FREQUENCY SYNTONIZATION
FOR DECENTRALIZED DISTRIBUTED PHASED ARRAYS

By

William Reinaldo Torres

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

Electrical and Computer Engineering—Master of Science

2025

ABSTRACT

This thesis presents the research and development of an alignment method for spatially diverse,

decentralized, distributed phased array systems. The challenge of limited access to reliable align-

ment signals from a destination or other external primary control system is a significant hurdle for

distributed antenna systems in the context of emerging 5G/6G technologies. Even without refer-

ence signals, the system is designed to remain fully functional. Decentralized and open-looped

wireless sensor networks (WSNs) coordination can mitigate these issues. Still, it necessitates a

high level of precision in signal agreement for proper signal operation, such as distributed antenna

array beam forming.

This work presents a wireless distributed system intended to complement a decentral-

ized distributed wireless antenna array for communication and remote sensing systems without

needing information from second or third-party entities to achieve consensus. The intent is also

to demonstrate the partial independence of the syntonization from the synchronization approach.

Through the sole use of software in software-defined radios (SDRs), I explore different parameter

estimation techniques designed to reduce the residual error when the system achieves frequency

consensus, presented as a proof of concept and initial research into such applications that leverage

the use of this technique in legacy architecture.

The system uses orthogonal frequency division multiplexing (OFDM) to identify trans-

mitters. It implements an average consensus approach at a system level, which requires minimal

prior information from neighboring nodes. This approach ensures that the frequencies converge

to the desired residual error tolerance of less than 18◦, ensuring constructive interference with a

relative power gain of 90% concerning an ideal constructively interfering set of signals.

Copyright by
WILLIAM REINALDO TORRES
2025

TABLE OF CONTENTS

CHAPTER 1

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.
1.1 Problem Overview .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Current Technologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Contribution of this Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1
1
2
5

CHAPTER 2

.
.
.

. .
. .
.
.

THEORY AND SIMULATION METHODS . . . . . . . . . . . . . .

.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.

7
7
2.1
Introduction . .
7
2.2 Network Model
2.3 Signal Model
. 10
.
2.4 Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.5 Average Consensus Algorithm Simulation . . . . . . . . . . . . . . . . . . . . 21
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.6 Simulation Results
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
. .
2.7 Conclusion . .

.
.

CHAPTER 3

. .

SYSTEM DESIGN . . . . . . . . . . . . . . . . . . . . . . . . . . .

. 30
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.1
.
Introduction . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.2 System Architecture .
3.3 System Components
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.4 Data Flow and Storage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
. 32
3.5 Technologies and Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6 Algorithms and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.7 Scalability and Performance
. . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.8 Testing and Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.9 Conclusion . .

. .

.

CHAPTER 4

Introduction . .

4.1
4.2 Results . .
. .
4.3 Discussion . .
4.4 Conclusion . .

EXPERIMENTAL RESULTS . . . . . . . . . . . . . . . . . . . . . . . 37
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
. 37
.
.
. 46
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. .
.
.
. .

. .
. .

.
.
.
.

CHAPTER 5

CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. 48

BIBLIOGRAPHY .

APPENDIX .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

iv

CHAPTER 1

INTRODUCTION

1.1 Problem Overview

In recent years, the scientific community has been drawn to imitate biological natural

phenomena such as the collective swarm intelligence of ants and birds for technological advances.

This has pushed researchers to develop distributed systems of smaller cooperating subsystems that

perform better than single platforms and monolithic counterparts [1, 2]. An increase in the need

for privacy, security, speed, and resilience on the radio frequency (RF) spectrum has drawn interest

from the RF and antenna design engineering community. It has brought interest in developing a

spatially distributed antenna system with these capabilities [3–7]. For the system to take full advan-

tage of spatial diversity, signal stability over long-distance separations and prolonged time must be

achieved. Hence, the system must actively achieve a signal’s wavelength stability without physical

cables linking the antennas. This gives rise to a strong interest in achieving signal alignment of

distributed systems utilizing RF signals [8].

Departing from classical antenna array theory, which considers a static set of antenna

array elements (nodes) in either of the three dimensions (linear, planar, and volumetric) static

typical distributions, each with either uniform, non-uniform, random spacing [9], signal coherence

is achievable by adjusting each antenna’s phase electrical state, to achieve coherence at the moment

of transmission or reception.

Extending the single-platform approach to a multi-platform system, each node’s fre-

quency [10, 11], phase [12, 13], and timing [14, 15] electrical state must be adequately adjusted to

cooperate [7, 16, 17]. In the scenario where the nodes’ frequencies are not aligned, a beat pattern

is generated, causing regions where the superposition of the signals results in destructive interfer-

ence (valleys) and other portions of constructive interference. Similarly, when the signals have a

phase difference, the summation can result in complete destructive interference. Last but not least,

signal or data transmission timing needs to be correctly synchronized to avoid generating a phase

difference and resulting in destructive interference, as mentioned previously.

1

The project detailed in this thesis was conceived with a significant goal: to contribute to

designing and developing a system that can stabilize the individual transmitting frequencies from

each node of a decentralized distributed phased array system. This system will enable cohesive and

constructive interference at any arbitrary destination, significantly advancing RF and antenna engi-

neering by taking advantage of the spectral and spatial diversity of multiple nodes. To achieve the

precision and tight requirement presented in [7,8], it is desired to have signals achieve a maximum

phase error of 18◦. Distributed antenna systems promise to enhance RF communications [18, 19]

and radar [20–25] operations in environments with limited access to reference signals, such as

global navigation satellite system (GNSS) [26] or a lack of second-party alignment signals.

1.2 Current Technologies

Current frequency stabilization technologies can be divided into closed and open-loop

architectures, each with its consensus method. Furthermore, there are centralized and decentral-

ized architectures, which bring advantages and drawbacks to the syntonization of the network of

antenna nodes. Each approach to signal stabilization has limitations that make it less viable for a

robust and resilient system, demonstrating the necessity of the presented approach.

1.2.1 Closed Loop Architectures

Closed-loop syntonization requires a third-party beacon that generates a reference signal

to which the nodes in the system will syntonize. This reference signal is transmitted to the system,

which aligns the electrical parameters as necessary [27–29]. This architecture is a closed-looped

centralized architecture because it depends on the third-party reference signal or signal information

to control the system. The alignment information can be transmitted to the system in two ways:

centralized and decentralized.

The centralized approach uses a leader node in the system to relay the information to

align with the other nodes. Thus, the beacon system only needs to constantly communicate with the

leader, and the leader with the followers. This method leaves a single point of failure, jeopardizing

the system. On the other hand, the decentralized approach requires communication between the

beacon and each node of the phased array. One beacon and multiple nodes mean that a schedule is

2

necessary for the communication among them, requiring high timing precision but also introducing

significant latency to the operation, which can affect the oscillator drifts [30–32].

The drawback of a closed-loop architecture is its inherent limitation on beam steering.

Since the nodes calibrate their parameters around the beacon, they must recalibrate themselves

when steering to a different destination, and this requires prior knowledge of the location of the

new destination, knowledge that is not always available to the system. Then, the system must steer

the antenna’s beam, limiting the direction in which the distributed system can focus the beam. This

enables the signals to constructively interfere in the beacon’s direction and nowhere else, limiting

the applications to wireless communications. Using Fig1.1 as an example, the four nodes in the

system can only align their parameters with respect to the destination, and nowhere else, because

they highly depend on it.

Figure 1.1 Four Node Closed Loop System [33]

3

1.2.2 Open Loop Architectures

As the name suggests, open-loop architecture does not rely on a second or third-party

beacon signal for the system to agree upon its consensus values; the calibration of the signal’s elec-

trical parameters happens locally among the different nodes that make up the system. The electrical

parameters are aligned locally at the system level. Because they do not depend on external signals,

modifying the parameters for beam steering, forming, or nulling at arbitrary locations is trivial.

With the ability to perform arbitrary beam-forming, radar and communications applications are

possible.

Also, reducing the dependency on a beacon synchronization signal, a single point of

failure, and scheduling the communications links between the beacon and nodes that make up the

system helps make the system more resilient in harsh environments. To put it in perspective, Fig1.2

shows a four-node system where the nodes align themselves without receiving any information

from a third-party beacon. This allows the system to beam steer, enabling more capabilities.

Figure 1.2 Four Node Open Loop System [33]

4

1.2.3 Centralized Architecture

The leader-follower dynamic is another topology for distributed phased arrays, as fore-

shadowed in 1.2.1. It is also called centralized architecture because it makes up a central node, a

leader, and the other nodes or subnetworks are followers. In this network setup, the leader, as part

of the system’s nodes, relays directly the values of the electrical parameters to the followers and

ensures they are correctly aligned. The system leader does the most significant portion of the work

because it needs to act as a relay between the destination and the other nodes, or if no feedback is

provided, it needs to determine and align the signals by itself, while the followers are just comply-

ing. A significant drawback of centralized architecture is the fragility of the single point of failure,

which is the leader node failing. In the case of a targeted attack or downtime of the leader node,

the entire system will be vulnerable and not operational. This would defeat the whole purpose of

distributing the phased array.

1.2.4 Decentralized Architectures

The decentralized architecture does not rely on a primary node that leads the other nodes

with the parameters alignment; the alignment occurs concurrently among the nodes, and they

attempt to align to an expected value. A decentralized network addresses the issue of a single

point of failure brought up in Sec1.2.3 because no single node is more important than the others;

therefore, the system can continue operating when one or several nodes have failed or are down

due to maintenance, without having a significant impact in the performance of the system.

A downside of a decentralized system is the limited information sharing among nodes.

Addressing the problem of limited information propagation by enforcing tighter constraints in the

network [34–38] (explained in more detail in Sec2.2) demonstrates the resilience to link or node

failure because no link or node is imperative.

1.3 Contribution of this Work

The work presented in this thesis focuses on contributing to the development of a physi-

cal demonstration of a distributed, decentralized, open-looped phased array system that is resilient

to single point failure, as mentioned in Sec1.2.4, and with both communication and radar capa-

5

bilities, as addressed in Sec1.2.2. This project aims to achieve inter-node signal syntonization of

two-tone waveforms with a residual error of less than 18◦ [7, 38–40] purely through software, uti-

lizing software-defined radios, which should allow for distributed beamforming in the future. It

also aims to demonstrate that it is possible to maintain consistently updated accuracy, as mentioned

earlier, with frequency orthogonal signals while frequency hopping to aggregate another level of

security from man-in-the-middle and interference attacks [41–45].

6

CHAPTER 2

THEORY AND SIMULATION METHODS

2.1

Introduction

Phased arrays are composed of multiple antennas operating coherently with transmission

and reception capability. When the antennas that compose the array are placed at a significant

distance from each other, it is not convenient to have them synchronized with extremely long

cables, which can limit the applications where the system can be used. This is when the interest in

a distributed system starts. Because the system is spatially distributed, each antenna has its power

source and local oscillator and must be independent of its neighbors but still contribute to the

system. To be able to achieve coherence, the signals must have a difference between their phases

of less than 18◦, or π

10 rad, to demonstrate the capabilities of stabilization of two-tone signals

for real-time applications, and without time alignment using a decentralized average consensus

approach.

2.2 Network Model

The antennas in the system are spatially separated; hence, the system can be described

using a graph, G = {V, E, A}, where V = {1, . . . , N } is the set of vertices corresponding to the

antennas node, and E represents the set of edges connecting the vertices, and the communication

links and line of sight of the transmitting antennas, and A is the adjacency matrix [46]. The graph

used to describe the system network can be directed [47] or undirected [39], greatly impacting the

necessary information and the node’s information from its neighbors. The graph that describes the

network must be strongly connected, which means that the information can flow from an origin

node to any other node in the system through intermediate nodes without getting stuck. The adja-

cency matrix, where the columns correspond to the nodes that are transmitting, and the rows the

receiving nodes, that best describe these requirements are 1) symmetrical over its diagonal, which

takes care of the undirected communication and robust connectivity A = AT . Another condition

that the connectivity matrix needs to comply with is 2) self-loop, with non-zero elements along the

7

matrix’s diagonal. Also, the network must be 3) decentralized, allowing for cells to be populated

with zeros for nonexistent links between nodes. As shown in table2.1, an adjacency matrix, A, de-

scribes the link connections between the nodes. Additionally, the adjacency matrix is adjusted to

RX \TX Node
0
1
2
3
4
5
6

0
1
1
0
0
1
0
1

1
1
1
0
0
1
0
1

2
0
0
1
0
1
0
0

3
0
0
0
1
1
0
0

4
1
1
1
1
1
0
1

5
0
0
0
0
0
1
1

6
1
1
0
0
1
1
1

Table 2.1 A sample strongly connected seven node adjacency matrix for fig 2.1

assign weights to the corresponding channel’s data and to compute the node’s local weighted arith-

metic mean. The only constrain for assigning the weights to the adjacency matrix and converting

to a mixing matrix, W, is that the matrix needs to be double stochastic when the network com-

munication links are undirected [39], and row stochastic when the network has directed links [47].

Focusing on an approach that enables local arithmetic averages with the neighboring nodes, we

use the Metropolis-Hasting algorithm [48] to generate the mixing matrix. In the interest of this

project, to test the scalability of orthogonal frequency signal syntonization, we assume simultane-

ous transmission, quasi-static nodes, and bidirectional communications that comply with all the

requirements for the system mentioned previously.

1
max{deg(i),deg(j)}+1 ,

if (i, j) ∈ E,






Wi,j =

0,

1 − (cid:80)

if (i, j) /∈ E and i ̸= j,

(2.1)

j:j̸=i wi,j,

if i = j

8

Figure 2.1 Strongly connected network with seven independent nodes with undirected/bidirectional
communication links among themselves.

RX \TX Node
0
1
2
3
4
5
6

0
23
60
1
4
0
0
1
6
0
1
5

1
1
4
23
60
0
0
1
6
0
1
5

2
0
0
5
6
0
1
6
0
0

3
0
0
0
5
6
1
6
0
0

4
1
6
1
6
1
6
1
6
1
6
0
1
6

5
0
0
0
0
0
4
5
1
5

6
1
5
1
5
0
0
1
6
1
5
7
30

Table 2.2 Sample mixing matrix, W, for fig 2.1

9

Figure 2.2 Fully connected, looped, network composed of four nodes with self-talk

2.3 Signal Model

As the name suggests, two-tone waveforms, described by equation 2.2, are composed of

the superposition of two single-tone signals at baseband with different frequencies, f1 and f2.

S(t) = A1(ej·2π·f1·t) + A2(ej·2π·f2·t)

(2.2)

This waveform is modulated onto a carrier signal. The advantages of using two-tone waveforms

have been extensively analyzed for electronic systems calibration techniques, simultaneous radar

and communications applications [49–51] with tracking capabilities down to 15 mm. Two-tone

waveforms have been used to syntonize nodes by modulating a single frequency of 10 MHz into the

signal’s bandwidth, and transmitting it to neighboring nodes with a self-mixing circuit which takes

advantage of products of sinusoids to demodulate the single frequency [52, 53]. The bandwidth is

the magnitude of the difference between the tones that compose the waveform, like in equation 2.3

∆f = |f2 − f1|

(2.3)

10

Node 0Node 3Node 1Node 2The downfall of these frequency-sharing approaches is that they require additional hardware for

proper demodulation of the reference frequency from the bandwidth, which significantly increases

the cost, limiting the speed of scaling to an extensive system, or an already deployed system, and

requiring scheduling the times each internode syntonization signal can be transmitted, affecting the

purity of the signal generated by the oscillators [30–32]. Contrary to the earlier work, this project

focuses on achieving a similar feat without requiring external hardware and purely on the digital

signal processing capabilities of off-the-shelf hardware such as software-defined radios.

Two essential characteristics of the two-tone waveform are the difference between the

two frequencies, the bandwidth, and the midpoint of the frequency values. Traditionally, the two

frequency values have been equally distant from 0 Hz as double sideband signals, but in the interest

of this project, the signal remains double sideband but not centered at the carrier; instead, they will

be centered arbitrarily, away from the carrier, in different channels of the transceivers instantaneous

bandwidths allowing simultaneous reception and transmission.

f1 + f2
2

̸= 0

(2.4)

2.3.1 Transmit Signal Model

Each node is scheduled to transmit simultaneously but spread across the spectrum in its

predetermined channel, obeying orthogonality conditions, leaving a band guard among tones, as

depicted in figure 2.3. The signals are assumed to be planar waves that can be modeled using

complex exponentials. As proof of concept and validation in simulation, it is believed that the

internode syntonization signals were initially transmitting pulses outside the detectable portion of

the spectrum that the nodes could scan; hence, there is an initial error in the waveform’s tones

and consequently in the bandwidth. Then, they all agree to syntonize in some predetermined por-

tion of the spectrum, such that their signals occupy the instantaneous bandwidth the nodes can

sample. For instance, an over-the-shelf software-defined radio, such as National Instruments (NI)

Ettus X310, with UBX160 daughter boards that can sample at 200 MSa/s, hence the transmitters

divide the portion of the spectrum among themselves leaving an approximately equally spaced

11

channel where the signals will coexist. To obey the Nyquist criterion [54], the signals can only

exist between the carrier frequency and 100 MHz away from the carrier frequency. After the sub-

channels are allocated, the two-tone central transmission value is determined at the middle of the

channel, separating the signal’s tones enough for orthogonality. Thus, equation 2.2, which de-

scribes a continuous two-tone waveform, is rewritten to equation 2.5, where fc corresponds to the

channelization offset which centers the two-tone’s frequencies away from the carrier frequency

and the other inter-node aligning signals and allows identification if frequencies hopping is nec-

essary [42–45, 55–57]. Also, ∆f corresponds to the two-tone waveform bandwidth as defined in

equation 2.3, where f1 is considered the higher frequency tone and f2 the lower frequency tone,

and A1 and A2 are their corresponding single tone amplitudes.

S(t) = A1 ej2π(fc+ ∆f

2 )t + A2 ej2π(fc− ∆f
2 )t

(2.5)

Figure 2.3 A graphical representation of a transmission schedule for two-tone signals within the
same portion of the electromagnetic spectrum, and its reference frequency demodulation

As part of a larger project working towards the electrical parameter alignment of dis-

tributed antenna array signals, it is necessary first to demonstrate syntonization independent of

signal synchronization and phase alignment, but not completely excluding them from the equa-

tion. This is why, to work along with a similar system designed in [58–60], the signals will be

pulsed for the security of internode aligning signals. This means that to go around the problem

of time misalignment among nodes and reduce the impact it may have on signal syntonization’s

proof of concept, the two-tone waveform is pulsed, and equation 2.5 is modulated onto a rectan-

gular wave of duration t. Because each node believes they are synchronized to have at least one

12

aligning iteration, the transmitting is appended to the leading and trailing zero pad to position the

signals in the middle of the transmit window, as described in equation 2.6.

S[n] =






0

if 0 ≤ n ≤ z

A1 ej2π

∆f
2

fc+
fs

n + A2 ej2π

∆f
2

fc−
fs

n

if z < n ≤ z + l

(2.6)

0

if N − z < n ≤ N

Equation 2.6 is the ideal discrete single-channel transmit signal, where fs is the sampling frequency

at the transmitter, N is the length of the transmit window, l is the length of the signal pulse, and z

is the total length of the both leading and trailing zero padding divided by two. To reiterate, and for

clarity, the reason for zero padding is to mitigate the impact of time misalignment of the system,

which is not the focus of the research. At a node level, the node passes equation 2.6 through a

digital-to-analog converter (DAC) and modulates that signal to a carrier signal.

2.3.2 Receive Signal Model

Each node receives its neighbor’s signal that has passed through a noisy channel. Ide-

ally, highly directive antennas would be used, but this would limit the speed and cost at which the

system can scale. Hence, the use and simulation of omnidirectional antennas are implemented to

achieve a similar result to a real-world scenario. This means that because transmission and re-

ception happen at the same internal clock tick of each node, the node will also receive its signal

from self-interference. In the scenario of a two-node system, equation 2.7 corresponds to the sin-

gle communication channel between two nodes without self-interference. In contrast, equation 2.8

encompasses the superposition of all the neighbor’s internode syntonization signals, including the

node’s local signal, where j is the identification index for transmitting nodes, and i is the identifi-

cation index for receiving nodes.

Sj[n] =






wi,j[n]

if 0 ≤ n ≤ z

B1 ej·2π·

∆f
2

fc+
fs

·n + B2 ej·2π·

∆f
2

fc−
fs

·n + wi,j[n]

if z < n ≤ z + l

(2.7)

wi,j[n]

if N − z < n ≤ N

13

Figure 2.4 A sample simulation, in the time domain of the pulsed digital signal, a single node will
send through the DAC and then modulate to the carrier signal.

For equation 2.7, wi,j[n] is assumed to be zero-mean additive white Gaussian noise inherited in

the channel the signal travels where i is the transmitting node, and j is the receiving node. It

is essential to highlight that although the nodes receive the same signals, each signal is traveling

through its own channel in the direction of their neighboring nodes; hence, w[n] in equation 2.7

is different for each transmitted signal and their corresponding receiving node, this is contrary to

the approach proposed by [39, 47] where they assumed that each node detected the same channel

noise, and interference among alignment signal because their approach depends on time division

multiplexing access.

Si,RX[n] =

J
(cid:88)

j=1

Sj[n]

14

(2.8)

020406080100120Time[S]1.000.750.500.250.000.250.500.751.00N.UNode 4Figure 2.5 A sample simulation, in the frequency domain of the pulsed digital signal, a single node
will send through the DAC and then modulate to the carrier signal.

2.4 Parameter Estimation

Now that the nodes have received the signals, it is necessary to evaluate the electrical

parameters of the tones individually and collectively concerning the two-tone waveform to which

they belong. As part of transitioning distributed phased array alignment to solely digital signal pro-

cessing, this work focuses on bandwidth syntonization; hence, it is crucial to look for single-tone

frequency estimation techniques and how they can be extended to real-time multi-tone estimation

techniques.

2.4.1 Cramer Rao Lower Bound

The Cramer-Rao lower bound (CRLB) is the variance of any unbiased estimator for a

parameter in a statistical model. Although many bounds exist, such as the Barankin bound [61,62],

15

0255075100Frequency (MHz)0.000.250.500.751.00N.UNode0 TX0255075100Frequency (MHz)0.000.250.500.751.00N.UNode1 TX0255075100Frequency (MHz)0.000.250.500.751.00N.UNode2 TX0255075100Frequency (MHz)0.000.250.500.751.00N.UNode3 TXFigure 2.6 A sample capture, in the time domain of the pulsed digital signal, each node will send
through the DAC and then modulate to the carrier signal.

in theory, it is the most obvious to determine if an estimator exists capable of attaining the bound.

In layman’s terms, the CRLB is the lower bound that an estimator can achieve. In the interest of

this project, it is necessary to assume that the single signal is deterministic and is submerged in

additive white Gaussian (AWG) noise. The derivation of the CRLB for a single sinusoidal signal is

well known and has been derived multiple times in the literature by [63, 64], and they determined

the CRLB, inequality 2.10, from equation 2.9.

x[n] = A cos(2πf0n + ϕ) + w[n]

n = 0, 1, . . . , N − 1

(cid:17)

(cid:16) ˆf0

≥

var

12 · f 2
s
(2π)2 N 3SN R

16

(2.9)

(2.10)

020406080100120Time[S]1.000.750.500.250.000.250.500.751.00N.UNode 1020406080100120Time[S]1.000.750.500.250.000.250.500.751.00N.UNode 2020406080100120Time[S]1.000.750.500.250.000.250.500.751.00N.UNode 3020406080100120Time[S]1.000.750.500.250.000.250.500.751.00N.UNode 4Figure 2.7 A sample capture, in the frequency domain of the pulsed digital signal, a single node
will send through the DAC and then modulate to the carrier signal.

In [65], Richards derives the CRLB assuming that the signal is a complex exponential in AWG

noise from equation 2.11 and ends up with the variance, inequality 2.12, half of that derived by

Van Tree and Kay. For both, equation 2.9, 2.11, f0 is the normalized frequency and is equivalent

to f
fs

, where f is the signal’s frequency and fs is the sampling frequency.

x[n] = A(ej·2πf0n+ϕ) + w[n]

n = 0, 1, . . . , N − 1

(cid:17)

(cid:16) ˆf0

≥

var

6 · f 2
s
(2π)2 N 3SN R

(2.11)

(2.12)

For equations 2.10 and 2.11, the CRLB is trivially obtained because the estimator of a single

sinusoid is unbiased but with the existence of multiple sinusoids obtaining a close form expression

17

0255075100Frequency (MHz)0.000.250.500.751.00N.UNode1 RX0255075100Frequency (MHz)0.000.250.500.751.00N.UNode2 RX0255075100Frequency (MHz)0.000.250.500.751.00N.UNode3 RX0255075100Frequency (MHz)0.000.250.500.751.00N.UNode4 RXfor the CRLB or Barankin bound is mathematically not feasable, and a further analysis will be

done on the phenomena of spectral leakage in section 2.4.3, which gives rise to the biased.

2.4.2 Methods for single tone estimation

The interest in parameter estimation has developed into a full area of research, and sev-

eral single approaches for parameter estimation have been developed. The most popular reference

algorithm researchers look into is the methods derived by Steven Kay [66], which are done in the

time domain. Contrary to Kay’s method of estimating the frequency in the time domain, [67] ap-

proaches the problem with an initial coarse frequency estimate and refinement using calculus and

geometry to interpolate the discrete Fourier domain and calculate the signal’s frequency. Similarly

to [67], [68] depends on the coarse frequency estimate and further refinement using the secant

method. [5, 69, 70] also achieve or closely approach the CRLB at high SNR when estimating a

single sinusoidal frequency. It is important to reiterate that these methods rely heavily on the ge-

ometry of the sinusoid and its inherited characteristics to achieve the CRLB, given that a single

sinusoid estimator is unbiased, as explained in section 2.4.1.

2.4.3 Parameter Estimation Extension

In the presence of multiple sinusoids with a finite duration and an unknown future update

value that may not lie at a frequency bin, mutual interference between the tones is inevitable. This

interference is known as a frequency or spectral leakage. As depicted in Fig.2.8, the a’s correspond

to the time domain transmit signal, while the b’s represent the Fourier domain. The summation

of two monotone signals generates a two-tone waveform. Figure 2.9 presents the superposition in

the Fourier domain of both single tones and their corresponding combination. Although it seems

that the plots align properly, there is an impact on the plotted sinc function, affecting the peak’s

location and, hence, any geometric estimate in the Fourier domain. The distortion becomes more

prominent with an increase in the number of frequencies present during the signal capture. The

estimation of multiple frequencies introduces bias to every tone; hence, a single estimator won’t

be able to achieve the CRLB directly. That is why isolating the tones from each other is neces-

sary to utilize the single-frequency estimation techniques. This means the signals must be passed

18

through filters to isolate the tones. No external hardware should be essential for this project; hence,

narrowbandpass finite impulse response (FIR) filters are considered. As demonstrated in [71], to

approximate the CRLB with estimators, it is necessary to make a two-pass filtering approach,

where the sole purpose of the first pass is to obtain a coarse estimate of the frequency to locate the

center of the filter. Once the filter is designed, and the signal is passed through the filter, another

coarse frequency estimate, followed by a refinement or interpolation approach, yields an estimate

that approaches the CRLB. This two-way approach is implemented in every single tone. Thus, the

time complexity of frequency estimation grows with the number of tones present. Another way

to estimate the frequencies is to use super-resolution estimation techniques, such as the Pisarenko

method [72], ESPRIT [73], and MUSIC [74]. The literature has analyzed all of these techniques,

and the consensus is that even though they significantly approach or achieve the CRLB estimation

error, the computation cost is too high because they all require Eigenvalue decompositions and

large matrix inversions to obtain the frequency estimate. Also, the nodes must know a priori the

quantity of frequencies they are estimating, which increases the computation time to determine

the value initially and limits the number of applications where it can be implemented. Because a

single computer controls the nodes for the coarse time alignment, it also means that the computer

does all the necessary processing, meaning that the time complexity of any estimator is scaled by

a factor of n, where n is the number of nodes that compose the system.

19

Figure 2.8 A simulation of the superposition of two single-tone waveforms to create a two-tone
waveform, both in time and frequency domain

20

210121.00.50.00.51.0Amplitudea.105101520250.000.050.100.150.200.25Magnitudeb.1210121.00.50.00.51.0Amplitudea.205101520250.000.050.100.150.200.25Magnitudeb.221012Time [s]21012Amplitudea.30510152025Frequency [Hz]0.000.050.100.150.200.25Magnitudeb.3Figure 2.9 A simulation comparison of two single-tone waveforms to a two-tone waveform

2.5 Average Consensus Algorithm Simulation

Once the nodes have calculated the received signal’s bandwidth and identified the trans-

mitting node, they can calculate the weighted arithmetic average of the bandwidths and their own

transmit bandwidth. A Python3 script was programmed for the system simulation, APPENDIX I.

Now that the network has been designed, the nodes must generate the signals they transmit. The

code assumes that all nodes transmit and receive simultaneously and that signal propagation is neg-

ligible. Although this assumption is not valid during the physical experiment, it greatly facilitates

the processing of the signals in both the simulation and the physical experiment. The simula-

tion also assumes that the nodes are identical, have the same transmit and receive duration, which

are not the same for both transmit and receive, and have the same sampling rate. For a realistic

approach, by default, the nodes transmit at the same carrier frequency and have a sampling rate

21

7.510.012.515.017.520.022.525.0Frequency [Hz]0.000.050.100.150.200.25Magnitude12132122Tone 1Tone 2Two-Tone Signalof 200 MSa/Sec; hence, the Nyquist criterion determines that the highest frequency component

cannot exceed 100 MHz. With the Nyquist rate in mind and leaving enough spectral space away

from the DC noise, the spectrum is divided equally among the number of nodes to determine the

channel at which they will transmit. After deciding which band of the spectrum they will be al-

lowed to transmit on, the midpoint of the band becomes the channelization offset, fc, as explained

in Section 2.3.1 equation 2.5, and section2.3.2 equation 2.7 and equation 2.8. After determining

the central frequency value, the transmission channel’s bandwidth, and the band guard bandwidth

for the worst-case scenario the bandwidth, the simulation generates random values for the initial

bandwidths from a random normal distribution with limits between half the channel’s bandwidth,

the highest frequency respecting the band guard, and 10% of the bandwidth. Then, the bandwidth

and central frequency values are assigned to their corresponding transmitter, and the two-tone time

complex sinusoid is generated in time domain signals for the designated pulse duration, which

should be less than or equal to the receiver capture window duration.

Once the signals have their independent noise, they are added to each other as explained

in equation 2.8. Now that the node has received the superposition of the signals, it will strip

away any zero padding because it contains channel noise, which will impact the estimation more

than desired. In the simulation, a threshold value cuts off the noise. Hence, the minimum SNR

considered viable for the experiment must be greater than 0 dB. Now that just the superpositions

of the signals are left, the estimation algorithm is implemented. As addressed in Section 2.4.2, any

geometric method to approximate the frequency estimation should be good enough [5, 67–69] for

the simulation results presented in Section 2.6, [5]’s quadrature least square (QLS) approach for

peak finding was chosen as the most efficient approach compared to the others. The advantage

of this technique is that it is a very similar estimation error to the other methods while being less

computationally expensive. The nodes estimate the frequency of tones they detect and store them

in a list, from which they find the difference of the sequential values to calculate the bandwidths.

The program simulates the down-converted two-tone pulse in a fully connected network

to characterize the system’s behavior through simulation. Then, it lets the mixing matrix place a

22

link connection and proper weight to these values when performing the proper multiplication. As

explained in section 2.3.2, equation 2.7, each internode syntonization link has its own independent

AWG noise, which leads to slightly different estimates. Extending the model proposed by [39,

47, 75–78], instead of 2.13, the algorithm is extended to equation 2.14. Where M is an n×n

matrix holding each node’s measured bandwidth in their corresponding columns, and diag(·) is

the diagonal function, which extracts only the diagonal elements of (·)

a1












a2

. . .












aN .

Because the simulation considers a fully connected network of measurements, M is a nonzero ma-

trix, and through the mixing matrix synthetically it maintains or drops the communication links.

For the simulation, T (k + 1) is a one-dimensional array that holds the values of the next itera-

tion’s bandwidth, and the process is repeated again until convergence is achieved. Equation 2.14

describes the system-level syntonization. At the node level, this is equivalent to performing a

dot product between the corresponding rows of W and M(k), and the results become the updated

bandwidth for the next iteration.

f (k + 1) = W × f (k)

T (k + 1) = diag(W × M(k))

(2.13)

(2.14)

The algorithm is executed until a stopping criterion is met which depends on a case by case scenario

depending on the system’s capabilities.

2.6 Simulation Results

The results of averaging 50 Monte Carlos simulation over 50 pulse duration from 10µS to

100µS are shown in Figure 2.12, Figure 2.13, and Figure 2.14 with four, five, and six node systems

respectively. These results show that increasing the pulse duration and system connectivity greatly

impacts the system’s performance. For this simulation, the stopping criterion was decided from

23

the average consensus algorithm iteration counter with a limit of 100 iterations. Then, the last 10

iteration’s residual values were averaged and presented.

24

Figure 2.10 Sample average consensus experiment of simulations 100 ¯sec

25

0255075100125150175200Average Consensus Iteration (k)1.52.02.53.03.5Bandwidth Estimates (MHz)Figure 2.11 Average residual bandwidth at consensus over multiple pulse durations for a four node
system with pulse duration of 100 ¯S

26

0255075100125150175200Average Consensus Iteration (k)103104105106Residual Bandwidth Estimates (Hz)Figure 2.12 Plot of sweeping 50 pulse duration between 10 ¯S to 100 ¯S over 50 Monte Carlos
simulation of 4-node system with connectivities of 66% and 100%

27

20406080100Signal Duration (S) 0.10.20.30.40.50.6Mean Residual Bandwidth (rad)18o mark4 nodes, c=14 nodes, c=0.6Figure 2.13 Average residual bandwidth at consensus over 50 pulse duration, for two different
topologies composed of 5 nodes

28

20406080100Signal Duration (S) 0.10.20.30.40.50.6Mean Residual Bandwidth (rad)18o mark5 nodes, c=15 nodes, c=0.6Figure 2.14 Average residual bandwidth at consensus over 50 pulse duration, for two different
topologies composed of 6 nodes

2.7 Conclusion

From the Monte Carlos simulations, it can be concluded that for a small network, no

larger than six nodes, the minimum pulse duration needed to ensure stability less than the 18◦

needs to be longer than 40 µs, with a capture window of 120 µs. It is important to reiterate that

these simulations are valid for small-size networks, each signal would have 30 dB SNR, but the

SNR as shown in Figure 2.10, and Figure 2.11, where the network is composed of over 10 nodes

and signal duration of 100 µs, the residual error yields a phase offset of approximately 36◦.

29

20406080100Signal Duration (S) 0.20.30.40.50.6Mean Residual Bandwidth (rad)18o mark6 nodes, c=16 nodes, c=0.6CHAPTER 3

SYSTEM DESIGN

3.1

Introduction

In this chapter, the SDRs experimental implementation to perform a wavelength level of

stability is discussed. The frequency stabilization can be divided into two significant components,

the system as a whole and the individual nodes. The node level is divided into the hardware’s

functionality, and its corresponding software. It is important to understand the building block of

the system, how they perform individually, collectively and as a whole unit. These are the pillars

of the innovation provided by this project.

3.2 System Architecture

No other research group or industry has fully explored the unique characteristic of dis-

tributed phased arrays: the open-loop and decentralized architecture concept. This innovation will

permit the technology to be mounted on spatially independent moving vehicles that can be sent to

remote areas where both sensing and communication applications are crucial for performing tasks,

but control of the units is limited.

As explained before, the system can be described as a graph with vertices corresponding

to the graph’s nodes and links connecting them. To develop distributed and open-loop capabili-

ties, the graph must be strongly connected; in other words, a path of links and nodes exists that

allows information to propagate to all the nodes, flooding the network. There are two ways to

achieve this, and the most convenient way is to assume the links are undirected, which means the

nodes have bidirectional communication. To achieve bidirectional communication, it is necessary

to share the resources available, and the use of OFDM significantly facilitates this. Time-division

multiplexing is another option, but this project intends to demonstrate full timing independence

while performing signal syntonization. Another approach is to use a different gossip-based net-

work flooding approach [34], such as push-sum protocol [47], which allows the network to have

directed communication links. Figure 2.1 is an example of a decentralized and strongly connected

30

network. The advantage of this kind of network is that information sharing occurs at the internode

level without external feedback from third parties. Implicit information sharing is also enabled,

enhancing the security and resilience of the system, which aligns with the project’s goal.

3.3 System Components

The system is composed of multiple nodes with simultaneous transmission and reception

capability. For the physical system SDRs take the place of the nodes and the link that allows for

inter-node communication are the two-tone signals transmitted wirelessly that implicitly convey

the bandwidth information. As an initial step, to demonstrate the viability of the approach taken

to stabilize the signal the SDRs are all connected through 10 Gbit/s Ethernet cables to a single

host computer unit running GNURadio Companion in Linux Ubuntu 20.04 operating system. The

reason for connecting the SDRs to a single computer is to partially address the timing mismatch of

the SDRs.This way they all activate, transmit, and receive relatively almost at the same time.

3.4 Data Flow and Storage

The system is initialized with each node agreeing upon the portion of the spectrum that

they will utilize to transmit their signals and identify their neighbors. The SDRs can be close to

each other and at line of sight. When necessary, they can be synthetically dropped even when

maintaining the spatial proximity for extreme cases where the estimates are unreliable. As a rule

of thumb, the nodes should transmit within the same instantaneous bandwidth that the nodes can

receive to avoid random oscillator drifts when updating the local oscillators to search for signals

through the spectrum. Also, the signals must be independent of each other. Hence, a wide enough

separation, band guard, exists between signals and coupled tones. Assuming the system has ob-

tained a coarse time alignment that allows each node to transmit relatively simultaneously, the time

delays related to the positioning of the nodes can be neglected. In the cases where the signals that

are being transmitted are pulsed, then a tighter condition for time alignment is necessary, but if all

the nodes are transmitting continues waves (CW) then time alignment is only required for updating

the tone’s frequencies in the transmitter end, but not as crucial in the receivers side of the node.

The receiving nodes do not explicitly receive the tone values or the bandwidth at which

31

the rest of the network is operating because of the difference between the nominal frequency value

inputted by the software and the actual transmission [30], hence the receiving nodes implicitly

receive the transmitting frequencies and are responsible of down-converting, detecting, identifying

the transmitter, estimating the received frequencies, calculate the bandwidth, compute their local

average and update for the next cycle of transmission.

This cycle of signal transmission, reception, estimation, average calculation, and trans-

mit updating continues indefinitely, syntonizing the local node transmission with the other nodes

in the system. Figure 3.1 summarizes the flow of the data and decision making by each node that

cumulatively achieves consensus at the node level and convergence at the system level.

Figure 3.1 Block diagram of information flow at a local node level.

Given the SDRs’ memory capabilities, there are two options for deciding the estimated

value used for calculating the local average: either the input value stored in memory or its own

self-estimated bandwidth value. For the sake of spectral awareness, it was decided to use the

current transmission estimation value but store the previous transmission estimation locally for

self-reporting and offline analysis.

3.5 Technologies and Tools

As stated multiple times previously, this project aims to demonstrate the capabilities and

limitations of designing a decentralized distributed phased array using software-defined radios that

can perform syntonization purely through software using signal processing approaches. This is

why the technology used is SDRs that run solely on software and omnidirectional antennas to

32

communicate locally among the SDRs.

3.5.1 Node Design

In the effort to demonstrate the capabilities of decentralized open loop network antenna

array stabilization, the average consensus algorithm was validated using National Instrument (NI)

Ettus Research USRP X310 software defined radios [79], with NI UBX160 daughter-boards [80].

The reason for the use of these software-defined radios is the multiple methods of programming

and interfacing to software the NI technology allows, ranging from high-level programming with

LabView, MATLAB Simulink, and GNU Radio Companion with Python to low-level program-

ming with Verilog.

Figure 3.2 National Instrument’s picture of USRP X310 Software Defined Radio.

We have decided to use the GNU Radio block diagram development environment, which

allows high-level programming languages such as C++ and Python to create the flow diagram that

manipulates data for processing and takes advantage of the plethora of signal processing libraries

these languages possess.

3.5.2 Antenna Selection

The Pulse Electronics Multi Band Swivel Mount Dipole SPDA24700/2700 antennas

were used to verify the validity of the proposed approach. Each SDR has two antennae connected

to their corresponding daughter board, and each daughter board is assigned the task of either trans-

33

Figure 3.3 National Instrument’s picture of UBX160 daughterboard for X310 SDR.

mitting or receiving. These are omnidirectional antennae; therefore, if the SDRs are placed close

to each other but far away from the near field of the antenna and the transmit frequencies, then

all the SDRs can communicate with each other, enabling a completely connected network to start

with, which estimated values can be zeroed out to generate smaller networks and different network

architectures synthetically. Another reason, besides being omnidirectional, to choose these anten-

nas for the setup is the bandwidth of operation of the antennas, which allowed for proper operation

with the X310 SDRs in a lab environment within Michigan State University.

3.6 Algorithms and Methods

The average consensus method of frequency stabilization described in section 2.2 is

utilized, and the different bandwidths calculated previously are averaged over the SDRs’ local

mean. This means that the values of each bandwidth are properly mapped to their corresponding

34

transmitting SDR and assigned the necessary weight, which is multiplied and consequently added

to compute the node’s local average. This operation is performed locally at the node level, and as

a composite, the system approaches a single bandwidth value.

3.7 Scalability and Performance

The reason for taking the approach of orthogonal frequency division multiplexing is to

be able to take advantage of spectral orthogonality to identify the signals and their corresponding

transmitting nodes, instead of time orthogonality, which would require a predetermined schedule

and protocol for new nodes to enter the network, as well as drop from the system. As far as the

author of this thesis is aware, techniques such as those demonstrated in [81] require a syntonized

system, which has only been achieved by connecting the SDRs through cables or using self-looped

external hardware.

As a proof of concept, we have experimented with a relatively small-scale network of

nodes. To sustain the assumption of coarse time alignment, we have utilized a centralized com-

putation center as depicted in section 3.2. The connection of four nodes to the same computation

center leaves the distribution of computation resources to the computer architecture, limiting the

system’s scalability. To allow for full scalability of the network and system, it would be neces-

sary to provide each node with its computational unit or directly program the field-programmable

gate array (FPGA) to perform frequency estimation, average calculation, and transmit frequency

update. This means that the node’s clock needs to be synchronized adequately to achieve complete

independence.

3.8 Testing and Validation

To define convergence, interest in the system-level performance is important. Taking

as reference [39, 40, 47, 52] the desired accuracy of the should be less than 18◦. The conversion

of the tolerable error for 90 % relative gain, of 18◦ is elaborated more in [7, 40]. That is why

although the goal of achieving a residual error of less than 10 Hz is a significant milestone that can

mimic monolithic arrays, it is possible with only under pulse duration that is longer than 150 µs.

To validate the system’s performance, it was decided to use the self-reported values and estimate

35

Figure 3.4 Experimental setup composed of four SDRs, with omnidirectional antennas, connected
to a single computing unit. The system was designed to wirelessly syntonize two-tone signals
without node synchronization. A coarse synchronization is achieved by connecting the SDRs to
the same computer and daisy-chaining a reference clock.

the residual error against the mean bandwidth value. Figure 3.4 is a picture of the setup of the

four SDRs with their corresponding antennas arranged in a square for line-of-sight communication

among themselves. The results of the different network architectures are presented in the following

chapter, Chapter 4, with a comparison against those simulated in the previous chapter, Chapter 2 .

3.9 Conclusion

The proposed system design for syntonizing a decentralized, distributed phased array

composed of SDRs is unique in its class because it does not require any external hardware con-

nected to the SDRs, and it aims to achieve syntonization solely through signal processing algo-

rithms. The integration of all system components enables the validation of the proposed method-

ologies. Under the constraints presented and addressed in this section, the system may be operable

and demonstrate promising results that need further validation.

36

CHAPTER 4

EXPERIMENTAL RESULTS

4.1

Introduction

The intent of Chapter 4 is to present the experimental results of the experiments per-

formed using the hardware setup and design presented in Chapter 3 in a controlled lab environment

as a proof of concept and compare these results with the simulations performed in Chapter 2. Un-

der these conditions, the experiment validates the minimum signal duration needed and is viable to

achieve the 18◦ residual phase error. Also, this chapter includes a comparison of the state-of-the-art

technology and the proposed approach.

4.2 Results

4.2.1 Fully Connected 4-Node Network With No Frequency Hopping

The first network experimented on was a fully connected network of four SDRs. This

network has a connectivity of 100 %; in essence, every node communicates directly with the other

nodes and itself. Theoretically, the system syntonizes perfectly to a single value if the nodes have

no local estimation error. In reality, each node is independently estimating the two-tone signal

bandwidths of the other nodes, and inherently, these estimations have errors associated with each

tone estimation that add up to the bandwidth estimation and impact the calculation of the local

average value of the node, preventing the system from agreeing upon to a single value. This is

demonstrated in the inlet of Fig.4.1, where the estimated transmission bandwidths are all oscillating

around the mean value of the initial bandwidths after k = 1. For this experiment, the measurement

matrix, M, was not modified to follow any hopping scheme. The average of 25 experiments with

four different signal durations is presented in table 4.1, summarizing the average central frequency

offset (CFO), accuracy with respect to the carrier frequency, and phase error.

37

Figure 4.1 Sample real-time experiments with no frequency hopping, fully connected network with
signals’ duration of 50 µs

Figure 4.2 Sample real-time experiments residual error with no frequency hopping, fully connected
network with signals’ duration of 50 µs

38

Table 4.1 Average CFO: 25 real-time experiments with no frequency hopping, fully connected
network

Pulse duration (µs) CFO(Hz) Accuracy (ppb) Phase Error (rad)

τ = 20
τ = 30
τ = 40
τ = 50

855.58
495.68
354.60
245.68

407
236
169
117

0.107515
0.093443
0.089120
0.077183

39

4.2.2 Loop Connection 4-Node Network With No Frequency Hopping

The second network experimented on was a looped network of four SDRs. This network

has a connectivity of 66.66 %; the nodes communicate with their nearest neighbor and themselves.

For this scenario, each node independently estimates the two-tone signal bandwidths, and inher-

ently, there is an estimation error associated with each tone estimation that adds up to the bandwidth

estimation and impacts the local arithmetic mean calculation. The simultaneous presence of mul-

tiple two-tone signals in the same part of the spectrum induces an estimation error larger than the

CRLB, as explained in 2.4.3. Even though the error is present in each bandwidth estimation that

affects their local average, the bandwidth values remain close to each other as shown in Fig.4.3 and

in more detail in the sample plot of the residual values, Fig.4.4. As explained in sec.4.2.1, because

no frequency hopping scheme has been implemented yet hence, the measurement matrix, M, is not

changed from one iteration to another.

Figure 4.3 Sample real-time experiments with no frequency hopping, looped network with signals’
duration of 50 µs

40

Figure 4.4 Sample real-time experiments residual error with no frequency hopping, looped network
with signals’ duration of 50 µs

Table 4.2 Average CFO: 25 Real-time experiments with no frequency hopping of a looped network

Pulse duration (µs) CFO(Hz) Accuracy (ppb) Phase Error (rad)

τ = 20
τ = 30
τ = 40
τ = 50

3460.50
1541.40
935.03
877.75

1648
734
445
418

0.447425
0.290547
0.234998
0.275753

41

4.2.3 Fully Connected 4-Node Network While Frequency Hopping

The third network experimented on was a fully connected network of four SDRs. This

network has a connectivity of 100 %; in essence, every node communicates directly with the other

nodes and itself. The mixing matrix, W, has all the elements with a weight of 0.25; the mea-

surement matrix, M, has all the entries with their corresponding measurement. For this set of

experiments, the measurement matrix changed according to the schedule described in fig.2.3. Ide-

ally, mapping the estimated bandwidth to the schedule, the system syntonizes perfectly to a single

value if the nodes have no local estimation error. The impact of tones and bandwidth estimation is

demonstrated in the inlet of Fig.4.5, where the estimated transmission bandwidths are all oscillating

around the mean value of the initial bandwidths after k = 1. The average of 25 experiments with

four different signal durations is presented in table 4.1, summarizing the average central frequency

offset (CFO), accuracy concerning the carrier frequency, and phase error.

Figure 4.5 Sample real-time experiments while frequency hopping, fully connected network with
signals’ duration of 50 µs

42

Figure 4.6 Sample real-time experiments residual error while frequency hopping, fully connected
network with signals’ duration of 50 µs

Table 4.3 Average CFO: 25 real-time experiments while frequency hopping, fully connected net-
work

Pulse duration (µs) Connected CFO(Hz) Accuracy (ppb) Phase Error (rad)

τ = 20
τ = 30
τ = 40
τ = 50

1458.76
520.00
424.36
247.99

695
248
202
118

1.816941
0.098017
0.106653
0.077908

43

4.2.4 Loop Connection 4-Node Network While Frequency Hopping

The fourth and final network experimented with was a looped, connected network com-

posed of four SDRs, operating in frequency hopping mode. This network has a 66.66 % connec-

tivity. After looking into the data, it was brought to the author’s attention that the behavior of the

experiment’s behavior is not the one expected because the mapping of the measurement matrix,

M, and connectivity matrix, W, did not happen simultaneously according to the schedule depicted

in fig2.3. The connectivity matrix trailed the measured matrix by one iteration update. Hence, the

behavior described in fig4.7 is different than the behavior in fig4.1, and consequently, the behavior

in fig4.8 is different from the behavior in fig4.4. Even though this error persists in every iteration of

this experiment, the network happens to approach the global mean value at the beginning, k = 0.

Still, contrary to the non-hopping experiment, the bandwidths do not follow each other but remain

close among themselves.

Figure 4.7 Sample real-time experiments while frequency hopping, looped network with signals’
duration of 50 µs

44

Figure 4.8 Sample real-time experiments residual error while frequency hopping, looped network
with signals’ duration of 50 µs

Table 4.4 Average CFO: 25 real-time experiments while frequency hopping, looped network

Pulse duration (µs) Connected CFO(Hz) Accuracy (ppb) Phase Error (rad)

τ = 20
τ = 30
τ = 40
τ = 50

1874.58
1606.40
1290.00
853.08

893
765
614
406

0.235566
0.302799
0.324212
0.268003

45

4.3 Discussion

Figure 4.9 Experimental and simulation results.

Figure 4.9 summarizes the results obtained during the experiment and are overimposed

with the simulated results, allowing for easy comparison between the theoretical and experimental

values. As expected from the simulation, longer signals can achieve better estimation, but longer

signals may result in higher residual phase error, given the instability of the oscillators. From sim-

ulations, it is clear that fully connected networks can go past the 18◦ set threshold with signals

with durations greater than 10 µs while looped networks need to last more than 30 µs. Experimen-

tal values were able to surpass the 18◦ marker from 20 µs disregarding the type of network. This

could have been because the initial errors introduced during the experiment differed from those

introduced for the simulations, given that the simulations’ initial errors were introduced from a

more significant standard deviation from a pseudo-random number generator. Although the exper-

imental results are better than the simulated values, this can be attributed to the randomness of the

46

20406080100Signal Duration (S) 05101520253035Mean Residual Bandwidth (deg)18o mark4 nodes, c=1, no hopping4 nodes, c=0.6, no hopping4 nodes, c=1, hopping4 nodes, c=0.6, hoppingtwo-tone signal’s bandwidth for each scenario. This proves that the estimation accuracy from the

beginning of the average consensus algorithm is crucial for the result of the consensus problem.

The results obtained from the experiments open the door for new research, such as in-

corporating simultaneous signals syntonization with synchronization, enabling other waveforms’

syntonization, and comparing them to figure out which one allows for better parameter estima-

tion and, consequently, more stable syntonization. Also, it would be interesting to experiment and

understand dynamic networks that incorporate or disable links depending on the quality of the

communication links among nodes.

4.4 Conclusion

In summary, the simulation and experimental results prove that to syntonize the two-

tone signals’ bandwidth, it is essential to have at least signal durations greater than 40 µs. These

results provide the necessary evidence to support this thesis while bringing new considerations

regarding the scalability of this approach as the electromagnetic spectrum becomes more crowded

and the signals interfere more and more with each other. The implications of these findings and

their alignment with existing literature should be explored in greater depth in the future.

47

CHAPTER 5

CONCLUSION

An attempt at two-tone bandwidth syntonization was presented to align the signals of a distributed

phased array. A small, four-SDR experimental system is presented, and the self-evaluated fre-

quency deviations are reported. This is the first attempt to achieve two-tone bandwidth syntoniza-

tion of pulsed signals solely through signal processing on off-the-shelf SDRs for a fully distributed

system without compensating for time misalignment. In the system, where all SDRs are connected

to the same computer and detect signals from all SDRs, experiments are conducted with both a

fully connected network and a looped network, presenting multiple pulse durations. Upon receiv-

ing the superposition of its neighboring node signal, down-converting and digitizing the analog

signals, the SDRs would estimate each tone frequency and bandwidth. Once the SDRs had a list of

detected bandwidths, they would calculate the weighted average of the bandwidths, which would

become their next iteration’s transmission bandwidth, until they asymptotically achieved a global

average. This experiment aimed to achieve an integral phase offset of less than 18◦ solely due to

the frequency mismatch. Both in simulation and experimentally, this was achieved.

Although the experimental measurements presented in this thesis were performed at a

carrier frequency of 2.1 GHz, it can be extended to higher carrier frequency signals and longer

pulse durations given that the SDRs can receive longer signal duration if using an input data buffer,

contrary to the work presented here. If the experiment is conducted at higher frequencies and di-

rectly at RF, then the amount of unnecessary data could exacerbate the problem of memory and

computing power that limits the system’s performance and improvement. This project demon-

strates that, in practice, syntonization remains independent even when there is no time alignment

in the system. Orthogonal signals can be used to synchronize the bandwidths of two-tone signals

for distributed phased array alignment. Further investigation is essential for developing an inde-

pendent computation and communication schedule for the nodes comprising the distributed phased

array to fully realize the potential of distributed phased array systems.

48

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APPENDIX

56

# %% [markdown]# # Estimation Techniques Vs. Residual Frequencies# ##### Description: Compare the performance of estimations techniques of each tone Vs. # the resulting residual BW after convergence, this techniques include coarse peak search, QLS(Praguer)# ##### Author: William R. Torres# ##### Last Day Modified: date: 07/08/24# %% [markdown]# ### Assumptions made for the simulation# - All signals arrive at the same time and there is no window slipping from time misalignment# %% [markdown]# # ## Import all libraries and necessary dependencies# %%import numpy as npimport random; random.seed(0)import matplotlib.pyplot as pltfrom scipy.signal import find_peaksfrom scipy.interpolate import CubicSplineimport scipy.signal as signalfrom scipy.fft import fftfrom scipy.fft import fftshiftfrom scipy.interpolate import CubicSplinefrom copy import deepcopy import mathimport timeimport networkx as nx """Call MATLAB API to have access to the metropolis hasting network generator provided """import matlab.engineeng = matlab.engine.start_matlab()hard_code_exponent_for_base2 = 17# Start timing the duration of the simulation for performance comparison purposestart_time = time.time()57

#---#---# Auxiliary Functions#---#---def networkPlotter(adjacency_matrix):    rows, cols = adjacency_matrix.shape    G = nx.DiGraph()    for i in range(rows):        for j in range(cols):            if adjacency_matrix[i][j] != 0:                G.add_edge(i,j)    #nx.draw(G)    return Gdef base2_size(array_length):    non_incremented_base = math.log2(array_length)    rounded_base = math.ceil(non_incremented_base)+4    new_length = 2**(rounded_base)    return new_lengthdef bandwidth_estimator(tones_estimates):    sorted_values = np.sort(tones_estimates)    bw_estimates = sorted_values[1::2] - sorted_values[0::2]    return bw_estimatesdef tone_value_generator(fs=200e6, number_of_nodes=4, bw = None):    """    Input(s):    fs(float) - Sampling frequency of the system;     number_of_nodes(int) - number of nodes/SDRs used in the simulation    bw(list/array or blank) - ordered list of the bandwidth of the two-tone waveform signals        *** If bw is not provided (it is the first iteration of the algorithm) or the length of the bw is             not the same as number_of_nodes            it will generate the values to match properly :)           return a list containig the tone's value within the instantaneous bandwidth     """    # Respect the nyquist rate    max_freq_dom_val = fs/2    # Calculate the max bandwidth of each channel (assume that they are the save size)58

    single_channel_bw = (max_freq_dom_val - 0)/number_of_nodes    # Determine the center point of a channels bandwidths, distance from left boundary    channel_bw_half_point = single_channel_bw /2    # Calculate the value of the center of each channel, which also corresponds to the center of the two-tone's frequencies    central_freq = ((np.arange(start=0, stop=number_of_nodes, step=1, dtype=int))* single_channel_bw)+ channel_bw_half_point    ## two-tone waveforms Bandwidth generator    # if input is not give, or the number of nodes does not correspond to the size of the bw list    if bw is None or len(bw) != number_of_nodes:        # Scale the channel bandwidth to prevent overlap and generate a bandguard, and preserve orthogonality        two_tone_bw_range = channel_bw_half_point * np.array([0.2,0.8])        # Generate the two-tone BWs values        bw = np.random.uniform(low=two_tone_bw_range[0], high=two_tone_bw_range[1], size=(number_of_nodes,))    # return a list of sorted values    return bw, central_freqdef zero_pad_truncation(raw_rx_time_capture):    ## Determine the threshold to detect    only_positive_vals = np.absolute(raw_rx_time_capture)    average_noise_mag = np. mean(only_positive_vals[1:200])    threshold_val = (np.max(only_positive_vals) - average_noise_mag)/2    # Find the index of the first non-zero element    first_nonzero_index = np.argmax(only_positive_vals >= threshold_val)    last_nonzero_index = len(only_positive_vals) - np.argmax(np.flip(only_positive_vals) >= threshold_val) - 1    truncated_time_array = raw_rx_time_capture[first_nonzero_index+1:last_nonzero_index-1]    return truncated_time_arraydef coarse_peak_finder(arr_of_mag, num_of_peaks):        first_peak_index,_= find_peaks(arr_of_mag/np.max(arr_of_mag), height=0.001, distance=10)    highest_peak_indices = np.argsort(arr_of_mag[first_peak_index])[-num_of_peaks:]    coarse_peaks_idx = first_peak_index[highest_peak_indices]     coarse_peaks_idx = np.sort(coarse_peaks_idx)      return coarse_peaks_idx59

def praguer_qls_frequency_estimate(array_of_mag, num_of_peaks, F, DF):            windowing_center_idx = coarse_peak_finder(array_of_mag, num_of_peaks)    freq_estimates = np.zeros(num_of_peaks,dtype=float)        d1 = coarse_peak_finder(array_of_mag, num_of_peaks)        y0 = np.log10(array_of_mag[np.array(d1-1)])    y1 = np.log10(array_of_mag[np.array(d1)])    y2 = np.log10(array_of_mag[np.array(d1+1)])        partial_idx = 1*((y2 - y0)/(2*y0 - 4*y1 + 2*y2))        full_F = F[d1]    partial_F = partial_idx*DF        freq_estimate = full_F - partial_F    return freq_estimatedef cubic_spline_frequency_estimate(array_of_mag, num_of_peaks, F, DF):        windowing_center_idx = coarse_peak_finder(array_of_mag, num_of_peaks)    freq_estimates = np.zeros(num_of_peaks,dtype=float)        for peak in range(0, num_of_peaks):        central_idx = windowing_center_idx[peak]        lower_bound = central_idx - 3        upper_bound = central_idx + 3+1        idx_range = np.arange(lower_bound, upper_bound)                cubicSplineFunction = CubicSpline(F[idx_range], array_of_mag[idx_range])                F_interp = np.linspace(F[lower_bound+0], F[upper_bound-0], num=150)        y_interp = cubicSplineFunction(F_interp)                    peak_index,_= find_peaks(y_interp/np.max(y_interp))        highest_peak_indices = np.argsort(y_interp[peak_index])[-1:]        60

                freq_estimates[peak] = F_interp[highest_peak_indices]    return freq_estimatesdef tx_ttwvfrm_generation(bw=500e3,fs=200e6, tt_center=0, pulse_duration=50e-6):    L = int(round(fs*pulse_duration))    pulse_time_axis = np.arange(0, L)*1/fs    pulse = np.zeros(L, dtype=complex)    tones = np.array([-bw/2, bw/2])+tt_center    phase = random.uniform(0, 2*np.pi)    for tone in tones:        pulse += (1/np.size(tones))*np.exp(1j*2*np.pi*pulse_time_axis*tone + phase)    return pulsedef band_limited_noise(min_freq: float, max_freq: float, samples=1024, samplerate=1):    """Generate band-limited white noise using Fourier method.    Based on: https://stackoverflow.com/a/35091295/2831057    Parameters    ----------    min_freq : float        Minimum noise frequency (Hz)    max_freq : float        Maximum noise frequency (Hz)        samples:    samples : int, optional        Lengh of vector to return (samples), by default 1024    samplerate : int, optional        Sample rate of noise vector (Hz), by default 1    Returns    -------    array_like        Vector or noise with properties matching specified parameters.    """     def fftnoise(f):        f = np.array(f, dtype="complex")        Np = (len(f) - 1) // 2        phases = np.random.rand(Np) * 2 * np.pi        phases = np.cos(phases) + 1j * np.sin(phases)        f[1 : Np + 1] *= phases61

        f[-1 : -1 - Np : -1] = np.conj(f[1 : Np + 1])        return np.fft.ifft(f, norm="ortho").real    freqs = np.abs(np.fft.fftfreq(samples, 1 / fs))    f = np.zeros(samples)    idx = np.where(np.logical_and(freqs >= min_freq, freqs <= max_freq))[0]    f[idx] = 1    noise = fftnoise(f)    return noisedef add_complex_noise(signal, fs=200e6, snr_db=30):    L = signal.size    snr = 10**(snr_db / 10)    signal_power = np.var(signal)    #print(f'Signal Power: {10 * np.log10(signal_power)} dBm')    noise = np.sqrt(signal_power / snr / 2) * band_limited_noise(0, fs/2, L, fs) + 1.0j * band_limited_noise(0, fs/2, L, fs)    #print(f'Noise Power: {10 * np.log10(np.var(noise))} dBm')    #print(f'SNR: {10 * np.log10(signal_power/np.var(noise))} dB')    rx = signal + noise        return rx#---#---#---# Actual Simulatin Starts Here#---#---#---# Monte Carlos Simulation Parameters and memory holdersmax_MC = 1000time_range = np.logspace(-5,0, endpoint=True)max_average_consensus_count = 50average_consensus_count = np.arange(0,max_average_consensus_count)# %% [markdown]# ## Network Generation # %%# the network will have this number of nodes(SDRs)numberOfNodes = 5# generate the network of nodes undirected communication channels62

# generate the network of nodes undirected communication channelsW = np.array(eng.generateW(numberOfNodes, 0.5))W_mixing = W.copy()transmit_pulse_duration = 80e-6# %%# Get memory space for values holdingholder_bws_for_experiment = np.zeros([max_average_consensus_count+1, numberOfNodes])holder_for_residual = np.zeros([max_average_consensus_count+1, 1])for k in average_consensus_count:    ## TX Parameters    # Sampling frequency of transmitter and receivers is going to be constant for all nodes    fs = 200e6 # 200MSa/s    # length of pulse      pulse_length = int(fs*transmit_pulse_duration)    DT = 1/fs    # Transmitter time axis    tx_time_axis = np.arange(0,pulse_length,1)*DT    ## RX Parameters    # Capture duration    capture_window_duration = 120e-6    # length of capture window    RX_window_length = int(fs*capture_window_duration)    # Receiver time axis    rx_time_axis = np.arange(0, RX_window_length)*DT    ## Tone Values Generation    # number of tones in the spectrum    numberOfTones = numberOfNodes*2 # two-tone waveform        if k==0:        vals = tone_value_generator(fs=fs, number_of_nodes=numberOfNodes, )        holder_bws_for_experiment[k,:] = vals[0]        holder_for_residual[k] = np.max(vals[0]) - np.min(vals[0])    else:        vals = tone_value_generator(fs=fs, number_of_nodes=numberOfNodes, bw = next_nominal_bw_holder)63

    # list of two-tone bandwidths    tt_bw = vals[0]    # list of central frequencies axis corresponding 1-to-1 mapping with the bandwidths    central_freq = vals[1]        ### Generate the noiseless case for the transmitted waves    pulse_holder = np.zeros([pulse_length,numberOfNodes], dtype=np.complex_)    # for every node in the system ...    for node in range(0,numberOfNodes):        # create a label to identify        #identifier_label = "Node"+str(node+1) + " TX"                # Generate and store the noiseless wave forms        pulse_holder[:,node] = tx_ttwvfrm_generation(tt_bw[node],fs, tt_center=central_freq[node],                                                      pulse_duration=transmit_pulse_duration)        ## =============================================================================================================    ## System wide holder    individual_node_rx = np.zeros([numberOfNodes,RX_window_length], dtype=complex)    # for each node ...    for node_rx in range(0,numberOfNodes):        # make a copy of ideal sinusodials for the node        list_of_pulses_for_node = deepcopy(pulse_holder)        # allocate memory for reception, (capture window)        zero_padded_holder = np.zeros([RX_window_length], dtype=complex)                # for each two-tone signal ...        zero_padding_length_one_way = int((RX_window_length-pulse_length)/2)        for node in range(0,numberOfNodes):        # add noise to the channel            noisy_pulse = add_complex_noise(list_of_pulses_for_node[:,node]/(np.max(list_of_pulses_for_node[:,node])),                                             fs=fs, snr_db=30)        64

            # append pre/pos zero pad            zero_padded_holder[zero_padding_length_one_way:-zero_padding_length_one_way] = noisy_pulse            individual_node_rx[node_rx] += zero_padded_holder    ## =============================================================================================================    # Estimation Starts From Here    #---    #---    #---    coarse_estimates = np.zeros([numberOfNodes, numberOfTones])    coarse_bw_estimates = np.zeros([numberOfNodes, numberOfNodes])    next_nominal_bw_holder = np.zeros((numberOfNodes,1))    for receiving_node in range(0,numberOfNodes):        #plt.subplot(int(np.sqrt(numberOfNodes)), int(np.sqrt(numberOfNodes)), receiving_node+1)        identifier_label = "Node"+str(receiving_node+1) + " RX"        time_rx = individual_node_rx[receiving_node]        just_pulse = zero_pad_truncation(time_rx)        # Convert to the fourier domain        new_fourier_dom_axis_size = base2_size(RX_window_length)        fourier_magnitude = np.abs(fftshift(fft(just_pulse, n= new_fourier_dom_axis_size)))/np.size(just_pulse)        fourier_domain_DF = fs/new_fourier_dom_axis_size        new_F = np.arange(-fs/2, fs/2, fourier_domain_DF)        # Coarse Frequency Finder [FFT index]        coarse_index = coarse_peak_finder(fourier_magnitude, numberOfTones)        coarse_estimates[receiving_node,:] = new_F[coarse_index]        """        Replace the estimation function below for each different approach        -----------------------------------------------------------------------------------------------------------        """        tones_estimates = praguer_qls_frequency_estimate(fourier_magnitude, numberOfTones, new_F, fourier_domain_DF)        bw_estimates = bandwidth_estimator(tones_estimates)65

                """        -----------------------------------------------------------------------------------------------------------        """                local_average = np.dot(W_mixing[receiving_node,:], bw_estimates)        next_nominal_bw_holder[receiving_node] = local_average    holder_bws_for_experiment[k+1,:] = next_nominal_bw_holder[:,0]    holder_for_residual[k+1,:] = np.max(next_nominal_bw_holder[:,0]) - np.min(next_nominal_bw_holder[:,0])# %%end_time = time.time()print("The time of execution of above program is :",      (end_time-start_time), "s")eng.quit()