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III I7 I.I.:.:"::I7 ).7,I':‘ : "II“I.." 7 ”I 7-7:I;'774I'.3;71I|10 7‘ 'I ‘I7 H 7 II I. ”but *7; i7 ”ii ”977': “7:177; it‘s (77 777777 “:6“ 77777;. 777,1(771 7:7 777777777II|I7717;7 7 777777777 I 77 77777;. 77777.77. M7 7777 .77 27 II|7III 7‘ 7777777 77777777 77:77 77 III *7 III III I.7 III" *7‘I ’77 I77’77I'7‘7‘7777777777777 77M I II '7 I7 "777 7(7'7I1I I77II7I 777777777 7-7777 1I {‘7 777M 7777 'I “II“"I‘ ., . III-”I III I 777777 7777777 777777777777 777 I'lH.J'I I“! 77,.”777717717777777 I '77“ i777IIIIIII7‘7h 77777.77 “71;.“ . 79595.1. -. 77777 7 7777777777 77777777 7777II M7 77 III II IIIII777 IIUIIII777777 7777777777 7777 7 770777 7' II777I77H 7"“ “w“ & THESIS W . . '3‘ LIBRARY Michigan State ' University This is to certify that the thesis entitled ULTRASONIC BLOOD FLOW IMAGING USING CORRELATION PROCESSING presented by Michael De Olinger has been accepted towards fulfilhnent of the requirements for Eh D degreein glee. Engr. & ys. Sci. 2214. .g/c/agz Major professor Date March 12, 1981 0-7639 01-; N 'xi"fiflflfis\ t. ‘3 ‘ ‘3}:351’ .,, lliflfllllllflflflflfli ' 0“ 'TJ'.‘ ’1 OVERDUE FINES: ' 25¢ per day per item RETURNING LIBRARY MATERIALS: Place in book return to remove charge from circulation records v ._... AA—‘4‘_ , ' 'fig—g—V ULTRASONIC BLOOD FLOW IMAGING USING CORRELATION PROCESSING By Michael De Olinger A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Electrical Engineering and Systems Science ABSTRACT ULTRASONIC BLOOD FLOW IMAGING USING CORRELATION PROCESSING By Michael De Clinger A general investigation of those fundamental processes affecting the design of ultrasonic blood flow imaging instru— ments is presented. First, known physical attributes of the various biological tissues are combined to formulate a simple model which describes the interaction of ultrasound with the human body. Using this model, it is shown that a periodic pulse train is the preferred transmitted signal and that significant performance improvement may be realized by imple- menting "true" correlation as opposed to the conventional correlation processing. The desirability of having the true correlation receiver operate in a near optimum manner is demonstrated and the con- straints imposed on signal bandwidth, effective signal dura- tion, achievable velocity resolution and transducer angle are presented. In order to present the complex interrelationships which exist between these constraints, a unique set of design curves were generated. These curves indicate those combina- tions which insure near optimality. Also, the constraints imposed by transducer limitations are discussed. The trans— ducers which are examined include the traditional piston type (both focused and unfocused) and the array type, as well as a newly developed dual element transducer. Finally, these results are utilized in an example system design where implementation difficulties and system limitations are discussed. To make this dissertation as self contained as practical, a rather complete tutorial on detection and estimation theory, as it applies to the blood flow measurement problem, is pre- sented in the Appendices. ACKNOWLEDGMENTS So many people have contributed in so many ways that it is difficult to acknowledge all of them. However, it is a pleasure to acknowledge those few individuals whose valuable contributions deserve special recognition. The academic guidance and technical critcism provided by Dr. Marvin Siegel is especially appreciated as is the contin- ual encouragement of Dr. Bong Ho. Much of the initial draft was typed by Cindy Turner while the final manuscript was type by Donna Ahrens. Their efforts are sincerely appreciated. Finally, I would like to thank the members of my family for their patience and support. Especially my mother, father, and my wife Marcia. ii TABLE OF CONTENTS CHAPTER I. INTRODUCTION CHAPTER II. BACKGROUND Transmission Type Ultrasonic Flowmeters Conventional Reflection Type Ultrasonic Flowmeters Conventional Pulsed Doppler Systems Random Signal System Conventional Reflection Type Visualization Systems Velocity Imaging Systems Early Experimental Results CHAPTER III. TARGET MODEL Measurement Objective General Measurement Method Physical Structure Target Definition and Characterization Simple Tissue Model CHAPTER IV. SIGNAL CONSIDERATIONS Desirable Signal Characteristics Signal Selection CHAPTER V. PROCESSING CONSIDERATIONS Transit Time Effects Derivation of the Optimum Detector in the Presence of Transit Time Effects iii \OOOVO‘GH 10 10 12 18 l9 I9 21 29 40 48 48 49 73 73 79 w.‘,. . Q... \- .‘._.‘ ‘\—“ . .~.~‘ ' ‘-‘\‘l .. 'bfl‘ iv Performance of the Optimum Detector in the Presence of Transit Time Effects Performance of the Correlation Detector in the Presence of Transit Time Effects Constraints on the Correlation Receiver for Near Optimality Resolving Capability of Both True and Conventional Correlation Receivers CHAPTER VI. RESOLUTION CONSTRAINTS AND RECEIVER PERFORMANCE Resolution Constraints 'Receiver Performance CHAPTER VII. TRANSDUCER CONSIDERATIONS CHAPTER VIII. SYSTEM IMPLEMENTATION, CHAPTER IX. SUMMARY AND CONCLUSIONS . APPENDIX A. Vector Representation of Signals APPENDIX B. Complex Signal Notation for Narrowband Processes Complex Representation of Transfer Functions APPENDIX C. Bandpass Random Processes Complex White Process Complex Gaussian Process APPENDIX D. Hypothesis Testing for Bloodflow Imaging APPENDIX E. Estimation Theory APPENDIX F. Detection of a Point Target APPENDIX C. Ambiguity Functions Generalized Ambiguity Functions DOppler Approximation Ambiguity Function APPENDIX H. Acoustic Wave Propagation 89 95 101 107 119 119 130 136 149 I61 168 171 177 179 184 185 189 207 210 216 216 218 222 229 APPENDIX I. Blood Flow in Cylindrical Vessels APPENDIX J. Computation of the Elements of A LIST OF REFERENCES GENERAL REFERENCES 236 239 243 247 Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure 9 U1 NNNNN U‘I-I-‘UJNH bbbbwwwwwww bWNl—‘NOU‘IPWNH LIST OF FIGURES Simplified Blood Flow Imaging System The Blood Flow Simulation System Experimental Velocity Measuring System Flow Profile Across Unobstructed 7mm Tube Actual Spectrum Analyzer Output at Selected Positions Along the Diameter of a 7mm Tube General Measurement Method Stratified Squamous Epithilial Tissue Muscle Tissue Typical Flow vs Time in the Aorta Target Definition System Geometry Typical Relative Range Scattering Function Ambiguity Function for a Single Pulse Ambiguity Function for Random Noise A Typical Feedback Shift Register Average Ambiguity Function for a PN Sequence Periodic Pulse Train Approximate Ambiguity Function for a Periodic Pulse Train Periodic Train of PN Sequences Geometry for Clutter Computations Target and Beam Geometry vi ll 13 14 15 16 20 22 22 27 32 33 45 51 53 54 56 58 60 62 65 75 Figure Figure Figure Figure Figure Figure Figure Figure: Figure Figure Figure Figuxng Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure CD N 0) OJ LHU'IU'IU'IU'IU'I (DCDVNNVVNVGO‘O‘CDG NHVONLfl-L‘LJJNi—‘me-L‘w VOW-PLUM vii Optimum Detector . A Approx1mate Ap(BT) Correlation Detector Correlated Energy Function Maximum.Norma1ized Eigenvalue of A Conventional Correlation Receiver Operation True Correlation Receiver Operation Typical Doppler Resolution Plot System Signal Processor Bloodflow D0pp1er Profile Target Interaction Design Curves for fc==5 MHz, D==1 cm Design Curves for fC = 3 MHz, D= 1 cm Design Curves for fc= 5 MHz, D= .5 cm Design Curves for fc = 3 MHz, D = .5 cm Probability of Error Curves Basic Piezoelectric Transducer Typical Beam Profile A Focused Beam Profile Dual Element Transducer Annular Ring Beam Pattern Typical Transducer Array Generalized Array Beam Pattern Expected Output of a Two Correlator System Effect of Doubling the Number of Correlators Bandpass Correlator Blood Flow Imaging System 80 84 95 99 103 108 113 116 120 123 128 128 129 ‘129 135 137 140 142 143 145 146 147 151 151 154 157 Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure FiSure Figure Figure A1. B1. B2. B3. B4. B5. B6. Cl. C2. C3. D1. D2. D3. D4. D5. D6. D7. D8. G1. G2. H1. I1. viii Typical Sample Function From a Random Process Typical Signal Spectrum Spectrum of Complex Envelope Sample On/Off Sequence Sample Binary-Phase Sequence Symmetric [S(f - fc)]LP Example of a Narrowband Transfer Function Example of a Complex Envelope Spectrum Example of Band Limited White Noise Spectrum of wB(t) Correlator Implementation Matched Filter Implementation Typical Receiver Operating Characteristics Optimum Detector Implementation Filter Correlator Optimum Receiver Optimum.Receiver for NO Near Zero Optimum Receiver for Random Amplitude and Phase Bandpass Processing for Case 2 Example of Compression in Binary Phase Signals Approximate General Ambiguity Surface One Quarter Wave Matching Layer Section of Cylindrical Vessel 168 173 174 175 176 176 177 180 184 184 191 191 195 202 203 205 206 206 219 226 234 236 Table4u1 LIST OF TABLES 8’(O,fT) For a Periodic Pulse Train ix 72 CHAPTER I INTRODUCTION Currently there is widespread interest in developing instrumentation capable of non-invasively measuring various cardiovascular system parameters. Included are such para- meters as mean volume flow rate, blood velocity, condition of the blood vessel walls and the presence of partial or complete occlusions. In the past, such techniques as Fick and other dilution methods, angiographic techniques, and attaching electromag- netic flowmeters directly to the vessel of interest have been employed. These methods have severe limitations since they are physically invasive, expensive to perform, limited in the number of repeated applications, and there exists an attendant risk. Ultrasound has been suggested as a non- invasive alternative to these methods. Specifically, an ultrasonic system capableof display- ing and/or measuring the velocity of the flowing blood as a function of position within the vessel could provide the phYSiCian with important quantitative, as well as qualita- tlve. information on the condition of the patient's cardio- VasCular system. Many ultrasonic systems which will be 1' 2 described in the next chapter have been proposed to provide this information. The primary difference between these systems is in the particular choice of transmitted signal, or in the technique employed to implement the signal pro- cessing. Fundamentally however, each of these systems employ some form of correlation detection. The research re- ported in this dissertation theoretically examines the feas- ibility of implementing a practical Blood Flow Imaging System employing correlation processing. In the large majority of cases, research can be grossly classified as basic or applied, with a poorly defined line dividing the two. Paradoxically, however, the research reported herein may be-best classified as both. In the most general terms, this research examines the fundamental nature of those physical attributes and signal processing techniques which significantly affect the design of a correlation type blood flow imaging system. Because this research covers relatively uncharted ground and because of its complex nature and scope, it is difficult to separate those processes and Procedures which should be investigated from those that ShOUId not. Therefore, in an effort to prevent the scope of this investigation from becoming unmanageable, the central theme of designing a practical system has been adopted. Although this is a highly visible theme in this dissertation, it ‘ . . 13 also superfic1al to a more subtle underlying purpose. Th . e true value of this research lies in the insight gained in t0 the fundamental processes affecting the system design 3 Sc) ‘that readers might be better equipped to make their own engineering judgement in designing their own system. The various phases of the research are listed below in tire: order that they are presented in the text. Each phase cxorlstitutes a major portion of work in the investigation of ultrasonic blood velocity imaging. 1. Background: This section presents a brief history of previous work relevant to the work presented in this dissertation. A simplified block diagram of the veloc- ity imaging system is presented and discussed in this section. 2. Development of a Mbdel: In this section a model for the flowing blood as a "target" is developed. Additionally, a characterization of the surrounding tissue (target environment) is presented. 3. Signal considerations: An investigation of the potential advantages and limitations of utilizing various large time bandwidth product signals is carried out in this section. Based on these results and prac- tical considerations, a particular signal is selected. 4. 'Processing Considerations: Under certain con- straints on the transmitted signal, the correlation receiver can be considered near optimum. These con- straints are derived and discussed in this chapter. 4 5. Resolution Constraints and Receiver Performance: Various complex resolution interrelationships which affect the design of an optimum flow measurement system are derived and discussed in this chapter. Additionally, theoretical receiver performance is evaluated. 6. Transducer Considerations: At this point, a discussion of various transducer designs is included. This discussion includes results of a specially designed dual element transducer developed by Jethwa, as well as more conventional focused and unfocused transducers. 7. System Implementation: In this chapter, real world considerations are included in an example system design. Mbst of the ideas presented in the example are the result of experience gained in the actual hardware construction of a similar system. 8. Finally, a brief conclusion section is included to summarize the research and comment on the implications Of some of the results. Additionally, the required detection/estimation theory back- grotfiui is developed in Appendices A through G. Since the the°ry is developed in the context of the blood flow imaging Problem, the reader is strongly advised to at least skim these Appendices before reading the main text. They are very 5 much a part of the total effort, and it will be assumed in the main body that the reader has developed the necessary Earn :1. liarity . CHAPTER II BACKGROUND In the past, several different techniques from plethys- mography to NMR (nuclear magneticresonance) have been used for the detection and measurement of blood flow. The major difficulty with these methods is the lack of ability to mea— sure flow in a specific region. Ultrasonic techniques do 1101: experience this problem and are only limited by the resolution and accuracy designed into the instrumentation and processing. Hence, ultrasound is ideally suited for Inet'lical diagnosis and treatment, as well as, medical research. A number of excellent surveys of various ultrasonic blood EIOWmeters have been published by Franklin,1 Wells,2 Baker,3 and Fryg. TI-\a-3'51smission Type Ultrasonic Flowmeters Many transmission-type flowmeters have been designed“)-18 These flowmeters are based on the principal that a sound beam passed diagonally through a blood vessel exhibits a difference in the time required to traverse the vessel alternately in the upstream and downstream directions. A major disadvantage associated with these flow detectors is that surgery must be peI‘formed to locate the transducer close to the blood vessel 7 beirn; examined. Thus, these flowmeters are not suitable for transcutaneous blood velocity measurements. anventional Reflection Type Ultrasonic Flowmeters The conventional reflection-type flowmeters, which are in common use, are based on the principle of the Doppler effect. When ultrasound is transmitted into the blood stream a portion of the signal is scattered by the moving blood particles and is Doppler shifted. The relationship between the average Doppler difference frequency fd and the average blood flow velocity v, is given by the following equationz’19 c fd chcos a + cos 8} 2.1 where c is the velocity of ultrasound in blood, fC is the transmitted ultrasonic frequency, and a and B are the angles between the transmitting and receiving ultrasonic transducers and the velocity vector of the blood stream. Reflection type devices may operate in the continuous wave mode or the more sopisticated pulsed Doppler mode. 20 Satomura was the first to demonstrate continuous wave Doppler motion detection in 1956. Franklin, et al21 used this technique for blood velocity detection in animals. 22 . . Baker developed a practical instrument for the transcuta- neous detection of bloow flow in humans. 8 gyrventional Pulsed Dgppler Systems The early continuous wave flowmeters were not capable of detecting the direction of the blood velocity vector with respect to the transducer. In 1966, McLeod23"24 used a ver- sion of quadrature phase detection to demonstrate the first successful direction sensing Doppler device. One of the shortcomings of the continuous wave (CW) sinusoidal systems is its inability to obtain a measurement of range. This lack of range sensitivity prevents the measurements of vessel diameter and velocity profiles. The current pulsed Doppler Systems by Baker, Perroneau, and others3_8 25 and the Random Signal Systems by Newhouse and Jethwa26 were devel- oped in an attempt to make transcutaneous blood flow measure- ments . In the pulsed Doppler system, a burst of ultrasound is transmitted. The distance to the moving target and the tar- get's velocity with respect to the ultrasonic beam are deter- mined by range-gating the return echoes. These received echoes are both phase and amplitude modulated by the moving target. This range-gated echo is then compared with a sample of the original transmitted signal in a phase detector to extract the Doppler components. A number of these systems have been built and used for measuring blood velocity}-8 As will be discussed in the next chapter, one of the concerns with a conventional Doppler system is providing sufficient energy in the pulse to enable targets of interest to be easily detected, simultaneously with providing suffi- cient range resolution (obtained by using shorter duration pulses). A conventional system increases transmitted energy by increasing the peak energy in each pulse, thus improving the system immunity to noise. However, there is presently some concern about the safe R‘Eik. intensity levels for ti ssue825’26. Ran dom Signal System One potential solution to the peak power problem associated with conventional Doppler systems lies in the use of random noise as the transmitted signal. Using random nOise, a duty cycle of 1.00% may be utilized without degrad- ing the systems range resolving capability. Also, the signal may be pulsed modulated and transmitted at an arbitrarily high repetition rate without introducing range ambiguities. (Those concepts will be treated in detail in Chapter Four.) The random signal system proposed by Newhouse and by Jet1‘1wa were attempts to utilize random noise in this fashion. Also, it was speculated by these researchers that the random signal system could theoretically provide arbitrarily good velocity and range resolution. 10 Conventional Reflection Type Visualization Systems Conventional reflection type ultrasonic visualization systems provide cross sectional presentations of static targets (tissue, muscle, bone, etc.) These systems transmit a series of narrow pulses of ultrasound. A portion of the transmitted energy is reflected from any acoustical impedance discontinuity (tissue interface) and received by the receiv- ing transducer. The received signal is amplified and envelope detected. An image is then formed on a CRT by scan- ning properly and intensity modulating according to the energy in the received signal. Eocity Imaging Systems A second form of a visualization system is the presenta- t123—on of dynamic (moving) targets, or Doppler visualization. In the special case where red blood cells are the moving tar- gets, the system will be referred to as a Blood Flow Imaging SYStem. A simplified block diagram of a Blood Flow Imaging SYS tem is shown in Figure 2.1. As shown in the block diagram, the signal to be transmitted is generated by the signal generator, amplified by the power amplifier, and converted to an acoustical signal by the transmitting transducer. The ultrasonic energy is then reflected by a continuous stream of moving red blood cells in the flowing blood as well as ll sur rounding tissue. The signal received by the receiving 1:1:23L118ducer is amplified and processed to provide a velocity estimate corresponding to each spatial location. Blood Vessel O Transmitting Transducer \ \_, \.-/ Power Receive zéunnplifier Receivinc . Amplifier g Transducer i V Signal Image ‘ H Signal Generator Display Processor Figure 2.1 Simplified Blood Flow Imaging System Some limited success has been reported in the area of B:l-Qod Flow Imaging27. However, very little work has been done in investigating fundamental limitations, system con- $1:zltfaints, appropriate transmitted signals, or optimum pro- c: , . Q S Sing and the assoc1ated performance. This investigation was an outgrowth of a desire to nt1<1erstand more clearly those fundamental processes involved 12 in Ultrasonic Blood Velocity Imaging. Bar 1} Experimental Results Initially, the purpose of this research was to exploit the suggested advantages of the random signal system in making blood velocity measurements. As a result, a flow measurement system was constructed which was capable of com— paring the performance of the random signal flowmeter with the more conventional pulsed sinusoid system. Additionally, a blood flow simulation system was constructed so that in- vitro measurements could be made. Figure 2.2 shows the physical construction of the blood flow simulation system. Seven millimeter dialysis tubing was used to simulate blood vessels of the size most likely to be encountered in practice. The length of the tube was cho sen as approximately 100 times the diameter of the tube so that laminar time independent flow could be guaranteed in the exposed section. Also, as shown, this construction al lowed for the introduction of obstructions into the tubing. C" E ~ Antifoam containing seven micron silicon particles was 111' lked with distilled water to simulate actual blood. The block diagram of the experimental system which was c0118 tructed is shown in Figure 2.3. As shown, the trans- Init":ed signal consisted of either a sample function from a ratItiom noise source or a pulsed sinusoid. A portion of the t: bahsmitted signal is delayed by the variable acoustic delay l3 Emumzm Gowumasafim 30am noon one N.N opnwwm toon wfiHDSH pmumHDEHm mfimhamwn wcHnsH - flflflU/ muwosq Te capm— _ .IHVI \\QV i) H Hi I.— — t . N C I .I .NV ,» _ \\xmflMWflWWHe J “ 1““:1‘ — \ // coHuoDHumno Hoonpmcmue mo>Hm> HmuwB poaaflumwo wswnsH comkH _ - . i u sum _ T2328 _ age l4 Transmitting Transducer Random Noise Gen. Power Rcvr Ampli- iAmpli- fier fier he: ———-1——— . Receiving ‘Wmnable Transducer Aammtic Delay Line Spectrum Low 1 lzer ' Pass y Filter Blood Vessel ‘IF'igure 2.3 Experimental Velocity Measuring System 15 line and then mixed with the received signal. The output of the mixer is low pass filtered and its frequency content analyzed by a digital spectrum analyzer. The range under examination is determined by the delay introduced into the reference signal by the delay line. Figure 2.4 shows a typical velocity profile reconstruc- ted from Doppler spectra obtained using the experimental system38'39. The actual spectrum analyzer output corres- ponding to a few selected positions across the vessel diame- ter is shown in Figure 2.5. 7 7mm Dialysis *’ Tube @ e gemeeeT if igure 2.4 Flow Profile Across Unobstructed 7mmTube 16 e # I'U'-‘ -"'-'|||' "Ii'- -‘I-ll-'|'|"' ‘v|"| "‘| I--- l"'|| "|'||l|"'l' "||| '-'|l|| '|-" ""'llll|' 'II""" I I I I l I I I I I I I I I I I I I I I IPosition 6 APosition 8 360 400 5(IO 600 700 8I)0 at Selected Positions Along the Diameter of a 7 mm Tube 280 fiPosition 1 JIPosition 4 -'-|-.-I‘i Figure 2.5 Actual DOppler Spectrum Analyzer Output 100 17 These spectra were obtained using random noise as the transmitted signal by averaging the output spectra over several seconds. When long time averaging was not used, the Doppler spectra were virtually uninterpretable. Even with averaging, the difficulty in reconstructing a profile from them is apparent. Clearly, long time averaging cannot be tolerated in-vivo where pulsatile flow exists. Nearly identical results were obtained when a pulsed sinusoid was used as the transmitted signal. These results indicated that the range and velocity resolutions of the random signal system were comparable to tho se of the existing conventional ultrasonic DOppler systems, but not superior as had been anticipated. Additionally, although the in-vitro results indicated that similar system Performance could be expected, in-vivo results were quite different. Using random noise, not a single meaningful mea- S"~-1?l1'.‘ement was ever accomplished. On the other hand, obtain- ing in-vivo measurements using the conventional system were a matter of routine. It was this combination of failures on the part of the l:.513'53.ciom signal system which was to become the seed of this res earch. It became apparent that a more fundamental under- S{tell'lding of the processes involved in making in-vivo blood f . 10w measurements was required. CHAPTER III TARGET MODEL In this chapter, a model for the flowing blood as a target is presented and discussed. In the sequel, the word model is intended to mean a mathematical characterization of a. physical target which is consistent with the measurement methods and objectives. From this characterization (model), various performance measures and design parameters may be analyzed and optimized according to the measurement objec- tiVes. However, before such a characterization is completed a Clear understanding and/or definition of the following is required: 1. .Measurement objective 2. General measurement method to be employed 3. Actual physical structure of the system upon which measurements are being made and its effects on the measurement method. Therefore, the development of an actual mathematical charac- te:C‘ization must be preceded by a discussion of each of the top ics listed above . I After defining the general measurement method and O - b3 ective as well as describing the important attributes of the physical system, this chapter addresses the problem of 18 l9 formulating a model for the flowing blood. Additionally, a simple model for the surrounding tissue is presented. Thus, the final two sections of this chapter are: 4. Target Definition and Characterization 5. Simple Tissue Model Me a surement Obj ective The objective of the ultrasonic flow visualization sys- tem is to display, as a function of time and as accurately as possible, the velocity of scatterers at each spatial location in the field of observation. Unfortunately, as will become apparent, the simultaneous achievement of good spatial, vel- ocity and time resolution are conflicting requirements. Therefore, it is difficult, if not impossible, to Specify a single criteria which provides a measure of overall System capability. However, a unique set of design curves will be presented in a later chapter which suggest a design methodology to meet the measurement objectives of any specific app lication. ere\rleral Measurement Method The general measurement method employed in this research is shown pictorially in Figure 3.1. As indicated, electrical Signals are converted to propagating acoustic waves by the transmitting transducer. These waves are then alerted (scat- t eJ'l‘ed, attenuated, distorted, etc.) by the internal structures 20 of the human body. A portion of the altered transmitted waves are received by the receiving transducer and con- verted to electrical signals. These signals are then pro- cessed in the true correlation receiver to provide informa- tion related to the measurement objective. The only further restriction imposed on the general method is to constrain the transmitting and receiving transducers to occupy the same spatial location and orientation. . —-——1 Corre- Range and r lation"""Velocity Rcvr Estimates Epidermis 'ITaransmit/Receive I I ' Trans duc er - fl 6 L —— Blood Vessel Figure 3.1 General Measurement Method It should be pointed out that this choice of general thee~Surement method is based on the author's, and other's, e)‘t‘p‘ariences as described previously in the introduction. And it is the purpose of this research to further exploit the a. c1\7€=1.ntages of this measurement scheme. 21 Physical Structure The actual physical structure which the measurement system must interface with is the human body. Simply stated, the human body consists of an almost endless number of micro- s copic cells, each performing specific functions which are integrated for maintenance of the entire being. Primarily, there are four types of cells, which group together in a regular fashion to form the four general tissue types: 1. Epithelial 2. Muscle 3. Connective 4. Nerve 5. Blood For example, Figure 3.2 shows how epithelial cells group to- gether to form stratified tissue, while Figure 3.3 shows how Smooth muscle cells group to form muscles. Since tissue is composed of similar cells organized in an orderly fashion, it is reasonable to expect the acoustic properties of a given tissue to be nearly homogeneous. (See Appendix H for a brief discussion of acoustic waves.) Also, one might expect the acoustic properties of dissimilar tissues to be dissimilar. In fact, all reflection type ultrasonic tis sue visualization systems are based on the fact that differ- ent tissue types provide different characteristic acoustic impedances and the fact that reflection occurs at the acoustic i . . . . . . mpedance discontinuities present at tissue interfaces. At iI?igure 3.2 Stratified Squamous Epithilial Tissue Figure 3.3 Muscle Tissue 23 bone/tissue or tissue/air interfaces, the characteristic impedance mismatch is so large that very little acoustic energy is transmitted (large reflection coefficient). These tissue types form the layers of the various mem- The membranes cover the body, line branes of the body. In addition, mus- various body cavities and enclose organs. cle surfaces, organ surfaces, and interfaces between various organ parts, nerves, bones, tendons, etc. , will all scatter ultrasonic energy. Signals returned to the receiving trans- ducer from these interfaces are unwanted signals and only interfere with our ability to make accurate velocity measure- These undesired returns are collectively known as men ts. the effects of these returns cannot c-‘-:L‘L:I.tter. Unlike noise, be reduced by increasing transmitter power. Only through appropriate signal design and/or sophisticated signal process- ing techniques can the effects of clutter be reduced. Another important characteristic of tissue is its a~t‘tenuation coefficient since it directly affects the design of an ultrasonic blood flow measuring instrument. With the e3'ttzzeption of bone and lung tissue, other soft tissues enhibit a-':tenuation coefficients ranging from .5 db per cm per MHz for adipose (fat) tissue, to over 2 db per cm per MHz for e . . triated' muscle tissue28. Bone and lung tissue exhibit ttenuation coeffiCients much greater than the attenuation QeffiCients listed above. For the remainder of this dis- % . . ertation a nominal value of 2 db per cm per MHz will be 24 used for acoustic waves in the one to ten MHz category. Also, attenuation losses have been shown to vary directly with the carrier frequency29 in biological tissue. Of course, for a blood flow measurement system, the tissue whose characteristics are of most interest is the blood. Blood is composed of two major components. The first component is the fluid portion or blood plasma in which solutes are dissolved. The second component is the collection of living cells suspended in the plasma. These cells may be classified in one of three categories: erythrocytes, leukocytes, and thrombocytes. Erythrocytes, or red blood cells, are approximately 7 microns in diameter, 1 - 2 microns thick, number 4.5 to 5.5 million, and occupy 40 to 507. of the total blood volume in a normal individual. The percentage of blood volume occupied by the red blood cells is known as the hematocrit. The total volume occupied by the smaller and less numer- (>118 thrombocytes (platelets) and the leukocytes (white blood Q311$) is approximately 17. of the total blood volume. Thus, it is not surprising that the red blood cell has proven to be the major source of ultrasonic scattering from the blood”. Blood is only one component of a larger system called 11% cardiovascular system. The cardiovascular system in- Q llIdes the heart, arteries, capillaries, and veins. The 25 heart creates the pressure differential which pumps the blood through the system. The arteries carry blood away from the heart to the capillary beds in the tissue, and the veins return blood from the tissue to the heart. The arteries and veins form a branching system of vessels which divide into more and more vessels of decreas- ing diameter as the capillary bed is approached. This branching occurs in such a way that the sum of the cross- sectional area of the two vessels following a branch is approximately 1.5 times the cross-sectional area of the Vessel before the branch. Thus, the blood velocity de- creases from an average value of 20 cm/sec in the aorta to less than .2 cm/sec in the capillary beds3l. This research dCbes not address flow measurement problems associated with I'I1i<:.rocirculation of the tissue. But rather, it is directed toward flow measurement in the major arteries having diam- eters greater than approximately .5 cm. The major arteries and veins of the body resemble thin walled distensible circular cylindrical tubes. Their walls are made up of layers of endothelial cells, elastin fibers, Smooth muscle, and colagenous fibers. Under control of the Qel'itral nervous system, these vessels expand and contract to QQIitrol blood flow and pressure. Also, blood flow in the Eaarteries is quite pulsitile due to pulsitile pumping of the 1-1Qai't and these arteries expand and contract with the time 26 varying pressures in them. The situation is quite different in veins where flow is nearly time independent. As derived in Appendix I, the flow profile for laminar flow in a circular cylindrical tube is parabolic. 2 - -E. V(r) - Vmax (1 a2) 3.1 where a is the radius of the vessel. The relationship between the average velocity across the cross section of the vessel and the maximum velocity in a Parabolic profile is also derived in Appendix I and is found t:1:) IDe Vmax = Z-VaVg 3.2 At high velocities, the pattern of laminar flow breaks dQWn and the fluid elements appear to flow in a random time varying fashion. The critical point at which this transition frOm laminar to turbulent flow occurs is defined by the so- Qa-lled Reynolds number, where Vav Zap Reynolds number = —-g——— 3.3 n where p = blood density = l gm/cm3 and n is the blood viscosity w . . . q I:j-lch is approximately .035 dyne sec/cmz’gz. Turbulence 27 usually occurs when the Reynolds number exceeds 2,300. C o ulter and Papenheimer33 found the critical Reynolds number for blood to be approximately 2,000. This indicates that, in the larger vessels, blood behaves much as a homogeneous fluid. With the exception of diseased vessels and flow at the root 0 f the aorta, Reynolds numbers are usually well below 2,000 in the circulatory system. Thus, for purposes of this research, laminar flow will be assumed. The design of any practical flow measuring instrument Will be highly dependent upon the dynamic range of velocities to be measured. Figure 3.4 shows a typical plot of average VOlume flow rate vs. time for blood flow in the aorta. Using ecluation 3.2 and an average value of 1.0 cm for the diameter of the aorta, it is seen that blood velocities in the-cardio- Vas cular system should normally lie in the range of 0 - 100 cm/ sec. 1500 A 1200 __ g 900 ._ \. B 600 «- 3001.. [4335195 from -5 to 1.0 seconds ’ I T: t Figure 3.4 Typical Flow vs. Time in the Aorta 28 Another very important parameter characterizing the blood, as well as the surrounding tissue, is the effective scattering cross section. The scattering cross section of continuous bi- ological media is defined as the fraction of incident acoustic power scattered in the direction of the receive transducer per unit solid angle per unit volume of media. A great deal of re- cent research has been directed toward measuring the scattering cross section of various tissues. Unfortunately, with only a few exceptions (whole blood for example), current measurements of absolute scattering cross sections provide little in the way of useful design data. However Baker, using the Duplex Scanner,27 measured the relative scattering cross section of blood.at 3sz to be 20 db below the scattering cross section of the vessel wall. In the analysis and design that follow, this number is used as the nominal relative scattering cross section of blood relative to Ell the surrounding tissue. 'Also, experiments34 have shown that the scattering cross section of blood increases as the fourth power of frequency and is nearly isotropic. This is consistent with Rayleigh scattering theory which states that given a Poisson distribu- tion of independent scatterers, each with dimensions much less than a wavelength, and assuming first order scattering, then the scattered energy will be isotropic and exhibit a fourth power dependence on frequency. Thus, treating red blood cells as independent Poisson distributed scatterers is not only intuitively pleasing, but also consistant with the above 29 mentioned experimental results. The optimum choice of carrier frequency depends on a trade-off between the back scattered energy which is proportional to m4 and attenuation losses which are proportional to w . Experience in building and working with C. W. Doppler flowmeters, Pulsed Doppler flow- meters, and Continuous Noise flowmeters has shown that 5 MHz is a reasonable choice for a blood flow imaging system. Target Definition and Characterization In Appendix F, a technique for using a correlation type receiver to estimate the range and range rate (or radial ve- locity) of a point target is introduced. It is shown that the maximum likelihood estimate of delay (E) and Doppler frequency (0)in the presence of white Gaussian noise is obtained by maximizing the square of the magnitude of the log likelihood function over all possible values of delay and Doppler frequency. That is: 1,03 = Eff»: |L(T,w) '2 3.4 €>r~l> where L(r,w) = _g”f(t)f* [(1 + a)t T]ejmt dt 3.5 In equation 3.4 r(t) is the complex envelope of the re- ceived signal and f [(1 + a)t-r] is a delayed,time scaled version of the transmitted envelope. 30 There are two apparent difficulties with maximizing lL(r,w)|2 as shown. One arises from the infinite limits of integration. The other because of the continuous nature of the candidate delay Doppler estimates. The first difficulty is easily removed since in any practical system the trans- mitted signal will be non-zero only for some finite time dur- ation and therefore the limits of integration are finite. One the other hand, there is no apparent means of arguing away the existence of infinite possibilities.for delay and Doppler estimates. A practical method for dealing with this difficulty con- sists of the following two steps. First, limit the allowable ranges of both delay and Doppler to values which can physi- cally occur. And secondly, divide the remaining ranges of r and w into a grid of discrete candidate delay/Doppler pairs. The pair which maximizes the magnitude of the log likelihood function is then chosen as the estimate of target delay and Doppler. However, this introduces a new difficulty. Namely that of choosing the delay and Doppler separation of these pairs. A logical first step in solving this problem would be to some- how determine the delay and Doppler resolving capability of the transmitted signal. Fortunately, this information is available by properly interpreting the signal ambiguity function discussed in Appendix G. In the Appendix it is shown 31 that for all cases of interest in blood flow measurement, the range resolving capability is approximately the inverse of the effective bandwidth of the transmitted signal. Also, it is shown that the Doppler resolving capability of a conventional correlation reCeiver is roughly the inverse of the effective time duration of the signal, and it is suggested that in most cases of interest the same criteria holds for the true corre- lation receiver. However, demonstrating the reasonableness of this suggestion is reserved for Chapter Four where specific signals and their attributes are discussed. Thus a reasonable choice for delay separation of the discrete delay/Doppler pairs is also the inverse of the effective bandwidth and a reason- able choice for Doppler separation is, for most signals, the inverse of the effective signal duration.’ As stated earlier, the objective of the blood flow imaging system is to estimate the velocity of the blood at a particular location within the vessel. That velocity estimate must be made by examining the sum of signals reflected from all red blood cells as they pass through the desired location. There- fore, since laminar flow has been assumed, the target will be defined as a "small" cylindrical subsection of the vessel parallel to the flow vector as shown in Figure 3.5. The range to the center of the target is assumed known. However, the approximate size and shape of the target cross section has not been specified. In general, the target shall consist of all red blood cells which by themselves would produce a significant 32 output from a correlation receiver perfectly matched in velocity and set at a known delay T. Based on the previous discussion concerning the range resolving capability of the transmitted signal, the target shall specifically consist of those red blood cells which pass through the ultrasonic beam Ultrasonic Beam Blood Vessel Beamwidth = B ‘\¥Transducer Angle = e .. + // + ‘1’ fix”: Jr r I: '- .‘- 2‘ ~I 1 :-' .- :2.‘ § 1. Diameter=D ~ 2 T // AX :- m COSG w arget U Figure 3.5 Target Definition . 1 at ranges corresponding to delays between I - ITBW' and r + l 27BW‘ The approximate dimensions and shape of a typical target are shown in the figure. Of course, the actual shape of the target depends on the specific signal and transducer beam pattern. Even when these are completely specified, the target shape is still somewhat arbitrary. Therefore, when specific target shapes and beam patterns are required, they will be chosen for mathematical convenience. 33 Having described and defined the physical target, a math- ematical characterization (or model) shall be developed next. Figure 3.6 shows the geometric relationships between the th target, the i RBC within the target, the ultrasonic beam, and the transducer. In addition to moving along the flow vec- h19.130 tor with velocity vi, the spatial orientation of the it is continually changing. Thus, each red blood cell in the target appears as a fluctuating point target. However, the assumption will be made that each RBC appears as a non-fluc- tuating point target during the time required to pass through the ultrasonic beam. Additionally, it will be assumed that the beam is sufficiently narrow and the transducer sufficiently far removed such that 6i is approximately equal to 0 when the associated RBC is within the illumination volume. For Ultrasonic Transducer ———" Target (See Figure 3.5) Figure 3.6 System Geometry 34 example, with a beam width of 1 mm and a target distance of approximately 2.5 cm, 9i would vary on the order of two degrees. The importance of this assumption is to allow all RBC's along a particular streamline to be associated with a unique Doppler frequency given by: Zlvilwc wd = ————-——-cos(€) 3.6 C Under these assumptions and temporarily neglecting the beam pattern, the signal received from the ith red blood cell is the same as that derived :flmr a point target in Appendix F (see equation F.l4). §i(t) = /t; BiE[(L+a(Ei))t-T(?i)1ejw(ri)°t 3.7 The effects of transducer losses, beam pattern, and two way propagation losses can be included by multiplying by an appropriate function of space T(fi) = T(fi)e3¢i. If phase changes associated with T(fi) are included in bi’ the actual th received signal from the i red blood cell may be written as: ri(t) -VEt-T(ri-+vit)bif[(14-0(ri))t T(ri)]e i 3.8 The total signal received from all RBC's in the target is the Sum of signals received from the individual red blood cells. i(t) = gEi(t) l 35 Next, the summation will be divided into two summations. This division is achieved by breaking the target volume into j' Each Vj will in turn con- tain some random number, Nj’ of red blood cells. The re- small elements of equal volume V ceived signal may now be rewritten as: -E[(1 + d(fij))t-T(fij)]ejw(rij)t . 3.9 If the volume elements are sufficiently small, then rij=rj and the expression for f(t) can be rearranged as follows: r = /E;T<£j+vjt>f[<1+a>t-:: b. . 3.11 . ._ i,j J i-l By assumption E{bij} = 0 so that Nj _ Nj E: b. =2.E{b..}=0 l=l 1'3 i=1 1’3 Therefore, 130:) = 0 ‘ 3.12 This simplifies the expression for the complex covariance function K(tl,t2). K E{[E(tl>-m11E1*} E{E(tl)E*1 +1 2 c r 2 i _ [FTI t only relative scattering intensities will be of the following chapters, the range scattering function may be normalized to IFTIZI -9-d1. The result of t 2 this normalization is the desired relative range scattering function. SR(1) SR(1) SRCT) = -2ac1 < < o 1 1C-1r 3.29 2 II‘Bl -2ac(1 -1 ) _ < + IT I To Tr T1c+1r 3.31 Using the assumed value of 10 db/cm for the attenua— tion of ultrasound in tissue at 5 Mhz, the value of a is found as a = 1.15 (refer to Appendix H for the conversion factor). As mentioned earlier, the relative reflection 44 coefficient of tissue was measured by Baker and found to be 20 db above that for whole blood. Thus the ratio IFBlz/IFTIZ equals .01. Substituting these values into equations 3.29 through 3.31, the final form of the relative range scattering function may be written. SR(T) e o<1<1C 1r 3.32 ‘ _ -n (1 -1 ) _ SR(1) - .Ole c r 1C 1r1C+1r 3.34 where n = 3.45 x 105. In the next chapter various signals will be evaluated in terms of their clutter rejection capabilities. To do this a typical vessel range of 2.25 cm will be assumed and the vessel will be assumed to have a diameter of .5 cm. In this case, equations 3.32 through 3.34 may be further simplified. 5 SR(T) = e"nT O3.67X10-5 3.37 Figure 3.7 shows a sketch of lO-log[SR(1)] described by equations 3.35, 3.36, and 3.37. Decibels 45 10 log (SR(1)) I 0 Blood Vessel .. (D = .5 cm) "201- : l | i : ' I 4— ' I i i g I -40__ : : . ' u -451 ................... -50-- -65‘ """"""""""" ——'-|' uh ' ' ' I I a I I i i i 1' i 'r :— 1 0 10 20 30 40 50 36.7 Microseconds Figure 3.7 Typical Relative Range Scattering Function 46 Another dimension may be added to the Relative Range Scattering function which describes the distribution of energy at each range as a function of the Doppler shift which occurs upon reflection. The resulting function will be called the Relative Range Doppler Scattering function, SRD(1,f). For purposes of the analysis conducted in this dissertation it is adequate to assume that tissue is sta- tionary. Also, if laminar flow is assumed, the DOppler within the vessel may be described as a function of range, fD(R). Note: In circular cylindrical vessels fD(R) will be parabolic. This information is easily incorporated into the clutter model in the following manner. SRD(1,f) = SR(1)l T (Tl—F s = l—IT—L e’j"fTsinc<f> 2 T-ITI 2 . 2 emf) = I¢| = —T— smc ((T-lrlm 4.1 where ]_ lxl < 1‘. MK) = { 2 0 elsewhere 51 In Figure 4.1, illustrating the ambiguity function for a single pulse, it is observed that the function has a single peak whose width along the delay axis is directly prOpor- tional to the pulse width and whose width along the frequency axis is inversely proportional to the pulse width. To achieve the velocity resolutions necessary for blood flow imaging (i.e., 100 hz - l Khz with a 5 Mhz carrier) signal durations on the order of one to ten milliseconds must be used. Since . this provides a range resolving capability in the 75 to 750 cm range, the single pulse is clearly unacceptable. 0(1,f) : I I, I I I ’ 2 / ¢-—- Sinc ((T-I1I)f) I -T l ‘4‘ T x ,v ‘_—_(1 IT!)2 L T “T Figure 4.1 Ambiguity Function for a Single Pulse 52 One method of improving the range resolving capability without disturbing the velocity resolution is to introduce complexities into the transmitted pulse. For example, if the basic pulse is multiplied by wideband noise, a large bandwidth long time duration signal is generated. The average or eXpected ambiguity function for such a signal is derived by computing '§(1,f) = E{¢(1,f) 0*(1,f)} 4.2 where the expectation is an ensemble average taken over all possible waveforms of the noise process. The ambiguity 36 function for random noise has been derived by COOper and is shown in Figure 4.2. As expected, the width along the delay axis is deter- mined by the signal bandwidth while the width along the frequency axis is determined by the signal duration. Of particular importance is the fact that there are no signif- icant ambiguous responses. There are difficulties in implementing a random signal system. One of the limiting devices in the hardware imple- mentation of this system is the delay line required to pro- duce a delayed version of the transmitted signal. This delay line must be capable of delaying a 1-3 Mhz bandwidth 53 ZBW 7r 2T 1 ,— --------- Figure 4.2 Ambiguity Function for Random Noise 53 23W __1___ 4-BW-T 1 .2 1 / Figure 4.2 Ambiguity Function for Random Noise 54 analog signal centered at 5 Mhz for up to 50 us. The require- ment for such a delay line is unique to systems using non- deterministic signals. In the past, analog acoustic delay lines were used in the experimental random signal system described in Chapter Two. However, these mechanically varied lines are much too slow for use in velocity imaging. Pseudo random codes (Barker codes, Gold codes, and other Pseudo Noise (PN) sequences) have been developed to approxi- mate a random pulse sequence. Actually, these codes are deterministic and therefore the use of a delay line to pre- serve the transmitted waveform is not required. There is a class of PN sequences which are particularly appealing. They are called maximal length linear shift register sequences. Briefly, these sequences are called shift register sequences because they can be generated by a feedback shift register. A typical feedback arrangement is shown in Figure 4.3. Shift Register Stages Ini- Ini- Ini- Out 0 o ' put r_____mtlaIIY'_______tially tlally Sequence equals equals equals one zero zero 0010111 <1; Figure 4.3 A Typical Feedback Shift Register 55 They are called maximal length because they are periodic on ZL-l clock pulses, where L is the number of shift register stages employed. For example, see the output sequence asso- ciated with the feedback shift register of Figure 4.3. Addi- tionally, PN sequences exhibit some very nice properties which make them nearly ideal substitutes for truly random sequences. Specifically, the number of ones exceeds the number of zeros by one, a run of length K occurs with proba- bility Z-K and the cyclic autocorrelation function has con- stant sidelobes equal to -l/N.. Unfortunately, the noncyclic autocorrelation function, 0(1,0), for PN sequences does not have uniformly flat sidelobes. However, it has been shown37 that the average sidelobe level for the ambiguity function (for PN sequences) along the delay axis, 0(1,0), is -l/N. Also, the average ambiguity function for PN sequences has been shown to be approximately that pictured for random noise in Figure 4.2. The only modifications necessary are to re- place bandwidth (BW) with l/TB (where TB is the bit period) so that T-BW is N, the length of the code in bits. Figure 4.4 illustrates the average sidelobe level of an ambiguity function for a PN sequence. Typically, these sequences are used to Binary phase modulate a carrier. With this modulation scheme, the trans- mitted signal exhibits most of the desirable signal charac- teristics mentioned earlier. Specifically, it can be a large 56 bandwidth long time duration signal with the highest possible energy for the lowest peak power. Figure 4.4 Average Ambiguity Function for a PN Sequence To evaluate the potential performance of a correlation receiver using this signal in the presence of clutter, the clutter model of equations 3.38, 3.39, 3.40 will be com- bined with the ambiguity function of Figure 4.4 through the relationship given by equation 3.41. Using these models, the average output of the correlation receiver due to target returns in the absence of noise can be reduced to: [‘11 I t - ftargetSR(1)0(1-1t,fD(1)-ft)d1 4,3 57 For a target perfectly matched in D0ppler to the correlator and having a range extent of A1, the average energy returned from the target is: (using equation 3.36) Et = A1 - 3x10“7 4,4 In deriving this expression, a uniform velocity profile within the target has been assumed. The effects of DOppler spreading due to non-uniform profiles will be considered in Chapter Six. Similarly, the average energy output due to clutter returns is found by substituting equations 3.38, 3.39, 3.40 into equation 3.41 and using equations 3.35, 3.36, and 3.37. E = 1' ‘ s (1)6(1-1 ,f (1)-f )d1 clutter R t D t -5 w 3.7xlO _ -n1 ;L_ -7 l — .{e 4N d1 + 1' _5 3x10 ZN d1 0 3x10 -7 l " AT ° 3X10 m- -7 ~ 7.2xlO _ 7N 4.5 For a typical delay resolution on the order of 1x10"6 the signal-to-clutter ratio is computed as follows: 58 -7 SCR 3 A1 3x10-7 7.2xlO N SCR = 4.2Nx10'7 4.6 This is an example of the signal-to-clutter ratio eXpected for a typical .5 cm diameter vessel located 2.25 cm from the transducer using a long time duration PN sequence. The correct interpretation of equation 4.6 is that increasing the number of bits in the sequence improves the signal-to- clutter ratio. In the case of blood flow imaging where 1-10 msec signals are required, N will range from 103 to 104 which results in an SCR in the range from .0004 to .004. This is not an acceptable value and other signals must be investigated. The simplest large time bandwidth product signal which combats clutter as well as or better than any other signal is the periodic pulse train shown in Figure 4.5. T :2 NT 1: P J WI T I:r D ’1 TB—* ‘— -—--- --——-—>-->———1- ——-m: Figure 4.5 Periodic Pulse Train 59 Insight into some of the properties of ambiguity func- tions can be gained by intuitively developing an approximate representation for the ambiguity function associated with a periodic pulse train. (A rigorous derivation is available in any standard radar text.) From Figure G2, it is observed that the area of the pedestal of a generalized ambiguity function is 2T°ZBW. For the periodic pulse train this area is approximately 2NTp %% . However, the ambiguity function is non-zero only in narrow strips parallel to the Doppler axis. The width of these strips is approximately TB so that the total area where the function takes on non-zero values is approximately 2NTp %% $§ . Since the volume of the ambiguity function must be unity, the average pedestal height is £§ . Using this and properties one and two in Appendix C, an approximate ambiguity function has been sketched in Figure 4.6. This approximation for the average pedestal height between peaks has neglected the volume under each peak. If this volume had been included, the derived value for the pedestal height would have been reduced slightly. The important features of this particular ambiguity function are l) the narrow central peak, 2) the presence of "clear" areas that are not affected by clutter, and 3) the presence of subsidiary peaks. Clutter present at these peaks can cause the receiver to make an incorrect decision. 60 Central Spike _____. Figure 4.6 Approximate Ambiguity Function for a Periodic Pulse Train Based on the clutter model of Chapter Three, it is reasonable to assume that clutter more than about 2.5 cm beyond the vessel is negligible. Therefore, to allow for the observation of vessels up to 3.5 cm deep the ambiguity peaks in delay must be greater than 47 us (the two-way prepagation time to a target 3.5 cm deep). This corres- ponds to a Pulse Repetition Frequency (PRF) of 21 Khz or less. To allow a margin of safety, a PRF of 10 KHz is selected. This choice of PRF generates ambiguous peaks in 61 Doppler separated by 10 Khz. Since the maximum expected Doppler is on the order of 5 Khz, this is completely accept- able. This signal choice provides the best immunity to clutter, and at the same time provides the worst ratio of total energy to peak power. However, it has been suggested that another class of signals exist which reach a reasonable compromise between the PN sequence and the pulsed sinusoid. These signals are formed by replacing each pulse of the periodic pulse train with a slightly longer time duration burst of a large band- width signal. One possible compromise would be to replace each pulse with a short, large bandwidth PN sequence, and observe the effect on the signal-to-clutter ratio. A typical signal sequence is illustrated in Figure 4.7. The sequence period is selected to be 10.4 seconds as it was for the periodic pulse train. The total signal dura- tion is T, (l-lO msec), and TB is the bit period, which is on the order of 1 us. 62 '1 'U .1. "MT " g: 1 :J' B 13 Figure 4.7 Periodic Train of PN Sequences This signal may be represented analytically as T r__ _ TP 1 \ ~ T f(t) = MTET< k: P(t-kTP)) 4,7 L J where P(t) is an M bit PN sequence. The first step in com- puting the ambiguity function is to compute the time fre- quency autocorrelation function ¢(1,f). ~ 4(1,f) =_f°°f(t) f*(t+T)ej2Trftdt 4.8 If TP is greater than 2MTB, then $(') contains strips of clear area for certain values of T. These strips of clear area lie midway between the subsidiary peaks of $(') along the delay axis. The centers of the clear areas nearest the origin correspond to the location of clutter 3.75 cm from the target. Since the greatest expected target depth is 3.5 cm, the only portion of the ambiguity function which is 63 of any interest lies between these first two clear areas. For [Tl Z MTB using 4.7 in 4.8, T ~ TP TP - 1 kTPi-MTB-|1[ ¢ B k=0 kTP P*(t-kTP+1)ej2"ft dt For [TI 3 MTB 4.9 Letting u = t - kTD 1 —— - 1 ~ TPTP 1 MTB-I1I ¢(T.f) "’ T1.- 2 MT f P(u) k=0 B 0 .P*(U+T)ej2nfudu ej2wfkTP 1 - 1 ~ 1P Tp jZkaTP ~ 4(r.f) = —— z e ¢P 6.7X10- I = 4N(2M-1) B 2 i (M-1)TB _7 -6 °3x10 (M-1)TB< 6.7x10 K 4N(2M-l) — -5 3x10 +MTB exp[-3.45x105°T] 1 d1 3x10‘5+1B 4N(ZM-l) I3 = r -11 < 2-3le (1-eXp1-3.45x105-(M-1)TBJ N(2M-1) < (101-1)1'3 > 6.7){10-6 0 (M-1)T < 6 7x10‘6 L B ‘ ' k It is easily verified that EC is a monotonically increasing function of M which is approximately flat for M2 4. For M2 4 the exponential in Ilis reasonably well approximated by the first two terms of a Taylor series (assuming TB to be on the order of 1x10-6). Thus, Il’ 12 and I3 are given approximately as: -11 I1 = 2'3X10 -3.45x10S (M-1)TB N(2M-l) = axlo‘6 T “‘1 4.15 B N(2M-l) 12 . 7.5x10‘8 -TB .4 “‘1 4.16 N(2M-l) I z 0 4.17 67 And the total energy is approximately I1' 3 '6 M-l Therefore, for M :_4, the signal-to-clutter ratio is: E T )c3x10-7 t B SCR = -—'= . EC M-1 T ~ 8x10'6 NEH-“71 B _ -2 (2Mal) N In major blood vessels such as the aorta where high velocities are encountered, the signal durations employed can be on the order of .001 seconds or less. (The reasons will become apparent in a later chapter.) Thus N can take on values less than or equal to approximately ten. Without resorting to a periodic pulse train by setting M equal to one, the maximum signal-to-clutter ratio is obtained for M equals two. Even for this signal, the signal-to-clutter ratio is approximately zero db and therefore not adequate for reliable blood flow imaging. The overwhelming effect of the vessel wall is obvious. In order for a signal to be usable, the vessel wall mpg; lie in a clear area of its ambiguity func- tion. Thus, for the instrument to make reliable velocity measurements over as much of the vessel as possible, the periodic pulse train appears to be the only reasonable signal choice. 68 All that remains to be accomplished in this chapter is to show that the Doppler resolution of the periodic pulse train using a true correlation receiver is approximately that indi- cated by the conventional ambiguity function. The transmitted signal shown in Figure 4.5 may be written analytically as g— - 1 .. TP P -kTP f(t) = T—T' Z 77 -'—T—— 4.20 B k=0 B where 0 IX! > % TT(x) = 1 1 IX] < '2- Along the DOppler axis (i.e., T=0) the generalized time frequency autocorrelation function is f . ~’ _ ~ 1 JZTl'f t f . ,” 2 JZWf t If f2 is assumed greater than fl’ and fl and f2 are both assumed small compared to fc, then by substituting equation 4.20 into 4.21 an expression for $’(o,fl,f2) can be derived. f f2 . _ l _ For convenience, a1 — f; and a2 - f;" T13 kTP4-jr kT 1+0:2 P T t"l-Fa ~, ~ P l l 4’ (O'fl'fz) ' Ti T; T 7‘ TB / kT - B P 7? I4-a1 kT t- P IH+a2 j2w(f1-f2)t -n T e dt 4.22 B Using the fact that eXp 1+0 is approximately exp(j2“(f1-f2)kTP) for all Dopplers of interest, $’(o,fl,f2) can be simplified as: (letting f = fl-fz) TIL" 1 ~ TP P kTP jZWkTPf (:0 (0.f)=‘=f" Z w --T_B:_fe 4.23. k=0 c where the function C(x) is defined as C(x)=0 for x i 0 and C(x)=x for x>>0. Now when TPf .[ g(t )g (t )dV 5.1 1 2 b t target 1 2 volume jw(r)t g(t) T(E+V(f)t)f[(l+a(r))t-T(r)]e Recall that the effective scattering cross section associated with each red blood cell is 0%. The density of red blood cells is p. Et is the energy in the transmitted signal whose normalized complex envelope is f(t). The Doppler frequency is represented by w(f) and is a function Of spatial location E. Also, a is directly related to m(f) by a = géEl where we is the carrier frequency of the transmitted signaI. And 1(f) is the two-way propagation time to the initial loca- tion of each differential volume element. The function which gives rise to the transit time effects is the beam pattern T(-). Notice that as each differential volume element of blood moves through space, the effective scattering is modu- lated by T(-). Since the intention is to determine the general effect that transit time has on true correlation receiver perfor- mance and not to determine the performance of a specific system, the choice Of beam and target shape is somewhat arbitrary. For mathematical convenience a somewhat unusual beam pattern is chosen. As shown in Figure 5.1, a Spatially hard limited beam with square cross section and beamwidth B has been chosen. 75 Ultrasonic Beam :L _ C - d-Z7§W»Sine4-B Cose [; / "J €71: ==7===r=~l__, f""f ............. fl 3 ,/ \. Target (subsection Of the vessel) Transducer Angle==e Figure 5.1 Target and Beam Geometry Temporarily neglecting the relative motion of the tar- get, this choice of beam pattern may be described by T(-) as follows. (The effects Of prOpagation losses are being included in T(o)). Z _ y---— T(x.y,z> = e “/2 Mg” gene 5.2 sine To be consistent with equation 3.21, the relative motion of the target is included by substituting x==x+th, y=y+Vyt, z=z+Vzt, 76 24-V t _ x+V t y+V t-——7=— T(x.Y.z) = e “/21: _fi—L «n YB tame 5.3 sine Additionally, a rectangular target is defined to match the beam pattern and range resolution of the signal. In order to facilitate the analysis, two assumptions shall be made. 1. The velocity of red blood cells within the target is not a function of spatial location. That is, V(E) = vy§. This also implies that m(f) = w and a(f) = a. 2. The DOppler shifted received signal was reflected from a stationary fluctuating point target such that T(E) = T. (As will be discussed later, this assumption is valid only when true correlation is employed.) The first assumption is reasonable since, in the center of the vessel where transit time effects are the greatest, the velocity profile is nearly uniform. In Chapter Six further design constraints will be imposed on ultrasonic blood flow measurement systems to account for non-uniform velocity profiles within the target. 77 Effects which can result from the non-zero range extent of scatterers have been minimized by the target definition. Thus, assuming true correlation processing, assumption two is also reasonable. Making these assumptions and substituting 5.3 into 5.1: $ _ 2 -nT Jf. ” ~* . K(t1,t2) - ZObEtpe g1(tl)gl(t2)dV target y+Vt-—£L 81(t) = fi(%)W ‘yB tane f[(1+a)t-T]e3wt 5.4 E336— Bringing variables which are not_functions of space outside the integral, 5.4 may be rewritten as: R(tl,t2) = ZogEtpe-UTE[(1+d)tl-T]ertl 5.5 ~I°f“[(1+a)t2-T]ejwt2 where I= ffffi(%)w) In the eXpansion of equation 5.13 the normalization of the standard Fourier coefficients was necessary to insure that A(O) = 1. By letting 8 equal avT, where T is the signal duration, then 8 represents the ratio of signal duration to transmit time. The coefficients Cn may now be rewritten to emphasize their dependence on B. 0 n = even Cn = 5.14 _£E§_. n = odd 84 5.15 __ ZNB _ 33 Also, to show explicitly the dependence on B, Ap(av1) may be written as Ap(B %) . A plot Of Ap(B %) using N= 2.5 and a six term expansion for various values of B is shown in Figure 5.3. Notice that a six term expansion implies that k equals 5 in equation 5.13. V H U1 t-3|>’ Figure 5.3 Approximate Ap(8 %) 85 A(av1), given by equation 5.13, is transformed to a separable form using the common trigonometric identity: cos(x-y) = cos(x)cos(y) + sin(x)sin(y) . fin Then, letting on = NT and A = t-u; k A(av(t-u)) = a0 + nil ancos(wnt)cos(wnu) 5.16 k + nil an31n(wnt)31n(wnu) And finally, the integral equation of 5.12 may be written in a separable form. T ~ 7 2k+l ~ ~ . ~ 1¢(t)=/ z Mj(t)Nj(u)q‘>(u)du 5.17 T j=l “2' where Ml(t) = 53K Ruejwt M2n(t) = /anK cos(%—1.’I.l-t)f(t)ejwt _ /_ . EIT- ~ jwt M2n+l(t) - vanK Sin(NT t)f(t)e {vim = 113:) 86 From this equation the Eigenvalues and Eigenfunctions of R(t,u) may be determined. First, the order of integration and summation are interchanged so that T ~ 2k+1 ~ 2-~ ~ M: (t) = 2 M.(t)f N.(u)¢ (u)du j=1 J T J '2' Letting the integral equal Cj ~ 2k+1 ~ - M (t) = z M.(t)-C. j=l J J Multiplying both sides by Ni(t) and integrating: T T 7 ~ ~ 2k+1 5 ~ ~ .. t N. t dt = M. t N. 1: (112°C. jf ¢() 1( ) jgl./p J( ) 1() ,J -2 -I. 2 2 T/2 ~ ~ Letting ai,j =-T§2 Mj(t)-Ni(t)dt ~ 2k+1 ~ AC. = Z 0!.- .C. l j=1 l,J J Or, in matrix notation: AE=AE Therefore, the Eigenvalues of the signal process Eigenvalues of the 2k+1 by 2k+1 matrix A. Also, 5.18 5.19 5.20 5.21 5.22 are the if the 87 Optimum receiver were to be implemented, then the Eigen- functions could be generated as follows (from equation 5.19): ~ ]_ 2k+1 .t=—-Z C..M.t .23 ¢1( > *1 j=l 1,3 J( > 5 Where Cij is the jth component Of the normalized Eigenvector associated with 2i“ The computation of the elements of A, aij’ is somewhat laborious and is carried out in Appendix J. In the deriva- ZET which is reasonable for tion it is assumed that Tp << all cases of interest. The elements of A are summarized below. all = a K 5.24 2K/aoan a1,2n = ' fn sin(fn/Z) 5.25 Kan a2n,2n = 2?;-(fn4-31n(fn)) 5.26 Kan a2n+l,2n+l = 2?; (nt'Sln(fn)) 5'27 5. fn+m . fn-m , Sln<‘—7‘) 51n("7—) a2n,2m = Kvanam f + f 5.28 n$m n+m n-m 88 emf?) emf—“52> O‘2n+1,2m+1 "' K/anam f m ‘ f 5'29 naém n n-m OL2n,l = O£1,2n 5'30 where _ = T = Signal Duration 8 _ avT ___—:l' Transit Time 5'31 [av] — El fn'_ N 5.32 The remaining entries are zero. Having determined the entries of the A matrix, the Eigenvalues and Eigenfunctions can be computed and the Optimum detector could be implemented. It is interesting to note, as can be seen by studying the Fourier coefficients, that only one parameter has any effect on the receiver structure. That parameter is B, the ratio of signal dura- tion to transit time. Later in this chapter an interesting interpretation of B will be given which will be the basis for comparing Doppler resolution Of the true correlation and conventional correlation receivers. 89 Performance of the Optimum Detector in the Presence of Transit Time Effects Begin by Observing equation D.15 and its implementation in Figure 5.2. If equal apriori probabilities are assumed, then for minimum probability of error y’ equals one and the receiver decision threshold becomes Y = Z NOQID N— “l" l 5.33 i=1 O This threshold is then used in the computation of the probability of detection (PD) and the probability Of false alarm (PF)’ "d II D f p(2[target is present) 5,34 Y . '11 II F fmp(£ltarget is absent) 5.35 Y 2k+1 2 = 2 Y1 (see Figure 5.2) ,5.36 i=1 Before the integrals of equations 5.34 and 5.35 can be completed, the density functions which describe 2 on both hypotheses must be determined. Since 1 is the sum of independent random variables, the characteristic function for p(2|target present) may be determined by multiplying the characteristic functions Of the individual random variables. 90 Before this can be completed, however, the probability density of yi, given that a target is present, must be determined. First, note that because f(t) is assumed to be a Gaussian process, then each Ei is an independent Gaussian random variable with mean and variance given as follows: Efri} = E{I(E(t)+w(t))$i(t)dt} = fE{E(t)+%(t)}$i(t)dt 5.37 = o E{EiEi} = E{I(E(t)+%(t))$i(t)dt -;>*5:du} = ffgict)[i (2‘ >> T—dlgil pyl Y1 " Ulril (Yi I y Making the appropriate substitutions: p(yi|target is present) = £4 e 3.42 i 92 Similarly, priItarget absent) may be obtained from P(Ifilltarget absent). IEiiZ ~ 2Ir.| - NO P(lrilltarget is absent) = N e Carrying out identical steps as for p(yiltarget present), the probability density of yi, given that the target is absent, can be derived. The result is: - (Ai+No)yl Ai+No NOAi p(yiitarget is absent)==N—x—— e 5.43 O i Both Of these are exponential densities of the general form: p(Yi) = g; e l 5.44 The characteristic function associated with p(yi) is W?- easily found by computing the expectation of e 1. yi 6P1 - E{e - g e E; e dyi 5.45 . 1 (JV"§T)yi 1 m 1 = §—'f' e dyi i o or d) = - ——1-. 1 5.46 ‘pi JVBi- 93 Now since the yi's are independent, the characteristic func- tion of their sum, 2, is the product of their characteristic functions. 2k+1 1 ¢ = 11 - ?———-j- p2 i=1 ( JVBi- > This in turn may be eXpanded as a partial fraction expansion of the form 2k+1 Pi 6 = z “—_1" _ 5.47 p2 i=1 JVBi where 2k+1 B. P. - H l , 5.4.8 1 j=1 Bi‘Bj jaéi Each term Of the expansion is of the same form as 6p 1 and therefore the probability density associated with 6 2 is determined by inspection. -3. 2k+1 Pi Bi P(QIH.) = X g— 5.49 3 i=1 i where On Hl:(target is present) 8. = A 5.50 94 On HO:(target is absent) l.N _ l 0 Bi ‘ IETFN— 5-51 0 Substituting 5.49 and 5.50 (5.51) into 5.34 (5.36) and carrying out the integration, the probability of detection (probability of false alarm) is easily determined. 2k+1 B. _ l The total probability of error (assuming equally likely events) is simply the average total error: P _ 1 E - 2 (l-PD+PF) 5.53 Notice that if the transit time of red blood cells were 1 T . for all A. In the Fourier series expansion of A(B,%), a infinity, then A(B j)would simply be a constant value of one 0 would equal one and all other coefficients would be zero. As a result, all entries of A would be zero except all which would equal K, the total eXpected received signal energy. Therefore, the first component of the associated Eigenvector would equal one, and all other components would be zero. Thus, in this degenerate case the correlation receiver is identically the Optimum receiver. 95 Performance of the Correlation Detector in the Presence of Transit Time Effects Figure 5.4 shows the Operations involved in a correla- tion type detector. As shown, the received signal consists of the reflected signal plus additive noise. Both f(t) and 6(2) are assumed to be sample functions from independent zero mean complex Gaussian processes. Additionally, w(t) is assumed white. {~(t) _—.®__/ Jl-I—L—qg 2‘ o \ “f*(t-1')e-jmt Figure 5.4 Correlation Detector In order to determine the performance capability of the correlation detector, two pieces of information are required: (1) the threshold, and (2) the probability density function governing 2. 96 Having already derived a probability of error expression for the optimum receiver simplifies the task Of deriving a similar expression for a correlation receiver perfectly matched to the target DOppler. Since many of the detailed computations are identical, they need not be repeated. Because f(t) and 6(2) are zero mean processes, f is a zero mean random variable with a mean value equal to zero. E{E} = E{r(}(t)+&(t))E*(t)e‘jwtdt} = ;E{£(t)+%(t)}E*(t)e‘3“tdt 5.54 = 0 ~ Finding the variance of r is somewhat more involved. E{Er*} = E{f(f(t)+w(t))f*(t)e-jwtdt -f(r(u)+w(u))*f(u)ejwudu} 5.55 E{EE*} = fff*(t)E{(r(t)+w(t))(f*(u)+w*(u))} -E(u)ejw(u't)dtdu 5.56 Because E{§(t)w(t)} = O, the expectation may be simpli- fied as follows: 97 E{(E(t)+&(t))(E*(u)+6*(u))} = E{E(t)E*(u)} + E{w(t)w*(u)} 5.57 The first expectation is just the covariance function which was originally defined in equation 3.13. Eventually, this covariance was determined to be as given by equation 5.11. Assuming the noise to be white, the second expectation is simply N06(t-u). Making these substitutions into equation 5.56, the expression for the variance of f becomes: E{EE*} = Kfff*(t)f(t)A(avl)f*(u)f(u)dtdu + fff*(t)N06(t-u)f(u)dtdu 5.58 Using the normality Of f(t) and the Fourier approximation for A(av(t-u)) given by equation 5.13, equation 5.58 expands to: E{EE*} = Kfflf(t)l2aoif(u)lzdtdu ~ k + Kfflf(t)|2[ Z ancos(wnt)cos(wnu)]lf(u)Izdtdu ~ 2k . . E de + Kff|f(t)l nElan51n(wnt)31n(wnu) 1(u)l t u + N 5.59 o 98 Each of these terms are directly related to the elements of the A matrix so that: E{fr*} = a1 1 + Z a .1.22i1.+ N 5_50 But, as shown in Appendix J, a1’2n+1==0. Also, defining E{rr*} in the absence of noise as Ec’ a measure of the correlated energy, the variance of r may be written as: --* _ - E{rr } - Ec + NO 5.61 where EC is given by: 2 - k a E — K a1 1 + Z 1,2n C : n=l ”1,1 k 2 = K a + Z a sine (21) 5.62 o n 1 n NT Figure 5.5 shows the dependence of EC on B, where B is the ratio of signal duration to transit time and provides the first indication as to the effect 8 has on correlation detector performance. As shown, the signal energy at the output Of the correlation receiver decreases as 8 increases. 99 OK 1.0 - ‘- «In-4h- W U) 1.0 2.0 3.0 Figure 5.5 Correlated Energy Function Before the implications of this relationship are discussed in the next section, the probability of error for the corre- lation detector is derived. The magnitude of [El has a Rayleigh density of the form _ Ii-IZ p([r||target is present)==‘3‘lrl e Ec+No 5,63 EC+NO The derivation of the density of p(lr|2) follows the same procedure as that used to derive equation 5.42. There- fore, the details of the derivation have been omitted. The result is: 100 2 p(2[target is present) = _ l e Ec+No 5.64 E2+N c O .2. 1 NO p(2]target is absent) = N—-e 5.65 o The likelihood ratio and the associated test for minimum probability of error is given by the following equation. To ensure results consistent with those for the Optimum detec- tor, equal apriori probabilities have been assumed. _ 2 - 1 e EO+N EO+NO > LRT = 2 < 1 5%- . N. O E 2 -c N0 N0 (Ec+No) _ e 2 1 5.66 EO+N By taking the logarithm Of both sides, an equivalent test is: No(Ec+NO) (E¢+No) 2n No 5.67 .< II E c 101 The probabilities of correlation detection and false alarm may now be computed. "d | CD — fmp(2ltarget present)d2 Y (- - l )dQ’ E-+N C O -C+NO N ) O meCQItarget absent)d2 Y m f Y 1 +N c o No exp -—— C PM exp 2n m: "U ll 061 2 f — exp (- —)d2 Y NO NO 173C+No EC+NO exp - _ 2n N 5.69 EC 0 And the total probability of correlator error is again com- puted as the average total error. P =%[1-P P 5.70 CE CD + CF] Constraints on the Correlation Detector for Near Optimality Returning to the expressions for the performance of the Optimum.detector, it is relatively easy to determine under what conditions the correlation detector behaves near optimum. 102 Since the sum of the Eigenvalues must equal the signal energy in the received signal (Property 2, Appendix D), if any one Eigenvalue approaches that value, then the remaining Eigen- values are necessarily small. Thus, as a single Eigenvalue tends to dominate, the threshold given in equation 5.33 approaches the threshold associated with that Eigenvalue alone. Recalling that K is the average signal energy in the received signal, 7E?“ "' ”Icing—”L 1) 5.71 0 Also as one Eigenvalue tends to dominate, a single Bi tends to dominate on each hypothesis (equations 5.50 and 5.51) and as a result one of the Pi coefficients (equation 5.48) tends to a value of one while all others tend to zero. Therefore, as all Eigenvalues except one tend to zero, the probabilities of detection and false alarm for the Optimum detector reduce to: NO K+N0 D EXP - 'R— 2n N ) 5 . 72 o K+N K+N ex --——2-2n -——2- 5 73 F P K NO ' These are seen to be nearly identical to those equa— "U ll "U ll tions 5.68 and 5.69 which describe the performance of a correlation detector. (For 8 equal to zero, they are iden- tical.) Maximum Normalized 103 Additionally, and possibly most importantly, as 8 becomes small, 21 tends to dominate, and the Optimum receiver structure tends to that of the correlation detector. Figure 5.6, which is a plot of the largest normalized Eigenvalue of A, reveals that for B 2 l the correlation detector is nearly identically the Optimum structure. 1.0‘ I .91-p ' I w I :3 .8-- : pa 3 l I 8 7-"- I w 1 £3 . I .6-1— 1 l O 1 1 1 l 1. C) 2.0 3.0 Figure 5.6 Maximum Normalized Eigenvalue of A Thus, for the correlation receiver performance to approximate that of the Optimum receiver, it is sufficient but not necessary that the signal duration be less than or equal to the correlation time of the scattering process. 104 For true correlation this is the transit time. A nice inter- pretation of this result will be given in the next section. The determination of the necessary condition is somewhat subjective and is partially based on the effect that transit time has on the resolving capability of the instrument. As previously discussed, a correlation detector is capable of resolving non-fluctuating targets separated in DOppler by approximately the inverse of the signal duration. As a result, the DOppler resolution could be improved without bound by increasing the effective signal duration. This is not the case when fluctuating targets are involved, as in the case of blood flow measurement, and has been a major misconception on the part of several researchers working the ultrasonic blood flow measurement problem. In the absence of noise, the signal received from the target portion of the vessel may be written to emphasize the modulation effect of finite transit time. Em -- 5mm) 5.74 The output of the integrator section of a correlation detector mismatched in Doppler by 2wf radians is simply the Fourier transform of the product of g(t) and the squared envelope of the transmitted signal. 105 He II f;(t)|E(t)|2ejz"ftdt F£§(t)-1§(t)lz} 5.75 Which in turn is the convolution of the Fourier trans- forms of 5(2) and lf(t)lz. E = F{5(t)}*F{[E(t)|2} 5.76 The squared magnitude of the Fourier transform of |f(t)l2 is just the correlation receiver response to a Doppler mismatched non-fluctuating point target and has an effective width of the inverse of the signal duration. Thus the effect of the time modulation 5(2) is to spread the cor- relation detector reSponse and thereby spoil the resolution of the instrument. Some feeling for the average spreading can be Obtained by recalling that the power Spectrum of g(t) is the Fourier transform of its autocorrelation function, which in this case is A(B%). 85(f) |F{A(8%)}{2 _ T. . 2 Tf 2 - E Sinc (7;J] 106 As indicated by this equation, the power spectrum of g(t) has its first zero at f equal the inverse of the red blood cell transit time. Thus for transit times less than the signal duration the resolution is limited primarily by the transit time and not by the signal duration. Similarly, when the transit time is greater than the signal duration, the DOppler resolution is determined by the signal duration. Thus the fundamental effect of finite transit time is to modulate the reflected signal which, in the frequency domain, has the effect of spreading the Doppler frequency spectrum. By limiting the signal duration to the transit time of the red blood cells, the Doppler spectrum has been confined to the central spike of the ambiguity function for a correlation receiver. In fact, any effect which spreads the DOppler spectrum more than the resolution of the instru- ment will cause the correlation detector to be non-Optimum. In addition to transit time, another effect which could cause this undesirable spreading is the non-uniformity of blood velocity across the target. One of the system constraints imposed in the next chapter will Specifically address this issue. There are at least two good justifications for trans- mitting shortest possible signal without degrading the instru- ment's resolution. The first can be extrapolated from Figure 5.5 and equations 5.68, 5.69, and 5.70. That is, for a 107 constant energy transmitted signal, the performance of the correlation detector decreases as the ratio of signal dura- tion to transit time increases. The second reason is that the shorter the time duration of the transmitted Signal, the better the instrument will follow the time varying DOppler associated with arterial flow. Additionally, since the DOppler resolution of the correlation detector is limited to the inverse of the correlation time of the scattering process, there are no apparent justifications for transmit- ting a fixed energy signal whose duration exceeds the transit time. Based on these reasons and the sufficiency already established, a reasonable requirement for Optimality of the correlation receiver in the presence of transit time effects is that the signal duration be less than or equal to the effective correlation time of the target. Resolving Capability of Both True and Conventional Correlation Receivers In this final section of the chapter, the effect of finite transit time on the resolution of true and conven- tional correlation receivers is investigated. This treatment provides a nice physical interpretation of many of the prin- ciples discussed thus far and is relatively intuitive in nature . 108 The discussion begins by postulating a conventional correlation receiver perfectly matched to the velocity of particles in the target. Figure 5.7 shows a schematic repre- sentation of the ultrasonic beam intersecting a target in the blood vessel. It will be assumed that the velocity is con- Stant across the target so that the target appears as a ran- dom distribution of particles moving with velocity v. Note: This assumption is particularly valid at the center of the vessel where the gradient of the velocity profile is the smallest. As before, each particle is considered a nonfluc- tuating point scatterer whose scattering prOperties are statistically independent of all other particles. Blood Ultrasonic fj/le Vesiel. Beam ; Beamwidth= B “ M: / )fl\ Sim—57.— 5235““) AR Target -—-I I4— AX = VT Vessel Diameter = D Figure 5.7 Conventional Correlation Receiver Operation 109 As shown in the figure, targets within region one contribute to the correlator output at time t1. All other particles do not contribute because they are either outside the beam or their range does not correspond to delays which lie under the central Spike of the ambiguity function. Each particle in this region Doppler shifts the carrier of the transmitted signal and returns a portion of its energy to the receiving transducer. As long as a particle can remain in this region returning energy on a carrier which is shifted by the apprOpriate DOppler frequency, it will con- tinue to increase the energy at the output of the correlator. Clearly this cannot continue indefinitely since, in order to appropriately DOppler shift the carrier, the particle must be traveling with a constant radial velocity toward or away from the receive transducer. Thus it will necessarily move out of the Spatial region of receiver sensitivity. The maxi- mum.time a particle could spend in the region would be less than or equal to d (the range extent of the region of receiver sensitivity) divided by the radial velocity of the particle. Assuming the delay resolution to be the inverse of the signal bandwidth and converting to spatial resolution, an expression for this time, T , can be derived. max d ._ AT'C _ c Tmax 1.6: — 2vr — vr-BW 5'78 110 or Tmax.BW : 2%?- r This is just the DOppler criteria derived in equation G.4. Thus anotherway to interpret the DOppler criteria is as follows: point targets can contribute to the output of a perfectly matched conventional correlation receiver for the entire duration of the signal if, and only if, the DOppler criteria is satisfied. At time t2 = tli-T the particles initially in region one would have traveled to region two. However, only those particles in the shaded region remain within the region of receiver sensitivity. Since the particles are independent scatterers, the value of the correlation function of the scattering process is just the ratio of the volume of the Shaded region to the volume of region one. This is easily Shown by the following simple derivation. The average number of particles in region one is pvl while the average number of particles in the shaded region is pvs. Also, the average number of "new" particles in region one at time t2 is p(V1-VS). Then the covariance function of the received sig- nal may be computed as follows: pvl ov K(tl,t2) = E :E b.f(tl) - .2 v s ‘0 A? .15) 1‘ he A\V 111 Now Since ~ 0 iaéj E{B.b$} = 1J 2 20b 1 = 3 Then pvs~ 2 ~ _ ”* K(t1,t2) - Etiil f(t1)20bf (t2) = E v f(t )202f*(t ) to s 1 b 2 - 2 2 E f(t )Zi f*(t ) ‘ 0pr1 t‘ 1 v1 2 = 202 v E f(t )R (T)f*(t ) bp 1 t 1 c 2 If a square beam shape is again assumed for mathematical simplicity, the correlation function of the scattering pro- cess reduces to a ratio of the areas shown. This is computed to be: ” _ (B-vrsine)(d-vrcose) Kc(l) ‘ 3.3 5.79 This expression can be simplified by expanding and defining two parameters: 112 1-avr -1[bv-Tabv2] IT |_<_min[%7— , 51?] Kc“) = 5.80 0 Otherwise where sine a —B b = cose The usage of the parameter a is consistent with previous usage and b is a newly introduced parameter. Next, the same line of reasoning will lead to a similar expression for the scattering covariance function when a "true" correlation receiver is employed. Remember that true correlation in this develOpment is taken to mean that the received waveform is correlated with a DOppler shifted, E193 scaled, delayed version of the transmitted signal. In effect, the true correlation receiver tries to follow the target in Space by introducing a delay rate in the reference Signal. Thus if a true correlation receiver is perfectly matched to a point target, not only is the reference carrier perfectly matched to the target DOppler, but the spatial region of receiver sensitivity travels through space at exactly the same rate as the radial velocity of the target. Clearly, this has the potential of allowing a longer correlated 113 observation time and therefore, ultimately better velocity resolution. Referring to Figure 5.8, red blood cells within region one contribute to the correlation output at time t1. At time t2 = tl-FT, only those red blood cells in the shaded region remain in the region of sensitivity. Beamwidth= B Blood Vessel T° Delay/ Rate 2 Taril////r1////IAR. Ad Sinai-B cose Ultrasonic Beam Figure 5.8 True Correlation Receiver Operation The primary difference between the operation Shown in this figure and that shown for the conventional receiver is that the region of sensitivity has maintained the best possible delay match with the original particles. 114 The covariance function of the scattering process may now be computed in the same manner as it was for the conven- tional receiver. By taking the ratio of shaded area to the area of region one; R(t1,t2) = ZonglEtf(tl)RT(T)f(t2) d(B-Vrsine) KT“) = B-d Vsine B = 1 - B lTl c _ 2 2 /D Tan (am 6.17 4 2 ax)--64°B fC Equation 6.17 is a constraint on bandwidth as a function of 6m with beamwidth as a parameter. By using the con- ax straint of equation 6.6, a similar constraint on bandwidth as a function of emax with N as a parameter is obtained. This is accomplished simply by solving equation 6.6 for B and substituting into equation 6.17. The result of this substitution is: 127 2N-c-f BW > C 6.18 — 2 4 2 2 /$ fc-4N C Tan (emax) Plots of the functions 6.17 and 6.18 are given in Figure 6.3, 6.4, 6.5, and 6.6. These curves are for carrier frequen- cies of 3 and 5 MHz and vessel diameters of .5 and 1.0 cm. These plots contain a great deal of information if inter- preted properly. Basically, they indicate what choice of parameters results in the Optimum use of the instrument's frequency resolving capability. For example, assume that a transducer with a beamwidth of .5 mm is to be employed in a system intended to visualize flow in a vessel 1 cm in diameter using a 5 MHz carrier. Also suppose that three gray levels are sufficient. Then, using the plot of Figure 6.3, the acceptable solution points are seen to be those in the inter- section of the regions above the B==.05 curve and above the N==3 curve. If minimum bandwidth were the requirement, then the point at the intersection of the two curves would be selected. A one megahertz signal would be utilized and the maximum acceptable transducer angle would be 50 degrees. If minimum bandwidth were not the requirement, then any other point in the intersection region may be used. Bandwi dth (MHz) 128 m V‘ O C I o 0 Ln KO 1‘ II II o. '. o. m m co I II II II 4_ N ,' Im an m o- ,_, ' III I 1 “ I o g a In. , 'm I II. I ll ' .' .II I m I Z" I [Z I 3- I ' v] 7 I I I III I I z N. I II I ’I I ' zl I I, I I : 2'- ’ I I I I ’ . o O I I ' a' I . I 1- II- " - ..... ’ ’ l I I I r I I ‘ 9 10° 20° 30° 40° 50° 60° 70° 80° 90° Figure 6.3 Design Curves for fc=5 MHz, D= 1 cm Bandwidth (MHz) .02 V C II m C II (D l 10° In \D O O o o I\ II II c: III II ' m ox O I00 - 0. II n In .-I II «I In I II ' NI 2' III I ' Z I i I ’ I I I I H 70° 60° 90° 1 l T l 20° 30° 40° 50° 80° Figure 6.4 Design Curves for fc= 3 MHz, D= 1 cm 129 m In '3' O. 8 II " I; " II. '7 3 S I m v m ‘1 ' I . <2 8 4 II ' II , m, . q m I n H 7 I III m I ll ' I Z, I m I I I ' N ’3 ’ I I II‘ I ’ I :3: 3- ' I 2' v I I I I I I. 1 4.5 I I I! E ’ . ’I '5 2.1 I I I: --’ ’ ’ m o’ I In ’.o l-I ‘ 6 I 10° 20" 36° 45° 505 60° 7o'° 80° 9° Figure 6.5 Design Curves for fC= 5 MHz, D= .5 cm B= .07 Bandwidth (MHz) fi l I l T I fi l 10° 20° 30° 40° 50° 60° 70° 80° 90° Figure 6.6 Design Curves for fc= 3 MHz, D= .5 cm 130 By carefully studying these plots, raising questions, rereading the derivation and studying them some more, a tre- mendous intuition about the complex interrelationships involved can be developed. These plots are quite important and quite unique to true correlation processing. Receiver Performance In this section the performance of the correlation type estimator shown in Figure 6.1 is evaluated. What follows is important in that it supports the possibility of blood flow imaging using a periodic pulse train as the transmitted sig- nal and it is in keeping with the most important goal of this dissertation; that is to provide insight into the fundamental processes involved and how they affect the performance of a correlation type blood flow measurement system. Before developing analytical expressions for receiver performance, a short discussion on the sources of error as. well as the differences between the function of the flow estimator of Figure 6.1 and the detector described in Chapter Five will be presented. The correlation detector shown in Figure 5.4 performs the specific task of deciding whether or not a target with the appropriate velocity existed at the specified range dur- ing the time of observation. The decision is made by 131 comparing the correlated energy with a threshold which is determined by the statistics of the scattering process and any additive noise. The estimator described in this chapter consists of a bank of correlators, each being similar to the correlation detector. However, the final decision criteria is quite different. For the estimator, the decision is to choose the correlator with highest correlated energy output. In general, there are three sources of correlated energy output for each correlator. l. Desired Signal Energy 2. Clutter Returns 3. Noise Since the periodic pulse train described in Chapter Three has been selected as the transmitted signal, clutter returns are negligible. However, at the output of each of the correla- tors there will be an additive noise component due to receiver noise, transducer noise, etc. If this noise is assumed to be white and Gaussian, and if the reference signals to the indi- vidual correlators are orthogonal, then the noise contribu- tions in each correlator are uncorrelated and therefore statistically independent. Actually, the signals to the individual correlators are not completely orthogonal, as is easily shown. 132 . T . .. .. / ~ -,.. qua-.5): - O fi(t)fj(t)e dt = $(O»wi-wj> 6.19 Thus the inner product of fi(t) and fj(t) (where fi(t) th and jth corre- and fj(t) are the reference signals to the i lator, respectively) is just the value of the time frequency autocorrelation function evaluated along the DOppler axis at wi-wj. As previously discussed, this is small but, in gen- eral, not zero. Thus the noise is only approximately uncor- related. Assuming one (and, of course, only one) of the correlators is perfectly matched to the target DOppler, and assuming that each of the correlators is equally likely to be matched, the probability of error is given by: PE = P(£m<£l or £m<:£2 or .... or £m<:2N) 6,20 Alternatively, one could compute the probability of error by subtracting the probability of being correct from one. PE = 1 - P 6.21 The probability of being correct is equal to the proba- bility that 2i is less than 1m for all i not equal to m. That is: 3,,,f: 2m an;‘uap 133 .2N)d£l .dfi dim 6.22 N Because of the near orthogonality of the reference sig- nals to the individual correlators, are nearly independent. With this assumption, the correlator outputs the proba- bility of being correct may be simplified to read: '2’ “N‘J. w m Pc = f 001m) f p(2,i)d$2,i dam 6.23 The density functions governing gm and 1i have already been determined in equations 5.64 and 5.65, respectively. Substituting these expressions into equation 6.26, . - “m ' 2,, -‘LiflN'l E'+N N P = f _ 1 e ° ° f Le ° 69. 6.24 c E +N No m 2m-0 c o li-O A After carrying out the integration in brackets this expres- sion reduces to: 6.25 Applying the binomial expansion theorem, this expression may be put in an integrable form. 134 2m ~i£ co - _ N-l N m P = f _ 1 e EC+N° E (N71>(-l)le 0 d2 6.26 C E +N . l m 2m=0 c o 1=0 After grouping terms and carrying out the necessary integra- tion, a rather simple expression for the probability of cor- rect decision results. N-l 1 PC = Z (N;1)(-1)1 f 6.27 i=0 1+ i(l+-I- T, and continuous on the interval 0 < t < T. A typical sample function is shown in Figure Al. Figure Al. Typical Sample Function From a Random.Process Valuable intuition may be gained by visualizing any function in this class as a vector in an infinite dimen- sional space with each point on the t axis (0 < t < T), representing a different coordinate direction in an infi- nite dimensional orthogonal coordinate system. The value of the function at each instant of time represents the 168 169 magnitude of the unit vector in that coordinate direction. With this conceptualization, the operations involving addi- tion and dot product on continuous time functions may be interpreted in the same manner as their finite dimensional counterparts. The following analogies demonstrate this for the dot product operation. Analogies m m m m T m t ’ A-B = zAiB§ f(t)-g(t) = ID f(t)g*(t)dt A.l IXIZ = zlAiIZ |%(t)|2 = I: f(t)rf*(t)dt A.2 The N over a function indicates that it is a complex func- tion and the asterisk indicates the complex conjugate operation. As with finite dimensional vectors, an infinite dimen- sional vector, f(t), may be represented by any complete orthonormal set of unit vectors (CON set). These unit vectors will be designated $1(t), 32(t), ...... $m(t). Notice that it takes an infinite set of unit vectors to span an infinite dimensional space. The definitions of orthonormal and complete are summarized as follows. 170 T o orthonormal => f m M" ¢i(t)¢j(t)dt 2 complete => §E$§[f(t)- "b ’L : fi¢i(t)]dt= 0 1 l for all f(t) with finite energy A where the coefficients, fi, are computed as follows: I: f(t)$§(t)dt = Iz¥i$i(t)$§(t)dt = in I: $i(t)$§(t)dt \—————~v»——-————/ l for i=j o for i#j f1 = I: f(t)$:(t)dt The well known Fourier Series is an example of one particular orthogonal function expansion, with the expansion functions being complex exponentials. A.3 A.4 A.5 A.6 Complex Signal Notation for Narrowband Processes APPENDIX B Let the transmitted carrier Ina cos(mct), then a trans- mitted signal with the most general kind of modulation can be written as st(t) = Let X(t) = Y(t) = Then st(t) may st(t) /7’M(t) cos(wct + ¢(t)) /7'M(t)[cos(mct) cos(¢(t)) - sin(wct) sin(¢(t))] M(t) COS(¢(t)) MCt) sin(¢(t)) be written as {I [X(t) cos(wct) - Y(t) sin(wct)] Ref /7 [X(t) + jY(t)][cos(wct) + j sin(mct)]} 171 3.1 3.2 3.3 172 Defining the complex envelope as: m X(t)+jY(t) = f(t) = complex envelope the transmitted signal is: jm t st(t) = Re{fff(t) e C} "4 Also, from equations B.4 and B.2, f(t) may be written as ?:‘(t) = M(t) ej¢(t) Thus for signals which are "narrowband" about some arbitrary carrier frequency we, the quadrature components X(t) and Y(t), the amplitude modulation M(t), and the phase modulation ¢(t), may be found by performing the operations indicated in equations B.7 through B.lO. The Operation designated as [-]LP implies an ideal low pass filter opera- tion and assumes the signal is sufficiently narrowband so that negligible aliasing occurs. X(t) [ /7 g(t) cos(m t)] C LP Quadrature Y(t) Components [ f7 s(t) sin(wct)]LP B.4 B.5 B.6 B.7 B.8 173 M(t) = [X2(t) + Y2(t)]% 3.9 ¢(t) = tan.1 §%%%- B.lO Let the Fourier transform of st(t) be St(f) = F{st(t)} B.ll where F{-} is the Fourier transform operation. Since st(t) is real: S(f) S*(—f) 3.12 or l8(f>l2 I'S<-f)l2 . 13.13 A typical signal spectrum is shown in Figure Bl. Figure Bl. Typical Signal Spectrum Tb Mm .4 ‘.~A ab 174 Note that in general ’b 1 W W Re{A} = 7-{A + A*} so that from equation B.5 jZWf t -j21rf t st(t) = % [f(t)e C +f*(t)e C] 3.14 and therefore the Fourier transform of st(t) may be written in terms of the Fourier transform of the complex envelope as: St(f) = F{st(t)} = .1. F(f-f ) + .£_.F*(-f-f ) 3.15 ./‘2' C a C As an example, the Spectrum of the complex envelope of the typical signal shown in figure Bl is shown in Figure BZ. ___________ 2A-------;;;;7----------- Figure B2. Spectrum of Complex Envelope A few'important examples of the complex envelope of typical signals are given below. 175 A. Amplitude Modulation: l) st(t) = /2 M(t) cos(ZNfCt) ’1; f(t) = M(t) 2) St(t) = /2 M(t) sin(2wfct) q, . TT f(t) = Mme'J 2 3) On/Off Sequence m(t) 1 0 pt Figure B3. Sample On/Off Sequence st(t) = /2 M(t) cos(2wfct) ’\.v f(t) = M(t) B. Phase Modulation: l) St(t) f2 cos[(2wfct) + ¢(t)] E0» = ejd) (t) .16 .17 .18 .19 .20 .21 .22 .23 176 2) Binary-Phase Sequence +1 fut) -1 Figure B4. Sample Binary-Phase Sequence St(t) /7 A(t) cos(znfct) 3.24 f(t) = ej"[l‘A(t)3/2 3.25 Notes: The choice of fc is somewhat arbitrary. ’b f(t) is real if we choose fc such that [S(f-fc)]LP is symmetric as illustrated in Figure B5. S(f) I I . f - c fc Figure B5. Symmetric [S(f-fc)]LP -j23f t 3. f(t) = {/2 st(t)e C ]LP B.26 177 Complex Representation of Transfer Functions By definition, the system transfer function, H(w), is the Fourier transform of the system impulse response, h(t). In this section it is assumed that the system transfer func- tion is narrowband about we, as illustrated in Figure B6. H(w) -(.0 0) C C Figure B6. Example of a Narrowband Transfer Function ’b The complex impulse response h(t), can be defined by the following relationship: m jw t h(t) = Re [2h(t)e C 1 3.27 where h(t) = hc(t) - jhs(t) hc(t) = [h(t) cos(mct)]LP hs(t) [h(t) sin(wctHLP 178 Notice that equation B.5 and equation B.27 are different by a factor of 2. This normalization is chosen so that equation B.28 will be consistent with conventional system theory. That is, the complex envelope of the response of ’b ’\1 h(t) to f(t) is the familiar convolution integral. Wt) ; 7fi%(:>dr A B.28 .00 and the actual response is written as jm t y(t) = Re{/2 3’7(t)e C} 3.29 APPENDIX C Bandpass Random Processes If n(t) is a random time function (Stochastic process) essentially band limited, it may be written in terms of its complex envelope (see Appendix B) as: .wt '[/2 n(t)e-J c 3(t) j Re{ /2'n(t)e C } n(t) As with deterministic signals, g(t) = nc(t) - jns(t) nc(t) = [/7'n(t) cos(wct)]LP nS(t) = {/2 n(t) sin(mct)]LP For stationary processes 53(f) = 2[Sn(f4-fc)]LP Figure Cl illustrates the relationship between the actual spectrum for a random process and its complex counterpart. 179 180 Sn(f) (I)? :38 -"f Figure C1. Example of a Complex EnveloPe Spectrum where § = Imfi’amz ’b n Two very important prOperties of 3(3) are: 1) 3{H(t1)3(t2)} = o for all t1 and t2 if fc > 3w [EN = bandwidth of s(t)] 2) F‘1{[2sn(f+fc)1LP} = §3(t1_t2)93{3(t)8*(t-(tl-t2))} where Rg(t1,t2) is by definition the correlation function for the complex process. 181 Also, the covariance function for the complex process will ’1: be defined as Kg(t1,t2). If the process is stationary ’1; ’1: R3(t1,t2) = Rg(r) r = t1 - t2 C.7 i? i"< 8 3(t1,t2) — 3(1) T — t1 - t2 C. If the process is also zero mean, the covariance function equals the correlation function. W ’b Kg(r) = RH for zero mean stationary C.9 processes Only zero mean stationary processes will be considered. This implies that all processing will occur during a time interval for which the process is stationary. (This will be an impor- tant constraint when working with time dependent blood flow and suggests extentions of the work reported herein.) Using the identity, Re{A} = (A + A*)/2, the relationship between §g(r) and Kn(r) can be derived as follows. 182 Kn(t,t-T) = E{n(t)n(t-T)} jZfif t -j23f t _ r /2'n(t)e c 4—/2'3*(t)e C — E1[ ] 2 jZWf (t-T) ‘_ -j23f (t-T) x {/2'3(t-T)e c. +/2 n*(t-T)e c 1} 2 q, jZTTf T = Re [K’b(r)e C 1 n j23f (t-T) + Re[E{3(t)3(t-T)e C 1] Using property one m m j2fif0(t-T) E{n(t)n(t-T)e ‘ } = O for fc > BW 0.10 Thus, q, jZTTfCT Kn(T) = Re[Kfi(r)e ] C.11 Now, since Sn(f) = F‘1{Kn(r)} 183 q, jZTTf T q, -j21Tf T -l Kg(r)e C 4-K3*(T)e c = F { - 2 Sm(f-f ) + Sm(-f -f ) s (f) = n C n C 0.12 n 2 The relationship eXpressed by equation C.12 is illus- trated in Figure C1. Included here are some additional comments on complex processes. 1. Eg(r) is real if Sg(f) is even. N jZchT 2. Kn(r) = Re[K3(T)e ] 3. KC(T) = E[nc(r)nc(t-T)] KC(T) = KS(T) = % Re[§g(r)] KC(T) and KS(T) are even. 4. K (r) -KCS(T) = -KSC(-T) = % Im[Eg(r)] SC sc(o) = O KSC(T) is odd. 184 5. All complex covariance functions are Hermitian;ine., ’b ’b K30) -- Kg*(-r) COmplex White Process White noise, w(t), has a two sided spectral height = No/Z, i.e., F{E[w(t)w(t+r)]} = 111529 6(1)} = 1329 0.13 Band limited white noise, wB(t), has a spectrum similar to that shown in Figure C2. E (f) ’b w3 $2311 __ No _. 23w __ 7f' -f f‘ "’ C C Figure C2. Example of Band Limited White Noise ’b . The Spectrum of wB(t) lS [ZSWB(f+fC)]LP E (f) "U W3 No f .- - EN EN Figure C3. Spectrum of wB(t) 185 and the complex covariance function of wB(t) is: m -1 K%B(T)' F {[ZSWB(f+fc)]LP} Sin(2"-BW-r) TI’T No If BW is larger than other bandwidths of the system, the process appears white. That is, '11 K%B(T) ; No 6(1) = 2KW(T) C.14 COmplex Gaussian Process Before describing a complex Gaussian process it is help— ful to be familiar with a real Gaussian process. The Stochas- tic process, x(t), is a Gaussian process if for any g(t), y is a Gaussian random variable, where: y = I: gx(t>dt 0.15 That is, _ (z-m) 2 1 202 = -——-- C.16 py(Y) #2302 e m = E{y} 02 = E{(y-m)2} ‘ 186 Though not a definition, for the purposes of this disserta- tion it will be sufficient to know that the complex envelope of any zero mean Gaussian process is a legitimate complex Gaussian process. This implies that the quadrature components of a Gaussian process are themselves Gaussian processes with identical mean and covariance functions. If the quadrature components of a zero mean Gaussian process are sampled at the same time, then those samples are jointly Gaussian random variables. But since by property 4 E[nc(t1)ns(t1)] = Kcs(o) = O the quadrature components are uncorrelated and therefore statistically independent such that: N. 2+11 2 _ [ c s 1 p(N N ) = 1 e 202 c 17 c’ s ' In other words, P1:'[Nc < r1C (t1) < Nc+dNC’NS < nS (t1) < NS+dNS] = p(NC,NS)dNCdNS 187 Converting to polar coordinates INl2 dN dN c s PN,¢(N9¢) where For white noise KC(0) Also, note that KgCO) E[nc2(t1)] E[n82(t1)] Kc(0) KS(0) 313(t1)21 ll 7:: m C.18 1 Km(o) c.19 C.20 202 = 2E{(n(t1))2} 0.21 It is evident from the form of p(N,¢) that o and N are inde- pendent random variables with probability density functions. 188 N2 p(N) = gg e i3? = Rayleigh C.22 E[N] = ".203 0.23 N E[N2] = K3(o) = 202 C.24 _ 1 P(¢) ‘ 2? ¢€[0,2W] C.25 . . . . ’b For future reference it 18 also worth noting that if x(u) is the complex envelope of the zero mean Gaussian process ’\.: . . . x(u), then y is a complex Gau581an random variable whenever related to x(u) through an integral of the form T .§ = f0 §§ T 1 f0 _—"-" r1 ¢1*(t) ”f(t) __Q?” I: __.. 3, ¢2*(t) 1 l a I I n u l ' | I __L_ I I? C... ¢m*(t) Figure Dl. Correlator Implementation An equivalent method using matched filters is shown in Figure D2. o”/” m——————-— g1 311*(T-t) 32*(1-12) __o/ .__.... 32 f(t).___ L__ Riga-c) __2/ o____...% Sample at time T Figure D2. Matched Filter Implementation 192 For notational convenience let Suppose that the vector R has been received and a deci- sion about the state of the transmitter must be made. To minimize the probability of error H1 will be chosen if it is more probable than H0; i.e., if P(HIIR‘) > P(HOIR’) P(H1,K) p(H0,R) > 1? (1‘1") p (R’) P(EIH1)P(H1) > P(R- HO)P(H0) _ H p(R H1) :1 P(HO) p(R H0) H0\¥_P(H1) _v xv; Likelihood Threshold.YI ratio A(R) Therefore this hypothesis test may be written as H1 > 1 MR) < Y D. Ho 3 193 Note that this simply is a special case of the Bayes test. For Bayes test the threshold is slightly more complicated to reflect the costs associated with making a decision. P(Ho)[C10"Coo] P(H1)[501"C111 where C00 = the COSt deciding the cost C? H H u deciding C01 = the COSt deciding is true C10 = the COSt deciding is true associated H0 is true associated H1 is true associated H0 is true associated H1 is true with with with when with when correctly correctly incorrectly in fact H1 incorrectly in fact Ho Since A(R)is a function of random variables, A is itself a random variable with probability density p(A) = P(AIH1)P(HI) +pdu1*1 i ij 0 t i o t j T2 m T” m E£f08t(t)¢§(t)dtf08§(u)¢j(u)du} _ Tm* T w m m xiaij - fo¢i(t) IOE{St(t)S§(u)}¢j(u)dudt L J mm . Ks(t,u) for zero mean TN _ Tm* m - fo¢i(t) IOK§(t,u)¢j(u)dudt For this equality to hold for a specific j and any i, it is sufficient that TV ”b _ m foK§(t,u)¢j(u)du — Aj¢j(t) D.8 This is a sufficient requirement because the set of func- tions,{¢i(t)}, which solve the above integral equation form an orthogonal set. Again note the similarity between the above integral "Eigenfunction" equation and the matrix Eigenvector equation: 197 X3 _, 7: L—J 9| u E K.. ¢. = loi for all i The set of functions {$(t)} are called Eigenfunctions of ’b K§(t,u) and the set {ii} are called the associated Eigen- values. Other important prOperties of the Eigenfunctions and ’1; Eigenvalues of K§(t,u) are: (1.! m m m l. K§(t,u) = X l.$.(t)¢*(u) [Mercers Theorem] i=1 1 1 1 2. Xi is the expected energy in the $i(t) component of ’1: '1: St(t) and the total expected energy in St(t) is Eifg§t(t) §§(t)dt} = f§X§(t,t)dt II II M 8 >19 "\2 3. Because K§(t,u) is Hermitian, the Eigenvalues, Ai’ are real (this is not surprising in light of 2). 6.. as already stated. 4. f$i(t)$§(t>dt 11 198 . ’\J 5. There eXists at least one ¢(t) and real I which satisfies 11(t) = I: 3§$du There may be only one. For example ’\1 ’1; ’b K§(t,u) = f(t)f*(u) 6. For a real bandpass process, the real Eigenvectors and Eigenfunctions are related to the complex Eigenvectors and Eigenfunctions as follows: X1 m jmct A1 = if- 41(t) = Re[/§ ¢1(t)e 1 i -j = Rat/f $2(t)e C 1 7 .3 $§$i = a 1—1 "I; The resulting expansion of St(t) in terms of the Eigenfunc- N. tions of K§(t,u) is known as the Karhunen-Loeve expansion. m °° mm S (t) = )3 S.¢.(t) D.9 Or equivalently, the real process may be expanded as in equation D.lO. oo S(t) = z: t i=1 Si¢i(t) D.lO Where the set,{¢i(t)}, is given in property 6 above. The coefficients are given below. 'h ’1: SI R8(Sl) , 32 3 1111(51) ’b ’h 33 Re(82) , sh = Im(Sz) etc. When H1 is true P(t) = “s’tm +9603 If f(t) : H1 is expanded in terms of the Eigenfunctions "a of K§(t,u), the resulting coefficients are zero mean indepen- dent complex Gaussian random variables with a joint proba- bility density function given as 200 ” 2 m _lril _ m lr.l 20 p(RIHl) = II 12 e D.12 =1 230 where 202 = Xi + N0 Similarly, on H0 %(t) = 360:) and '§.2 _ °° lg-l 'No p 'b m T , 1 A2 H1 r(t)—1 lo I Pa 4, 3 Y ' ' (12+N0)No H0 : 13 l | I l I 3' / . T.__—.... a - °° f I F ,, ° (AQ+N0)N0 $§ Figure D4. Optimum Detector Implementation From equation D.15, 2 may be rearranged as follows. .. 11 % IL = Z W Iri 2 1:1 (Xi+N0)N0 on '1” = 2 1 [f:¥(t)$§(t)dt111:?(u)$§(u)du1* i=1 (’i’i +N0)N0 203 'b = fgr*(u) x§3 1 z m i $§3 Figure D5. Filter Correlator Optimum Receiver Thus the optimum receiver as shown in Figure D5 takes the form of a correlation receiver which correlates the complex envelope of the received signal with the minimum mean square 204 ’1: error estimate of St(t). Actually it is possible to solve '\a for ho(t,u) without explicitly finding Eigenfunctions and 'b Eigenvalues. With a lot of work it can be shown that h0(t,u) solves: N h (t u) + 1T3 (t 2)Em(z u)dZ = §m(t u) D 17 O O 3 o o D S l S 9 ’ At this point a few examples might help build up a little intuition. Two examples of particular interest will be used. 1. The transmitted signal is completely random.and there is no additive white noise (No a 0). 2. The transmitted signal is deterministic except for an unknown amplitude and phase. ’b S%(t) = bf(t) Case 1: With N0 + O 1'”: (t u) = ”m -1- 32° xi Wt)" (u) 0 ’ N0+O N0 i=1 Ai+No Ci ¢i =1 lim 1. m m m _ lim 1. _ N0+0 N"; iil ¢i(t)¢i(u) " N0'>O N? 5(t 11) 205 This is just the transfer function for a resistance— less wire. f(t)——— Iconjugate X f 0H T1:7 N0 —..J Figure D6. Optimum.Receiver for No Near Zero Note that for any finite signal energy, as NO + O, 2-+m. This means that in the absence of noise if there is any energy in the received signal, H1 is chosen (i.e., decide the transmitter was turned on). Case 2 §t(t) = 8%(t) 31321 = 20% m . q, '1. m E[St(t) S’E(u)] = K§(t,u) == Zogf(t)f*(u) The Eigenfunctions solve 13(t) = rgzogict>*f*$ _. 'V‘b jmc(t-T) Re[¢2Et bf(t-T)e I Re[/2E:'b e C I F(jm)e jw(t-T)dm 2F] F.6 If the target remains within the transducer beam for time t, and has radial velocity Vr’ its range is given by: R(t) = R0 - Vr-t F.7 212 If a particular part of the received signal is received at time t, it was transmitted at time t-T(t), where r(t) is the two-way propagation delay of that part of the signal received at time t. Therefore, that portion of the signal received at time t was reflected from the target at time [t-T(t)/2]. At that time, the target's range was R(t - léEl) = R0 - Vr-(t - lgEl) F.8 and therefore, since r(t) = 2R(t)/C, where C is the velocity of sound in tissue, 2R(t - T(D) 2[R -V -(t-T(t))] r(t) = C 2 = ° C C 2 3.9 or solving for r(t) 2Vr-t T03) = -—v—— " -—V- F.10 1-+7§- 1+~€§ In the case of blood flow measurement, Ill V 2 10 cm/sec max 0 2 1.5 x 105 cm/sec 3 3 S .67 x 10‘ << 1 III 113 x 10‘ 01<: 213 So that, 2R0 2Vr°t T(t) " T - C F.11 If T is defined as 2R0 T = 73"” Then delay as a function of time may be written as 2Vr°t “f(t) = T - T F.12 and the received signal may be written as l: «.4. 2V jock-1+2; t)] SrCt) = Re VZE; bf(t -T-+7T-t)e F . ' . 2v . N“ 2V JO)c IT'C JC”ct? = Re /2'/E£ bf(t'T4'TT t)e e F.13 \__ —~vr J L '” 4 3,40 Thus the complex envelOpe of the received signal with additive noise may be written as; jw t “f(t) = 7’3: 3¥1(1+e(ed))t-T1e C +9.3(t) 3.14 where e = 731 C 214 Notice that this is very similar to the problem discussed in the previous appendices. Here the complex envelope of the received signal consists of the sum of a sample function from a complex Gaussian process and a sample function from a complex Gaussian white noise process. If the Eigenfunctions of m§m(t,u) are chosen as the expansion functions for f(t), then th: individual coefficients will be uncorrelated and therefore statistically independent (because Gaussian). Thus the 31's and 11's must solve the following integral equation. 00 A 35 (t) = f E” (t u)$ (u)du i i _ s; ' i where _ m ___WW jwdt Km (t,u) = E[/E bf (l+«)t-T]e 8% t ___ mm -jwdu ° /Et bf*[(l+«)u-T e ] 'wdt -jwdu J Et20§f[(1+a)t-T]e f*[(1+¢)u-T]e F.15 As demonstrated in a previous example, there is only one ’b Eigenfunction and one Eigenvalue associated with K§m(t,u). r They are: jwdt $(t) %[(1+m(wd))t-T]e >2 II 2 - _ 215 Equation E.4 states that for a ML estimate, the following log likelihood ratio is to be maximized over all Cd and T. p(Flwd’T) in _ = £nA(wd,T) P(rIHo) Noting the one-to-one correSpondence between £n[A(wd,T)]and £nA(R) implemented in equation D.15, the log likelihood ratio may be written immediately as E 2n[A(wd,T)] = N1— R [L(T.wd)l2 F.17 R -jm t C dt F.18 °° m L(T.wd) = I%:[<1+«)t-r1e An equivalent estimation procedure would be to simply ignore the constant in equation F.l7 and maximize [L(T,wd)l2 over all T and md. APPENDIX C Ambiguity Functions Generalized Ambiguity Function From equations F.l4 and F.18, L(1,w) may be written as .. jw t x 1/3;3¥[(1+«(ea))t-rale a +9é 3w ; —1_—6- = thz 10 Figure Gl. Example of Compression in Binary Phase Signals 220 In this example, = 4 => T << 4 us RIH BW-T << That is, for the Doppler approximation to be valid, T must be much less than 4 us. In general, for binary phase modulated phase signals, if the bit period for the transmitted signal is TB’ then the bit period for the received signal is I+oc TB G.5 The difference is: 1 _ o: 33-st - TB (11:) G-6 ’b ’1: At the end of N bit periods, f(t) and f((l+a)t) would differ by: NoTB (IE?) seconds G.7 for a difference of 1 bit period, N43011:) = TB G.8 221 Solving for N, N = —— C.9 Thus, for f(t) to remain nearlyix1phase with f((l+«)t), the constraint on N is 1+« c: N<< G.10 The following argument shows that this intuitively derived constraint for B phase signals is consistent with the criterion required for the DOppler approximation to be valid. 1+a TB N=T << TB :— G.11 1 1 + CC T << Ew- T 6.12 T C+2V ,_ C BW'T << —2_V_ - W +1 G.13 But it was previously assumed that C 2T].>>], And, as anticipated, BW-T << (§%) 6.14 222 T.us, equation G.lO is consistent with our previous result and provides interesting insight into applying the Doppler approximation to B phase code modulated waveforms. Ambiguity Function If it can be assumed that the Doppler approximation is valid (for most of the signals employed in this research it cannot), then q, q, jfl) t ’ f(t) = b/Eg'f(t-r)e C + w(t) 9.15 and mm j(ma-w)t 6(T,w) = If f(t-Ta)f*(t-T)e dtl2 G.l6 Let t’ = t-Ta t=t’+'ra w’ = wa-w Z = Ta-T I m "b ’ ’b » jm’t » 2 6(Z,m ) = If f(t )f*(t +Z)e dt I 6.17 l¢(Z.W’)I2 223 Using more conventional variables 00 (\I (\I o $(T,w) A f f(t)f*(t-I)eJCtdt C.18 ¢(1,w) is the time frequency auto correlation function and 6(T,w) is the conventional ambiguity function. 6(T,w) = |$(T,w)12 G.19 Even though the Doppler approximation is not strictly valid for many of the signals and physical situations of importance in this research, studying and being 2251 familiar with 8(T,w) will provide valuable insight into fundamental limita- tions of a blood flow imaging system as well as providing_ guidance in choosing an appropriately transmitted signal. Many properties of 6(I,m) have been derived. However, only six of the most useful ones will be listed. 00 f f(t)f*(t)ejmtdt = F{[f(t)|2} -(X) 1. $(o,w) = Fourier transform of the squared magnitude of the envelope 224 oo ’b ’1: 'b 2. ¢(1,o) = f f(t)f*(t+1)dt ’1; = auto correlation of f(t) 3. e(o,o) = ¢(o,o) = l 4. ff 0(T,f)dde = ff 6(T,w)d1 §¥.= 1 co 5. f 6(T,f)dT f 6(T,O)e-j2wadT 6. f 8(T,f)df = f 6(o,f)e12Cdef As an example, these prOperties will be used to investigate the shape of 6(1,w) for a pseudorandom binary phase sequence. a. Along the f = 0 axis (PrOperty 2) $(T,o) = ff(t)f*(t+r)dt 225 b. Along the T = 0 axis (Property 1) $(o,f) = F{f(t)12} = sinc(Tf) 1 1(0.f) height = 1 1’ l 'T T emf) = Wrmlz For a pseudorandom sequence: e(r,o>= meow who) height = l t ~ I ’1' ‘13 T3 3 1 T3"‘3'13 where BW is the one-sided bandwidth in hertz. _ 2_.2 e(o,f)-le(o.f)l "SlnC (1f) 16(0.f) height = l \ + I I n I I I r—JIH hip-41 226 Also, the average width of the central spike of 9(o,f) is approximately 1/2T while the average width of 9(T,O) is LLZBW. 6(o,£f) .81 e(z%—,o) = .66 Therefore, it appears that a system using a binary phase sequence as the transmitted signal can "resolve" targets r. separated in delay by approximately 7%? or in DOppler by approximately —%— . These results are approximately true ‘ for all transmitted signals and a typical ambiguity function takes on the approximate shape and dimensions shown in‘ L: Figure G2. G(T,f) height==1 (by prOperty 3) WT (by prOperty 4) 2T 2W (by prOperty 6) l (by prOperty 5) Figure G2. Approximate GeneraI Ambiguity Surface 227 It is important to emphasize that the ambiguity function shown in Figure G2 represents the response of a conventional correlation receiver to a point target mismatched in delay and Doppler under noiseless conditions when the Doppler approxima- tion is valid. When the DOppler approximation is not valid and a conventional correlation receiver is used, then the ambiguity function may not be a reasonable measure of the F signal's ability to resolve targets. However, if a true correlation receiver is used as previously described, and if' «(D) is small for all expected target velocities (which is the case with blood flow imaging), then the assertion is L made that the conventional ambiguity function still provides a valid measure of the signal's range resolving capability. The reasonableness of this assertion is made by the following argument. Along the m = 0 axis, that is, when Ca = w, from equations G.2 and G.3, the output of the true correlation receiver in the absence of noise may be written as: [L(T,O)|2 = 1; %[(1+a)t-Ta]f*[(1+a)t-T]dt[2 c.2o . ’0 Equation G.20 describes the autocorrelation of f[(1+«)t]. "I 0 Clearly, if u << 1, then the bandwidth of f(t) is approx1- mately the bandwidth of f[(l+¢)t]- Thus, the ability 0f the true correlation receiver to resolve targets separated in 228 initial delay is nearly identical to that given by the con- ventional ambiguity function. Unfortunately, the same assertion cannot be made for the DOppler resolving capability of a true correlation receiver without considering the specific signal to be trans- mitted. The Doppler resolving capability of specific signals or classes of signals using true correlation processing must be dealt with on an individual basis. F" .a« QB APPENDIX H Acoustic Wave Propagation The purpose of this appendix is to familiarize the reader wi th the terms and equations used to describe simple acoustic wave propagation. It is not intended as a rigorous treatment F of physical acoustics and the reader interested in a more complete treatment should consult an apprOpriate text. The discussion will start by considering propagation in IT Perfectly homogeneous, lossless material. Homogeneous implies that all macroscopic physical attributes associated with any tWO differentialvolume elements are identical. Lossless imp lies that the material simply supports acoustic wave I33:.(313agation and does not extract energy from that acoustic Wave (i.e. , the presence of acoustic waves in the material does not raise its temperature). This is certainly an ideal- ized- material which does not exist in reality. However, st"*-"-C137ing wave propagation in such a simple media provides valuable insight into propagation in more complex media such as - tissue. The propagation of acoustic waves in lossless, isotropic In - edla may be described by 229 230 f— K+4G p; F o .— __= VVot-GV-V'C H1 at? 9o where K = adiabatic bulk modules A_V 5%- G = modules of sheer rigidity p = average density of the media nl u ins tantaneous di sp lacement For fluids, the modulus of sheer rigidity, G, equals zero. Since tissue (including blood) is composed largely of water, it will be assumed that the body cannot support sheer waves. Thus, the vector wave equation for displacement becomes: 2— ._ ii = _IS—VVo C H.2 In One dimension, this wave equation reduces to 2 2 at _ 1 32: H.3 3t2 poCS 3x2 “Here the parameter 83’ has been introduced. By definition 3 - s 18 the adiabatic compressability of the medium, which is t: he reciprocal of K, the bulk modulus. The velocity of a plane wave satisfying equation H.3 is well known to be 231 V = -———-—- . H.4 If the coordinate origin is chosen appropriately, the solution to H.3 takes the form c: = cmax cos(kx - out) H.5 _ 2_Tr = a k 7 1 V The bulk modulus, K, is a constant which relates the fractional change in volume to the applied pressure. _ K AVO‘lume _VEIEEE' H.6 For plane wave propagation, as described by equation H.5, equation H.6 reduces to (using AVolume = dy-dz-dg; Volume 2 dy-dz-dx) d p = -K ai- H.7 Using equation H.5, it is seen that there exists a propagating pressure wave in spatial quadrature with the propagating dis- placement wave. P = 'K g%' = pmax Sin(kX-wt) H.8 232 where pmax = K k Cmax = (D 0 0V Cmax H ° 9 The time derivative of H.5 yields C = Cmax Sin(kx-mt) H.10 no II max (1) Cmax Also, it is seen that E and p are related as follows p = 9 V; H.ll pmax = poVCmax .Additionally, the wave intensity I is defined as the time average acoustic wave power per unit area and is related to Cmax and pmax as I: p _ max max i 20 V _ max 0 I - 2 H.13 2 I = pmax 23:V_' H.14 233 Thus, an analogy exists between the electric field E and pressure p, between the magnetic field intensity H and time rate of displacement i, and between the characteristic wave impedance Z0 and poV. In fact, if the identification is made, Z = 90V H.15 the analogy continues. At the interface of two dissimilar tissue types, the reflection coefficient is defined as _ 21 -22 The reflection coefficient in equation H.16 is the ratio of p of the wave reflected from the tissue discontinuity max to pmax.°f the incident wave. The reflected intensity is _ 2 Ireflected _ R Iincident H-17 while the transmitted intensity equals I (I H.18 transmitted incident - Ireflected) 21 2 231T Iincident 234 where 222 T = H.19 erz: and is the ratio of pmax of the transmitted wave to pmax of the incident wave. In most biological tissues, p0 is nearly constant so that the reflection coefficient R reduces to V - V 1 2 __._—__. H.20 00 V1 + V2 The reflection of acoustic energy at a material inter- face may be minimized by sandwiching a third layer between the two media. Matching Material Ultrasonic 1 Source Biological Tissue A Sound Absorbing Backing Material 22 pa ‘ Z iv ‘“fi Ultrasonic Transducer Figure H1. One Quarter Wave Matching Layer 235 One area where quarter wave matching is used extensively is to maximize energy transfer between the ultrasonic trans— ducer and biological tissue as shown in Figure Hl.' If the characteristic impedance of the matching material Z1 is between 20 and 22, then the optimum width of the matching layer is one quarter wavelength. In addition, if Z1 may be selected such that Zl =- #2220 , then there will be 100‘7o trans- mission of acoustical energy. APPENDIX I Blood Flow in Cylindrical Vessels Figure 11 shows a length of cylindrical vessel and a cylbndrical subsection of that vessel. The pressures of interest exerted on this cylindrical subsection are the sheer stress and fluid pressure at either end of the cylinder, as shown. Figure 11. Section of Cylindrical Vessel In the steady state, the sum of the forces acting on the subsection must equal zero. Thus, 2.. 2- = Pour PLnr Trz 23rL 0 1.1 where the sheer stress Trz is the z-directed sheer force per unit area exerted by the fluid at r on the fluid at r + dr. 236 r. .5. 237 Solving equation 1.1 for Trz results in the relationship shown in equation I.2. Po - P1 - r 1.2 rz 2L Substituting for Trz by using the defining relation for the'viscosity1lof a Newtonian fluid given in equation 1.3 fr- results in a differential equation for the velocity of the fluid in the vessel as shown in equation I.4. Trz 3 '1‘ Fr— I ' 3 I dvz - (Po - P1) I 4 3‘: ‘ ‘ 2111. r - ° The solution to equation 1.4 is given as P - 31 O 1 Z - K-Tr 1.5 By applying the boundary condition at the vessel wall, namely that the fluid velocity is zero at r = a, the unknown con- Stant may be determined and equation 1.5 reduces to equation I.6. (Po - P1) Vz(r) = - AUL (r2 - a2) 1.6 238 Also, by identifying the velocity at the center of the vessel as the maximum.velocity, V , equation 1.6 may be further max simplified. r 2 Vz(r) = Vmax1l - (E) 1 1.7 where 2 a (Po - Pl) vmax _ 4uL 1‘8 The total volume flow rate, Q, is easily found by integrating the differential flow rate over the vessel cross section as follows: Q =xsv-as R271 1‘2 = ’0 fovmax(l - g2 ) rdrde I19 Therefore, 2 11a Va Q = 211‘" 1.10 And finally, the average blood velocity is determined by dividing the total volume flow rate by the cross sectional area of the vessel. V _ = max Vavg . _Q2 _77_ I’ll 11a APPENDIX J Computation of the Elements of A A symmetric unit energy periodic pulse train is assumed to be the transmitted signal. T CT? ~ Tp t-MTP f(t)= FT; 21‘ TIT) Ma'Q—I: J. For every Cij’ there will be a computation which takes the form of 1 TP 2 u-kTP a: T711; {2 g(wnU)h(me.) .2 V(T) dt 2 . = 12. “n NT where g(-) and h(-) are either sin(:) or cos(-). Now, 2T since TP << 7? , the effect of the pulse train is to sample the product g(mn)h(wmu). That 15, ~ . 1.44.4443 T. - e;— 239 J. l .2 3 240 2 O O 0 Because TP << 51 this summation may be approximated as an n integral in the following manner. Since I 2 , ; 7 EST g(mnu)h(3mu)du i g(wnkTP)h(3nkTP)TP J.4 '2' then T ~ 1 2 ,. e: = f [T g(wnu)h(mmu)du J: ’2' With this approximation, the computation of the elements of A is straightforward. 1 “11 1 T . _ B1 - Cij - I; EDIT aokdu — aok J. '2' 2' cc1,2n T e = k/T . .1. II cos (m u) du 1,2n Con T_‘T_ 111' 2 1 2k /a a 2 = T C n . §%-[sin (%%10] o 2Nk /a a _ o n . nn c£1,2n _ nw, Cln (2N) J. “1,2n+1 “1,2n+1 cc2n,2n C2n,2n 2n,2n cc2n+1,2n+1 “2n+1,2n+1 cc2n+1,2n+1 C2n,2m C2n,2m = 241 0 T k 2 f2 o 2 (CC u)d an T o c 3 NT' u %§_ 2 NT 1 l . kan TI HF [E-x # Z-31n 2x]o 2Nka n n 1 . n n .ZN'+ Z Clu