as well as the simulation, assume far field operation and symmetric motion between both elements. For this experimental setup, this is not the case. The primary element remains stationary relative to the positive angles while the secondary element performs the motion towards the negative angles. Although this mismatch is present, a relatively good matching between measured, simulated, and estimated results can be seen providing approximately a BER of 10−5 at broadside and roughly 0.5 at off angle directions and therefore, proves that CDAs present a viable solution to secure sensing while maintaining near maximum power transfer. 142 (a) (b) Figure 5.9: (a) Comparison of simulated, measured, and calculated estimation of the BER for the linear case. (b) Comparison of simulated, measured, and calculated estimation of the BER for the sinusoidal case. 143 CHAPTER 6 CONCLUSION In this work I have designed and experimentally validated a method to perform phase alignment that is more conducive to scalability, simpler system implementation, lower cost, and distributed beamforming applications than what has been demonstrated in the literature. To overcome the lack of positional information inherent in closed-loop and retrodirective methods, the use of internode range to enable phase-based beamsteering operation is implemented. I derived and experimentally evaluated novel ranging waveforms utilizing time domain duxplexing, frequency division multi- plexing, and pulse encoding scalablity approaches to operate with a variety of distributed systems and applications including remote sensing as well as communication. These waveforms include: a near optimal range and velocity estimation waveform; a waveform to perform joint ranging and frequency transfer; a ranging waveform using existing IEEE 802.11 preamble formats; and highly scalable pulse encoded waveform based on a spectrally-sparse two-tone waveform that monochro- matically increases its frequency on pulse by pulse basis. Using the range estimations provided from the TTSFW waveform, an experimental demonstration of phase alignment supporting up to 120 nodes with three nodes transmitting in a dynamic open-loop distributed beamforming array with greater than 90% coherent gain using the primary-secondary hierarchical topology was presented. Beamforming in the L-band region was proven to be obtainable with no more than 12.5 MHz bandwidth allocated to the range estima- tion waveform making applications such as L-band SAR for vegetation and forestry measurements as well communications at the L-band frequency range obtainable using off-the-shelf equipment. Furthermore, this method is directly scalable to larger arrays. I demonstrated an implementation of distributed beamforming phase alignment of a dy- namic coherent open-loop CDA with designated element flight paths to improve the security of wireless operations. A wireless beamforming experiment was shown to have approximately 80% maximum power transfer, which can be improved upon by allocating more bandwidth to the esti- 144 mation process or alternate flight paths, while corrupting data to the point of becoming unrecov- erable at off-angle directions. This method has applications that are resistant to eavesdropping on transmit and resistant to jamming on receive. This is made possible by the dynamics of the array changing the electrical baselines of the synthesized array and thus changing the locations of constructive interference in space while maintaining coherence at the desired target destination. This dissertation has thus laid the ground work for open-loop distributed beamform- ing applications by addressing the most challenging coordination aspect to enable open-loop dis- tributed beamforming, phase alignment of the beamforming signals. As many wireless distributed systems require flexibility to support various applications, I have addressed phase alignment in a highly scalable manor for a variety of applications and, due to the pairwise structure of the primary- secondary topology, total system failure can be avoided from individual element failure. Wireless coherent beamforming operations in the microwave region were performed with relatively low cost off-the-shelf equipment making this method highly repeatable and easily implemented for fu- ture works in distributed beamforming from aerial platforms. The introduction of array dynamics, shown in this work, make applications consisting of mobile nodes such as drone swarms or satellite constellations performing coherent operations feasible using this technique for application spaces of communication and remote sensing, which has previously been unobtainable for closed-loop architectures. 145 APPENDICES 146 APPENDIX A MATLAB FORMULATION OF BASEBAND PSWF c l e a r c l o s e a l l a l l c l c %%%%%%%%%%%%%%%%%%%%%%%%%%% c =3* p i ; o r d e r = 0 : 7 ; M= c e i l ( 1 . 1 * c+max ( o r d e r ) + 1 0 0 ) ; f o r i _ o r d =1: l e n g t h ( o r d e r ) %%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%% %A m a t r i x A= z e r o s (M,M) ; f o r i_N =1:M A( i_N , i_N ) = ( i_N − 1 ) * ( ( i_N − 1 ) + 1 ) + ( 2 * ( i_N − 1 ) * ( ( i_N − 1 ) + 1 ) − 1 ) . . . / ( ( 2 * ( i_N − 1 ) + 3 ) * ( 2 * ( i_N −1) −1))* c ^ 2 ; i f i_N+1 t o p s _ t ) = [ ] ; t o p =( mean ( t o p s _ f )+2* s t d ( t o p s _ f ) ) ; b o t =( mean ( t o p s _ f ) −2* s t d ( t o p s _ f ) ) ; t o p s _ f ( top < t o p s _ f ) = [ ] ; t o p s _ f ( bot > t o p s _ f ) = [ ] ; d a t a _ t ( i_dB )= v a r ( t o p s _ t ) ; d a t a _ f ( i_dB )= v a r ( t o p s _ f ) ; temp= s o r t ( e i g ( cov ( s i g ) ) , ’ descend ’ ) ; sig_w=mean ( temp ( 5 :MC) ) ; Ps =1/MC* ( temp (1) − sig_w ) ; 161 s n r _ t ( i_dB )=10* lo g10 ( abs ( Ps / sig_w ) ) + 1 0 * lo g10 ( smpps *10 e − 6 ) ; s n r _ f ( i_dB )=10* lo g10 ( abs ( Ps / sig_w ) ) ; temp= s o r t ( e i g ( cov ( s i g _ f ) ) , ’ descend ’ ) ; sig_w=mean ( temp ( 5 :MC) ) ; Ps =1/MC* ( temp (1) − sig_w ) ; s n r _ f ( i_dB )=10* lo g10 ( abs ( Ps / sig_w ) ) ; % % % % % c l c end f i g u r e ( ) s e m i l o g y ( s n r _ t , ( 3 e8 ) ^ 2 / 4 * d a t a _ t , ’ − o ’ , ’ Linewidth ’ , 1 . 5 ) h o l d on s e m i l o g y ( snr , c r l b _ t , ’ Linewidth ’ , 1 . 5 ) g r i d on % a x i s ( [ 4 5 85 2 . 5 e −8 0 . 5 e − 3 ] ) x l a b e l ( ’ P o s t p r o c e s s i n g SNR ( dB ) ’ , ’ F o n t s i z e ’ , 1 8 ) y l a b e l ( ’ V a r i a n c e (m^ 2 ) ’ , ’ F o n t s i z e ’ , 1 8 ) l e g e n d ( ’ S i m u l a t i o n ’ , ’CRLB’ ) f i g u r e ( ) s e m i l o g y ( s n r _ t , d a t a _ f , ’ − o ’ , ’ Linewidth ’ , 1 . 5 ) h o l d on s e m i l o g y ( snr , c r l b _ f , ’ Linewidth ’ , 1 . 5 ) g r i d on % a x i s ([ −5 35 0 . 5 e2 1 e6 ] ) 162 x l a b e l ( ’SNR ( dB ) ’ , ’ F o n t s i z e ’ , 1 8 ) y l a b e l ( ’ V a r i a n c e ( Hz ^ 2 ) ’ , ’ F o n t s i z e ’ , 1 8 ) l e g e n d ( ’ S i m u l a t i o n ’ , ’CRLB’ ) s a v e ( ’ s i m u l a t i o n _ r e s u l t s . mat ’ , ’ d a t a _ f ’ , ’ d a t a _ t ’ , ’ s n r _ t ’ , ’ s n r _ f ’ ) 163 APPENDIX D MATLAB PSWF MEASUREMENT PROCESSING c l e a r c l o s e a l l a l l c l c %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% l o a d opt_wf . mat t e m p l a t e = opt_wf ; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% dB=[0 3 6 10 13 16 2 0 ] ; f i l t =100 e6 ; MC=1000; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% f o r i_dB =1: l e n g t h ( dB ) d i s p ( i_dB ) l o a d ( [ num2str ( dB ( i_dB ) ) ’ _dB . mat ’ ] ) i n d =1; f o r i_MC =1:MC d i s p ( i_MC ) t 1 = l i n s p a c e ( 0 , 1 0 e −6 , l e n g t h ( waveform ( : , i_MC ) ) ) ; temp_wf=waveform ( : , i_MC ) ’ . * exp ( j *2* p i *3 e9 * t 1 ) ; smpps = 1 / ( t 1 (2) − t 1 ( 1 ) ) ; waveform1 ( : , i_MC)= l o w p a s s ( temp_wf , f i l t , smpps ) ; 164 % % % % % % % % % % % % % % % % % % % % waveform1 ( : , i_MC)= temp_wf ; t 1 = t 1 ( 1 : 5 0 : end ) ; r e t u r n P1= abs ( f f t s h i f t ( f f t ( waveform1 ( : , i_MC ) ) ) ) ; f s = l i n s p a c e ( − smpps / 2 , smpps / 2 , l e n g t h ( P1 ) ) ; f i g u r e ( ) p l o t ( fs , P1 ) r e t u r n f i g u r e ( ) s u b p l o t ( 2 , 1 , 1 ) p l o t ( t1 , waveform1 ( : , i_MC ) ) s u b p l o t ( 2 , 1 , 2 ) p l o t ( t1 , t e m p l a t e ) r e t u r n [ c o r r , l a g ]= x c o r r ( temp_wf , t e m p l a t e ) ; [ ~ , lmxs ]= f i n d p e a k s ( abs ( c o r r ( f l o o r ( 0 . 2 5 * l e n g t h ( c o r r ) ) . . . : f l o o r ( 0 . 7 5 * l e n g t h ( c o r r ) ) ) ) , ’ S o r t S t r ’ , ’ descend ’ ) ; f i g u r e ( ) p l o t ( l a g / smpps , abs ( c o r r ) ) r e t u r n waveform2 ( : , i_MC)= c o r r ; [ ~ , lmxs ]= f i n d p e a k s ( abs ( c o r r ) , ’ S o r t S t r ’ , ’ descend ’ ) ; 165 i f i_MC==1 [ ~ , pp ]= min ( abs ( l a g ( lmxs ) / smpps −3.87 e − 8 ) ) ; e l s e end [ ~ , pp ]= min ( abs ( l a g ( lmxs ) / smpps − t o p s _ t ( 1 ) ) ) ; %%−−−−−−−−−−−−−−−−−−−−−−−−−−−−− %i n t e r p i n t r p =3; r e d i c =10000; newtau= l a g ( lmxs ( pp ) − i n t r p ) / smpps : 1 / ( smpps * r e d i c ) . . . : l a g ( lmxs ( pp )+ i n t r p ) / smpps ; Vs= i n t e r p 1 ( l a g ( ( lmxs ( pp ) − i n t r p ) : ( lmxs ( pp )+ i n t r p ) ) . . . / smpps , abs ( c o r r ( ( lmxs ( pp ) − i n t r p ) : ( lmxs ( pp )+ i n t r p ) ) ) . . . , newtau , ’ s p l i n e ’ ) ; [ ~ , I 1 ]=max ( Vs ) ; t o p s _ t ( i_MC)= newtau ( I 1 ( 1 ) ) ; f i g u r e ( ) p l o t ( l a g / smpps , abs ( c o r r ) ) h o l d on s c a t t e r ( t o p s _ t , max ( abs ( c o r r ) ) ) s c a t t e r ( l a g ( 2 0 0 2 ) / smpps , max ( abs ( c o r r ) ) ) r e t u r n % % % % % % %%_−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− 166 % % % % % Y = f f t ( [ c o r r z e r o s ( 1 , 1 0 * l e n g t h ( c o r r ) ) ] ) ; Y = f f t ( c o r r ) ; Y = f f t ( waveform ( : , i_MC ) ) ; L = l e n g t h (Y ) ; P2 = abs (Y/ L ) ; P1 = P2 ( 1 : L / 2 + 1 ) ; P1 ( 2 : end −1) = 2*P1 ( 2 : end − 1 ) ; f s = l i n s p a c e ( 0 , smpps / 2 , L / 2 + 1 ) ; f i g u r e ( ) p l o t ( fs , P1 ) r e t u r n [ ~ , lmxs ]= f i n d p e a k s ( P1 , ’ S o r t S t r ’ , ’ descend ’ ) ; i f i_MC==1 [ ~ , pp ]= min ( abs ( f s ( lmxs ) −59.2 e6 ) ) ; e l s e end [ ~ , pp ]= min ( abs ( f s ( lmxs ) − t o p s _ f ( 1 ) ) ) ; %%−−−−−−−−−−−−−−−−−−−−−−−−−−−−− %i n t e r p i n t r p =3; r e d i c =1000; newtau= f s ( lmxs ( pp ) − i n t r p ) : ( f s (2) − f s ( 1 ) ) . . . / r e d i c : f s ( lmxs ( pp )+ i n t r p ) ; Vs= i n t e r p 1 ( f s ( ( lmxs ( pp ) − i n t r p ) : ( lmxs ( pp )+ i n t r p ) ) . . . 167 , P1 ( ( lmxs ( pp ) − i n t r p ) : ( lmxs ( pp )+ i n t r p ) ) . . . , newtau , ’ s p l i n e ’ ) ; % % % % % f i g u r e ( ) p l o t ( newtau , Vs ) r e t u r n [ ~ , I 1 ]=max ( Vs ) ; t o p s _ f ( i_MC)= newtau ( I 1 ( 1 ) ) ; s c a t t e r ( 1 : i_MC , t o p s _ f ) h o l d on end t o p =( mean ( t o p s _ t )+2* s t d ( t o p s _ t ) ) ; b o t =( mean ( t o p s _ t ) −2* s t d ( t o p s _ t ) ) ; temp1= f i n d ( top < t o p s _ t ) ; i f ~ i s e m p t y ( temp1 ) t o p s _ t ( temp1 ) = [ ] ; t o p s _ f ( temp1 ) = [ ] ; waveform ( : , temp1 ) = [ ] ; end temp2= f i n d ( bot > t o p s _ t ) ; i f ~ i s e m p t y ( temp2 ) t o p s _ t ( temp2 ) = [ ] ; 168 t o p s _ f ( temp2 ) = [ ] ; waveform ( : , temp2 ) = [ ] ; end c l e a r temp1 temp2 t o p =( mean ( t o p s _ f )+2* s t d ( t o p s _ f ) ) ; b o t =( mean ( t o p s _ f ) −2* s t d ( t o p s _ f ) ) ; temp1= f i n d ( top < t o p s _ f ) ; i f ~ i s e m p t y ( temp1 ) t o p s _ f ( temp1 ) = [ ] ; t o p s _ t ( temp1 ) = [ ] ; waveform ( : , temp1 ) = [ ] ; end temp2= f i n d ( bot > t o p s _ f ) ; i f ~ i s e m p t y ( temp2 ) t o p s _ f ( temp2 ) = [ ] ; t o p s _ t ( temp2 ) = [ ] ; waveform ( : , temp2 ) = [ ] ; end f i g u r e ( ) s c a t t e r ( 1 : l e n g t h ( t o p s _ t ) , t o p s _ t ) r e t u r n 169 % % % % d a t a _ t ( i_dB )= v a r ( t o p s _ t ) ; d a t a _ f ( i_dB )= v a r ( t o p s _ f ) ; c l c temp= s o r t ( e i g ( cov ( waveform1 ) ) , ’ descend ’ ) ; sig_w=mean ( temp ( 2 : l e n g t h ( waveform1 ( 1 , : ) ) ) ) ; Ps =1/ l e n g t h ( waveform1 ( 1 , : ) ) * ( temp (1) − sig_w ) ; s n r _ t ( i_dB )=10* lo g10 ( abs ( Ps / sig_w ) ) + 1 0 * lo g10 (2* f i l t *10 e − 6 ) ; s n r _ f ( i_dB )=10* lo g10 ( abs ( Ps / sig_w ) ) ; end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% f i g u r e ( ) s e m i l o g y ( s n r _ t , ( 3 e8 ) ^ 2 / 4 * d a t a _ t , ’ − o ’ , ’ Linewidth ’ , 1 . 5 ) h o l d on s e m i l o g y ( snr , c r l b _ t , ’ Linewidth ’ , 1 . 5 ) g r i d on % a x i s ( [ 4 5 85 2 . 5 e −8 0 . 5 e − 3 ] ) x l a b e l ( ’ A t t e n u a t i o n ( dB ) ’ , ’ F o n t s i z e ’ , 1 8 ) y l a b e l ( ’ V a r i a n c e (m^ 2 ) ’ , ’ F o n t s i z e ’ , 1 8 ) % l e g e n d ( ’ S i m u l a t i o n ’ , ’CRLB’ ) f i g u r e ( ) s e m i l o g y ( s n r _ t , d a t a _ f , ’ − o ’ , ’ Linewidth ’ , 1 . 5 ) h o l d on s e m i l o g y ( snr , c r l b _ f , ’ Linewidth ’ , 1 . 5 ) g r i d on 170 % a x i s ([ −5 35 0 . 5 e2 1 e6 ] ) x l a b e l ( ’ A t t e n u a t i o n ( dB ) ’ , ’ F o n t s i z e ’ , 1 8 ) y l a b e l ( ’ V a r i a n c e ( Hz ^ 2 ) ’ , ’ F o n t s i z e ’ , 1 8 ) % l e g e n d ( ’ S i m u l a t i o n ’ , ’CRLB’ ) s a v e ( ’ r e s u l t s . mat ’ , ’ d a t a _ f ’ , ’ d a t a _ t ’ , ’ s n r _ t ’ ) 171 APPENDIX E MATLAB NETWORK SYSTEM SIMULATION c l e a r c l o s e a l l a l l c l c %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% f s =160 e6 ; dB = 0 : 2 : 1 5 ; MC=1000; %%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− %v h t ( ac ) l o a d 160MHz_SFW . mat preamble =SFW; %%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− % t = ( ( 0 : l e n g t h ( vhtWaveform ) − 1 ) / f s ) ; t = ( ( 0 : l e n g t h ( preamble ) − 1 ) / f s ) ; % preamble =vhtWaveform ( 1 : ( f s *8e − 6 ) ) ; % t 1 = ( ( 0 : l e n g t h ( preamble ) − 1 ) / ( f s ) ) ; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % p e a m b l e _ i n t e r p =[ preamble ; z e r o s (2* l e n g t h ( preamble ) , 1 ) ] ; % yy= f f t s h i f t (10* lo g10 ( abs ( f f t ( p e a m b l e _ i n t e r p ) ) ) ) ; % f f = f s * ( 0 : l e n g t h ( yy ) − 1 ) / l e n g t h ( yy ) ; %%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− f o r i_dB =1: l e n g t h ( dB ) d i s p ( i_dB ) 172 f o r i_MC =1:MC d i s p ( i_MC ) waveform_noise =awgn ( preamble , dB ( i_dB ) ) ; s i n g ( : , i_MC)= waveform_noise ; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% [ Xs , l a g ]= x c o r r ( preamble , waveform_noise ) ; [ ~ , lmxs ]= f i n d p e a k s ( abs ( Xs ) , ’ S o r t S t r ’ , ’ descend ’ ) ; pp =1; i f i_MC>1 [ ~ , pp ]= min ( abs ( l a g ( lmxs ) / fs − t o p s ( 1 ) ) ) ; end i n t r p =3; r e d i c =1000; newtau= l a g ( lmxs ( pp ) − i n t r p ) / f s : 1 / ( f s * r e d i c ) . . . : l a g ( lmxs ( pp )+ i n t r p ) / f s ; Vs= i n t e r p 1 ( l a g ( ( lmxs ( pp ) − i n t r p ) : ( lmxs ( pp )+ i n t r p ) ) . . . / fs , abs ( Xs ( ( lmxs ( pp ) − i n t r p ) : ( lmxs ( pp )+ i n t r p ) ) ) . . . , newtau , ’ s p l i n e ’ ) ; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% [ ~ , I 1 ]=max ( Vs ) ; t o p s ( i_MC)= newtau ( I 1 ( 1 ) ) ; end t t _ s t d = s t d ( t o p s ) ; t t _ m e a n =mean ( t o p s ) ; t o p s ( t o p s <( tt_mean −2* t t _ s t d ) ) = [ ] ; 173 t o p s ( t o p s >( t t _ m e a n +2* t t _ s t d ) ) = [ ] ; v a r i a n c e ( i_dB )= v a r ( t o p s ) ; s i n g = s i n g ( : , 1 : 1 0 0 0 ) ; temp= s o r t ( e i g ( cov ( s i n g ) ) , ’ descend ’ ) ; sig_w=mean ( temp ( 5 : 1 0 0 0 ) ) ; Ps = 1 / 1 0 0 0 * ( temp (1) − sig_w ) ; s n r _ c r l b ( i_dB )=10* lo g10 ( ( abs ( Ps / sig_w ) ) * max ( t ) / 2 * 1 6 0 e6 ) ; end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% f i g u r e ( ) s e m i l o g y ( s n r _ c r l b , v a r i a n c e ) variance_SFW= v a r i a n c e ; snr_SFW= s n r _ c r l b ; s a v e ( ’ s i m _ r e s u l t s . mat ’ , ’ snr_SFW ’ , ’ variance_SFW ’ , ’ − append ’ ) 174 APPENDIX F MATLAB NETWORK SYSTEM MEASUREMENT PROCESSING c l e a r c l o s e a l l a l l c l c l o a d vht_160 . mat cd 11 _18_19 cd 160MHz_2 %%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− %802.11 ac f s =160 e6 ; % S e t s a m p l i n g f r e q u e n c y = c h a n n e l bandwidth smpps =500 e6 ; dB=[0 3 6 10 13 16 2 0 ] ; % dB=[16 20 26 30 36 40 46 5 0 ] ; %%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− % cfgVHT = wlanVHTConfig ; % cfgVHT . ChannelBandwidth =[ ’CBW’ num2str ( f s / 1 e6 ) ] ; % b i t s = [ 1 ; 0 ; 0 ; 1 ; 1 ] ; % vhtWaveform = wlanWaveformGenerator ( b i t s , cfgVHT , . . . % ’ NumPackets ’ , 1 , ’ IdleTime ’ , 2 0 e − 6 ) ; %%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− t = ( ( 0 : l e n g t h ( vhtWaveform ) − 1 ) / f s ) ; t 1 = 0 : 1 / smpps : max ( t ) ; vhtWaveform= i n t e r p 1 ( t , vhtWaveform , t 1 ) ; preamble =vhtWaveform ( 1 : ( smpps *8e − 6 ) + 1 ) ; 175 [B ,A]= b u t t e r ( 1 0 , ( 8 0 e6 ) / ( smpps / 2 ) ) ; preamble = f i l t e r (B , A, preamble ) ; preamble = preamble −mean ( preamble ) ; % p e a m b l e _ i n t e r p =[ preamble ] ; % y= f f t ( p e a m b l e _ i n t e r p ) ; % L= l e n g t h ( p e a m b l e _ i n t e r p ) ; % % P2=10* lo g10 ( abs ( y / L ) ) ; % P1=P2 ( 1 : ( L / 2 ) + 1 ) ; % P1 ( 2 : end −1)=2* P1 ( 2 : end − 1 ) ; % f f =smpps * ( 0 : ( L / 2 ) ) / L ; % % f i g u r e ( ) % p l o t ( f f , P1 ) % r e t u r n f o r i_dB =1: l e n g t h ( dB ) d i s p ( i_dB ) % % % % % % l o a d ( [ num2str ( dB ( i_dB ) ) ’dB . mat ’ ] ) f i g u r e ( ) p l o t ( ( 1 : l e n g t h ( waveform ( : , 1 ) ) ) / smpps , waveform ( : , 1 ) ) h o l d on p l o t ( t1 , p r e a m b l e _ i n t e r p ) r e t u r n y= f f t ( waveform ( : , 1 ) ) ; 176 % % % % % % % % % % % % % L= l e n g t h ( waveform ( : , 1 ) ) ; P2=10* lo g10 ( abs ( y / L ) ) ; P1=P2 ( 1 : ( L / 2 ) + 1 ) ; P1 ( 2 : end −1)=2* P1 ( 2 : end − 1 ) ; f f =2* f s * ( 0 : ( L / 2 ) ) / L ; f i g u r e ( ) p l o t ( f f , P1 ) r e t u r n f o r i t =1:100 d i s p ( i t ) waveform ( : , i t )= f i l t e r (B , A, waveform ( : , i t ) ) ; waveform ( : , i t )= waveform ( : , i t ) − mean ( waveform ( : , i t ) ) ; [ Xs , l a g ]= x c o r r ( waveform ( : , i t ) , preamble ) ; f i g u r e ( ) p l o t ( l a g / smpps , abs ( Xs ) ) r e t u r n [ ~ , lmxs ]= f i n d p e a k s ( abs ( Xs ) , ’ S o r t S t r ’ , ’ descend ’ ) ; pp =1; i f i t >1 [ ~ , pp ]= min ( abs ( l a g ( lmxs ) / smpps − t o p s ( 1 ) ) ) ; end %%−−−−−−−−−−−−−−−−−−−−−−−−−−−−− 177 %i n t e r p i n t r p =3; r e d i c =10000; newtau= l a g ( lmxs ( pp ) − i n t r p ) / smpps : 1 / ( smpps * r e d i c ) . . . : l a g ( lmxs ( pp )+ i n t r p ) / smpps ; Vs= i n t e r p 1 ( l a g ( ( lmxs ( pp ) − i n t r p ) : ( lmxs ( pp )+ i n t r p ) ) . . . / smpps , abs ( Xs ( ( lmxs ( pp ) − i n t r p ) : ( lmxs ( pp )+ i n t r p ) ) ) . . . , newtau , ’ s p l i n e ’ ) ; [ ~ , I 1 ]=max ( Vs ) ; t o p s ( i t )= newtau ( I 1 ( 1 ) ) ; end t o p s ( t o p s >( mean ( t o p s )+2* s t d ( t o p s ) ) ) = [ ] ; t o p s ( t o p s <( mean ( t o p s ) −2* s t d ( t o p s ) ) ) = [ ] ; v a r i a n c e ( i_dB )= v a r ( t o p s ) ; temp= s o r t ( e i g ( cov ( waveform ) ) , ’ descend ’ ) ; sig_w=mean ( temp ( 5 : 1 0 0 ) ) ; Ps = 1 / 1 0 0 * ( temp (1) − sig_w ) ; s n r ( i_dB )=10* lo g10 ( abs ( Ps / sig_w ) * f s *8e − 6 ) ; c l c end %%%%%%%%%%%%%%%%%%%%%%%%%%%% [ ~ , l o c ]= s o r t ( snr , ’ descend ’ ) ; s n r = s n r ( l o c ) ; v a r i a n c e = v a r i a n c e ( l o c ) ; 178 m e a s u r e d _ v a r i a n c e = v a r i a n c e ; m e a s u r e d _ s n r = s n r ; cd . . % cd . . % s a v e ( ’ a c _ 1 6 0 _ d a t a . mat ’ , ’ m e a s u r e d _ v a r i a n c e ’ , ’ measured_snr ’ ) f i g u r e ( ) s e m i l o g y ( snr , v a r i a n c e , ’ − o ’ ) f i g u r e ( ) s e m i l o g y ( dB , v a r i a n c e ) 179 APPENDIX G MATLAB TTSFW SIMULATION c l e a r c l o s e a l l a l l c l c %%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− BW=5 e6 ; NN= 1 : 2 5 ; f s =25 e6 ; MC=1000; %%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− %waveform f o r i_NN =1: l e n g t h (NN) d i s p ( i_NN ) N=NN( i_NN ) ; t = 0 : 1 / f s : 1 e − 3 / ( 2 *N ) ; waveform =0; f o r i_N =1:N d f =BW/ ( 2 *N− 1 ) ; Df=N* d f ; S =[ exp ( j *2* p i *(1 e6 +( i_N −1)* d f ) * t )+ exp ( j *2* p i * . . . ( 1 e6+Df +( i_N −1)* d f ) * t ) z e r o s ( 1 , l e n g t h ( t ) ) ] ; waveform= c a t ( 2 , waveform , S ) ; 180 end waveform=waveform ( 2 : end ) ; f o r i_MC =1:MC d i s p ( i_MC ) waveform_noise =waveform+wgn ( 1 , l e n g t h ( waveform ) , − 2 0 ) ; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% s i n g ( : , i_MC)= waveform_noise ; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% [ Xs , l a g ]= x c o r r ( waveform , waveform_noise ) ; [ ~ , lmxs ]= f i n d p e a k s ( abs ( Xs ) , ’ S o r t S t r ’ , ’ descend ’ ) ; pp =1; i f i_MC>1 [ ~ , pp ]= min ( abs ( l a g ( lmxs ) / fs − t o p s ( 1 ) ) ) ; end i n t r p =3; r e d i c =2000; newtau= l a g ( lmxs ( pp ) − i n t r p ) / f s : 1 / ( f s * r e d i c ) . . . : l a g ( lmxs ( pp )+ i n t r p ) / f s ; Vs= i n t e r p 1 ( l a g ( ( lmxs ( pp ) − i n t r p ) : ( lmxs ( pp )+ i n t r p ) ) . . . / fs , abs ( Xs ( ( lmxs ( pp ) − i n t r p ) : ( lmxs ( pp )+ i n t r p ) ) ) . . . , newtau , ’ s p l i n e ’ ) ; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% [ ~ , I 1 ]=max ( Vs ) ; t o p s ( i_MC)= newtau ( I 1 ( 1 ) ) ; end 181 % % % % t t _ s t d = s t d ( t o p s ) ; t t _ m e a n =mean ( t o p s ) ; t o p s ( t o p s <( tt_mean −2* t t _ s t d ) ) = [ ] ; t o p s ( t o p s >( t t _ m e a n +2* t t _ s t d ) ) = [ ] ; v a r i a n c e ( i_NN )= v a r ( t o p s ) ; temp= s o r t ( e i g ( cov ( s i n g ) ) , ’ descend ’ ) ; sig_w=mean ( temp ( 5 : 1 0 0 0 ) ) ; Ps = 1 / 9 9 5 * ( temp (1) − sig_w ) ; s n r ( i_NN ) = ( abs ( Ps / sig_w ) ) * 1 0 ^ ( 3 / 1 0 ) * ( 0 . 5 e −3*12.5 e6 ) ; TTSF_var ( i_NN ) = 1 . / ( 4 * s n r ( i_NN ) * ( p i ^ 2 . * Df ^ 2 + . . . ( 2 . * p i . * d f ) ^ 2 . / N. * (N*(2*N^2 −3*N + 1 ) / 6 ) ) ) ; TT_var ( i_NN ) = 1 . / ( 4 * s n r ( i_NN ) * ( p i ^ 2 . * Df ^ 2 ) ) ; TT_var ( i_NN ) = 1 . / ( 4 * s n r ( i_NN ) * ( p i ^ 2 . *BW^ 2 ) / 3 ) ; c l e a r s i n g c l c end f i g u r e ( ) p l o t ( lag , abs ( Xs ) ) %%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− %F o u r i e r % y= f f t ( waveform ) ; % P2=10* lo g10 ( abs ( y / l e n g t h ( waveform ) ) ) ; 182 % P1=P2 ( 1 : ( l e n g t h ( waveform ) / 2 ) + 1 ) ; % P1 ( 2 : end −1)=2* P1 ( 2 : end − 1 ) ; % f = f s * ( 0 : ( l e n g t h ( waveform ) / 2 ) ) / l e n g t h ( waveform ) ; %%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− % f i g u r e ( ) % p l o t ( 1 : l e n g t h ( waveform ) , waveform ) % f i g u r e ( ) % p l o t ( f , P1 ) f i g u r e ( ) s e m i l o g y (NN, v a r i a n c e ) h o l d on p l o t (NN, TTSF_var ) 183 APPENDIX H MATLAB DISTRIBUTED BEAMFORMING MEASUREMENT PROCESSING c l e a r c l o s e a l l a l l c l c %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% s t e p = 1 : 9 ; f o r i _ s t e p =1: l e n g t h ( s t e p ) l o a d ( [ ’ s t e p _ ’ num2str ( s t e p ( i _ s t e p ) ) ’ . mat ’ ] ) ch1 =0; ch2 =0; comb =0; f o r i =1:100 temp1= f i n d p e a k s ( waveform ( : , i ) ) ; ch1 =[ ch1 temp1 ’ ] ; end ch1_mean ( i _ s t e p )= mean ( ch1 ( 2 : end ) ) ; end % cd . . % l o c k _ c h 1 =ch1_mean ; % l o c k _ c h 2 =ch2_mean ; % lock_comb=comb_mean ; 184 % s a v e ( ’ r e s u l t s . mat ’ , ’ lock_ch1 ’ , ’ lock_ch2 ’ , ’ lock_comb ’ ) cd . . n o _ l o c k =ch1_mean ; s a v e ( ’ r e s u l t s . mat ’ , ’ no_lock ’ , ’ − append ’ ) r e t u r n %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% f i g u r e ( ) p l o t ( s t e p , ch1_mean , ’ − − r ’ ) h o l d on p l o t ( s t e p , ch2_mean , ’ − − r ’ ) p l o t ( s t e p , ch1_mean+ch2_mean , ’ − − r ’ ) p l o t ( s t e p , 0 . 9 * ( ch1_mean+ch2_mean ) , ’ − − k ’ ) p l o t ( s t e p , comb_mean ) 185 MATLAB SECURE TRANSMISSION SIMULATION FOR THE LINEAR CASE APPENDIX I c l e a r c l o s e a l l a l l c l c %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ang= p i / 1 8 0 * ( − 9 0 : 0 . 1 : 9 0 ) ; lambda_f = 0 . 9 3 9 8 / 0 . 2 ; lambda_n = 0 . 1 9 6 1 / 0 . 2 ; num_bits =300000; SNR= 1 2 . 6 ; AP= z e r o s ( num_bits , l e n g t h ( ang ) ) ; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %l i n e a r motion d= f l i p l r ( l i n s p a c e ( lambda_n , lambda_f , num_bits ) ) ; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %a r r a y p a t t e r n f o r i _ d =1: l e n g t h ( d ) d i s p ( i _ d ) AP( i_d , : ) = 1 + exp ( j *2* p i *d ( i _ d ) * s i n ( ang ) ) ; end AP=AP . / max ( max ( abs (AP ) ) ) ; 186 c l c %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %d a t a b i t s _ c l e a n = r a n d s a m p l e ([ −1 1 ] , num_bits , ’ t r u e ’ ) ; % n o i s e =wgn ( 1 , num_bits , −SNR ) ; n o i s e =awgn (AP ( : , 9 0 1 ) , SNR) −AP ( : , 9 0 1 ) ; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %n o i s e PSD N = l e n g t h ( n o i s e ) ; x d f t = f f t s h i f t ( f f t ( n o i s e ) ) ; psdx = ( 1 / ( num_bits *N) ) * ( abs ( x d f t ) . ^ 2 ) ; n o i s e _ e n e r g y =sum ( psdx ) ; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %r a d i a t e d d a t a f o r i _ a p =1: l e n g t h (AP ( 1 , : ) ) d a t a = abs (AP ( : , i _ a p ) ) . * b i t s _ c l e a n ’+ n o i s e ; i f ang ( i _ a p )== p i /180* −20 f i g u r e ( ) p l o t ([ −2 2 ] , [ 0 0 ] , ’ k ’ , ’ Linewidth ’ , 1 . 5 ) h o l d on p l o t ( [ 0 0 ] , [ − 2 2 ] , ’ k ’ , ’ Linewidth ’ , 1 . 5 ) a x i s ([ −2 2 −2 2 ] ) s c a t t e r ( d a t a ( 1 : 5 0 0 ) , z e r o s ( 1 , 5 0 0 ) , 6 0 , ’ F i l l e d ’ ) s c a t t e r ([ −1 1 ] , z e r o s ( 1 , 2 ) , 6 0 , ’ r ’ , ’ F i l l e d ’ ) 187 x l a b e l ( ’ I ’ , ’ F o n t s i z e ’ , 2 2 ) y l a b e l ( ’Q’ , ’ F o n t s i z e ’ , 2 2 ) x t i c k s ([ −1 1 ] ) y t i c k s ( [ ] ) e l s e i f ang ( i _ a p )==0 f i g u r e ( ) p l o t ([ −2 2 ] , [ 0 0 ] , ’ k ’ , ’ Linewidth ’ , 1 . 5 ) h o l d on p l o t ( [ 0 0 ] , [ − 2 2 ] , ’ k ’ , ’ Linewidth ’ , 1 . 5 ) s c a t t e r ( d a t a ( 1 : 5 0 0 ) , z e r o s ( 1 , 5 0 0 ) , 6 0 , ’ F i l l e d ’ ) s c a t t e r ([ −1 1 ] , z e r o s ( 1 , 2 ) , 6 0 , ’ r ’ , ’ F i l l e d ’ ) x l a b e l ( ’ I ’ , ’ F o n t s i z e ’ , 2 2 ) y l a b e l ( ’Q’ , ’ F o n t s i z e ’ , 2 2 ) a x i s ([ −2 2 −2 2 ] ) x t i c k s ([ −1 1 ] ) y t i c k s ( [ ] ) e l s e i f ang ( i _ a p )== p i /180*60 f i g u r e ( ) p l o t ([ −2 2 ] , [ 0 0 ] , ’ k ’ , ’ Linewidth ’ , 1 . 5 ) h o l d on p l o t ( [ 0 0 ] , [ − 2 2 ] , ’ k ’ , ’ Linewidth ’ , 1 . 5 ) s c a t t e r ( d a t a ( 1 : 5 0 0 ) , z e r o s ( 1 , 5 0 0 ) , 6 0 , ’ F i l l e d ’ ) s c a t t e r ([ −1 1 ] , z e r o s ( 1 , 2 ) , 6 0 , ’ r ’ , ’ F i l l e d ’ ) x l a b e l ( ’ I ’ , ’ F o n t s i z e ’ , 2 2 ) y l a b e l ( ’Q’ , ’ F o n t s i z e ’ , 2 2 ) a x i s ([ −2 2 −2 2 ] ) x t i c k s ([ −1 1 ] ) 188 y t i c k s ( [ ] ) end b i t _ e n e r g y ( i _ a p )= abs ( mean (AP ( : , i _ a p ) ) ) ^ 2 ; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %d e m o d u l a t e d a t a ( d a t a <0)= −1; d a t a ( d a t a > 0 ) = 1 ; %BER BER( i _ a p )= sum ( abs ( b i t s _ c l e a n − d a t a ’ ) / 2 ) / num_bits ; end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% d1= l i n s p a c e ( 0 , 5 , num_bits ) ; f o r i _ d =1: l e n g t h ( d1 ) d i s p ( i _ d ) %l i n e a r AP( i_d , : ) = 1 / ( max ( d1 ) − min ( d1 ) ) * ( 1 + exp ( j *2* p i . . . *d1 ( i _ d ) * s i n ( ang ) ) ) ; end AP=AP . / max ( max ( abs (AP ) ) ) ; 189 AP_mean=mean (AP ) ; BER_thry1 =1/2* e r f c ( s q r t ( abs ( AP_mean ) . ^ 2 / ( 2 * n o i s e _ e n e r g y ) ) ) ; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ang1=ang ; %f i g u r e s f i g u r e ( ) s e m i l o g y ( ang1 *18 0/ pi , BER, ’ Linewidth ’ , 1 . 5 ) h o l d on % c l e a r BER ang % l o a d r e s t u l t s . mat % % s e m i l o g y ( ang ( 1 : 1 3 ) , BER ( 1 : 1 3 ) , ’ g−o ’ , ’ Linewidth ’ , 1 . 5 ) s e m i l o g y ( ang1 *18 0/ pi , BER_thry1 , ’ − −k ’ , ’ Linewidth ’ , 1 . 5 ) a x i s ([ −90 90 − i n f 1 ] ) x l a b e l ( ’ Angle ( deg ) ’ , ’ F o n t s i z e ’ , 1 8 ) y l a b e l ( ’BER’ , ’ F o n t s i z e ’ , 1 8 ) g r i d on % l e g e n d ( ’ S i m u l a t e d ’ , ’ Measured ’ , ’ E s t i m a t i o n ’ ) l e g e n d ( ’ S i m u l a t e d ’ , ’ E s t i m a t i o n ’ ) 190 APPENDIX J MATLAB SECURE TRANSMISSION MEASUREMENT PROCESSING FOR THE LINEAR CASE c l e a r c l o s e a l l a l l c l c %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ang = − 6 0 : 1 0 : 6 0 ; t e m p l a t e = l o a d ( ’ mls . t x t ’ ) ; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% f o r i _ a n g =1: l e n g t h ( ang ) d i s p ( i _ a n g ) temp= l o a d ( [ ’ waveform_ ’ num2str ( ang ( i _ a n g ) ) ’ . t x t ’ ] ) ; temp=temp ( ( end − 4 0 0 0 ) : end , : ) ; temp1=max ( max ( temp ) ) ; temp2=min ( min ( temp ) ) ; i f abs ( temp1 ) < abs ( temp2 ) temp=−temp ; end [ row , c o l ]= f i n d ( temp==max ( abs ( [ temp1 temp2 ] ) ) ) ; [ ~ , l o c ]= f i n d ( temp ( row , : ) > max ( abs ( [ temp1 temp2 ] ) ) / 2 ) ; temp =[ temp ( : , l o c ( 1 ) + 1 + 1 0 : end ) temp ( : , 1 : l o c ( 1 ) ) ] ; % f i g u r e ( ) 191 % % % % % p l o t ( 1 : l e n g t h ( temp ( 1 , : ) ) , temp ( row , : ) ) r e t u r n f o r i _ i t =1: l e n g t h ( temp ( : , 1 ) ) d i s p ( i _ i t ) p l a c e = 5 : 1 0 : l e n g t h ( temp ( i _ i t , : ) ) ; f o r i_demod =1: l e n g t h ( p l a c e ) wf_demod ( i_demod )= mean ( temp ( i _ i t , p l a c e ( i_demod ) . . . −4: p l a c e ( i_demod ) + 4 ) ) ; end wf_demod ( wf_demod > 0 ) = 1 ; wf_demod ( wf_demod <1)= −1; i f ang ( i _ a n g )==0 wf_demod (1)= − wf_demod ( 1 ) ; end temp_BER ( i _ i t )= sum ( abs ( wf_demod ’ − t e m p l a t e ( 2 : end ) ) / 2 ) . . . / ( l e n g t h ( t e m p l a t e ) − 1 ) ; end BER( i _ a n g )= mean ( temp_BER ) ; temp= s o r t ( e i g ( cov ( temp ’ ) ) , ’ descend ’ ) ; sig_w=mean ( temp ( 2 : l e n g t h ( temp ( : , 1 ) ) ) ) ; Ps =1/ l e n g t h ( temp ( : , 1 ) ) * ( temp (1) − sig_w ) ; 192 s n r ( i _ a n g )=10* lo g10 ( abs ( Ps / sig_w ) ) ; c l e a r temp c l c end f i g u r e ( ) s e m i l o g y ( ang , BER) s a v e ( ’ r e s t u l t s . mat ’ , ’BER’ , ’ ang ’ ) 193 MATLAB SECURE TRANSMISSION SIMULATION FOR THE SINUSOIDAL CASE APPENDIX K c l e a r c l o s e a l l a l l c l c %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ang= p i / 1 8 0 * ( − 9 0 : 0 . 1 : 9 0 ) ; lambda_f = 0 . 9 3 9 8 / 0 . 2 ; lambda_n = 0 . 1 9 6 1 / 0 . 2 ; num_bits =300000; SNR=12; AP= z e r o s ( num_bits , l e n g t h ( ang ) ) ; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %s i n u s o i d a l motion d= cos ( l i n s p a c e ( 0 , 2 * pi , 2 * num_bits ) ) * ( lambda_f −lambda_n ) / 2 . . . +( lambda_n+ lambda_f ) / 2 ; d=d ( num_bits +1: end ) ; % f i g u r e ( ) % p l o t ( 1 : num_bits , 0 . 2 * d ) % r e t u r n %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %a r r a y p a t t e r n 194 f o r i _ d =1: l e n g t h ( d ) d i s p ( i _ d ) AP( i_d , : ) = 1 + exp ( j *2* p i *d ( i _ d ) * s i n ( ang ) ) ; end AP=AP . / max ( max ( abs (AP ) ) ) ; c l c %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %d a t a b i t s _ c l e a n = r a n d s a m p l e ([ −1 1 ] , num_bits , ’ t r u e ’ ) ; % n o i s e =wgn ( 1 , num_bits , −SNR ) ; n o i s e =awgn (AP ( : , 9 0 1 ) , SNR) −AP ( : , 9 0 1 ) ; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %n o i s e PSD N = l e n g t h ( n o i s e ) ; x d f t = f f t s h i f t ( f f t ( n o i s e ) ) ; psdx = ( 1 / ( num_bits *N) ) * ( abs ( x d f t ) . ^ 2 ) ; n o i s e _ e n e r g y =sum ( psdx ) ; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %r a d i a t e d d a t a f o r i _ a p =1: l e n g t h (AP ( 1 , : ) ) d a t a = abs (AP ( : , i _ a p ) ) . * b i t s _ c l e a n ’+ n o i s e ; b i t _ e n e r g y ( i _ a p )= abs ( mean (AP ( : , i _ a p ) ) ) ^ 2 ; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 195 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %d e m o d u l a t e d a t a ( d a t a <0)= −1; d a t a ( d a t a > 0 ) = 1 ; %BER BER( i _ a p )= sum ( abs ( b i t s _ c l e a n − d a t a ’ ) / 2 ) / num_bits ; end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% d1= l i n s p a c e ( lambda_n , lambda_f , num_bits ) ; d1=d1 ( 2 : end − 1 ) ; f o r i _ d =1: l e n g t h ( d1 ) d i s p ( i _ d ) %c o s i n e AP( i_d , : ) = 1 / ( p i * s q r t ( 1 − ( ( d1 ( i _ d ) . . . −(( lambda_n+ lambda_f ) / 2 ) ) . . . / ( ( lambda_f −lambda_n ) / 2 ) ) ^ 2 ) ) * . . . (1+ exp ( j *2* p i *d1 ( i _ d ) * s i n ( ang ) ) ) ; end AP_mean=mean (AP ) ; BER_thry1 =1/2* e r f c ( s q r t ( abs ( AP_mean ) . ^ 2 / ( 2 * n o i s e _ e n e r g y ) ) ) ; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ang1=ang ; 196 %f i g u r e s f i g u r e ( ) s e m i l o g y ( ang1 *18 0/ pi , BER, ’ Linewidth ’ , 1 . 5 ) h o l d on c l e a r BER ang l o a d r e s t u l t s . mat s e m i l o g y ( ang ( 1 : 1 3 ) , BER ( 1 : 1 3 ) , ’ g−o ’ , ’ Linewidth ’ , 1 . 5 ) s e m i l o g y ( ang1 *18 0/ pi , BER_thry1 , ’ − −k ’ , ’ Linewidth ’ , 1 . 5 ) a x i s ([ −90 90 − i n f 1 ] ) x l a b e l ( ’ Angle ( deg ) ’ , ’ F o n t s i z e ’ , 1 8 ) y l a b e l ( ’BER’ , ’ F o n t s i z e ’ , 1 8 ) g r i d on l e g e n d ( ’ S i m u l a t e d ’ , ’ Measured ’ , ’ E s t i m a t i o n ’ ) 197 APPENDIX L MATLAB SECURE TRANSMISSION MEASUREMENT PROCESSING FOR THE SINUSOIDAL CASE c l e a r c l o s e a l l a l l c l c %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ang = − 6 0 : 1 0 : 6 0 ; t e m p l a t e = l o a d ( ’ mls . t x t ’ ) ; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% f o r i _ a n g =1: l e n g t h ( ang ) d i s p ( i _ a n g ) temp= l o a d ( [ ’ waveform_ ’ num2str ( ang ( i _ a n g ) ) ’ . t x t ’ ] ) ; temp=temp ( ( end − 5 5 0 0 ) : end , : ) ; temp1=max ( max ( temp ) ) ; temp2=min ( min ( temp ) ) ; i f abs ( temp1 ) < abs ( temp2 ) temp=−temp ; end [ row , c o l ]= f i n d ( temp==max ( abs ( [ temp1 temp2 ] ) ) ) ; [ ~ , l o c ]= f i n d ( temp ( row , : ) > max ( abs ( [ temp1 temp2 ] ) ) / 2 ) ; temp =[ temp ( : , l o c ( 1 ) + 1 + 1 0 : end ) temp ( : , 1 : l o c ( 1 ) ) ] ; % f i g u r e ( ) 198 % % % % % p l o t ( 1 : l e n g t h ( temp ( 1 , : ) ) , temp ( row , : ) ) r e t u r n f o r i _ i t =1: l e n g t h ( temp ( : , 1 ) ) d i s p ( i _ i t ) p l a c e = 5 : 1 0 : l e n g t h ( temp ( i _ i t , : ) ) ; f o r i_demod =1: l e n g t h ( p l a c e ) wf_demod ( i_demod )= mean ( temp ( i _ i t , p l a c e ( i_demod ) . . . −4: p l a c e ( i_demod ) + 4 ) ) ; end wf_demod ( wf_demod > 0 ) = 1 ; wf_demod ( wf_demod <1)= −1; i f ang ( i _ a n g )==0 wf_demod (1)= − wf_demod ( 1 ) ; end temp_BER ( i _ i t )= sum ( abs ( wf_demod ’ − t e m p l a t e ( 2 : end ) ) / 2 ) . . . / ( l e n g t h ( t e m p l a t e ) − 1 ) ; end BER( i _ a n g )= mean ( temp_BER ) ; temp= s o r t ( e i g ( cov ( temp ’ ) ) , ’ descend ’ ) ; sig_w=mean ( temp ( 2 : l e n g t h ( temp ( : , 1 ) ) ) ) ; Ps =1/ l e n g t h ( temp ( : , 1 ) ) * ( temp (1) − sig_w ) ; 199 s n r ( i _ a n g )=10* lo g10 ( abs ( Ps / sig_w ) ) ; c l e a r temp c l c end f i g u r e ( ) s e m i l o g y ( ang , BER) s a v e ( ’ r e s t u l t s . mat ’ , ’BER’ , ’ ang ’ ) 200 BIBLIOGRAPHY 201 BIBLIOGRAPHY [1] [2] [3] [4] [5] [6] [7] [8] [9] P. 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