Frequency Domain Method for Resolution of Two Overlapping ... · Frequency Domain Method for Resolution of Two Overlapping Ultrasonic Echoes Chi-Hang Kwan Doctor of Philosophy Department

Frequency Domain Method for Resolution of Two Overlapping Ultrasonic Echoes

by

Chi-Hang Kwan

A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy

Department of Mechanical and Industrial Engineering University of Toronto

© Copyright by Chi-Hang Kwan 2017

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Frequency Domain Method for Resolution of Two Overlapping

Ultrasonic Echoes

Chi-Hang Kwan

Doctor of Philosophy

Department of Mechanical and Industrial Engineering

University of Toronto

2017

Abstract

The ability to identify and resolve overlapping echoes is crucial to the enhancement of axial scan

resolution in ultrasonic testing. Overlapping echoes are frequently encountered in the inspection

of shallow and/or short cracks in Time-of-Flight Diffraction and normal incidence reflection

inspection of near surface flaws. Dictionary-based parametric representation (DBPR) has been

proposed as a powerful framework to separate overlapping echoes of different shapes. However,

the large solution space in DBPR renders the optimization process difficult. We propose a new

echo separation method named Trigonometric Echo Identification (TEI) that exploits the

consistent frequency domain amplitude and phase relationships of two overlapping ultrasonic

echoes to reduce the number of optimization parameters.

In TEI, frequency amplitude profiles are entered as inputs and the corresponding set of frequency

phase profiles are reconstructed as outputs. The optimality of the output phase profiles is then

used as a metric to determine the accuracy of the trial amplitude inputs. By reconstructing the

phase information instead of explicitly specifying the phase profiles, we can reduce the number

of unknowns in the problem of identifying two overlapping ultrasonic echoes. Compared to

DBPR, TEI can describe more complex ultrasonic echoes using the same number of optimization

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parameters. In addition, since the phase profiles are reconstructed using the acquired data, TEI

would perform more reliably in the presence of noise.

Simulation tests were conducted to assess the relative performance of TEI and DBPR. Echo

parameters including center frequency, phase shift and relative amplitudes were systematically

varied to yield different test configurations. The standard deviation of timing errors obtained

from TEI were 50% lower compared to DBPR. The difference in algorithm performance is

especially evident in low SNR signals and signals containing echoes of complex shapes. The TEI

algorithm was also verified on experimental ultrasound testing data containing overlapping

echoes. The echo arrival times extracted using TEI agree with the values obtained using

geometric calculations.

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Acknowledgments

Firstly, I would like to express my gratitude to my supervisor, Prof. Anthony Sinclair, for his

guidance and support throughout the course of my thesis project. Thank you for placing your

confidence in me to pursue my own research directions.

Secondly, I am grateful to the Natural Sciences and Engineering Research Council of Canada

(NSERC), Ontario Graduate Scholarship (OGS) and Olympus NDT Canada for sponsoring my

research. I am fortunate to have the opportunity to work on various interesting industrial research

projects at Olympus NDT and use their laboratory facilities to conduct my experimental work.

I would also like to express my gratitude to my colleagues at Ultrasonic Nondestructive

Evaluation Laboratory (UNDEL) and Olympus NDT for their collaboration and the valuable

discussions we’ve had together. Many of the ideas pursued in this research project stemmed

directly or indirectly from our many long conversations.

Finally, I would like to thank my friends and family for their encouragement, patience and love

during this long and at times arduous journey. This thesis would not have been possible without

your support.

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Table of Contents

Acknowledgments.......................................................................................................................... iv

Table of Contents .............................................................................................................................v

List of Tables ............................................................................................................................... viii

List of Figures ................................................................................................................................ ix

List of Symbols ............................................................................................................................ xiv

List of Appendices ....................................................................................................................... xvi

Chapter 1 Introduction ..............................................................................................................1

1.1 Introduction and Motivation ................................................................................................1

1.2 Thesis Objectives .................................................................................................................3

1.3 Thesis Overview ..................................................................................................................4

Chapter 2 Background and Literature Review .........................................................................6

2.1 Ultrasonic Inspection System ..............................................................................................6

2.1.1 Pulser-Receiver ........................................................................................................6

2.1.2 Piezoelectric Transducers ........................................................................................8

2.1.3 Ultrasonic Testing Data Representation ................................................................11

2.1.4 Resolution Limits in Ultrasonic Testing ................................................................14

2.2 Modeling of Ultrasonic Echoes .........................................................................................15

2.2.1 One-Dimensional Piezoelectric Transducer Models .............................................15

2.2.2 Complete Transfer Function Modeling of Ultrasonic Echoes ...............................19

2.3 Single Reference Deconvolution .......................................................................................23

2.3.1 Basic Assumptions .................................................................................................23

2.3.2 Direct Deconvolution Schemes ..............................................................................24

2.3.3 Iterative Deconvolution Schemes ..........................................................................25

2.3.4 Technique Limitations ...........................................................................................26

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2.4 Dictionary-based Parametric Representation .....................................................................27

2.4.1 Mathematical Formulation .....................................................................................27

2.4.2 Sparsity-Promoting Algorithms .............................................................................28

Chapter 3 Basis and Assumptions of TEI Algorithm .............................................................30

3.1 Frequency Domain Assumptions .......................................................................................30

3.1.1 Amplitude Profile Assumption ..............................................................................30

3.1.2 Phase Profile Assumption ......................................................................................31

3.1.3 Justification of Amplitude Profile Assumption .....................................................32

3.1.4 Applicability Limits of Echo Assumptions............................................................36

Chapter 4 Trigonometric Echo Identification Algorithm .......................................................40

4.1 Algorithm Overview ..........................................................................................................40

4.2 Trigonometric Phase Profile Reconstruction .....................................................................41

4.3 Components of TEI Algorithm ..........................................................................................43

4.3.1 Echo Optimality Metrics ........................................................................................43

4.3.2 Determination of the Correct Set of Phase Profiles ...............................................45

4.3.3 Phase Slope Inequality Constraint .........................................................................46

4.4 Implementation as Constrained Optimization Problem .....................................................46

4.4.1 Constrained Optimization Formulation .................................................................46

4.4.2 Implementation Details ..........................................................................................48

4.5 Summary of Novelty and Advantages of the TEI Algorithm ............................................50

Chapter 5 Results and Discussions .........................................................................................51

5.1 Simulation Tests and Comparison Benchmark ..................................................................51

5.2 Synthetic Echoes with Symmetric Envelope .....................................................................53

5.2.1 Echo Parameter Tests .............................................................................................53

5.2.2 Signal to Noise Ratio Tests ....................................................................................61

5.2.3 Results Summary and Discussion ..........................................................................66

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5.3 Synthetic Echoes with Asymmetric Envelope ...................................................................69

5.3.1 Echo Parameter Tests .............................................................................................70

5.3.2 Signal to Noise Ratio Tests ....................................................................................78

5.3.3 Results Summary and Discussion ..........................................................................83

5.4 Experimental Verification ..................................................................................................84

5.4.1 TOFD Test on Notched Sample .............................................................................84

5.4.2 Phased Array Test on Side-Drilled Hole Sample ..................................................90

Chapter 6 Conclusions ............................................................................................................95

6.1 Thesis Summary.................................................................................................................95

6.2 Research Findings ..............................................................................................................96

6.3 Future Work .......................................................................................................................97

References ....................................................................................................................................100

Appendix 1: Two-way Impulse Response of Van-Dyke Model ..................................................105

Appendix 2: KLM Model of Broadband Transducer ..................................................................107

Appendix 3: Source Code of KLM model ...................................................................................109

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List of Tables

Table 3.1: Properties for broadband KLM simulation .................................................................. 34

Table 4.1: Phase reconstruction chart ........................................................................................... 43

Table 5.1: Baseline parameters for symmetric echoes .................................................................. 54

Table 5.2: Performance table (phase difference vs time separation for symmetric echoes) ......... 56

Table 5.3: Performance table (frequency difference vs time separation for symmetric echoes) .. 58

Table 5.4: Performance table (amplitude ratio vs time separation for symmetric echoes) ........... 60

Table 5.5: Baseline parameters for asymmetric echoes ................................................................ 70

Table 5.6: Performance table (phase difference vs time separation for asymmetric echoes) ....... 74

Table 5.7: Performance table (center frequency difference vs time separation for asymmetric

echoes) .......................................................................................................................................... 75

Table 5.8: Performance table (amplitude ratio vs time separation for asymmetric echoes) ......... 77

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List of Figures

Figure 1.1: Acoustic travel paths for two adjacent defects ............................................................. 1

Figure 1.2: Configuration of a TOFD scan ..................................................................................... 2

Figure 2.1: Schematic diagram of ultrasonic inspection system..................................................... 6

Figure 2.2: Two models of Pulser-Receiver ................................................................................... 7

Figure 2.3: Voltage pulse of analog pulser ..................................................................................... 7

Figure 2.4: Voltage pulse of digital pulser...................................................................................... 8

Figure 2.5: Schematic diagram of a single element piezoelectric transducer (courtesy of [11]) .... 9

Figure 2.6: Steering of phased array transducers .......................................................................... 10

Figure 2.7: Focusing of phased array transducers ........................................................................ 10

Figure 2.8: A-scan representation from TOFD data ..................................................................... 11

Figure 2.9: TOFD B-scan containing 4 flaws ............................................................................... 12

Figure 2.10: C-scan of back surface of a coin (from [16]) ........................................................... 13

Figure 2.11: S-scan of three side-drilled holes (from [17]) .......................................................... 13

Figure 2.12: Lateral resolution in ultrasound imaging .................................................................. 14

Figure 2.13: Van Dyke approximate transducer model ................................................................ 16

Figure 2.14: Frequency amplitude response predicted by Van Dyke model ................................ 17

Figure 2.15: Schematic diagram of KLM model .......................................................................... 17

Figure 2.16: Transmission matrix model of transducer (Operated as transmitter) ....................... 19

Figure 2.17: Thevenin's equivalent circuit .................................................................................... 20

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Figure 2.18: Two-way impulse and frequency response for two points in a pressure field ......... 21

Figure 2.19: Single reference convolution .................................................................................... 23

Figure 3.1: Asymmetric Q-Gaussian distribution ......................................................................... 31

Figure 3.2: First harmonic impulse response of KLM model ....................................................... 34

Figure 3.3: KLM model of broadband transducer ........................................................................ 34

Figure 3.4: Pitch-catch backwall echo acquisition configuration ................................................. 35

Figure 3.5: Experimental pitch-catch backwall echo .................................................................... 36

Figure 3.6: Fourier transform of experimental pitch-catch backwall echo ................................... 36

Figure 3.7: Echo distortion due to wavefield diffraction .............................................................. 38

Figure 4.1: Flowchart of TEI algorithm ........................................................................................ 40

Figure 4.2: Vector representation of overlapping echoes ............................................................. 41

Figure 4.3: Alternative vector addition configuration .................................................................. 42

Figure 4.4: 50% taper Tukey window........................................................................................... 49

Figure 5.1: Baseline configuration for symmetric echoes ............................................................ 55

Figure 5.2: Percentage timing error (phase difference vs time separation for symmetric echoes)56

Figure 5.3: Percentage reconstruction error (phase difference vs time separation for symmetric

echoes) .......................................................................................................................................... 56

Figure 5.4: Percentage timing error (frequency difference vs time separation for symmetric

echoes) .......................................................................................................................................... 57

Figure 5.5: Percentage reconstruction error (frequency difference vs time separation for

symmetric echoes) ........................................................................................................................ 58

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Figure 5.6: Percentage timing error (amplitude ratio vs time separation for symmetric echoes) . 59

Figure 5.7: Percentage reconstruction error (amplitude ratio vs time separation for symmetric

echoes) .......................................................................................................................................... 60

Figure 5.8: Overlapped echoes at SNR = 40 dB (symmetric echoes) .......................................... 61

Figure 5.9: Percentage timing error (40 dB for symmetric echoes) ............................................. 62

Figure 5.10: Overlapped echoes at SNR = 25 dB (symmetric echoes) ........................................ 62

Figure 5.11: Percentage timing error (25 dB for symmetric echoes) ........................................... 63





Figure 5.16: Performance vs SNR (symmetric echoes) ................................................................ 66

Figure 5.17: Overlapped signal with time separation of 0.2 µs .................................................... 68

Figure 5.18: Quadratic modulation frequency .............................................................................. 71

Figure 5.19: Baseline configuration for asymmetric echoes ........................................................ 71

Figure 5.20: Phase slope difference of two asymmetric echoes (nominal time separation at 0.2

µs) ................................................................................................................................................. 72

Figure 5.21: Percentage timing error (phase difference vs time separation for asymmetric echoes)

....................................................................................................................................................... 73

Figure 5.22: Percentage reconstruction error (phase difference vs time separation for asymmetric

echoes) .......................................................................................................................................... 73

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Figure 5.23: Percentage timing error (center frequency difference vs time separation for

asymmetric echoes) ....................................................................................................................... 75

Figure 5.24: Percentage reconstruction error (center frequency difference vs time separation for

asymmetric echoes) ....................................................................................................................... 75

Figure 5.25: Percentage timing error (amplitude ratio vs time separation for asymmetric echoes)

....................................................................................................................................................... 76

Figure 5.26: Percentage reconstruction error (amplitude ratio vs time separation for asymmetric

echoes) .......................................................................................................................................... 77

Figure 5.27: Overlapped echoes at SNR = 40 dB (asymmetric echoes) ....................................... 78

Figure 5.28: Percentage timing error (40 dB for asymmetric echoes) .......................................... 79







Figure 5.35: Performance vs SNR (asymmetric echoes) .............................................................. 83

Figure 5.36: Test sample containing vertical notches ................................................................... 85

Figure 5.37: TOFD configuration for notch sample ..................................................................... 85

Figure 5.38: B-scan of notch sample TOFD scan ......................................................................... 86

Figure 5.39: Overlapping echoes in TOFD scan of notch sample ................................................ 86

Figure 5.40: Notch sample time series data analyzed by TEI and DBPR .................................... 87

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Figure 5.41: Reconstructed echoes for notch sample (TEI using phase linearity condition) ....... 88

Figure 5.42: Reconstructed echoes for notch sample (DBPR) ..................................................... 88

Figure 5.43: Frequency phase profiles of TEI reconstructed echoes (notch sample) ................... 88

Figure 5.44: Lateral wave reference echo for TEI ........................................................................ 90

Figure 5.45: Reconstructed echoes for notch sample (TEI using cross-correlation condition) .... 90

Figure 5.46: Test sample for pitch-catch matrix probe scan ......................................................... 91

Figure 5.47: Phased array pitch-catch testing of SDH sample ..................................................... 91

Figure 5.48: Indirect path for SDH ............................................................................................... 92

Figure 5.49: Overlapping echoes for SDH pitch-catch test .......................................................... 92

Figure 5.50: SDH sample time series data analyzed by TEI and DBPR ...................................... 93

Figure 5.51: Reconstructed echoes for SDH sample (TEI) .......................................................... 93

Figure 5.52: Reconstructed echoes for SDH sample (DBPR) ...................................................... 94

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List of Symbols

Symbol Definition

𝐴 Amplitude scaling parameter in time domain echo model

𝐶 Capacitance (F)

𝐷 Diameter of transducer (m)

𝐹 Force output from transducer (N)

𝐺(𝜔) Frequency domain Wiener filter

𝐻𝑥(𝜔) Transfer function of system x (variable)

𝐼 Electrical current (A)

𝐿 Electrical Inductance (H)

𝑀𝐴(𝜔) Frequency amplitude profile of constituent echo A

𝑀𝐵(𝜔) Frequency amplitude profile of constituent echo B

𝑀𝑇(𝜔) Frequency amplitude profile of total signal

N Number of data points

𝑁(𝜔) Frequency noise

𝑃(𝜔) Frequency pressure response (N/m2)

𝑅 Electrical resistance (Ω)

𝑅𝐸𝐹(𝜔) Frequency domain reference signal

𝑆 Amplitude scaling parameter for frequency domain Q-Gaussian model

𝑆𝐼𝐺(𝜔) Fourier transform of total signal

𝑆𝑁𝑅(𝜔) Frequency domain signal-to-noise ratio

𝑇𝑥 Transfer matrix of layer x in KLM model (variable)

𝑉𝑥(𝜔) Frequency domain voltage response of system x (V)

𝑍 Electrical or acoustic impedance (Ω or Rayl)

𝑎 Exponential time decay parameter (1/s)

𝑏 Width parameter for frequency domain Q-Gaussian model (s2)

𝑐 Speed of sound (m/s)

𝑒(𝑡) Reconstruction error

𝑒𝑐ℎ𝑜(𝑡) Echo waveform in the time domain

𝑒𝑛𝑣(𝑡) Echo amplitude envelope in the time domain

𝑓 Frequency (Hz)

ℎ𝑥(𝑡) Impulse response for system x (variable)

𝑘 Wavenumber (1/m)

𝑙 Thickness (m)

𝑚 Order for series in mathematics (positive integer)

𝑛(𝑡) Time domain noise

𝑝𝑥(𝑡) Time domain pressure response of system x (N/m2)

𝒑 Vector of optimization parameters used in Dictionary-based Parametric

Represenation (DBPR) model

𝑞 Tail-heaviness parameter for frequency domain Q-Gaussian model

𝑟 Radial direction (m)

𝑟𝑒𝑓(𝑡) Time domain reference signal

𝑠𝑖𝑔(𝑡) Total ultrasound time series signal

𝑡 Time (s)

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Δ𝑡 Arrival time difference between two ultrasound echoes (s)

𝑢 Frequency power in attenuation for materials

𝑣(𝑡) Time domain voltage signal (V)

𝑤𝑗 Weights used in auto-regressive extrapolation model

𝒙 Vector of optimization parameters used in Trigonometric Echo Identification

(TEI) model

𝑧 Axial distance (m)

𝛼(𝜔) Interior angle opposite of 𝑀𝐴(𝜔) in phase reconstruction triangle (rad)

𝛽(𝜔) Interior angle opposite of 𝑀𝐵(𝜔) in phase reconstruction triangle (rad)

𝛾(𝜔) Interior angle opposite of 𝑀𝑇(𝜔) in phase reconstruction triangle (rad)

𝜃𝐴(𝜔) Frequency phase profile of constituent echo A (rad)

𝜃𝐵(𝜔) Frequency phase profile of constituent echo B (rad)

𝜃𝐵(𝜔) Frequency phase profile of total signal (rad)

𝜅(𝜔) Transformer ratio in KLM model (V/N)

𝜆 Wavelength (m)

𝜇 Penalty parameter in Augmented Lagrangian (ALAG) constrained

optimization algorithm

𝜉𝑠𝑝𝑟𝑒𝑎𝑑 Beam spread angle of ultrasound transducer (rad)

𝜌 Envelope asymmetry parameter in time domain echo model

𝜎 Sparsity control parameter in Basis Pursuit (also known as L1-norm

deconvolution)

𝜏 Time shift parameter in time domain echo model (s)

𝜙 Constant phase shift parameter in time domain echo model (rad)

𝜒 Lagrange multiplier in Augmented Lagrangian (ALAG) constrained

optimization algorithm

𝜓 Quadratic modulation frequency variation parameter in time domain echo

model (1/s3 )

𝜔 Angular frequency (rad/s)

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List of Appendices

Appendix 1: Two-way Impulse Response of Van-Dyke Model ................................................. 105

Appendix 2: KLM Model of Broadband Transducer ................................................................. 107

Appendix 3: Source Code of KLM model .................................................................................. 109

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Chapter 1 Introduction

1.1 Introduction and Motivation

Ultrasonic testing is a Non-Destructive Testing (NDT) method to characterize the internal

structure of a test sample using high frequency sound waves. The typical frequencies employed

in ultrasonic inspection systems range from 200 kHz up to 100 MHz [1]. An advantage of

ultrasonic testing is that sound waves can propagate in a multitude of solids and liquids.

Consequently, ultrasonic testing can be performed on test samples made of metals, plastics,

ceramics, polymers, composite materials and biomedical materials [2].

In ultrasonic testing, a voltage waveform originating from an ultrasonic wave scattered by a

discontinuity in the test sample is called an echo. The shapes and time durations of ultrasonic

echoes are determined by the design of the transducer, the characteristics of the electronics of the

inspection system and the characteristics of the defect present in the test sample [1]. If two

defects in the test sample are located adjacent to each other, as shown in Figure 1.1, the

difference in acoustic travel times for the two acquired echoes might be shorter than the time

duration of the individual echoes. In such situations, the two echoes will overlap in the time

domain and it would be difficult to accurately determine the arrival times of each echo. Although

Figure 1.1 shows that separate transducers are used for transmitting and receiving the acoustic

waves, there are many NDT applications where a single transducer is used for both roles, in what

is known as a “pulse-echo” configuration.

Figure 1.1: Acoustic travel paths for two adjacent defects

Overlapping ultrasonic echoes are frequently encountered in applications where the examined

features have characteristic dimensions comparable to the wavelength in the material. NDT

examples of such applications include characterization of shallow and/or short cracks in Time-

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of-Flight Diffraction (TOFD) studies [3], testing of adhesive bonds between thin structures [4]

and normal incidence inspection of subsurface corrosion [5]. In these applications, multiple

overlapping echoes with similar frequency content may be picked up by the receiving transducer.

As will be explained in Section 2.1, overlapping echoes is the limiting factor for the axial

resolution in ultrasonic cross-section imaging techniques. In addition, the presence of

overlapping echoes can also directly affect the accuracy of time-of-flight based ultrasonic testing

measurements. As an illustrative example, consider the TOFD scan for a weld sample containing

a vertical crack shown in Figure 1.2. In Figure 1.2, the cone in the center represents the weld area

and each coloured line represents the ray path of an ultrasonic wave travelling from transmitter

to receiver, and is assigned a name indicative of the path followed by the wave.

Figure 1.2: Configuration of a TOFD scan

For the test configuration shown in Figure 1.2, we expect to obtain four return echoes

corresponding to the lateral wave, the top tip diffracted echo, the bottom tip diffracted echo and

the back-wall reflection echo. If the speed of sound in the sample is known, then simple

trigonometry will yield the vertical position and size of the cracks from the arrival times of each

echo at the receiving transducer. However, if the vertical extent of the crack is small and/or if the

crack is located close to the top or bottom surface of the sample, overlapping echoes would be

acquired and it would not be possible to obtain accurate estimate of the arrival time of each echo,

nor the location and size of the crack.

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For the reasons listed above, a method to separate overlapping ultrasonic echoes would be highly

beneficial for accurate location and sizing of small defects. High-frequency high-bandwidth

ultrasound transducers have been designed to reduce the time duration of the ultrasonic echoes in

order to minimize the problem of overlapping echoes[6]. However, such transducers have weak

output and limited penetration depth since acoustic attenuation increases with wave frequency

[2]. Consequently, hardware solutions to mitigate overlapping echoes are limited for many NDT

applications.

Due to the limitation of hardware solutions, a software solution to separate overlapping echoes is

proposed to enhance the axial resolution in ultrasonic imaging and provide an improved estimate

of the size and location of any defects present. In this thesis, we will present a novel post-

processing algorithm designed to separate two overlapping echoes that are present in ultrasonic

testing time series data.

1.2 Thesis Objectives

The major objectives of this research project are introduced in the section.

Development of Trigonometric Echo Identification (TEI) Algorithm

The main objective of this thesis is to develop a novel algorithm for separation of two

overlapping ultrasonic echoes. The name of the proposed algorithm is called Trigonometric Echo

Identification (TEI). The proposed algorithm is designed to identify and separate two ultrasonic

echoes that overlap partially in time and also possess similar frequency content. (If two

ultrasonic echoes have distinctly different spectral content, conventional time-frequency

transform methods such as the continuous wavelet transform [7] can be used to separate the two

echoes.)

Since the shapes of ultrasonic echoes are highly dependent on the configurations of the

inspection system, it is not feasible to develop a generic algorithm that can successfully process

echoes acquired from all possible test configurations. Consequently, the proposed algorithm is

targeted to separate echoes acquired from bulk (longitudinal and/or shear) wave inspection of

metallic samples using high bandwidth piezoelectric transducers. The proposed algorithm

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should also be sufficiently flexible to handle two overlapping echoes that have differently shaped

amplitude envelopes and different phase shifts.

It should be stressed that goal of this research is not to develop a new ultrasound imaging

technique. The proposed echo separation algorithm is a signal processing tool that can be

incorporated in existing ultrasound testing methods to improve the resolution in defect size and

location estimates.

Evaluation of Algorithm Performance

After the development of the echo separation algorithm, its performance is to be evaluated using

simulation and experimental tests. The performance of the proposed algorithm will also be

compared to that of an existing state-of-the-art echo separation algorithm. Simulation tests are

valuable because we have exact information of the properties of the individual echoes.

Simulation tests also allow us to vary the shapes of the individual echoes and the signal-to-noise

ratio (SNR) level of the input signals to obtain statistical metrics of algorithm performance.

Experimental tests will also be conducted on the proposed algorithm to verify that the

assumptions made during the algorithm development process are actually applicable for real

world NDT applications. Results obtained from simulation and experimental tests will allow us

to determine the advantages and limitations of the proposed algorithm.

1.3 Thesis Overview

In this section, we present an outline of the material that will be presented in the remaining

chapters of the thesis.

In Chapter 2, we present a literature review of the relevant background topics. The chapter

begins with a description of ultrasonic inspection systems and the different representations of

ultrasonic testing data. The importance of separation of overlapping echoes for axial resolution

enhancement is also discussed. The chapter then introduces linear models that can be used to

predict the voltage-to-voltage frequency response of ultrasonic inspection systems. Next, two

main categories of conventional echo separation algorithms are reviewed: Single Reference

Deconvolution and Dictionary-based Parametric Representation (DBPR).

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In Chapter 3, we introduce the assumptions used in our new TEI algorithm. Since TEI is a

frequency-domain method, the algorithm assumptions are expressed in terms of the frequency

amplitude and phase profiles. The justifications for these assumptions are then described based

on both empirical data and theoretical models.

Chapter 4 is dedicated to the detailed presentation of the TEI algorithm. The chapter begins with

a high-level overview of the complete algorithm. The chapter then introduces a trigonometry-

based method to recover the phase profiles from the amplitude profiles of two overlapping

echoes. This phase profile reconstruction method is an integral part of the TEI algorithm and

contributes to its unique properties. Next, details of the different components of the TEI

algorithm are described. Using the described components and assumptions presented in Chapter

3, TEI is then formulated as a constrained-optimization problem. This problem formulation

allows TEI to be solved using existing optimization methods. The chapter concludes with a

summary of the novel ideas and advantages of the TEI algorithm.

In Chapter 5, we present results from simulation and experimental evaluations of the TEI

algorithm. The echo separation performance of TEI is compared to that of DBPR, which we

select as the benchmark method. For the simulation tests, the percentage timing and

reconstruction errors are used as performance metrics. Signal parameters including phase shift,

frequency difference, amplitude ratio and noise level are varied to obtain different test

configurations. For the experimental tests, we evaluate the applicability of the TEI algorithm for

processing of ultrasound testing data acquired from two NDT applications. The echo separation

performance of TEI is assessed by comparing its extracted arrival time difference between the

two echoes with the arrival time difference estimated using geometric calculations.

Thesis conclusions are presented in Chapter 6, which begins with a review of the major tasks

completed in the research project. This review is followed by a summary of the most important

research findings. The thesis concludes with a list of suggestions for future research directions.

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Chapter 2 Background and Literature Review

2.1 Ultrasonic Inspection System

A schematic drawing of a typical ultrasonic inspection system is shown in Figure 2.1. In this

schematic drawing, the computer controls the pulser which sends a high voltage pulse to the

transmitting transducer. The voltage pulse is transformed into a mechanical vibration in the

transmitting transducer and leads to a propagating ultrasonic wave being sent into the sample. If

a flaw or discontinuity is present in the sample, a portion of the propagating ultrasonic wave

would be reflected or scattered, and a portion of these deflected waves could then be captured by

the receiving transducer. The receiver then amplifies the output voltage signal, and sends the

analog waveform signal to the oscilloscope. Finally, the oscilloscope converts the analog signal

into digital data and sends the data to the computer for further processing and storage. Although

in Figure 2.1 separate transducers are used for the transmission and reception paths, in many

NDT applications a single transducer can be used in pulse-echo mode to both transmit and

capture the reflected ultrasonic wave.

Figure 2.1: Schematic diagram of ultrasonic inspection system

2.1.1 Pulser-Receiver

The Pulser-receiver is an electronic device used for both the creation of a voltage drive pulse for

the transmitting transducer and the reception and amplification of the voltage signal from the

receiving transducer. Since the drive voltage is usually many orders of magnitude stronger than

the received signal (hundreds of volts compared to millivolts), protection circuits must be in

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place to prevent voltage cross-talk between the two compartments [8]. Two different models of

pulser-receivers are shown in Figure 2.2.

Figure 2.2: Two models of Pulser-Receiver

The top pulser-receiver shown in Figure 2.2 is a newer model with digital drive pulse control

while the bottom model uses analog control circuits. In pulsers that use analog control circuits,

the voltage drive pulse is created from a sudden release of electrical energy stored in a capacitor.

Consequently, the voltage waveform would follow an exponential decay as shown in Figure 2.3.

Figure 2.3: Voltage pulse of analog pulser

From Figure 2.3, we see that the voltage pulse has a characteristic decay time which is controlled

by the amount of electrical energy stored in the capacitor and the amount of damping in the

circuit. This decay time has a significant impact on the transducer pressure waveform output.

Using a linear model, the pressure waveform transmitted from the transmitting transducer can be

modeled as the convolution of input voltage pulse with the transducer voltage-to-pressure

impulse response [1]:

Decay Time

8

𝑝𝑡𝑟𝑎𝑛𝑠𝑑𝑢𝑐𝑒𝑟(𝑡) = 𝑣𝑝𝑢𝑙𝑠𝑒𝑟(𝑡) ∗ ℎ𝑡𝑟𝑎𝑛𝑠𝑑𝑢𝑐𝑒𝑟(𝑡) (2.1)

In pulsers with analog control circuits, there are typically energy and damping settings which can

changed independently to modify the shape of the output echo. However, in practice it is often

difficult to use these settings to obtain an output voltage pulse that matches with the bandwidth

of the transducer.

In contrast, for newer pulsers with digital pulse control, the output voltage signal is a square

pulse as shown in Figure 2.4. In addition, both the pulse amplitude and width can be specified.

Consequently, digital ultrasound pulsers offer a more powerful method to fine-tune the output

pressure waveform of a transducer. For this reason, digital pulsers can be used to drive different

transducers across a wide range of design center frequencies.

Figure 2.4: Voltage pulse of digital pulser

2.1.2 Piezoelectric Transducers

Despite recent developments in electromagnetic[9] and capacitive [10] ultrasound transducers,

piezoelectric transducers based on the direct and inverse piezoelectric effects remain the most

commonly used type of transducers used for ultrasonic testing. There exist two main types of

piezoelectric transducers: single element transducers and phased array transducers.

Single Element Transducers

Single element transducers are the simplest ultrasonic transducers; they consist of only one

active piezoelectric element used to transmit and/or receive pressure waves. A schematic

diagram of the components of a single element transducer is shown in Figure 2.5.

Pulse width

9

Figure 2.5: Schematic diagram of a single element piezoelectric transducer (courtesy of [11])

In Figure 2.5, we see that there is an electrical connector that sends and receives electrical signal

to and from the piezoelectric element. The piezoelectric element is usually in the shape of a disc;

it is in contact with a backing element on one face and a matching layer on the other. The

purpose of the backing element is to attenuate excessive ringing from the piezoelectric element

to increase the frequency bandwidth of the transducer. The purpose of the matching layer is to

maximize the wave energy transfer from the piezoelectric element to the test sample. Usually a

quarter-wavelength matching layer design is employed [12].

The center frequency of the transducer is controlled by the thickness of the piezoelectric element.

The thickness of the element is typically selected to be 1/2 of the wavelength at the design center

frequency. The beam spread of the transmitted pressure wave can be related to the diameter of

the piezoelectric element using the following formula [13]:

sin(𝜉𝑠𝑝𝑟𝑒𝑎𝑑) =1.22𝑐

𝐷𝑓

(2.2)

In Eq. (2.2), 𝜉𝑠𝑝𝑟𝑒𝑎𝑑 is the beam divergence angle from transducer centerline to point where

signal is at half strength, 𝑐 is the speed of sound in the propagation medium, 𝐷 is the transducer

active diameter and 𝑓is the pressure wave frequency. From Eq. (2.2), we see that transducer

beam spread can be reduced by increasing the active element diameter and/or the transducer

center frequency.

10

Phased Array Transducers

Phase array transducers are constructed from arranging multiple active piezoelectric elements in

a geometrical array. Even though rectangular matrix [14] and annular [15] array transducers have

been tested, linear arrays where the active elements are arranged along a single direction remain

the most popular phased array transducer design. A great advantage of phased array transducers

is that the steering angle and the focal depth of the output ultrasonic wave can be changed by

controlling the relative firing time delays of the individual elements. Figure 2.6 and Figure 2.7

demonstrate the time delay patterns used to achieve beam steering and focussing. Steering and

focussing can also be performed simultaneously by combining the two time delay patterns.

Figure 2.6: Steering of phased array transducers

Figure 2.7: Focusing of phased array transducers

11

Compared to single element transducers, phased array transducers offer much more flexibility.

Different areas of the test sample can be scanned without physically moving the transducer by

electronically changing the steering angles. In addition, the effective aperture size of the

transducer can be changed by altering the number of firing elements. Despite these advantages,

phased array transducers have yet to replace single element transducers in many NDT

applications due to their increased equipment cost, larger physical size and increased complexity

in data processing.

2.1.3 Ultrasonic Testing Data Representation

In this section, we will introduce different representation methods used to display the data

collected from ultrasound testing. The most basic data representation method used in ultrasound

testing is the A-scan, which is simply a 1D plot of the receiving transducer’s output voltage

signal as a function of time. Figure 2.8 shows an example of an A-scan using the data collected

from a TOFD experiment featuring a test piece with a vertical crack.

Figure 2.8: A-scan representation from TOFD data

Looking at Figure 2.8, we see that there are four return echoes which correspond to the lateral

wave, top tip diffracted echo, bottom tip diffracted echo and the specular backwall reflection

echo. The presence of these echoes corresponds well with the expected signal from a TOFD scan

of a vertical crack shown in Figure 1.2. In Figure 2.8, we also see that the lateral wave echo

overlaps with the diffracted echo from the top tip of the crack. If the two echoes interfere with

each other, then it becomes impossible to visually determine the exact temporal location of the

two echoes such that the TOFD technique cannot yield a good estimate of the crack size. In such

Lateral wave

Top tip Bottom tip

Back wall

12

situations, echo identification algorithms can be employed to separate the two echoes and

determine the time difference between them.

Another form of ultrasonic testing data representation is the B-scan. B-scan representations are

created by stacking A-scans line-by-line adjacent to each other in order to create a rudimentary

2D image. Figure 2.9 is an example of a B-scan obtained from translation of a pair of TOFD

probes along the direction of the weld. In Figure 2.9, the horizontal axis represents the probe

translation direction and the vertical axis is the time axis of the stacked A-scans. Consequently,

the A-scans obtained along the probe translation direction are stacked column by column in

Figure 2.9.

From Figure 2.9, we see that the lateral wave and back wall echoes are continuous along the scan

direction. This is expected as the weld sample has continuous top and bottom surfaces. There are

also four distinct echoes in Figure 2.9; these echoes correspond to localized flaws along the

length of the weld. From this example, we see that B-scans can be used to locate both the lateral

and axial (along the sound propagation path) locations of a flaw.

Figure 2.9: TOFD B-scan containing 4 flaws

Another form of ultrasonic testing data representation is the C-scan. C-scans are 2D maps of a

test sample, where the color of each pixel represents the arrival time of the echo or the strength

of the reflected signal. C scans are obtained by mechanically translating a single element

Scan Distance [mm]

Scan T

ime [

us]

B-scan of TOFD scan

0 50 100 150 200 250 300

0

0.5

1

1.5

2

2.5

3

3.5

4

Lateral wave

Flaw echoes Back wall

13

transducer over the scan region. Each time the transducer is moved to a new (x, y) co-ordinate,

an A-scan is performed and the return echo time of flight or return echo amplitude is recorded.

Figure 2.10 shows a C-scan from recording the echo amplitude reflected from the back surface of

the coin. Since the front surface topology also affects the strength of the transmitted signal,

features on both the top and bottom of the coin can be seen.

Figure 2.10: C-scan of back surface of a coin (from [16])

The final ultrasonic data representation method that we introduce in this section is the S-scan.

Similar to the B-scan, S-scan produces a 2D slice image that shows both the lateral and axial

locations of any discontinuities. However, S-scans are obtained by electronically changing the

beam steering angle of a phased array transducer instead of mechanically moving the probe.

Figure 2.11 is an example of a S-scan performed for a test sample containing three side-drilled

holes. Compared to the B-scan, S-scans are more convenient to acquire since it does not require

physical repositioning of the transducer.

Figure 2.11: S-scan of three side-drilled holes (from [17])

14

2.1.4 Resolution Limits in Ultrasonic Testing

As seen in Section 2.1.4, B-scan and S-scan are the two most commonly used representations to

obtain 2D slice images of the test sample. In ultrasound images, resolution is defined as the

minimum spatial separation of two flaws that can be clearly identified as two distinct features.

For both the B-scan and the S-scan, the resolution in the lateral direction is limited by the width

of the acoustic beam that is used to illuminate the flaw. This concept is shown in Figure 2.12. In

Figure 2.12, due to spreading of the beam, a point defect would be detected over a finite lateral

displacement. This displacement constitutes the lateral imaging resolution of the scan

configuration.

Figure 2.12: Lateral resolution in ultrasound imaging

The lateral resolution of an ultrasonic scan can be improved by focusing of the probe. A recent

development for the improvement of lateral scan resolution is the Total Focusing Method (TFM)

[18]. TFM uses post-processing to focus at every point within a desired scan region by summing

delayed unfocussed A-scans acquired from a phased array transducer. However, the size of the

focal zone in TFM is still limited by physical laws. The theoretical minimum size of the focal

zone of a transducer is determined by the Abbe diffraction limit [19]:

𝑑𝑖𝑓𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛 𝑙𝑖𝑚𝑖𝑡 =2(1.22 𝜆 𝑧𝑓)

𝐷

(2.3)

In Eq. (2.3), 𝜆 is the wavelength in the medium, 𝑧𝑓 is the focal depth and 𝐷 is the diameter of the

aperture of the transducer. In actual applications, the focal zone of a phased array transducer is

usually much larger than the theoretical limit expressed in Eq. (2.3) due to time-delay

quantization errors and non-uniform performance of the active piezo elements [20].

15

In contrast to the lateral resolution which is limited by the size of the focal zone, the axial

resolution of both B-scans and S-scans is primarily limited by the ability to separate two defect

features in the time domain A-scan signal. The time-duration of an ultrasonic echo is determined

by the bandwidth of the transducer and cannot be reduced through beam focusing [2]. Efforts

have been made to design high-frequency high-bandwidth transducers to reduce the time

duration of the output echo in order to improve the axial resolution [6]. However, such

transducers have weak amplitude output and have limited penetration depth since acoustic

attenuation increases with wave frequency [2]. Consequently, hardware solutions to improve the

axial resolution of ultrasound images are limited for many NDT applications. For these

applications, a post-processing algorithm to separate overlapping echoes is the most viable

method to enhance the axial resolution in ultrasonic testing and provide an improved estimate of

the height and depth of any defects present.

2.2 Modeling of Ultrasonic Echoes

In this section, we will review various physical models designed to analyze the shapes of

ultrasonic echoes. Some of these models will be used in Chapter 3 and 4 for the development of

the Trigonometric Echo Identification (TEI) algorithm

2.2.1 One-Dimensional Piezoelectric Transducer Models

As mentioned in Section 2.1.2, piezoelectric transducers are the most commonly used type of

ultrasonic transducers in industrial NDT applications. For this reason, we will tailor the TEI

algorithm for separation of echoes acquired from piezoelectric transducers.

The voltage-to-voltage two-way impulse response of piezoelectric transducers is often modelled

using one-dimensional equivalent circuit models [1], [2], [6]. The one-dimensional

approximation is valid if the thickness of piezoelectric element is much smaller than its lateral

dimensions. For typical piezoelectric transducers used in NDT applications, the thickness of the

piezoelectric element is of the order of 0.5 mm while the lateral dimensions are of the order of 10

mm. Consequently, the one-dimensional assumption can be applied.

For lightly loaded piezoelectric transducers, the Van Dyke approximate model can be used [2],

[6]. In the Van Dyke equivalent circuit model, a transformer is used to transform the electrical

voltage into mechanical force in the acoustic path. In the acoustic path, the transducer is

16

modelled by an RLC circuit. The capacitance C is inversely proportional to the stiffness of the

piezoelectric material; the inductance L is proportional to the vibration mass and the resistance R

is proportional to damping of the transducer. A schematic diagram of the Van Dyke approximate

model is shown below:

Figure 2.13: Van Dyke approximate transducer model

As shown in Appendix 1, when an impulse voltage is applied to the transducer, the face velocity

(analogous to electrical current) of the transducer takes on the general form of an exponentially

enveloped sinusoid:

𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦(𝑡) = 𝑐𝑜𝑛𝑠𝑡 ∙ 𝑒−𝑎𝑡 cos(𝜔𝑑𝑡 + 𝜙) (2.4)

Here 𝜙 is a constant phase shift. The exponential decay rate 𝑎 and the damped frequency 𝜔𝑑 are

dependent on the 𝑅𝐿𝐶 parameters:

𝑎 =𝑅

2𝐿; 𝜔𝑜 =

1

√𝐿𝐶; 𝜔𝑑 = √𝜔𝑜

2 − 𝑎2 (2.5)

As derived in Appendix 1, the frequency domain amplitude profile of the two-way voltage-to-

voltage transfer function predicted by the Van Dyke model can be expressed as:

|𝑉𝑜𝑢𝑡(𝜔)

𝑉𝑖𝑛(𝜔)| =

𝑐𝑜𝑛𝑠𝑡

1 +1

𝑎2 (𝜔 − 𝜔𝑑)2 (2.6)

Looking at Eq. (2.6), we see that the predicted amplitude profile of the two-way transducer

transfer function is a symmetric distribution with its peak located at 𝜔𝑑, the damped oscillation

frequency. The bandwidth of the distribution is determined by the decay rate 𝑎. The larger the

17

value of 𝑎, the wider the frequency bandwidth of the amplitude profile. Figure 2.14 shows

examples of this amplitude profile with 𝑐𝑜𝑛𝑠𝑡 =1, 𝜔𝑑 = 3 [rad/s] and 𝑎 = 0.5 [1/s] and 1.0 [1/s].

Figure 2.14: Frequency amplitude response predicted by Van Dyke model

For transducers that are coupled to acoustic media which have acoustic impedance values

comparable to the piezoelectric element, the lightly-loaded assumption of the Van Dyke model is

no longer valid. For these transducers, the exact KLM one-dimensional model can be used [2],

[6]. A schematic drawing of the KLM model is shown in Figure 2.15.

Figure 2.15: Schematic diagram of KLM model

In Figure 2.15, 𝐶𝑜 and 𝐶′ are the input capacitances and 𝜅(𝑓) is the ratio of the electro-

mechanical transformer that converts electrical voltage and current into mechanical forces and

velocities. The definitions of these parameters can be found in [2] and [6]. In addition, 𝐹1 and 𝐹2

18

are respectively the forces present at the front and back faces of the transducer. The mechanical

ports (the front and back faces of the transducer) are connected to the center transformer through a

pair of quarter-wave transmission lines. The lengths of these transmission lines are determined by the

thickness of the piezoelectric element.

Although the KLM model shown in Figure 2.15 does not include matching layers, the KLM model

can be extended using the method of transmission matrices [21]. In this method, all components in

the KLM model are replaced by a 2×2 transmission matrix. The definitions of these transmission

matrices are summarized in Eq. (2.7).

𝐷𝑖𝑠𝑐𝑟𝑒𝑡𝑒 𝑠𝑒𝑟𝑖𝑒𝑠 𝑖𝑚𝑝𝑒𝑑𝑎𝑛𝑐𝑒: [1 𝑍𝑠𝑒𝑟𝑖𝑒𝑠

0 1]

𝐷𝑖𝑠𝑐𝑟𝑒𝑡𝑒 𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙 𝑖𝑚𝑝𝑒𝑑𝑎𝑛𝑐𝑒: [1 0

1/𝑍𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙 1]

𝑇𝑟𝑎𝑛𝑠𝑚𝑖𝑠𝑠𝑖𝑜𝑛 𝑙𝑖𝑛𝑒: [

cos(𝑘𝑙) 𝑗𝑍𝑎 ∙ sin(𝑘𝑙)sin(𝑘𝑙)

𝑍𝑎cos(𝑘𝑙)

]

(2.7)

In the definitions shown above, 𝑍𝑎, 𝑘 and 𝑙 are respectively the impedance, angular wavenumber

and length of the transmission line. Note that for all three types of transmission matrices, the

determinant of the matrix is equal to one. Consequently, these matrices have reciprocal

properties and can be used in both the transmission and reception paths. Figure 2.16 shows a

schematic representation of how the method of transmission matrices can be used to model the

transmission path of a transducer with the addition of electrical and acoustic matching.

19

Figure 2.16: Transmission matrix model of transducer (Operated as transmitter)

The overall transmission matrix can be found by multiplying the transmission matrices shown in

Figure 2.16. The voltage-to-force frequency transfer function of the transmission is equal to the

inverse of the (1, 1) element of the overall transmission matrix:

[𝑇𝑡𝑟] = [1 𝑍𝑠

0 1] [𝑇𝑒𝑙𝑡][𝑇𝐶𝑜][𝑇𝐶′

][𝑇𝑥𝑓][𝑇𝑃][𝑇𝑇][𝑇𝑀] [1 0

1/𝑍𝑇 1]

𝐻𝑡𝑟(𝜔) = 1/𝑇11𝑡𝑟

(2.8)

Similar transmission matrix multiplication procedures can be conducted to find the reception

force-to-voltage frequency response of the piezoelectric transducer. Finally, multiplying the

transmission and reception transfer functions would yield the two-way voltage-to-voltage

frequency response of the transducer.

2.2.2 Complete Transfer Function Modeling of Ultrasonic Echoes

In the previous section, we examined in detail two different models that can be used to predict

the frequency response of a piezoelectric ultrasound transducer. Although modeling the

transducer frequency response is crucial to predicting the expected echo shape, other factors such

as wave diffraction and defect scattering can also greatly influence the echo shapes in NDT

ultrasonic testing.

20

According to [1] and [22], the frequency response of each echo can be expressed as a cascade

multiplication of frequency transfer functions:

𝐻𝑒𝑐ℎ𝑜(𝜔) = 𝐻𝑒𝑙𝑒𝑐(𝜔)𝐻𝑑𝑖𝑓𝑓(𝜔)𝐻𝑎𝑡𝑡(𝜔)𝐻𝑑𝑖𝑠𝑝(𝜔)𝐻𝑠𝑐(𝜔) (2.9)

Here 𝐻𝑒𝑙𝑒𝑐(𝜔) is the total electrical transfer function including the piezoelectric transducers and

the pulser/receiver system; 𝐻𝑑𝑖𝑓𝑓(𝜔) is the transducer diffraction transfer function; 𝐻𝑎𝑡𝑡(𝜔) is

the attenuation transfer function, 𝐻𝑑𝑖𝑠𝑝(𝜔) is the dispersion transfer function and 𝐻𝑠𝑐(𝜔) is the

defect scattering transfer function. In this section, we will provide an overview of how these

factors can be modelled.

Electrical System Transfer Function

The models introduced in the previous section can be used to predict the frequency response of

the piezoelectric transducers. However, the frequency response of the pulser/receiver system is

usually measured experimentally. Although the pulser/receiver circuits contain nonlinear

elements, they can be approximated by a Thevenin equivalent circuit shown in Figure 2.17 [23].

Figure 2.17: Thevenin's equivalent circuit

The Thevenin equivalent voltage source 𝑉𝑡ℎ(𝜔) and equivalent resistance 𝑅𝑡ℎ(𝜔) can be

experimentally determined using two simple measurements [23]. However, it should be noted

that both 𝑉𝑡ℎ(𝜔) and 𝑅𝑡ℎ(𝜔) can vary with energy and gain settings of the pulser/receiver

system. Once 𝑉𝑡ℎ(𝜔) and 𝑅𝑡ℎ(𝜔) are determined, the Thevenin’s equivalent circuit can be

incorporated in the KLM model introduced in the previous section to obtain the complete

transfer function of the electrical system.

Transducer Wavefield Diffraction Transfer function

21

The transducer wavefield diffraction transfer function describes the pressure field radiated into

an acoustic medium from an ultrasound transducer. For an idealized circular piston transducer,

exact analytical expressions of the pressure field have been solved in the time domain using the

impulse response method [24]. According to [24], the resultant pressure field is axially

symmetric and therefore only dependent on the axial distance 𝑧 (measured from the plane of

transducer) and the radial distance 𝑟 (measured from the central-axis of the transducer).

In Figure 2.18, we plot the normalized two-way impulse response and its Fourier transform for

two points in the pressure field using the expressions developed in [24]. For these calculations,

the radius of the circular transducer is set at 4 mm while the axial distance 𝑧 is set at 60 mm. The

radial distance away from the central axis of the transducer are set at 𝑟 = 0 and 𝑟 = 15 𝑚𝑚.

These observation points are located in the far-field for frequencies below 21 MHz.

Figure 2.18: Two-way impulse and frequency response for two points in a pressure field

Looking at Figure 2.18, we see that as we move laterally away from the central-axis, the impulse

response becomes broader. It is also time delayed because the point is located further away from

the transducer. From the frequency plot, we see that the 𝑟 = 15 𝑚𝑚 response has a much

smaller passband compared to 𝑟 = 0. These results are consistent with the well-known acoustic

property that the beam spread of a transducer is inversely proportional to its center frequency. As

we move away from the central-axis of the transducer, the transducer diffraction transfer

function suppresses the spectral content of the higher frequencies.

22

Attenuation Transfer Function

Over the typical frequency range used in ultrasonic testing (~from 1 MHz to 20 MHz), the

attenuation coefficient of most materials follows an approximate power law relationship with

frequency [1].

𝑎𝑡𝑡𝑒𝑛𝑢𝑎𝑡𝑖𝑜𝑛(𝑓) = 𝑐𝑜𝑛𝑠𝑡1 + 𝑐𝑜𝑛𝑠𝑡2𝑓𝑢 (2.10)

According to [25], the frequency power 𝑢 varies from 1.8 to 2.2 for different grades of low-

carbon steel. Since attenuation is frequency dependent, the Fourier transform of ultrasonic

echoes are in general asymmetric with respect to the center frequency.

Dispersion Transfer Function

The effects of dispersion can be safely neglected for bulk wave ultrasonic testing measurements.

Dispersion effects can arise either by material property of the acoustic medium or by the mode of

wave propagation. In contrast to plate waves such as lamb wave or the shear-horizontal wave,

bulk shear or longitudinal wave propagation is not inherently dispersive [26]. Consequently, any

dispersion effects present must be attributed to the material property of the acoustic medium.

Acoustic attenuation and dispersion of a medium can be related by the Kramers-Kronigs

equations [2]. From the Kramers-Kronigs equations, it can be shown that materials which follow

a quadratic attenuation curve are not dispersive. Since the attenuation power of steel varies from

1.8 to 2.2 in the frequency range from 1 to 20 MHz, we can conclude that its dispersion effects

are negligible. This theoretical conclusion is also corroborated by experimental results [1].

Scattering Transfer Function

The scattering coefficients for simple defect geometries such as cylindrical and spherical voids,

point reflectors, cracks and flat surfaces have been investigated in [1] and [27] using ray

methods. In general, the scattering coefficient of a defect is both frequency-dependent and angle-

dependent. For example, for Rayleigh scattering of small particles, the amplitude of the

scattering coefficient is proportional to the fourth power of frequency.

However, there also exists defects which have frequency-independent scattering responses.

Examples of these defects include diffraction from sharp crack tips and specular reflection from

23

flat surfaces [28], [27]. For such cases, the scattering transfer function would be a constant and

therefore would not introduce any shape distortion to the ultrasonic echoes.

2.3 Single Reference Deconvolution

Before we begin development of the Trigonometric Echo Identification (TEI) algorithm, it is

necessary to first examine the existing techniques that have been investigated for identification

of overlapping ultrasonic echoes. Among the different techniques, single reference

deconvolution is one of the most commonly investigated methods [29],[30]. In this section, we

will describe the assumptions of this technique and its limits of applicability.

2.3.1 Basic Assumptions

In single reference deconvolution, it is assumed that each return echo can be modelled by a time-

shifted and amplitude-scaled version of a reference echo [29]. Mathematically, this assumption

can be expressed as:

𝑠𝑖𝑔(𝑡) = ∑ 𝑒𝑐ℎ𝑜𝑖(𝑡) + 𝑛(𝑡) = 𝑟𝑒𝑓(𝑡) ∗ ℎ(𝑡) + 𝑛(𝑡) (2.11)

In Eq. (2.11), 𝑟𝑒𝑓(𝑡) is the reference echo, ℎ(𝑡) is the scattering impulse response of the defects

present in the test sample, 𝑛(𝑡) is the noise present in the system and ∗ is the convolution

operator in the time domain. A schematic representation of a convolution operation without the

addition of noise is shown in Figure 2.19.

Figure 2.19: Single reference convolution

From Figure 2.19, it is clear that if ℎ(𝑡) is recovered, we would obtain information regarding the

location and scattering strength of each defect present. Using the convolution-multiplication

24

duality property of the Fourier transform [31], the equivalent frequency domain expression of

Eq. (2.11) can be written as:

𝑆𝐼𝐺(𝜔) = 𝑅𝐸𝐹(𝜔)𝐻(𝜔) + 𝑁(𝜔) (2.12)

Using Eq. (2.12), we see that the scattering response 𝐻(𝜔) can be estimated using a simple

frequency domain division operation:

𝐻𝑒𝑠𝑡(𝜔) =𝑆𝐼𝐺(𝜔)

𝑅𝐸𝐹(𝜔)

(2.13)

Note that 𝐻𝑒𝑠𝑡(𝜔) is different from the true scattering response 𝐻(𝜔) because it neglects the

effect of the noise term in the signal. Eq. (2.13) is the fundamental single reference

deconvolution equation and in the next sub-section we will introduce various modifications that

have been investigated to improve the performance of this technique.

2.3.2 Direct Deconvolution Schemes

Direct deconvolution schemes are modifications made to the spectral division equation expressed

in Eq. (2.13) to improve its performance. One of the earliest modifications introduced is the

Wiener deconvolution [32]. The Wiener deconvolution is designed to minimize the impact of

deconvolved noise at frequencies with low SNR.

In Wiener deconvolution, the scattering response is estimated using the following formula:

𝐻𝑒𝑠𝑡(𝜔) = 𝐺(𝜔)𝑆𝐼𝐺(𝜔) (2.14)

Here 𝐺(𝜔) is the Wiener filter and is defined as:

𝐺(𝜔) =1

𝑅𝐸𝐹(𝜔)[

|𝑅𝐸𝐹(𝜔)|2

|𝑅𝐸𝐹(𝜔)|2 +1

𝑆𝑁𝑅(𝜔)

]

(2.15)

Looking at Eq. (2.15), we see that when the 𝑆𝑁𝑅(𝜔) is low, the denominator in the square

bracket would have a high value and therefore 𝐺(𝜔) would reduce the contribution from these

frequencies. Conversely, when 𝑆𝑁𝑅(𝜔) approaches infinity, 𝐺(𝜔) would approach 1/𝑅𝐸𝐹(𝜔)

and we would revert to the basic spectral division equation of Eq. (2.13).

25

For Wiener deconvolution to work effectively, we need to have an accurate estimate of the noise

spectral density |𝑁(𝜔)| or equivalently the signal-to-noise ratio 𝑆𝑁𝑅(𝜔). Although 𝑆𝑁𝑅(𝜔) is

in theory frequency-dependent, in practice it is often replaced by a constant SNR value [33] since

the frequency dependence of the noise level is difficult to estimate.

Another direct deconvolution scheme investigated by researchers is Auto-Regressive Spectral

Extrapolation [34], [35]. From Eq. (2.13), we can see that the functional bandwidth of 𝐻𝑒𝑠𝑡(𝜔) is

limited by the frequency bandwidth of 𝑅𝐸𝐹(𝜔). At frequencies where |𝑅𝐸𝐹(𝜔)| is small,

𝐻𝑒𝑠𝑡(𝜔) cannot be accurately determined even with the adoption of the Wiener deconvolution

scheme. Auto-Regressive Spectral Extrapolation is designed to address this problem.

In Auto-Regressive Spectral Extrapolation, it is assumed that 𝐻𝑒𝑠𝑡(𝜔) can be modeled by a sum

of complex sinusoids [35]. If this assumption is valid, the spectral content of 𝐻𝑒𝑠𝑡(𝜔) at

frequencies where the SNR is low can be extrapolated from a weighted sum of the spectral

content of 𝐻𝑒𝑠𝑡(𝜔) at frequencies where the SNR is deemed to be sufficiently large.

Mathematically, this can be expressed as:

𝐻𝑒𝑠𝑡(𝜔) = ∑ 𝑤𝑗𝐻𝑒𝑠𝑡(𝜔𝑖−𝑗)𝑚

𝑗=1

(2.16)

In Eq. (2.16), 𝑤𝑗 are the weights of each frequency point and 𝑚 is the order of the auto-

regressive process. Both 𝑤𝑗 and 𝑚 are parameters that need to be optimized. A successful

application of the Auto-Regressive Spectral Extrapolation method can extend the useful

bandwidth of 𝐻𝑒𝑠𝑡(𝜔) and therefore sharpen the time-domain scattering response ℎ𝑒𝑠𝑡(𝑡). A

sharpened time-domain scattering response can lead to a more accurate estimate of the locations

of each defect.

2.3.3 Iterative Deconvolution Schemes

Iterative deconvolution schemes are not based on the spectral division operation of Eq. (2.13).

Instead, an initial guess of ℎ𝑒𝑠𝑡(𝑡) is made and subsequently improved upon as an optimization

problem. A major advantage of iterative deconvolution schemes is that the solution can be

optimized to better satisfy the preconceived assumptions of ℎ𝑒𝑠𝑡(𝑡). However, iterative

deconvolution schemes are more computation-intensive compared to direct methods.

26

One of the most commonly investigated iterative deconvolution schemes is L1-norm

deconvolution. This scheme is designed to recover a ℎ𝑒𝑠𝑡(𝑡) that consists of sparse spike train

[36]. A sparse spike train scattering response is ideal because it provides accurate timing

information for all identified defects.

L1-norm deconvolution is mathematically formulated to minimize the following expression:

𝐦𝐢𝐧ℎ𝑒𝑠𝑡(𝑡)

[∑|𝑠𝑖𝑔(𝑡) − 𝑟𝑒𝑓(𝑡) ∗ ℎ𝑒𝑠𝑡(𝑡)|2

𝑡

+ 𝜎 ∑|ℎ𝑒𝑠𝑡(𝑡)|

𝑡

] (2.17)

From this equation, we see that there is a sum of two terms that needs to be minimized. The first

term is the L2-norm of the deconvolution error. By minimizing this term, we can obtain a

convolved response 𝑟𝑒𝑓(𝑡) ∗ ℎ𝑒𝑠𝑡(𝑡) that best approximates the observed signal 𝑠𝑖𝑔(𝑡). The

second term of Eq. (2.17) is the scaled L1-norm of the scattering response. By minimizing the

second term we can obtain a ℎ𝑒𝑠𝑡(𝑡) that is sparse (contains a small number of non-zeros values).

Since Eq. (2.17) has two conflicting minimization criteria, it is not possible to obtain a solution

that is optimal for both terms for real signals that contain some noise. By adjusting the value of

the 𝜎 in Eq. (2.17), we can vary the relative importance of the two terms. Details for choosing

the value of 𝜎 can be found in [37].

2.3.4 Technique Limitations

Despite the many improvements made to the single reference deconvolution technique, its

application is still limited by the fundamental assumption that all return echoes can be modelled

by a scaled and time-delayed copy of a reference echo.

As explained in Section 2.2.2, differences in defect location and scattering properties can yield

ultrasonic echoes with different center frequencies, envelopes and phase shift. Consequently,

single reference deconvolution often performs poorly in configurations where the ultrasonic

echoes are of significantly different shapes. In the next section, we will introduce a parametric

model approach that addresses this major limitation of single reference deconvolution.

27

2.4 Dictionary-based Parametric Representation

To separate overlapping echoes with different shapes, researchers have developed the

Dictionary-based Parametric Representation (DBPR) approach. We will begin a review of this

technique with an overview of its mathematical formulation.

2.4.1 Mathematical Formulation

In DBPR, it is assumed the acquired signal can be represented as a sum of echoes. In addition,

each echo is modelled by a parametric mathematical expression whose parameter values can be

adjusted [38], [39]. Mathematically, this can be expressed as:

𝑠𝑖𝑔(𝑡) = ∑ 𝑒𝑐ℎ𝑜(𝒙𝑖 , 𝑡) + 𝑒(𝑡)

𝑖

(2.18)

In Eq. (2.18), 𝑒(𝑡) is the reconstruction residual error. The notation 𝑒𝑐ℎ𝑜(𝒙𝒊 , 𝑡) indicates that

while each echo is expressed in the time-domain, its shape is controlled by the parameter vector

𝒙𝒊. The values of the parameters in each 𝒙𝒊 are optimized by minimizing the amplitude of the

reconstruction error 𝑒(𝑡):

𝐦𝐢𝐧𝒙𝑖

|𝑒(𝑡)|2 =𝐦𝐢𝐧

𝒙𝑖[∑ |𝑠𝑖𝑔(𝑡) − ∑ 𝑒𝑐ℎ𝑜(𝒙𝑖 , 𝑡)

𝑖

|

2

𝑡

]

(2.19)

For DBPR to be effective, it is necessary to use parametric mathematical models that accurately

describe the shapes of actual ultrasound echoes. The most commonly used parametric model for

ultrasound signals is the Gabor dictionary [38], [39], [40]. In a Gabor dictionary, each echo is

modelled as a Gaussian enveloped oscillation:

𝑒𝑐ℎ𝑜(𝒙𝑖 , 𝑡) = 𝐴 exp[−𝑎2(𝑡 − 𝜏)2] cos[2𝜋𝑓𝑐(𝑡 − 𝜏) + 𝜙] (2.20)

Looking at Eq. (2.20), we see that each parameter vector in the Gabor dictionary contains 5

different variables 𝒙𝑖 = [𝐴, 𝑎, 𝜏, 𝑓𝑐 , 𝜙]. These five parameters respectively control the amplitude,

width, time shift, oscillation frequency and the constant phase shift of the echo. The Gabor

dictionary is chosen because it is empirically found to be an adequate model of the backscattered

echo from a flat surface reflector in pulse-echo ultrasonic testing [39].

28

Despite the popularity of the Gabor dictionary, it often does not perform adequately in situations

where it is necessary to obtain an accurate time difference measurement between two

overlapping echoes [41]. This is because an accurate reconstruction of the echo envelope is

critical for timing measurements. The Gabor dictionary which uses Gaussian-enveloped

oscillations is often inadequate for this task. To address this problem, researchers have developed

more complicated parametric models to describe ultrasonic echoes. For example, the asymmetric

Gaussian chirplet model has been proposed [42]:

𝑒𝑐ℎ𝑜(𝒙𝒊 , 𝑡) = 𝐴 ∙ 𝑒𝑛𝑣 (t − τ)cos[2𝜋𝑓𝑐(𝑡 − 𝜏) + 𝜓(𝑡 − 𝜏)2 + 𝜙]

𝑒𝑛𝑣(𝑡) = exp[−𝑎2(1 − 𝜌 tanh(𝑚𝑡))𝑡2]

(2.21)

Note that in Eq. (2.21), 𝑚 is a fixed positive integer that determines the rate of transition in the

hyperbolic tangent function. Neglecting the predetermined parameter 𝑚, each parameter vector

now contains 7 parameters 𝒙𝒊 = [𝐴, 𝑎, 𝜏, 𝑓𝑐 , 𝜙, 𝜌, 𝜓 ]. The additional two parameters, 𝜌 and 𝜓

respectively control the asymmetry of the echo envelope and the frequency chirp factor. Since

the value of tanh(𝑚𝑡) varies from -1 to +1, the envelope function can also be expressed as:

𝑒𝑛𝑣(𝑡) = exp[−𝑎2(1 + 𝜌)𝑡2], 𝑡 < 𝜖

𝑒𝑛𝑣(𝑡) = exp[−𝑎2(1 − 𝜌)𝑡2], 𝑡 > 𝜖

(2.22)

Here 𝜖 is the transition period of the hyperbolic tangent function and is determined by the choice

of 𝑚. The asymmetric Gaussian chirplet model allows one to model ultrasonic echoes that are

both asymmetric and have a non-constant modulation frequency. Despite the enhanced

modelling flexibility, the asymmetric Gaussian chirplet model has not been widely adopted for

DBPR because the increased number of parameters makes it more difficult to obtain a stable

solution for the optimization problem shown in Eq. (2.19).

2.4.2 Sparsity-Promoting Algorithms

Looking at Eq. (2.19), we see that the reconstruction error 𝑒(𝑡) can be progressively reduced by

increasing the number of parametric echoes used to represent the signal 𝑠𝑖𝑔(𝑡). Although such

an approach is useful for ultrasonic data compression applications [38], it is not appropriate for

NDT applications because it can lead to the detection of extraneous echoes. False positive echoes

in NDT testing incur time and monetary costs as a more thorough scan or a destructive test

would need to be conducted to assess the condition of the test specimen. Consequently, for NDT

29

applications it is crucial to obtain a sparse solution which suppresses the occurrence of

extraneous echoes.

One of the methods to obtain a sparse solution is the L1-norm minimization approach introduced

in Section 2.3.3. When L1-norm minimization is used to solve for the parameters in DBPR, the

technique is also known as Basis Pursuit [37], [43]. The minimization objective of Basis Pursuit

can be expressed as:

𝐦𝐢𝐧𝒙𝑖

[∑ |𝑠𝑖𝑔(𝑡) − ∑ 𝑒𝑐ℎ𝑜(𝒙𝑖 , 𝑡)

𝑖

|

2

𝑡

+ 𝜎 ∑|𝐴𝑖|

𝑖

]

(2.23)

In Eq. (2.23), |𝐴𝑖| is the absolute value of the amplitude parameter of each echo identified. By

minimizing the sum of the amplitude parameters, one can reduce the number of echoes that are

used to represent the signal. Once again, the relative importance of the two terms in the

minimization objective can be adjusted by changing the value of 𝜎.

Another sparse-solution promoting algorithm is the Matching Pursuit technique [40], [44], [45].

At the first iteration of the matching pursuit algorithm, the 𝑒𝑐ℎ𝑜(𝒙1 , 𝑡) that best matches the

obtained signal is found by maximizing the inner product between the two:

𝐦𝐚𝐱𝒙𝒊

[∑𝑒𝑐ℎ𝑜(𝒙1 , 𝑡) ∙ 𝑠𝑖𝑔(𝑡)

|𝑒𝑐ℎ𝑜(𝒙1 , 𝑡)|2

𝑡

] (2.24)

The division in Eq. (2.25) is needed to normalize the energy of the echo. Notice that we use the

subscript 1 for the parameter vector because it is the first echo identified. After this echo is

found, it is subtracted from the signal and a residual signal 𝑒1(𝑡) remains.

𝑒1(𝑡) = 𝑠𝑖𝑔(𝑡) − 𝑒𝑐ℎ𝑜(𝒙1 , 𝑡) (2.25)

The process is then repeated by finding the next echo that has the largest inner product with the

residual echo. The Matching pursuit algorithm ends when the L2-norm the residual 𝑒𝑖(𝑡) is

below a threshold value. Compared to Basis Pursuit, Matching Pursuit is less computationally

expensive and is guaranteed to converge since it is always possible to find an echo that reduces

the residual signal. However, the algorithm is also “short-sighted” (only one echo is identified at

each iteration) and therefore its performance is often inferior than that of Basis Pursuit [37].

30

Chapter 3 Basis and Assumptions of TEI Algorithm

3.1 Frequency Domain Assumptions

During the development of an echo separation algorithm, it is necessary to introduce assumptions

of the expected properties of an ultrasonic echo. These assumptions enable us to correctly

decompose the ultrasound signal into its constituent echoes. In this section, we will first

introduce the frequency domain assumptions used in the TEI algorithm. The justifications and

the limits of applicability of these assumptions will be detailed in Sections 3.1.3 and 3.1.4.

3.1.1 Amplitude Profile Assumption

As will be shown in chapter 4, TEI is a frequency-domain algorithm where we iteratively

improve our estimates of the frequency amplitude profiles of the two constituent echoes. In order

to formulate the algorithm as an optimization problem, the amplitude profiles must be first

described as a parametric mathematical model.

After analyzing both the theoretical and experimental frequency response of piezoelectric

transducers, it was decided to model the amplitude response of each echo as an asymmetric Q-

Gaussian distribution. Compared to the standard Gaussian distribution, the Q-Gaussian has an

extra degree of freedom which allows it to vary the decay rate at the tails of a distribution [46].

This extra degree of freedom is important for modeling of the frequency amplitude profile of an

ultrasonic echo.

Amplitude Profile Assumption: The frequency amplitude profile of an ultrasonic echo can be

adequately modelled by an asymmetric Q-Gaussian distribution expressed as:

𝑓𝑜𝑟 𝜔 < 𝜔𝑐: 𝑀(𝜔) = 𝑆[1 − (1 − 𝑞1)𝑏1(𝜔 − 𝜔𝑐)2]1

1−𝑞1

𝑓𝑜𝑟 𝜔 > 𝜔𝑐: 𝑀(𝜔) = 𝑆[1 − (1 − 𝑞2)𝑏2(𝜔 − 𝜔𝑐)2]1

1−𝑞2

(3.1)

In Eq. (3.1), 𝑆 is the amplitude scaling parameter; 𝜔𝑐 is the center frequency; 𝑏1 and 𝑏2 are the

width scaling parameters and 𝑞1 and 𝑞2 are the “tail-heaviness” control parameters. When the

value of 𝑞 is less than 1, the distribution is less tail-heavy than the normal distribution and vice-

versa. The normal distribution is recovered when 𝑞 approaches 1.

31

By introducing independent width and tail-heaviness parameters below and above the center

frequency 𝜔𝑐, the mathematical expression in Eq. (3.1) can be used to model asymmetric

frequency-domain amplitude responses. With this definition, the frequency amplitude

information of each echo can be fully defined using six parameters 𝒑 = (𝑆, 𝜔𝑐, 𝑏1, 𝑏2, 𝑞1, 𝑞2).

Figure 3.1 shows an example of a Q-Gaussian distribution with 𝑆 = 1, 𝜔𝑐 = 4 [rad/s], 𝑏1 = 3 [s2],

𝑏2 = 2.5 [s2], 𝑞1 = 1, 𝑞2 = 2.

Figure 3.1: Asymmetric Q-Gaussian distribution

3.1.2 Phase Profile Assumption

In the frequency domain, the TEI algorithm assumes that one echo has an earlier arrival time for

all frequencies in which both echoes have significant spectral content. If the assumption is

satisfied, then one can unambiguously identify which echo arrives at an earlier time. This

assumption is important because for a non-dispersive medium the arrival time difference of two

echoes is directly proportional to the difference in their acoustic path travel distance. This time

difference can therefore be used to identify the exact location and size of a material flaw.

Using the well-known Fourier transform property that a positive time delay corresponds to an

increase of negative phase slope in the frequency domain [31], the assumption of sequential

spectral arrival time can be expressed as a phase slope inequality condition.

32

Phase Profile Assumption: Considering the frequency domain phase profiles of two echoes that

overlap in the time domain, the earlier arriving ultrasonic echo has a less negative phase slope

for all frequencies in which both echoes have significant spectral content:

𝑖𝑓 𝑒𝑐ℎ𝑜 𝐴 𝑎𝑟𝑟𝑖𝑣𝑒𝑠 𝑓𝑖𝑟𝑠𝑡:𝜕𝜃𝐴(𝜔)

𝜕𝜔>

𝜕𝜃𝐵(𝜔)

𝜕𝜔+ ∆𝑡𝑚𝑖𝑛

𝑖𝑓 𝑒𝑐ℎ𝑜 𝐵 𝑎𝑟𝑟𝑖𝑣𝑒𝑠 𝑓𝑖𝑟𝑠𝑡:𝜕𝜃𝐴(𝜔)

𝜕𝜔<

𝜕𝜃𝐵(𝜔)

𝜕𝜔− ∆𝑡𝑚𝑖𝑛

(3.2)

In Eq. (3.2), 𝜃𝐴(𝜔) and 𝜃𝐵(𝜔) are the frequency domain phase profiles of echo A and B and

∆𝑡𝑚𝑖𝑛 is the minimum allowable time separation between the two echoes. ∆𝑡𝑚𝑖𝑛 is added in Eq.

(3.2) because from a practical standpoint it is extremely difficult to separate two echoes which

have a time separation that can be infinitely small. In the simulation tests of the TEI algorithm

that will be presented in Chapter 5, the value of ∆𝑡𝑚𝑖𝑛 is set to be 1/2 of the inverse of the

estimated center frequency of the two echoes and represents half of a period of the characteristic

modulation frequency of the echoes. In other words, we assume that the smallest defect feature

size that we can detect is half of the wavelength of the center frequency of the transducer.

In actual applications, the phase slope inequality assumption presented above will be satisfied if

there exists sufficient time separation between the two echoes. In Chapter 5, we will explore how

the performance of the TEI algorithm is affected when the time separation between the two

echoes approaches the minimum time-separation ∆𝑡𝑚𝑖𝑛.

3.1.3 Justification of Amplitude Profile Assumption

Theoretical Justifications

In Section 2.2.1, we introduced two one-dimensional piezoelectric models. For a lightly loaded

transducer, the Van Dyke approximate model can be used to obtain an analytical expression of

the two-way voltage-to-voltage transducer transfer function. The transfer function expression

was shown in Eq. (2.6) and will be repeated here:




1 +1

𝑎2 (𝜔 − 𝜔𝑑)2

(3.3)

33

Comparing Eq. (3.3) with the asymmetric Q-Gaussian distribution shown in Eq. (3.2), it can be

easily seen that the Van Dyke transfer function can be exactly modelled by a symmetric Q-

Gaussian distribution with 𝑏1 = 𝑏2 =1

𝑎2 and 𝑞1 = 𝑞2 = 2. Therefore, we can conclude that the

two-way transfer function of a lightly-loaded transducer can be exactly described by our

amplitude profile model.

For highly-damped broadband transducers typically used in arrival time sensitive NDT

applications, the Van Dyke approximate model is no longer adequate and the KLM model can be

used. As detailed in Section 2.2.1, the two-way transfer function of the KLM model can be

calculated through a cascade multiplication of transmission matrices. Since many of these

transmission matrices contain complex frequency-dependent terms, a convenient analytical

transfer function expression cannot be obtained except for the simplest configurations. However,

one can still evaluate the validity of the asymmetric Q-Gaussian amplitude profile model using a

demonstrative model of a broadband piezoelectric transducer.

For this demonstrative model, we used the extended KLM model described in Section 2.2.1 to

model the frequency response of a broadband piezoelectric transducer. The geometric and

material properties of the model are shown in Table 3.1. The values of the parameters in Table

3.1 were selected based on piezoelectric transducer design guidelines [12]. The center frequency

of the transducer is designed to be at 2 MHz. By substituting these properties into the KLM

model, we can obtain its two-way transfer function. The fundamental harmonic impulse response

of the KLM model is shown in Figure 3.2 and its Fourier transform is shown in Figure 3.3. Only

the fundamental harmonic is displayed because in typical NDT applications, the higher

harmonics of the transducer are suppressed by the bandwidth of the excitation pulse and

frequency-dependent attenuation effects in the propagation path. Details of implementation of

this model can be found in Appendix 2 and 3.

34

Property Value

Impedance of Piezoelectric Material PZT-5H: Z = 34.6 MRayl

Material Coupling Factor 0.49

Piezo Clamped Permittivity 1.3 × 10-8 F/m

Piezo Density 7500 kg/m3

Piezo Active Area 1 cm2

Piezo Thickness 1 mm

Backing Material Assumption Perfectly Matched Layer

Matching Layer Acoustic Impedance Z = 40 MRayl

Matching Layer Density 9000 kg/m3

Matching Layer Thickness 550 µm

Impedance of Propagation Medium Steel: Z = 46 MRayl

Electrical Impedance of Pulser 50 Ω

Electrical Impedance of Receiver 50 Ω Table 3.1: Properties for broadband KLM simulation

Figure 3.2: First harmonic impulse response of KLM model

Figure 3.3: KLM model of broadband transducer

35

Looking at Figure 3.3, we see that the amplitude profile of a simple one-dimensional transducer

as simulated by the KLM model can be accurately approximated by an asymmetric Q-Gaussian

distribution. We can also see that the phase profile predicted by the KLM model is

approximately linear in the bandwidth of the first harmonic response.

Experimental Verification

In addition to the theoretical justifications introduced above, it is also worthwhile to verify the

validity of our frequency domain assumptions with experimental data. For this purpose, we

examine the backwall obtained from a TOFD experiment. In this experiment, we used a pair of 5

MHz, 3 mm diameter piezoelectric transducers attached to 60° Rexolite1 wedges. The small

active area diameter is chosen to provide divergent beams that cover a wide scan area. The test

setup schematic is shown in Figure 3.4. Following TOFD protocols [47], the transducers are

spaced horizontally such that the intersection of the central propagation axes of the transducers

occurs at the bottom 1/3 of the sample thickness as shown in Figure 3.4. The shaded region in

Figure 3.4 indicates the -6dB beam spread at 5 MHz.

Figure 3.4: Pitch-catch backwall echo acquisition configuration

The acquired backwall echo and its Fourier transform are shown in Figure 3.5 and Figure 3.6.

From Figure 3.6, we see that the amplitude peak of the backwall echo is located at approximately

3.8 MHz. This is lower than the expected center frequency of 5 MHz and is likely caused by

1 Rexolite is a trademark plastic made by C-Lec Plastics Inc. It is a material often used for acoustic lenses due to its

low acoustic attenuation coefficient and stable chemical properties.

36

frequency downshifting due to off-axis diffraction (see Section 2.2.2 for details). Despite the

influence of diffraction effects, we see that the amplitude profile can still be accurately modelled

by an asymmetric Q-Gaussian distribution. This experimental result is consistent with the

frequency domain assumptions employed by the TEI algorithm. In Figure 3.6, we also see that

the phase profile is again approximately linear within the -6dB bandwidth of the signal.

Figure 3.5: Experimental pitch-catch backwall echo

Figure 3.6: Fourier transform of experimental pitch-catch backwall echo

3.1.4 Applicability Limits of Echo Assumptions

The assumption justifications provided in the previous sections are mainly based on the transfer

function of the piezoelectric transducer. However, as explained in Section 2.2.2, the complete

transfer function model of an ultrasonic echo includes other contributing factors such as

37

attenuation, dispersion, transducer wavefield diffraction and flaw scattering. In this section, we

will discuss the applicability limits of our echo assumptions with respect to these contributing

factors.

Attenuation

As explained in Section 2.2.2, ultrasound attenuation in steel follows a near quadratic frequency

dependence in the Rayleigh scattering regime. Consequently, higher frequencies would be

attenuated at a greater rate and the resultant frequency amplitude profile would be asymmetric.

This type of smooth amplitude profile asymmetry can be adequately modelled by an asymmetric

Q-Gaussian amplitude model having independent width and tail-heaviness parameters below and

above the center frequency. Since most engineering materials including metals, polymers and

plastics demonstrate a smooth power law acoustic attenuation frequency dependence [2],

acoustic attenuation should not be a limiting factor in the application of the TEI algorithm.

Dispersion

TEI is designed as an algorithm to enhance the axial resolution in ultrasonic time-of-flight based

size estimates of defects and assumes that the speed of sound in the test sample is constant. If the

speed of sound were not constant, there would be distortion of the waveform group delay and it

would be difficult to relate the arrival time of the echo to the physical location of a flaw. In

Section 2.2.2, we showed that dispersion effects in bulk wave ultrasound testing of low-carbon

steel specimens are negligible. Consequently, TEI is applicable for longitudinal and shear wave

inspection of steel test pieces. However, the constant speed of sound assumption of TEI will not

be applicable for inspection of highly dispersive materials or for guided wave applications where

the wave propagation mode is inherently dispersive [26].

Transducer Wavefield Diffraction

As shown in Figure 2.18 of Section 2.2.2, the frequency response of transducer wavefield

diffraction can act as a low-pass filter if the point of observation is displaced laterally from the

central axis of the transducer. If the bandwidth of the transducer is only marginally higher than

the pass band of the diffraction low-pass filter, transducer wavefield diffraction will cause a

downshift of the central frequency of the transducer as observed in the pitch-catch backwall echo

38

of Section 3.1.3. For such cases the frequency assumptions of the TEI algorithm can still be valid

as demonstrated in the experimental backwall echo demonstrated in Section 3.1.3.

However, if the center frequency of the transducer is much higher than the pass band of the

diffraction filter, the wavefield diffraction response would cause significant distortions to the

phase and amplitude profiles of the ultrasound echo. To demonstrate this idea, in Figure 3.7 we

multiply a simulated transducer response which has a center frequency of 4 MHz with the

waveform diffraction frequency response shown in in Figure 2.18 (calculated using a = 4 mm, z

= 60 mm, and r = 15 mm).

Figure 3.7: Echo distortion due to wavefield diffraction

In Figure 3.7, we see that wavefield diffraction has created a double peak in the resultant echo

amplitude profile. In addition, the resultant echo phase profile is also heavily distorted.

Consequently, in such cases the asymmetric Q-Gaussian amplitude model of TEI would not be

able to accurately portray the echo amplitude profile. Distortion due to wavefield diffraction can

be reduced by using a small aperture transducer which has a wide beam spread pattern.

Flaw Scattering

The geometry of the scattering defect can have a significant impact on the shape of the return

echo. For example, for a cylindrical or spherical void with a radius comparable to the wavelength

39

of the incoming ultrasonic wave, the scattered wave response would have a creeping wave

component that has travelled around the round defect [1]. Such scatterers would cause a

substantial distortion in both the amplitude and phase profiles of the return echo. Consequently,

the frequency assumptions of the TEI algorithm are only applicable for the ultrasonic testing of

scattering defects that have an approximately frequency-independent scattering response within

the bandwidth of the transducer. Examples of frequency independent scattering defects include

sharp cracks and plane reflectors [28], [27]. NDT applications that are expected to contain these

defects include among others Time of Flight Diffraction [3] and normal-incidence testing of

adhesive bonds between thin structures [4].

40

Chapter 4 Trigonometric Echo Identification Algorithm

4.1 Algorithm Overview

TEI is an iterative algorithm where the amplitude control parameters are repeatedly updated in

order to obtain optimal phase response. The TEI algorithm begins with an initial estimate of the

amplitude control parameters of the two echoes. These amplitude control parameters are then

substituted into Eq. (3.1) to create the initial trial amplitude profiles. Next, the corresponding

phase profiles are reconstructed according to the phase reconstruction procedure that will be

introduced in Section 4.2. After the phase profiles are calculated, they are evaluated for violation

of the phase assumption that was introduced in Chapter 3. The recovered echoes are also

evaluated according to additional optimality metrics that will be introduced in this Section 4.3.1.

If the phase profiles are optimal according to these criteria, the algorithm is considered

converged. If the phase profiles are suboptimal, the amplitude control parameters of the two

echoes are updated, and so begins the second iteration of the algorithm. A flowchart of the TEI

algorithm is shown in Figure 4.1.

Figure 4.1: Flowchart of TEI algorithm

41

4.2 Trigonometric Phase Profile Reconstruction

At each frequency, the spectral information of a signal can be represented by a complex vector

with a length equal to its amplitude and an orientation equal to its phase value. Using this

concept, the Fourier transforms of two overlapping echoes (𝐸𝐶𝐻𝑂𝐴(𝑓), 𝐸𝐶𝐻𝑂𝐵(𝑓)) and the

resultant total signal 𝑆𝐼𝐺(𝑓) can be graphically represented by a vector addition diagram as

shown in Figure 4.2. Note that in this section we do not explicitly show the frequency

dependence of the amplitude and phase values for convenience in representation.

Figure 4.2: Vector representation of overlapping echoes

In Figure 4.2, 𝑀𝐴 and 𝑀𝐵 are the amplitudes of the overlapping echoes while 𝑀𝑇 is the

amplitude of the total signal. Similarly, 𝜃𝐴 and 𝜃𝐵 are the phases of the overlapping echoes while

𝜃𝑇 is the phase of the total superimposed signal. Note that we adopt the convention that a

positive phase change is an angular displacement in the counter-clockwise direction. The values

𝑀𝑇 and 𝜃𝑇 are obtained by the Fourier transform of the recorded time-domain ultrasonic data

and therefore are known for all frequencies within the bandwidth of the total signal. It should be

stressed that Figure 4.2 only shows the magnitude and phase information of the two echoes and

the total signal at one particular frequency. As one sweeps through the frequency range, the

amplitude (length) and phase (orientation) values of each component would vary.

From Figure 4.2, we can see that if the values of 𝑀𝐴 and 𝑀𝐵 are known (or estimated), the

interior angles (𝛼, 𝛽, 𝛾) of the vector addition triangle can be solved using the cosine law. Once

42

the interior angles are solved, the phase angles of the two echoes can be easily calculated. For

example, in the configuration shown in Figure 4.2, 𝜃𝐴 = 𝜃𝑇 + 𝛽 and 𝜃𝐵 = 𝜃𝑇 − 𝛼.

However, the vector addition orientation shown in Figure 4.2 is not unique. For a given set of

amplitude values (𝑀𝐴, 𝑀𝐵, 𝑀𝑇), there exists two possible phase configurations. Figure 4.3 shows

an equally valid vector addition configuration with the same set of component amplitude values

(𝑀𝐴, 𝑀𝐵, 𝑀𝑇). In this phase configuration, it can be seen that 𝜃𝐴 = 𝜃𝑇 − 𝛽 and 𝜃𝐵 = 𝜃𝑇 + 𝛼.

Figure 4.3: Alternative vector addition configuration

In order to select the correct vector addition configuration, one also needs to know the relative

rotation of the echo phasor vectors (𝐸𝐶𝐻𝑂𝐴(𝜔), 𝐸𝐶𝐻𝑂𝐵(𝜔)) and whether the interior angle

between the two vectors is increasing or decreasing. These two attributes along with the vector

amplitude values (𝑀𝐴, 𝑀𝐵, 𝑀𝑇) are sufficient to define a unique vector addition configuration. To

determine the change in the interior angle, one can use trigonometry to calculate the value of 𝛾

for the frequency range of interest and then calculate its derivative ∂γ

∂ω. In contrast, the relative

rotation of the echo phasor vectors can be determined using the sequence of arrival time of the

two echoes.

From the phase assumption expressed in Eq. (3.2), we see that the phase slope of the earlier

arriving echo is less negative than the phase slope of the second echo. Since the phase slope is by

definition the rate of change of the phase angle, in the complex plane it can be represented by the

rate of rotation of the complex phasor vector in the counter-clockwise direction. Consequently,

43

the earlier arriving echo would have a phasor vector that rotates counter-clockwise relative to the

phasor vector of the second echo.

Summarizing the concepts described above, we can develop the phase profile reconstruction

chart shown in Table 4.1.

Echo A arrives earlier Echo B arrives earlier

∂γ

∂ω> 0

𝜃𝐴 = 𝜃𝑇 − 𝛽

𝜃𝐵 = 𝜃𝑇 + 𝛼

𝜃𝐴 = 𝜃𝑇 + 𝛽

𝜃𝐵 = 𝜃𝑇 − 𝛼

∂γ

∂ω< 0

𝜃𝐴 = 𝜃𝑇 + 𝛽

𝜃𝐵 = 𝜃𝑇 − 𝛼

𝜃𝐴 = 𝜃𝑇 − 𝛽

𝜃𝐵 = 𝜃𝑇 + 𝛼

Table 4.1: Phase reconstruction chart

From Table 4.1, we see that the two separate sets of reconstructed phase profiles are obtained

depending on which echo arrives first. Since it is not possible to know in advance the sequence

of echo arrival, we need to examine the two sets of reconstructed phase profiles to select the

correct set of phase profiles. The phase profile selection procedure will be described in Section

4.3.2.

4.3 Components of TEI Algorithm

4.3.1 Echo Optimality Metrics

As seen from the flowchart shown in Figure 4.1, one needs to assess both the optimality of the

reconstructed ultrasonic echoes and their violation of the phase slope inequality in order to

determine the state of convergence at each iteration of the TEI algorithm. Assessment of echo

optimality is needed because the phase slope inequality condition imposes only a restriction on

the relative arrival times of the two echoes. In order to obtain echoes with the desired shapes, it is

necessary to introduce additional echo optimality conditions.

The exact form of the echo optimality metric employed is chosen depending on the prior

knowledge available for the return ultrasonic echoes. If the approximate echo shape for one or

both of the return echoes is known, a cross-correlation based method can be used to assess the

similarity between the recovered and reference echoes. Mathematically this can be formulated

as:

44

optimality = −max [𝐶𝐶(𝑟𝑒𝑓1(𝑡), 𝑒𝑐ℎ𝑜1(𝑡))

√∑ 𝑒𝑐ℎ𝑜1(𝑡)2 ∙ ∑ 𝑟𝑒𝑓1(𝑡)2𝑡𝑡

] −max [𝐶𝐶(𝑟𝑒𝑓2(𝑡), 𝑒𝑐ℎ𝑜2(𝑡))


] (4.1)

In Eq. (4.1), the subscripts 1 and 2 stand for the time order of echo arrival and

𝐶𝐶(𝑟𝑒𝑓(𝑡), 𝑒𝑐ℎ𝑜(𝑡)) is the cross-correlation operation between the reference echo and the

recovered echo. The denominators in Eq. (4.1) are required to normalize the energy of the cross-

correlation. We take the maximum value of each of the normalized cross-correlation functions,

which corresponds to the time-shift between the reference and the recovered echo. A negative

sign is required because optimization problems are typically formulated as minimization

problems. Although Eq. (4.1) is shown to use two references, the cross-correlation optimality

metric can also be applied if only one reference is known (either the first or the second arriving

echo).

If reference waveforms estimates are not available, more general echo optimality metrics can be

used. From Figure 3.3 and Figure 3.6, we see that both the simulation and experimental Fourier

transforms of broadband piezoelectric transducers exhibit near linear phase responses within the

transducer bandwidth. This is not a mere coincidence but in fact a conscious design goal of

piezoelectric transducer designers to obtain a near linear phase response to reduce echo shape

distortions [6]. For this reason, the linearity of the echo phase response could be used as an echo

optimality metric. The linearity of the phase response can be evaluated using the following

statistical measurement of the standard deviation of the phase slope:

𝑛𝑜𝑛𝑙𝑖𝑛𝑒𝑎𝑟𝑖𝑡𝑦(𝜃(𝜔)) =∫|𝜃′(𝜔) + 𝑡|𝑀(𝜔)𝑑𝜔

∫ 𝑀(𝜔)𝑑𝜔

(4.2)

In Eq. (4.2), 𝑀(𝜔) is the amplitude profile of the echo and 𝑡 is negative of the spectrally-

averaged phase slope and can also be interpreted as the spectrally-averaged echo arrival time:

𝑡 = −∫ 𝜃′(𝜔)𝑀(𝜔)𝑑𝜔

∫ 𝑀(𝜔)𝑑𝜔

(4.3)

Consequently, an alternative optimality metric can be formulated as the sum of the phase non-

linearity of the two phase profiles:

optimality = 𝑛𝑜𝑛𝑙𝑖𝑛𝑒𝑎𝑟𝑖𝑡𝑦(𝜃𝐴(𝜔)) + 𝑛𝑜𝑛𝑙𝑖𝑛𝑒𝑎𝑟𝑖𝑡𝑦(𝜃𝐵(𝜔)) (4.4)

45

The echo optimality metrics differ from the amplitude and phase assumptions listed in chapter 3

in that they are not conditions that are strictly enforced. The first priority of the TEI algorithm is

to satisfy the amplitude and phase assumptions listed in chapter 3. Once these conditions are

satisfied, TEI would attempt to minimize the selected optimality metric in order to recover

echoes with the desired shapes.

4.3.2 Determination of the Correct Set of Phase Profiles

As shown in Table 4.1 of Section 4.2, there exists two possible sets of reconstructed phase

profiles depending on which of the two echoes arrives first. Since the order of echo arrival

cannot be determined a priori, we must examine the two sets of reconstructed phase profiles

after each iteration to determine which set should be selected.

The TEI algorithm selects the set of phase profiles which has the smallest violation of the phase

slope inequality assumption that is used for its reconstruction. For example, if the first set of

phase profiles are reconstructed using the assumption that echo A arrives first, then from Eq.

(3.2) we see that the phase profile of echo A should have a less negative phase slope compared to

the phase profile of echo B. If the reconstructed phase profiles show that echo A has a more

negative phase slope than echo B, then the assumption used for the reconstruction of the phase

profiles is violated.

For any one set of phase profiles, the violation of its reconstruction phase slope inequality

assumption can be calculated using the following metric:

𝑉𝐼𝑂𝐿 = ∑ 𝑤𝑖𝑛𝑑𝑜𝑤(𝜔) ∙ |max [0, (𝜕𝜃2(𝜔)

𝜕𝜔−

𝜕𝜃1(𝜔)

𝜕𝜔)]|

0

𝜔

(4.5)

In Eq. (4.5), 𝜃1(𝜔) and 𝜃2(𝜔) are the unwrapped phase profile of first and second echoes. The

max[ ] function is used to avoid penalizing frequency points that satisfy the phase slope

assumption. The presence of a window function is needed to limit the applicability of the

constraint to frequencies for which the two echoes overlap. We also take the L0 norm of the

max[ ] function because the number of violation points is a more stable measurement of the

phase slope violation. (If we instead took the L2 norm of the max[ ] function in Eq. (4.5), the

stability of the metric would be greatly affected by the unwrapping errors that occur near the 0

and 2π phase crossover points.)

46

By selecting the set of phase profiles with the smallest violation, we can determine which arrival

time assumption is actually correct (i.e. which echo arrives earlier).

4.3.3 Phase Slope Inequality Constraint

To ensure that the phase slope inequality assumption of Eq. (3.2) is satisfied, we need to enforce

a constraint such that the violation of Eq. (3.2) must be less than a small tolerance value:

𝑉𝐼𝑂𝐿[𝜃1(𝜔), 𝜃2(𝜔), ∆𝑡𝑚𝑖𝑛 ] < 𝑡𝑜𝑙𝑒𝑟𝑎𝑛𝑐𝑒 (4.6)

Where the violation of Eq. (3.2) can be calculated as:

𝑉𝐼𝑂𝐿 = ∑ 𝑤𝑖𝑛𝑑𝑜𝑤(𝜔) ∙ |max [0, (𝜕𝜃2(𝜔)

𝜕𝜔−

𝜕𝜃1(𝜔)

𝜕𝜔) + ∆𝑡𝑚𝑖𝑛]|

0

𝜔

(4.7)

Note that the violation metric of Eq. (4.7) is a modified version of Eq. (4.5) with the addition of

the minimum allowable time separation ∆𝑡𝑚𝑖𝑛. The reason why ∆𝑡𝑚𝑖𝑛 is included in Eq. (4.7)

but not in Eq. (4.5) is because these two phase slope violation metrics serve different purposes.

Equation (4.5) is used to select the correct set of phase profiles and in the process also determine

which of the two echoes arrive first. After we establish the time order of echo arrival, Eq. (4.7)

can then be used to verify whether the reconstructed phase responses actually satisfy the TEI

phase profile assumptions.

4.4 Implementation as Constrained Optimization Problem

4.4.1 Constrained Optimization Formulation

Having introduced the phase slope inequality violation constraint of Eq. (4.6), we are finally in

the position to formulate the TEI algorithm as a constrained-optimization problem to satisfy all

frequency-domain assumptions while minimizing the echo optimality metric. A flexible method

to solve a constrained optimization problem is the Augmented Lagrangian (ALAG) method.

ALAG transforms a constrained problem into a series of unconstrained optimization problems

through the use of additional penalty terms that are proportional to the violation of any

constraints [48].

47

With the adoption of ALAG, the TEI algorithm is divided into inner and outer loops. In the inner

loop, we solve an unconstrained optimization problem with a cost function that contains both the

echo optimality metric and the violation of the phase slope inequality:

𝐶𝑜𝑠𝑡 = optimality −χ2

4𝜇+ 𝜇 [max (0, 𝑉𝐼𝑂𝐿 +

𝜒

2𝜇)]

2

(4.8)

In Eq. (4.8), 𝑉𝐼𝑂𝐿 is the phase slope inequality violation metric expressed in Eq. (4.7) and 𝜒 and

𝜇 are respectively the Lagrange multiplier and penalty parameters. The values of 𝜒 and 𝜇 do not

change within the inner loop. Once the inner loop is converged, the ALAG algorithm would

check for the value of 𝑉𝐼𝑂𝐿. If the value of 𝑉𝐼𝑂𝐿 is less than the tolerance, the outer loop and

hence the entire TEI algorithm is considered converged. Otherwise, the values of 𝜒 and 𝜇 will be

adjusted in the outer loop and we will go back inside the inner loop to solve a new unconstrained

optimization problem with an adjusted cost function.

Since any inner loop solution that has a 𝑉𝐼𝑂𝐿 value less than the tolerance is considered the final

solution, it is important that we initialize the values of 𝜒 and 𝜇 to small values. This can ensure

we do not place too large an initial penalty on 𝑉𝐼𝑂𝐿 and obtain a suboptimal solution that

prematurely ends the ALAG algorithm. By using small initial values, we can allow the ALAG

algorithm to update the values of 𝜒 and 𝜇 and obtain a more optimal solution that minimizes the

optimality metric while ensuring 𝑉𝐼𝑂𝐿 is less than the tolerance.

The following pseudo-code shows the constrained-optimization implementation of the TEI

algorithm using ALAG. Again note that the inner while-loop is the unconstrained optimization

problem and the outer while-loop updates the 𝜒 and 𝜇 parameters.

48

# Begin Procedure

1 Initialize 𝒑𝑨, 𝒑𝑩, 𝜒, 𝜇

2 While (𝑉𝐼𝑂𝐿 > tolerance)

3 While (not converged)

4 Generate 𝑀𝐴(𝜔), 𝑀𝐵(𝜔) by substituting 𝒑𝑨, 𝒑𝑩 into Eq. (3.1)

5 Reconstruct [𝜃𝐴(𝜔), 𝜃𝐵(𝜔)](1) and [𝜃𝐴(𝜔), 𝜃𝐵(𝜔)](2)

6 Select [𝜃𝐴(𝜔), 𝜃𝐵(𝜔)](𝑐𝑜𝑟𝑟𝑒𝑐𝑡) using 𝑉𝐼𝑂𝐿[𝜃𝐴(𝜔), 𝜃𝐵(𝜔)] of Eq. (4.5)

7 Calculate optimality using Eq. (4.1) or Eq. (4.4)

8 Calculate 𝑉𝐼𝑂𝐿[𝜃𝐴(𝜔), 𝜃𝐵(𝜔), ∆𝑡𝑚𝑖𝑛 ] using Eq. (4.7)

9 Calculate 𝐶𝑜𝑠𝑡[optimality, 𝑉𝐼𝑂𝐿, 𝜒, 𝜇 ] using Eq. (4.8)

10 Check convergence

11 Update 𝒑𝑨, 𝒑𝑩

12 End While

13 Update 𝜒, 𝜇 using 𝑉𝐼𝑂𝐿[𝜃𝐴(𝜔), 𝜃𝐵(𝜔), ∆𝑡𝑚𝑖𝑛 ] in ALAG

14 End While

# End Procedure

4.4.2 Implementation Details

The final TEI algorithm is implemented in the MATLAB programming environment. Since

MATLAB is optimized for matrix and vector operations, code vectorization is employed

extensively in the implementation in order to improve the performance of the TEI algorithm.

In the time domain, each simulated test signal is composed of 2048 data points with a time step

of 10 ns. With these time sampling settings, the corresponding Fourier transform would have a

frequency resolution of 48.8 kHz and a maximum frequency of 51 MHz. As will be shown in

Chapter 5, the simulated echoes have center frequencies near 3 MHz and -6dB percentage

bandwidths of approximately 50%. Consequently, the chosen sampling settings ensure that we

can utilize approximately 100 frequency domain data points to represent the amplitude and phase

profiles.

For experimental signals, the sampling time is determined by the data acquisition rate of the

hardware. Since the TEI algorithm can only handle two overlapping echoes, experimental signals

must be time-windowed to remove additional echoes. In the current implementation, a Tukey

window with 50% taper width (shown in Figure 4.4) is chosen for windowing the two

overlapping echoes. A Tukey window is used because it can suppress transition side-lobes

without affecting the amplitude at the center of the signal. After windowing the overlapping

echoes, the signal is zero-padded so that it contains 2048 data points. Zero-padding is used to

ensure that there are sufficient data points in the frequency domain to perform the TEI algorithm.

49

Figure 4.4: 50% taper Tukey window

For transforming the time-domain data into the frequency domain, we use the built-in Fast

Fourier Transform (FFT) function of MATLAB. The FFT algorithm in MATLAB is internally

based on the FFTW library [49]. The FFTW library automatically chooses the Fourier transform

method which is expected to provide the best performance depending on the processing

hardware and the length of the time series N. However, all Fourier methods employed by the

FFTW library have computational complexity of O(N log N).

To solve the unconstrained optimization problem in the inner-loop of the TEI algorithm, we use

the Covariance Matrix Adaptation Evolution Strategy (CMA-ES) optimization method [50].

CMA-ES is chosen as the optimization method because it does not depend on the gradient of the

cost function. Since the cost function shown in Eq. (4.8) consists of both the echo optimality

metric and the phase slope inequality violation, the gradient may not be continuous in the

solution space. Another advantage of CMA-ES is that it is a population-based approach, meaning

at each iteration the cost function is evaluated at multiple points. For this reason, CMA-ES is less

sensitive to the initial guess of the optimization parameters.

However, the use of CMA-ES must be conducted carefully. Since CMA-ES is a probability-

based algorithm, the exact search locations at each iteration are randomly distributed based on a

calculated multi-variate probability density distribution. Although this randomness allows the

CMA-ES algorithm to search through a multi-dimensional space efficiently [51], it can also lead

to different solutions for the same input parameters. In chapter 5 we will discuss the

methodology we employed to improve the robustness of our solutions.

50

4.5 Summary of Novelty and Advantages of the TEI Algorithm

To our knowledge, TEI is the only echo separation algorithm that calculates the phase

information using trial amplitude profiles and the frequency transform of the acquired signal.

Due to this important distinction, TEI possesses the following advantages:

• Compared to DBPR using the same number of optimization parameters, TEI would have

smaller echo reconstruction errors because the phase information is calculated from the

frequency transform of the acquired signal

• Compared to DBPR using the same number of optimization parameters, TEI can describe

more complex ultrasonic waveforms because the variable phase profiles offer extra

degrees of freedom

• Since phase information is adapted to the acquired data, TEI performs more reliably

when the echo waveforms are not perfectly described by the mathematical form of the

model (i.e. the parametric model used to describe the frequency domain amplitude

profiles)

However, when compared to DBPR, the new TEI algorithm also possesses two disadvantages:

• Since TEI is formulated as a constrained optimization problem, for the same number of

optimization parameters it would require more iterations to converge compared to

DBPR, which for the special case of two overlapping echoes is formulated as an

unconstrained optimization problem. However, the difference in convergence time can

be reduced by having suitable initial values of the Lagrange multiplier and penalty

parameters 𝜒 and 𝜇.

• Since TEI relies on phase reconstruction using trigonometric relationships, the algorithm

would not work properly if the spectral content of the two echoes are vastly different

from each other. However, for such situations a time-frequency transform such as the

continuous wavelet transform [7] can be easily applied to separate the two echoes.

51

Chapter 5 Results and Discussions

5.1 Simulation Tests and Comparison Benchmark

Once the TEI algorithm was implemented in MATLAB, simulation tests were conducted to

assess the performance of this novel technique. Since simulation signals were created with

known parameters, it was possible to determine precisely both the timing and reconstruction

errors of the algorithm. To benchmark the performance of the TEI algorithm against existing

techniques, the DBPR technique was also implemented in MATLAB. In particular, we

implemented a DBPR model that is a modified version of Eq. (2.21).

𝑒𝑐ℎ𝑜(𝒙 , 𝑡) = 𝐴 ∙ 𝑒𝑛𝑣 (t − τ)cos[2𝜋𝑓𝑐(𝑡 − 𝜏) + 𝜙]


(5.1)

This particular model was chosen because it uses six parameters, 𝒙 = [𝐴, 𝑎, 𝜏, 𝑓𝑐, 𝜙, 𝜌 ], to

describe each echo. Using the same number of parameters for both TEI and DBPR allows a more

direct comparison of the two techniques. Looking at Eq. (5.1), we see that this particular DBPR

model can describe ultrasonic echoes with asymmetric envelopes but cannot accurately model

echoes with non-constant modulation frequency.

It should be noted that DBPR is a time-domain based algorithm while TEI is a frequency-domain

based algorithm. However, the relative performance difference between the two algorithms can

still be assessed with the use of appropriate performance metrics. In this study, percentage echo

timing and reconstruction errors were selected as the critical performance metrics. To obtain the

arrival time of the extracted echoes, we used the spectrally-averaged arrival time as defined in

Eq. (4.3). Using the spectrally-averaged arrive time, we calculated the percentage echo timing

error as:

%𝐸𝑟𝑟𝑜𝑟(𝑡𝑖𝑚𝑖𝑛𝑔) =(|𝑡2 − 𝑡1|𝑐𝑎𝑙𝑐. − |𝑡2 − 𝑡1|𝑎𝑐𝑡𝑢𝑎𝑙)

|𝑡2 − 𝑡1|𝑎𝑐𝑡𝑢𝑎𝑙×100

(5.2)

In Eq. (5.2), 𝑡1 and 𝑡2 are respectively the spectrally-averaged arrival times of the first and

second echoes. To calculate the percentage reconstruction error, we used the following

definition:

52

%𝐸𝑟𝑟𝑜𝑟(𝑟𝑒𝑐𝑜𝑛𝑠𝑡. ) =

∑ [𝑠𝑖𝑔(𝑡) − (𝑒𝑐ℎ𝑜1(𝑡) + 𝑒𝑐ℎ𝑜2(𝑡))𝑐𝑎𝑙𝑐.

]2

𝑡

∑ 𝑠𝑖𝑔(𝑡)2𝑡

×100

(5.3)

The denominator of Eq. (5.3) is needed to normalize the reconstruction error by the energy of the

total signal.

In the implementation of DBPR, we use the Eq. (2.19) as the optimality metric in the resultant

unconstrained optimization problem. To ensure that the TEI and DBPR algorithms were

compared objectively, we used the same CMA-ES optimization solver for solving the

optimization problems in both algorithms. In addition, we selected the same population size

(500) and the same maximum number of iterations (350) for both techniques. In CMA-ES,

population size refers to the number of search locations at every iteration. Since TEI is solved

using the ALAG approach, the maximum number of iterations refers to the number of iterations

inside the inner loop.

As mentioned in Chapter 4, CMA-ES is a probability-based algorithm and therefore the

converged solution can vary even if the same starting parameters are used. To improve the

quality of the converged solutions, we ran the CMA-ES solver 6 times for each test configuration

and chose the solution with the lowest value for the cost function of Eq. (4.8). The reason why

we selected the best solution is that we are interested in the finding the global minimum in the

parameter solution space. If we instead take the average solution, we would be averaging a

number of local minima which may not provide us with physically meaningful results.

At each restart of the CMA-ES solver, we created a vector of random numbers within the search

space to use as the initial values for the optimization parameters. Random numbers were used

because CMA-ES is a population-based probabilistic optimization method that is not sensitive to

the initial guess for the parameters. For TEI the optimization parameters were the frequency

domain Q-Gaussian distribution amplitude parameters for the two echoes 𝒑 =

[(𝑆, 𝜔𝑐, 𝑏1, 𝑏2, 𝑞1, 𝑞2)𝐴, (𝑆, 𝜔𝑐, 𝑏1, 𝑏2, 𝑞1, 𝑞2)𝐵]; for DBPR the optimization parameters were the

time domain echo parameters 𝒙 = [(𝐴, 𝑎, 𝜏, 𝑓𝑐, 𝜙, 𝜌 )𝐴, (𝐴, 𝑎, 𝜏, 𝑓𝑐, 𝜙, 𝜌 )𝐵]. Since we used random

vectors as the initial parameter values, the converged solutions for each run can be considered

statistically-independent observations of a random variable. In other words, the converged

solution from one run is independent of the results from another run. For this reason, each

53

converged solution would have a 50% chance of being better than the mean of the solutions. By

repeating the same solver 6 times, the probability of obtaining a converged solution that is better

than the mean solution can be calculated using the binomial distribution:

𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑡𝑦(𝑠𝑜𝑙. > 𝑚𝑒𝑎𝑛) = 1 − 0.56 = 98.4% (5.4)

The probability of obtaining a better solution of course increases with the number of restarts.

However, setting the number of restarts to 6 offers a good compromise between the

computational time and the quality of the solution.

5.2 Synthetic Echoes with Symmetric Envelope

For the first set of simulation tests, we used synthetic echoes that are of the form:

𝑒𝑐ℎ𝑜(𝒙 , 𝑡) = 𝐴 exp[−𝑎2(𝑡 − 𝜏)2] cos[2𝜋𝑓𝑐(𝑡 − 𝜏) + 𝜙] (5.5)

Consequently, each echo for this set of simulation experiment was a constant frequency

oscillation multiplied by a Gaussian envelope. Comparing Eq. (5.5) with Eq. (5.1), we see that

the Gaussian modulated echoes used are simply a subset of the DBPR model. For this reason, in

theory DBPR should be able to identify the two echoes perfectly without the presence of noise.

In addition, the Gaussian modulated echoes of Eq. (5.5) should also perfectly satisfy the

frequency domain assumptions of the TEI algorithm. The Fourier transform of Eq. (5.5) can

easily be calculated to be [31]:

𝐸𝐶𝐻𝑂(𝒙 , 𝑓) =√𝜋

𝑎𝐴 ∙ exp [−

𝜋2(𝑓 − 𝑓𝑐)2

𝑎2] ∙ exp[−𝑗 (2𝜋(𝑓 − 𝑓𝑐)𝜏 + 𝜙)]

(5.6)

Since the amplitude profile of Eq. (5.6) is another Gaussian distribution, it can be fully described

by the Q-Gaussian distribution amplitude model. In addition, the phase response of Eq. (5.6) is

linear, consequently the phase nonlinearity measurement of Eq. (4.4) can be used as the TEI

optimality metric.

5.2.1 Echo Parameter Tests

The phase shift, center frequency difference and amplitude ratio between the two echoes and the

overall SNR of the signal can all affect the performance of both the TEI and DBPR algorithms.

54

For this reason, we varied the shape parameters of the individual echoes in order to obtain

statistically meaningful comparisons of algorithm performance.

Since there are many echo shape parameters that control the overall composition of the

overlapping echoes, it is not practical to examine all possible combinations of shape parameters.

Instead, for each test we only varied two parameters simultaneously and the other shape

parameters were fixed at their baseline values. The baseline echo parameter values for the

symmetric envelope synthetic echoes are summarized in Table 5.1. A multiplication factor of

106 is present in the envelope width parameter 𝑎 because the time shift is in the order of µs.

Parameter Echo A Echo B

Center frequency 𝑓𝑐 3.0 MHz 2.76 MHz

Amplitude scaling 𝐴 1.0 0.7

Phase shift 𝜙 0 1.0π

Time shift 𝜏 3.0 µs 3.6 µs

Envelope width 𝑎 3.2×106 [1/s] 2.8×106[1/s]

Table 5.1: Baseline parameters for symmetric echoes

The overlapping echoes created from the parameter values listed in Table 5.1 are shown in

Figure 5.1. The linear phase echo optimality metric was used for TEI in this section. In addition,

the value of ∆𝑡𝑚𝑖𝑛 in the phase slope inequality condition of Eq. (3.2) was set to be 0.17 µs

because it is approximately half of the time period at 3 MHz. Lastly, for all the tests in this

section, the SNR level was set to be 100 dB to simulate noise-free test cases.

55

Figure 5.1: Baseline configuration for symmetric echoes

Phase Difference vs Time Separation

In this test, we varied the phase difference (𝜙𝐴 − 𝜙𝐵) and the time separation (𝜏𝐴- 𝜏𝐵) between

the two echoes. The values of (𝜙𝐴 − 𝜙𝐵) were varied from 0 to 1.8π in increments of 0.2 π; the

values of (𝜏𝐴- 𝜏𝐵) were varied from 0.2 µs to 1.0 µs in increments of 0.08 µs. Consequently, a

total of 10×11 = 110 different signal configurations were examined. When the time separation

between the two echoes is only 0.2 µs, we are approaching the ∆𝑡𝑚𝑖𝑛 value that we was set for

the phase inequality condition.

Graphical representations of the results are shown in Figure 5.2 and Figure 5.3 where we

summarize the results of all 110 test configurations into color-coded image plots. Each square

represents a different configuration with its x-coordinate being the time separation and its y-

coordinate being the phase difference. The color of each square represents the best solution

obtained after 6 runs of the CMA-ES solver.

In addition, we also calculated the means and standard deviations of the 110 data points shown in

Figure 5.2 and Figure 5.3 and summarized the results in Table 5.2. Table 5.2 provides a

statistical summary of the results obtained for this test and allows for quantifiable comparisons of

the two echo separation algorithms.

56

Figure 5.2: Percentage timing error (phase difference vs time separation for symmetric echoes)

Figure 5.3: Percentage reconstruction error (phase difference vs time separation for symmetric echoes)

TEI DBPR

Mean (Timing Error) 14.39 % 18.33 %

Standard Deviation (Timing Error) 17.59 % 29.41 %

Mean (Reconstruction Error) 0.67 % 1.05 %

Standard Deviation (Reconstruction Error) 0.87 % 2.28 %

Table 5.2: Performance table (phase difference vs time separation for symmetric echoes)

Looking at Figure 5.2, we see that no apparent trend existed between the phase difference and

the percentage timing error. However, it can be seen that TEI performed more reliably compared

to DBPR. In this noise-free test, the performance of DBPR was bimodal, either there was perfect

reconstruction (zero timing error) or there was a large percentage timing error.

57

From Figure 5.3, we see that for TEI the reconstruction error actually decreased with decreasing

time separation between the two echoes. In fact, this result can also be observed for the

subsequent test cases that will be presented in this section. The cause of this seemingly counter-

intuitive observation will be explained in Section 5.2.2.

The reliability and performance advantage of TEI is clearly shown in Table 5.2. The standard

deviation of timing errors was approximately 40% lower than DBPR while the standard

deviation for reconstruction errors was 60% lower. For this test, the mean of both timing and

reconstruction errors were also lower for TEI. Lower timing errors indicates a more accurate

estimate of the location and size of a defect.

Frequency Difference vs Time Separation

In this test, we varied the frequency difference (𝑓𝑐,𝐴 − 𝑓𝑐,𝐵) and the time separation (𝜏𝐴- 𝜏𝐵)

between the two echoes. The values of (𝑓𝑐,𝐴 − 𝑓𝑐,𝐵) were varied from -0.6 MHz to +0.6 MHz in

increments of 0.12 MHz; the values of (𝜏𝐴- 𝜏𝐵) were varied from 0.2 µs to 1.0 µs in increments

of 0.08 µs. Consequently, a total of 11×11 = 121 different signal configurations were

examined. Graphical representations of the results are shown in Figure 5.4 and Figure 5.5. In

addition, a table of the means and standard deviations of the timing and reconstruction errors is

shown in Table 5.3.

Figure 5.4: Percentage timing error (frequency difference vs time separation for symmetric echoes)

58

Figure 5.5: Percentage reconstruction error (frequency difference vs time separation for symmetric echoes)

TEI DBPR





Table 5.3: Performance table (frequency difference vs time separation for symmetric echoes)

Looking at Figure 5.4 and Figure 5.5, we see that DBPR performed marginally better when there

was a large difference between the center frequencies of the two echoes. In contrast, such a trend

was not observed for TEI. This result may be explained by the fact that the trigonometric phase

reconstruction algorithm of TEI can only be applied at frequencies where both echoes contain

significant spectral content. If two echoes have a large difference in center frequency, the

spectral overlap between the echoes would be limited and TEI would be forced to use fewer

frequency data points in the optimization algorithm. In contrast, a large center frequency

difference would lead to rapid cycling of constructive/destructive oscillation interference in the

time domain signal. This rapid cycling allows DBPR to identify the time region of echo overlap

and therefore DBPR can separate the two echoes more effectively for such configurations.

In Figure 5.4 and Figure 5.5 we also observe occasional outlier points where the timing and

reconstruction errors are much larger than the adjacent points. These outlier points are present

due to the probabilistic CMA-ES solver used to solve the optimization problems. The probability

59

of the occurrence of outliers can be reduced by repeating the solver many times at the same test

configuration but can never be eliminated.

From Table 5.3, it is again evident that the performance TEI was more consistent within the

range of time separation tested. The standard deviation of the timing errors was approximately

50% lower for TEI compared to DBPR. The standard deviation for the reconstruction error was

also 12% lower for TEI compared to DBPR.

Amplitude Ratio vs Time Separation

In this test, we varied the amplitude ratio (𝐴𝐵/𝐴𝐴) and the time separation (𝜏𝐴- 𝜏𝐵) between the

two echoes. The values of (𝐴𝐵/𝐴𝐴) were varied from 0.4 to 1.0 in increments of 0.06; the values

of (𝜏𝐴- 𝜏𝐵) were varied from 0.2 µs to 1.0 µs in increments of 0.08 µs. Consequently, a total of

11×11 = 121 different signal configurations were examined. Graphical representations of the

results are shown in Figure 5.6 and Figure 5.7. In addition, a table of the means and standard

deviations of the timing and reconstruction errors is shown in Table 5.4.

Figure 5.6: Percentage timing error (amplitude ratio vs time separation for symmetric echoes)

60

Figure 5.7: Percentage reconstruction error (amplitude ratio vs time separation for symmetric echoes)

TEI DBPR





Table 5.4: Performance table (amplitude ratio vs time separation for symmetric echoes)

Looking at the results presented above, we see that the performances of both DBPR and TEI

were both less affected by changing the amplitude ratio compared to varying the frequency

difference and phase shift. The mean timing errors for this test was roughly 11% for both

techniques. This error percentage was lower than both the phase shift variation test (13% for

TEI; 16% for DBPR) and the center frequency variation test (14% for TEI; 18% for DBPR). This

result may be explained by the lack of change in echo oscillation interference when only the

amplitude ratio of the two echoes is varied. When the phase and frequency difference of the two

echoes are varied, the oscillation patterns formed by the two echoes are shifted and therefore the

amount of constructive and destructive interference is affected.

Even though the mean timing errors were similar for the TEI and DBPR methods, again it is

evident that TEI performed more consistently. The standard deviation of the timing errors for

TEI was 50% lower compared to DBPR while the standard deviation for the reconstruction

errors was 70% lower.

61

5.2.2 Signal to Noise Ratio Tests

In order to test the performance of each echo separation method in the presence of noise, we

repeated the amplitude ratio test at four different SNRs. The SNR was varied by adding white

Gaussian noise to the signal containing the overlapping echoes. For these simulation tests, the

percentage reconstruction error was mainly dominated by the noise variance and therefore these

metrics did not provide useful information regarding the relative performances of the TEI and

DBPR algorithms. For this reason, the color-coded image plots for the reconstruction errors are

not presented in this section.

SNR = 40 dB

At 40 dB, the average SNR amplitude ratio is equal to 100 and therefore the additive noise is

hardly visible from a visual inspection of the time domain signal containing the overlapped

echoes. A plot of a representative test configuration with (𝐴𝐵/𝐴𝐴) set at 0.7 is shown in Figure

5.8. The percentage timing error comparison between TEI and DBPR is shown in Figure 5.9.

Figure 5.8: Overlapped echoes at SNR = 40 dB (symmetric echoes)

62

Figure 5.9: Percentage timing error (40 dB for symmetric echoes)

Comparing Figure 5.9 with the noise-free case shown in Figure 5.6, we see that the addition of a

negligible level of noise was sufficient to influence the performance of DBPR. The timing error

of DBPR has noticeably increased at small time separation values compared to the noise-free

test. In contrast, the timing error of TEI was not noticeably influenced.

SNR = 25 dB

At 25 dB, the average SNR amplitude ratio is equal to 17.78; at this noise level the additive noise

can be visually detected in the simulated signal. A plot of a representative test configuration with

(𝐴𝐵/𝐴𝐴) set at 0.7 is shown in Figure 5.10. The percentage timing error comparison between

TEI and DBPR is shown in Figure 5.11.


63


Comparing Figure 5.11 with Figure 5.9, we see that the timing errors of DBPR at small time

separations have significantly increased while the timing errors of TEI were largely unaffected

by the decrease in SNR. With the introduction of noise, the performance of DBPR became less

bimodal but showed a gradual degradation in time difference estimation accuracy with

decreasing time separation between the two echoes.

SNR = 15 dB


significantly affects the oscillation waveform in the simulated signal. A plot of a representative

test configuration with (𝐴𝐵/𝐴𝐴) set at 0.7 is shown in Figure 5.12. The percentage timing error

comparison between TEI and DBPR is shown in Figure 5.13.


64


It is clear from inspection of Figure 5.13 that TEI had lower timing errors compared to DBPR at

small time separations. This result is drastically different from the noise-free case shown in

Figure 5.6 where DBPR was capable of obtaining near zero timing errors at small time

separations. This observation indicates that the TEI algorithm is more robust in the presence of

noise.

SNR = 10 dB


causes severe distortion of the oscillation waveform in the simulated signal. A plot of a

representative test configuration with (𝐴𝐵/𝐴𝐴) set at 0.7 is shown in Figure 5.14. The percentage

timing error comparison between TEI and DBPR is shown in Figure 5.15.


65


From Figure 5.15, we see that the timing errors for both TEI and DBPR have increased

compared to results obtained at higher SNR levels. However, the degradation in the performance

of DBPR was much more significant compared to TEI. This again shows that TEI is more robust

to the addition of noise.

SNR Tests Summary

To summarize the results in this section, we plot the mean and standard deviation of the

percentage timing error as a function of SNR in Figure 5.16. The 100 dB data points are plotted

using the results from the amplitude test in Section 5.2.1. From Figure 5.16, it is evident that the

performance of TEI was almost independent of SNR within the noise range that was examined.

The mean timing error remained constant at approximately 12% and the timing error standard

deviation was close to 15%. In contrast for DBPR, we see a sharp increase in both the mean and

standard deviation of the percentage timing errors when the SNR was decreased below 40 dB.

66

Figure 5.16: Performance vs SNR (symmetric echoes)

5.2.3 Results Summary and Discussion

One of the most significant trends observed from the results presented in sections 5.2.1 and 5.2.2

is that the TEI algorithm performed much more consistently than to DBPR. The standard

deviations of timing errors of TEI was approximately 40% lower than those of DBPR. This trend

can be explained by understanding the difference between the working principles of the two

algorithms.

From Eq. (2.19), we see that DBPR obtains the optimal echo parameters by minimizing the

reconstruction residual error between the total signal and the sum of parametric echoes.

Consequently, if the optimization solver fails to find the correct value for one or more of the

parameters, the other echo parameters would need to adjust in incorrect ways to reduce the

residual error. Since it is highly possible to obtain a local minimum of the residual error with

incorrect time shift parameters, we see many outlier points in the output of DBPR where the

timing errors are very large.

In contrast, for the simulation tests in sections 5.2.1 and 5.2.2, TEI adapts the amplitude profiles

in order to obtain phase profiles that are as linear as possible. If the optimization solver fails to

find the correct value for one or more of the amplitude parameters, the other amplitude

parameters would need to adjust in incorrect ways to maximize the linearity of the reconstructed

phase profiles. However, it is in general difficult to reconstruct near-linear phase profiles which

have phase slope values that are drastically different from the correct ones. For this reason, even

if the converged amplitude profiles are not strictly correct, the reconstructed phase profiles often

67

have phase slopes values that are close to the correct ones. Since the arrival time of an echo is

determined by its phase slope, TEI is therefore less likely to produce outliers in the estimate of

echo arrival time difference.

Another major trend that we can observe from Section 5.2.2 is that the performance of TEI was

less influenced by decreasing SNR compared to DBPR. For DBPR there was a sharp increase in

both the mean and standard deviation of the timing errors when the SNR was lowered below 40

dB. In comparison, the performance metrics of TEI remained approximately constant when SNR

was decreased. This trend can also be explained by understanding the optimization goals of each

algorithm.

DBPR aims to reduce the L2 norm of the difference between the total signal and the sum of

parametric echoes. Any noise present in the total signal would be directly entered into the

residual metric calculation as shown Eq. (5.7):

[∑ |𝑠𝑖𝑔(𝑡) + 𝑛(𝑡) − ∑ 𝑒𝑐ℎ𝑜(𝒙𝒊 , 𝑡)

𝑖

|

2

𝑡

]

(5.7)

In Eq. (5.7), the echo parameters need to be adjusted to compensate for the noise and minimize

the residual. Consequently, additive noise has a direct impact on the ability of DBPR to recover

the correct parametric echoes.

In contrast, TEI aims to reduce the nonlinearity in the phase profiles 𝜃𝐴(𝜔) and 𝜃𝐵(𝜔). Using

the phase reconstruction table shown in Table 4.1, we can decompose the noisy phase profiles

𝜃𝐴,𝑛𝑜𝑖𝑠𝑦(𝜔) and 𝜃𝐵,𝑛𝑜𝑖𝑠𝑦(𝜔) into the following components:

𝜃𝐴,𝑛𝑜𝑖𝑠𝑦(𝜔) = 𝜃𝐴(𝜔) + 𝑛𝑜𝑖𝑠𝑒[𝛽(𝜔)] + 𝑛𝑜𝑖𝑠𝑒[𝜃𝑇(𝜔)]

𝜃𝐵,𝑛𝑜𝑖𝑠𝑦(𝜔) = 𝜃𝐵(𝜔) + 𝑛𝑜𝑖𝑠𝑒[𝛼(𝜔)] + 𝑛𝑜𝑖𝑠𝑒[𝜃𝑇(𝜔)]

(5.8)

Where 𝜃𝑇(𝜔) is the phase profile of the total signal and 𝛼(𝜔) and 𝛽(𝜔) are the interior angles

calculated from the vector addition triangle shown in Figure 4.2. In Eq. (5.8), 𝑛𝑜𝑖𝑠𝑒[𝜃𝑇(𝜔)]

represents the phase noise present in the total signal. In contrast, 𝑛𝑜𝑖𝑠𝑒[𝛼(𝜔)] and 𝑛𝑜𝑖𝑠𝑒[𝛽(𝜔)]

stem from the amplitude noise 𝑛𝑜𝑖𝑠𝑒[𝑀𝑇(𝜔)] because the amplitude of the total signal 𝑀𝑇(𝜔) is

used in the calculation of 𝛼(𝜔) and 𝛽(𝜔).

68

When one measures the nonlinearity of Eq. (5.8), 𝑛𝑜𝑖𝑠𝑒[𝜃𝑇(𝜔)] adds a baseline level of

nonlinearity to the phase profiles. However, since the phase noise is common to all reconstructed

phase profiles, its influence on the selection of the most linear phase profile is minimal. In

addition, the 𝑛𝑜𝑖𝑠𝑒[𝛼(𝜔)] and 𝑛𝑜𝑖𝑠𝑒[𝛽(𝜔)] are not directly proportional to 𝑛𝑜𝑖𝑠𝑒[𝑀𝑇(𝜔)]

because the trial amplitudes 𝑀𝐴(𝜔) and 𝑀𝐵(𝜔) are also used in the calculation of 𝛼(𝜔) and

𝛽(𝜔). For example, using the vector addition triangle shown in Figure 4.2, the interior angle

𝛼(𝜔) can be calculated using the cosine law as follows:

𝛼(𝜔) = cos−1 [𝑀𝐴(𝜔)2 + 𝑀𝐵(𝜔)2 − 𝑀𝑇(𝜔)2

2𝑀𝐴(𝜔)𝑀𝐵(𝜔)]

(5.9)

Looking at Eq. (5.9), we see the influence of 𝑛𝑜𝑖𝑠𝑒[𝑀𝑇(𝜔)] is reduced by the trial amplitude

profiles of the two echoes. For the reasons explained above, we can understand why TEI is less

sensitive to the decreasing SNR compared to DBPR.

Yet another trend that we can observe from the results presented in Section 5.2.1 is that for TEI

the timing error increased while the reconstruction error decreased with decreasing time

separation. This apparently contradictory trend can be explained by examining two

representative echoes that are spaced 0.2 µs apart (the other echo parameters follow the baseline

parameter values shown in Table 5.1).

Figure 5.17: Overlapped signal with time separation of 0.2 µs

69

Looking at Figure 5.17, we see that the overlapped signal at a small time separation strongly

resembles a single echo. The frequency amplitude profile closely follows a Gaussian distribution

and the phase profile is close to being linear. Since the overlapped signal already satisfies the

assumptions of TEI, it is relatively easy to decompose the signal into two arbitrary echoes that

also satisfy the TEI assumptions. This is the reason why the reconstruction error is small for

small echo time separations. However, the decomposed echoes may not actually be the correct

ones as multiple valid solutions exist. This is the reason why the timing error increases with

decreasing echo time separation.

From this explanation, we can deduce that the time difference estimate performance of TEI

decreases when the signal containing the overlapped echoes resembles a single echo. This result

is consistent with our intuitive understanding of the echo separation problem.

5.3 Synthetic Echoes with Asymmetric Envelope

For the second set of analyses of simulated signals, we employed echoes with asymmetric

envelopes and non-constant modulation frequencies. Each asymmetric echo used in this study is

mathematically described by the following expression:

𝑒𝑐ℎ𝑜(𝒙 , 𝑡) = 𝐴 ∙ 𝑒𝑛𝑣 (t − τ)cos[2𝜋𝑓𝑐(𝑡 − 𝜏) + 2𝜋𝜓(𝑡 − 𝜏)3 + 𝜙]


(5.10)

The asymmetric echoes described above present a challenging case for both the TEI and DBPR

models. Comparing Eq. (5.10) with Eq. (5.1), we see that Eq. (5.10) has an extra 𝜓 parameter

multiplied by a cubic time delay. Since the instantaneous modulation frequency is proportional to

the time derivative of the argument of the cosine function, the modulation frequency is not

constant but rather a quadratic function of time. Consequently, in this test the constituent echoes

are no longer perfectly described by the DBPR mathematical model.

These asymmetric echoes also present multiple challenges for TEI. Firstly, using the

multiplication-convolution duality property, the Fourier transform of Eq. (5.10) is the frequency

domain convolution of the Fourier transform of envelope function 𝐴 ∙ 𝑒𝑛𝑣(𝑡 − 𝜏) with the

Fourier transform of the nonlinear oscillation. In general, an analytical expression cannot be

obtained for the Fourier transform of an oscillation with a non-linear frequency and numerical

70

methods (such as the FFT) are used to estimate it. Consequently, it is clear that the frequency

amplitude profile of the constituent echoes will not be perfectly described by the Q-Gaussian

distribution model shown in Eq. (3.1). Secondly, since the time envelope of Eq. (5.10) is

asymmetric, its phase profile in the frequency domain will be non-linear [31]. Consequently, the

linear phase assumption of Eq. (4.4) used for the echo optimality metric will not be strictly valid.

The purpose of this set of simulation tests is to compare the performance of the echo

identification algorithms in situations where the mathematical models employed do not perfectly

describe the actual echoes. It is important to analyze the performance of the algorithms in such

situations because real life ultrasonic echoes cannot be perfectly described by simple parametric

mathematical models.

5.3.1 Echo Parameter Tests

The asymmetric echo simulation tests closely followed the procedure outlined in Section 5.2.1.

In each test, we only varied two parameters while the other parameters were held constant at

their baseline values. The baseline values for the echo parameters are shown in Table 5.5:

Parameter Echo A Echo B

Center frequency 𝑓𝑐 3.0 MHz 2.76 MHz

Amplitude scaling 𝐴 1.0 0.7

Phase shift 𝜙 0 1.0π

Time shift 𝜏 3.0 µs 3.6 µs

Envelope width 𝑎 3.2×106 [1/s] 2.8×106[1/s]

Envelope asymmetry 𝜌 0.5 0.6

Frequency nonlinearity 𝜓 −0.1×1012[1/𝑠2] ∙ 𝑓𝐶,𝐴 −0.1×1012[1/𝑠2] ∙ 𝑓𝐶,𝐵

Table 5.5: Baseline parameters for asymmetric echoes

Looking at Table 5.5, we see that the nonlinear modulation frequency factor was set at the value

−0.1×1012[1/𝑠2] ∙ 𝑓𝐶. Since the instantaneous frequency is defined as 1/2𝜋 multiplied by the

time derivative of the argument of the cosine function, the modulation frequency of each echo

described in Eq. (5.10) would have the form:

𝑓(𝑡) = 𝑓𝑐 − 0.3×1012[1/𝑠2] ∙ 𝑓𝑐(𝑡 − 𝜏)2

= 𝑓𝑐(1 − 0.3×1012[1/𝑠2] ∙ (𝑡 − 𝜏)2)

(5.11)

71

In Eq. (5.11), the modulation frequency is a quadratic function of time. The instantaneous

frequency is highest at 𝑡 = 𝜏 and is reduced when we move away from the center of the echo. A

factor of 1012 is needed in the definition of 𝜓 because the time shift 𝜏 is in the order of µs and

the center frequency is in the order of MHz. A demonstrative frequency profile for 𝑓𝑐 = 3.0 MHz

and 𝜏 = 3.0 𝑢𝑠 is shown below in Figure 5.18. The overlapping echoes created from the

parameter values listed in Table 5.5 are shown in Figure 5.19.

Figure 5.18: Quadratic modulation frequency

Figure 5.19: Baseline configuration for asymmetric echoes

From Figure 5.19, we see that the effects of variation in modulation frequency is moderate since

the durations of the echoes were short. The linear phase echo optimality metric was again used

for TEI even though this assumption was not perfectly satisfied by the asymmetric echoes. In

addition, the value of ∆𝑡𝑚𝑖𝑛 in the phase slope inequality condition of Eq. (3.2) was set to be

72

0.17 µs because it is approximately half of the time period at 3 MHz. Once again, for all the tests

in this section, the SNR level was set to be 100 dB to simulate noise-free test cases.

Phase Difference vs Time Separation

In this test, we varied the phase difference (𝜙𝐴 − 𝜙𝐵) and the time separation (𝜏𝐴- 𝜏𝐵) between

the two echoes. The values of (𝜙𝐴 − 𝜙𝐵) were varied from 0 to 1.8π in increments of 0.2 π; the

values of (𝜏𝐴- 𝜏𝐵) were varied from 0.2 µs to 1.0 µs in increments of 0.08 µs. Consequently, a

total of 10×11 = 110 different signal configurations were examined.

For all asymmetric echo tests in this section, we set the ∆𝑡𝑚𝑖𝑛 value for Eq. (3.2) in the TEI

algorithm as 0.17 µs, which is approximately half the time period at 3 MHz. However, since the

phase profiles of the non-asymmetric echoes are non-linear, the value of the phase slope

difference, 𝑑(𝜃𝐴−𝜃𝐵)

𝑑𝜔, may be lower than 0.17 µs at some frequencies. As an illustrative example,

when we set the nominal time shift difference between the two echoes at 0.2 µs and retain all

other echo parameters at their baseline values listed in Table 5.5, we can obtain the phase slope

difference profile shown in Figure 5.20.

Figure 5.20: Phase slope difference of two asymmetric echoes (nominal time separation at 0.2 µs)

From Figure 5.20, we see that there are frequencies at which the value of 𝑑(𝜃𝐴−𝜃𝐵)

𝑑𝜔 is below the

value of ∆𝑡𝑚𝑖𝑛 set at 0.17 µs. Consequently, the phase slope assumptions of Eq. (3.2) are no

longer satisfied at all frequencies where both echoes have significant spectral content. In this

section, we will examine how this violation of the phase slope assumption would affect the

timing and reconstruction errors of the TEI algorithm. In addition, since the phase slope

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difference is non-constant, it is important that we use the spectrally-averaged arrival time defined

in Eq. (4.3) to calculate the arrival-time difference between the two echoes. If we instead use the

nominal time shift difference as the “true” arrival time difference, it would lead to inaccuracies

in the estimation of the timing errors for the two algorithms.

Graphical representations of the results of varying phase difference against time separation are

shown in Figure 5.21 and Figure 5.22. In addition, a table of the means and standard deviations

of the timing and reconstruction errors is shown in Table 5.6.

Figure 5.21: Percentage timing error (phase difference vs time separation for asymmetric echoes)

Figure 5.22: Percentage reconstruction error (phase difference vs time separation for asymmetric echoes)

74

TEI DBPR





Table 5.6: Performance table (phase difference vs time separation for asymmetric echoes)

Looking at Figure 5.21 and Figure 5.22, we see that TEI had much smaller timing and

reconstruction errors compared to DBPR. This observation is confirmed in Table 5.6 where TEI

outperformed DBPR in every statistical performance metric. For this test, there was not an

apparent trend between the percentage timing error and the phase difference between the echoes.

However, it is apparent that percentage timing error increased with decreasing time separation

between the two echoes. This result is expected because a small timing error can produce a large

percentage timing error at small time separations. In addition, as shown in Figure 5.20, the phase

slope inequality assumption used by TEI in this test is not strictly satisfied when the time

separation is only 0.2 µs. Consequently, by enforcing the phase slope assumption we can

introduce errors in the echo reconstruction process.

Comparing Table 5.6 with the symmetric echo results summarized in Table 5.2, we see that the

performance of DBPR has deteriorated significantly. The mean and standard deviation of the

timing errors have increased from 18% and 29% to 31% and 37% respectively. In comparison,

the timing errors of TEI have actually decreased. The mean and standard deviation of the timing

errors were approximately 12% and 14%; these values compare well with the previous values of

14% and 18% observed for the separation of symmetric echoes. This observation suggests that

TEI is more robust than DBPR in situations where the actual echo shapes are not perfectly

described by the mathematical forms of the chosen model.

Frequency Difference vs Time Separation

In this test, we varied the center frequency difference (𝑓𝑐,𝐴 − 𝑓𝑐,𝐵) and the time separation (𝜏𝐴-

𝜏𝐵) between the two echoes. The values of (𝑓𝑐,𝐴 − 𝑓𝑐,𝐵) were varied from -0.6 MHz to +0.6 MHz

in increments of 0.12 MHz; the values of (𝜏𝐴- 𝜏𝐵) were varied from 0.2 µs to 1.0 µs in

increments of 0.08 µs. Consequently, a total of 11×11 = 121 different signal configurations

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were examined. Graphical representations of the results are shown in Figure 5.23 and Figure

5.24. In addition, a table of the means and standard deviations of the timing and reconstruction

errors is shown in Table 5.7.

Figure 5.23: Percentage timing error (center frequency difference vs time separation for asymmetric echoes)

Figure 5.24: Percentage reconstruction error (center frequency difference vs time separation for asymmetric echoes)

TEI DBPR





Table 5.7: Performance table (center frequency difference vs time separation for asymmetric echoes)

76

Looking at Figure 5.23 and Figure 5.24, we see that DBPR performed better when there was a

large difference between the center frequencies of the two echoes. In contrast, such a trend was

not observed for TEI. The explanation for this result had already been presented in Section 5.2.1.

In this test, it is again evident that TEI had smaller timing and reconstruction errors than the

DBPR method.

Comparing Table 5.7 with the symmetric echo results summarized in Table 5.3, we see that the

performance of DBPR has deteriorated significantly. The mean of the timing errors has increased

from 16% to 26%. In comparison, the mean timing error of TEI has only increased marginally

from 13% to 15%. Changing the center frequency was a challenging test for both TEI and

DBPR. As shown in Eq. (5.11), the time variation in modulation frequency is designed to be

proportional to the center frequency. Consequently, the rate of modulation frequency variation is

not constant for two overlapping echoes when they possess different center frequencies.

Amplitude Ratio vs Time Separation

In this test, we varied the amplitude ratio (𝐴𝐵/𝐴𝐴) and the time separation (𝜏𝐴- 𝜏𝐵) between the

two echoes. The values of (𝐴𝐵/𝐴𝐴) were varied from 0.4 to 1.0 in increments of 0.06; the values

of (𝜏𝐴- 𝜏𝐵) were varied from 0.2 µs to 1.0 µs in increments of 0.08 µs. Consequently, a total of

11×11 = 121 different signal configurations were examined. Graphical representations of the

results are shown in Figure 5.25 and Figure 5.26. In addition, a table of the means and standard

deviations of the timing and reconstruction errors is shown in Table 5.8.

Figure 5.25: Percentage timing error (amplitude ratio vs time separation for asymmetric echoes)

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Figure 5.26: Percentage reconstruction error (amplitude ratio vs time separation for asymmetric echoes)

TEI DBPR





Table 5.8: Performance table (amplitude ratio vs time separation for asymmetric echoes)

From the results presented above, we see that the percentage timing error of DBPR was smaller

for the amplitude ratio test compared to changing frequency difference and phase shift. The

mean percentage timing error of this test for DBPR was 21%, which was smaller than 26% for

the center frequency variation test and 31% for the phase difference test. This trend was also

observed for the symmetric echo tests presented in Section 5.2.1 and may be explained by the

lack of change in echo oscillation interference when only the amplitude ratio of the two echoes is

varied. In comparison, the mean percentage timing error for TEI was relatively constant for all

three tests. This suggests that the performance of TEI is more robust to variation in echo shape.

Despite the fact that DBPR was less affected by change in amplitude ratio, its mean and standard

deviation of the percentage timing errors (21% and 28%) were still much larger compared to TEI

(14% and 14%).

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5.3.2 Signal to Noise Ratio Tests

In order to test the echo separation performances of TEI and DBPR for asymmetric echoes in the

presence of noise, we followed the test procedure in Section 5.2.2 and repeated the amplitude

ratio test at four different levels of SNR. The SNR level was again varied by adding white

Gaussian noise to the signal containing the overlapping echoes. Once again, the reconstruction

error for these tests were dominated by the noise variance and did not provide useful information

regarding the relative performances of the TEI and DBPR algorithms. For this reason, the color-

coded image plots for the reconstruction errors are not be presented in this section.

SNR = 40 dB

At 40 dB, the average SNR amplitude ratio is equal to 100 and therefore the additive noise is

hardly visible from a visual inspection of the time domain signal containing the overlapped

echoes. A plot of a representative test configuration with (𝐴𝐵/𝐴𝐴) set at 0.7 is shown in Figure

5.27. The percentage timing error comparison between TEI and DBPR is shown in Figure 5.28.

Figure 5.27: Overlapped echoes at SNR = 40 dB (asymmetric echoes)

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Figure 5.28: Percentage timing error (40 dB for asymmetric echoes)

Comparing Figure 5.28 with the noise-free case shown in Figure 5.25, there was not any

significant difference in performance for both methods. This indicates that at this SNR level

noise is not an important factor in the echo identification performance of both methods.

SNR = 25 dB


can be visually detected in the simulated signal. A plot of a representative test configuration with

(𝐴𝐵/𝐴𝐴) set at 0.7 is shown in Figure 5.29. The percentage timing error comparison between

TEI and DBPR is shown in Figure 5.30.


80


Comparing Figure 5.30 with Figure 5.28, we see that there was a small increase in timing error

for both the TEI and DBPR methods. For TEI, there was a clear increase of percentage timing

error with decreasing time separation. For DBPR, the overall trend was more random although

larger percentage timing errors occurred more frequently at smaller time separations.

SNR = 15 dB


significantly affects the oscillation waveform in the simulated signal. A plot of a representative

test configuration with (𝐴𝐵/𝐴𝐴) set at 0.7 is shown in Figure 5.31. The percentage timing error

comparison between TEI and DBPR is shown in Figure 5.32.


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Comparing Figure 5.32 with Figure 5.30, we see that there was an increased in percentage timing

error for both TEI and DBPR. The maximum timing errors for TEI and DBPR have risen to

115% and 140% at echo time separation of 0.2 µs. Once again, for TEI there was a gradual

transition to larger timing errors with decreasing time separation whereas the DBPR timing

errors were more randomly distributed.

SNR = 10 dB


causes severe distortion of the oscillation waveform in the simulated signal. A plot of a

representative test configuration with (𝐴𝐵/𝐴𝐴) set at 0.7 is shown in Figure 5.33. The percentage

timing error comparison between TEI and DBPR is shown in Figure 5.34.


82


Comparing Figure 5.34 with Figure 5.32, we see that the timing errors have increased again with

decreasing SNR level. The maximum timing error for TEI and DBPR were respectively 140%

and 240%. Both maximum timing errors occurred at an echo time separation of 0.2 µs. The large

percentage error at 0.2 µs is expected for TEI since the phase slope difference assumption is not

strictly satisfied as shown in Figure 5.20.

SNR Tests Summary

To summarize the results in this section, we plot the mean and standard deviation of the

percentage timing error as a function of SNR in Figure 5.35. The data points corresponding to a

SNR of 100 dB are plotted using the noise-free results summarized in Table 5.8. From Figure

5.35, we can see that the timing error means and standard deviations for both TEI and DBPR

increased significantly when the SNR was decreased below 40 dB. However, the time difference

estimation performance of TEI was still superior to DBPR for all noise levels tested.

It is interesting to note that the performance of TEI at a SNR level of 40 dB was marginally

better than the results obtained from 100 dB. This is likely caused by small random fluctuations

in the solutions obtained from the CMA-ES solver.

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Figure 5.35: Performance vs SNR (asymmetric echoes)

5.3.3 Results Summary and Discussion

One of the significant trends observed from Section 5.3.1 is that TEI outperformed DBPR

significantly for the separation of two asymmetric echoes with non-constant modulation

frequencies. The means of the timing errors (13-15% compared to 21-31%) and the

reconstruction errors (~1% compared to ~4%) were both significantly lower for TEI. This result

can be explained by understanding the parametric modeling aspects of the two algorithms.

From Eq. (5.1), we see that this implementation of DBPR uses 6 parameters to describe the

shape of each echo. In contrast, from Eq. (3.1) we see that TEI uses all 6 parameters to describe

the frequency domain amplitude profile of each echo, and then uses trigonometry to solve for the

phase profiles. Consequently, echoes described by the TEI algorithm can have more complex

amplitude profile shapes compared to DBPR. In addition, since the phase profiles of the TEI

echoes are reconstructed using the total signal amplitude 𝑀𝑇(𝜔) and the total signal phase

𝜃𝑇(𝜔), the phase profiles can adapt to the acquired data and are not governed by fixed

mathematical expressions. Due to these two unique aspects of the algorithm, TEI can describe

more complex ultrasonic echoes using the same number of modeling parameters compared to

DBPR and adapt the shapes of the echoes to fit the acquired data. This is the primary reason why

TEI outperforms DBPR in this set of simulation experiments.

Another important trend we can observe from Section 5.3.2 is that the performance of TEI was

no longer independent of SNR level for the separation of asymmetric echoes. This is different

from our observation in Section 5.2.3 where the performance of TEI was not affected by change

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in SNR level for the separation of symmetric echoes. This result can be explained using the

following argument.

For an asymmetric echo with a non-constant modulation frequency, its frequency phase profile is

in general non-linear even without the presence of noise. Consequently, the phase linearity

metric employed in this section is only an approximate measurement for the optimality of the

reconstructed echoes. When noise is added to the signal, there will be fluctuations added to the

reconstructed phase profiles as explained in Section 5.2.3. Since it is not possible to differentiate

between the fluctuations introduced by noise and the inherent non-linearity in the reconstructed

phase profiles, it is likely for TEI to converge to a suboptimal solution when the SNR is

decreased. This is the primary reason why the timing estimation performance of TEI deteriorated

with decreasing SNR.

5.4 Experimental Verification

Having compared the echo separation performance of TEI and DBPR for various simulated

echoes, in this section we evaluate the performance of the two algorithms for the separation of

ultrasonic echoes in signals obtained from experiments. These experimental results will verify

whether the assumptions of the TEI algorithm are applicable for actual NDT applications.

5.4.1 TOFD Test on Notched Sample

Test Configuration

For the first experimental test, we seek to separate two overlapping echoes obtained from TOFD

inspection of a sample containing a vertical notch. A photograph of the notched sample is shown

in Figure 5.36. The test sample is made of low carbon steel and has a thickness of 0.5”. There are

four vertical notches cut into the sample which are 0.3”, 0.2”, 0.1” and 0.05” deep. The TOFD

measurement was conducted on the 0.3” deep notch because the small distance between the

notch tip and the top surface leads to the creation of overlapping echoes in the A-scan data. The

adjacent notches are spaced sufficiently far apart that they did not interfere with the TOFD

inspection.

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Figure 5.36: Test sample containing vertical notches

A schematic diagram of the TOFD scan configuration is shown in Figure 5.37. From Figure

5.37, we see that the beam entry points of the two probes were spaced 30 mm apart such that the

intersection of the central propagation axes of the transducers occurred at the bottom 1/3 of the

sample thickness. This configuration was chosen according to standard TOFD measurement

protocol [47]. The design center frequency of the probes used was 5 MHz.

Figure 5.37: TOFD configuration for notch sample

During the TOFD acquisition, we translated the two probes parallel to the direction of the notch

to obtain the B-scan image shown in Figure 5.38. In Figure 5.38, the x-axis represents the

position of the inspection system in the scan direction and the y-axis represents time in the

individual A-scans. From this figure, we can see that there was an overlap between the lateral

wave and the notch tip diffraction echoes in the central portion of the scan along the x-axis. An

A-scan extracted from the scan location of 35 mm showing these overlapping echoes is shown in

Figure 5.39.

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Figure 5.38: B-scan of notch sample TOFD scan

Figure 5.39: Overlapping echoes in TOFD scan of notch sample

From the configuration of the inspection system, we know that the probe separation is 30 mm,

the notch tip distance from the surface is 5.08 mm and the speed of sound in the steel sample is

5890 m/s. Using these parameters, the theoretical time difference between the two echoes can be

estimated based on simple trigonometry:

∆𝑡𝑔𝑒𝑜𝑚𝑒𝑡𝑟𝑦 =

30𝑚𝑚 − 2√(15𝑚𝑚)2 + (5.08𝑚𝑚)2

5890 𝑚/𝑠= 0.284 𝑢𝑠

(5.12)

Results using Phase Linearity as Optimality Metric

Since TEI is designed to separate two overlapping echoes, the first processing step was to crop

the A-Scan signal so that only two overlapping echoes remained in the time series data. For this

reason, we cropped the A-Scan data as shown in Figure 5.39 from 1.7 µs to 3.1 µs. In addition,

we also multiplied the cropped signal by a Tukey window in order to reduce transition effects

Lateral wave Notch tip Back wall

Lateral wave

Notch tip

diffraction

Extracted A-scan

87

when we transform the time-series data into the frequency domain. The cropped signal and the

Tukey window are shown in Figure 5.40.

Figure 5.40: Notch sample time series data analyzed by TEI and DBPR

The cropped and windowed time series data was then passed into the TEI and DBPR algorithms

to separate the two echoes. For DBPR, we again used the parametric model shown in Eq. (5.1).

Therefore, each echo was described using six different shape parameters. For TEI, we first used

the phase linearity measurement of Eq. (4.4) as the echo optimality metric. In addition for TEI,

we set the value of ∆𝑡𝑚𝑖𝑛 to be 0.14 µs in the phase slope inequality constraint of Eq. (4.7). This

value was chosen because it was approximately half of the time difference between the apparent

peaks of the two echoes. Although currently the value of ∆𝑡𝑚𝑖𝑛 is chosen heuristically, in the

future the value of ∆𝑡𝑚𝑖𝑛 should be calculated automatically using the estimated bandwidth and

center frequency of the overlapped signal.

The echo separation results for TEI and DBPR are shown respectively in Figure 5.41 and Figure

5.42. In addition, the frequency phase profiles of the TEI reconstructed echoes are shown in

Figure 5.43.

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Figure 5.41: Reconstructed echoes for notch sample (TEI using phase linearity condition)

Figure 5.42: Reconstructed echoes for notch sample (DBPR)

Figure 5.43: Frequency phase profiles of TEI reconstructed echoes (notch sample)

89

Looking at the results of Figure 5.41 to Figure 5.43, we can see that both algorithms were able to

reconstruct echoes with a distinct order of arrival times. Using the reconstruction error formula

defined in Eq. (5.3), the percentage reconstruction error for TEI was 1.5% which is lower than

the percentage reconstruction error of DBPR at 7.5%. This indicates that the echoes

reconstructed from TEI in fact better described the input signal.

From Figure 5.43, we see that the TEI reconstructed phase profiles were near linear. This

suggests that the phase linearity assumption is appropriate for the piezoelectric transducers used

in this experiment. Using the spectrally averaged arrival time formula shown in Eq. (4.3), the

estimated time differences between the two separated echoes were 0.30 µs for TEI and 0.29 µs

for DBPR. Both time difference estimates were in line with the value of 0.284 µs obtained in Eq.

(5.12).

Results using Cross-Correlation as Optimality Metric

From the B-Scan shown in Figure 5.38, we see that the A-scans near the edges of the sample had

lateral wave echoes that did not overlap with the notch tip diffracted echo. Consequently, we

were able to use the clean lateral wave echo as a reference for the cross-correlation optimality

metric shown in Eq. (4.1). However, since we only had a reference for the lateral wave which is

the first echo, we had to modify Eq. (4.1) so that it only maximized the cross-correlation of that

single reference with the first echo:

optimality = −max [𝐶𝐶(𝑟𝑒𝑓1(𝑡), 𝑒𝑐ℎ𝑜1(𝑡))


] (5.13)

By maximizing the cross-correlation between the first echo and the reference echo, we are in

essence performing an adaptive background subtraction to remove the lateral wave echo.

The lateral wave reference echo used for Eq. (5.13) is shown in Figure 5.44. We also multiplied

the reference echo with a Tukey window to reduce the edge effects in the FFT. The reconstructed

echoes determined by the TEI algorithm using the optimality constraint of Eq. (5.13) are shown

in Figure 5.45.

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Figure 5.44: Lateral wave reference echo for TEI

Figure 5.45: Reconstructed echoes for notch sample (TEI using cross-correlation condition)

Despite the use of different optimality metrics, the reconstructed echoes shown in Figure 5.45

were similar to the ones obtained in Figure 5.41. The percentage reconstruction error of this test

was 2.1%, indicating the sum of the two reconstructed echoes accurately described the acquired

signal. The spectrally averaged time difference between the two echoes was 0.29 µs; once again

this extracted time difference value agreed with the theoretical value of 0.284 µs obtained in Eq.

(5.12) based on trigonometry.

5.4.2 Phased Array Test on Side-Drilled Hole Sample

Test Configuration

For the second experimental test, we seek to separate two overlapping echoes obtained from a

pitch-catch phased array scan. The engineering diagram for the test sample used is shown in

Figure 5.46.

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Figure 5.46: Test sample for pitch-catch matrix probe scan

From Figure 5.46, we see that the test sample has two machined side-drilled holes (SDH). The

top SDH is spaced 5.0 mm from the top surface while the bottom SDH is spaced 2.5 mm from

the bottom surface. The top SDH yields an echo that overlaps with the lateral wave echo; the

bottom SDH yields an echo that overlaps with the backwall echo. For this test, we used a pair of

phased array probes to transmit and receive a propagating wave that covered the bottom SDH. A

schematic drawing of the phased array testing configuration is shown in Figure 5.47.

Figure 5.47: Phased array pitch-catch testing of SDH sample

From Figure 5.47, we see that the first 12 elements of each phased array transducer were used for

transmission and reception. The element timing delays were adjusted such that the refracted

angle of each beam inside the sample was 63° with respect to the surface normal. Note that the

ultrasound wave propagated into the sample was a mode-converted shear wave because the angle

of incidence at the wedge/sample interface was greater than the critical angle for longitudinal

92

waves. The active aperture for each phased array probe was 12 mm × 10 mm and the design

center frequency of each active element was 5 MHz.

Using a ray tracing program, the round trip travel time difference from the center of the aperture

to the SDH and to the back wall was calculated to be 0.71 µs.

Echo Separation Results

Using the test configuration shown in Figure 5.47, we obtained the A-scan data shown in Figure

5.49. From Figure 5.49, we see that there were three overlapping echoes. The third overlapping

echo was created by the propagating wave travelling in a “W” path reflecting from the bottom

surface twice as shown in Figure 5.48. Since the TEI algorithm can only separate two

overlapping echoes, we had to crop the time series so that the input signal to the echo separation

algorithms contained only the direct SDH and backwall echoes. The cropped time series is

shown in Figure 5.50. Once again, we had to multiply the cropped signal with a Tukey window

to minimize transition edge effects when performing the FFT.

Figure 5.48: Indirect path for SDH

Figure 5.49: Overlapping echoes for SDH pitch-catch test

SDH

Backwall

SDH (“W” path)

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Figure 5.50: SDH sample time series data analyzed by TEI and DBPR

For DBPR, we again used the parametric model shown in Eq. (5.1). Therefore each echo was

described using six different shape parameters. For TEI, we used the phase linearity

measurement of Eq. (4.4) as the echo optimality metric. In addition for TEI, we set the value of

∆𝑡𝑚𝑖𝑛 to be 0.3 µs in the phase slope inequality constraint of Eq. (4.7). This value was chosen

because it was approximately half of the time separation between the apparent peaks of the two

echoes. In the future, the value of ∆𝑡𝑚𝑖𝑛 should be calculated automatically using the estimated

bandwidth and center frequency of the overlapped signal. The echo separation results for TEI

and DBPR are shown respectively in Figure 5.51 and Figure 5.52.

Figure 5.51: Reconstructed echoes for SDH sample (TEI)

94

Figure 5.52: Reconstructed echoes for SDH sample (DBPR)

Looking at results plotted above, we see that the echoes reconstructed by TEI only overlapped

partially in time while the echoes restricted by DBPR overlapped throughout the entire time

duration of Echo B. This result is not surprising as TEI actively enforces a phase slope difference

between the two echoes. In addition, the percentage reconstruction error of TEI was 2.5% which

was much lower than the 25.4% reconstruction error attained by DBPR. The spectrally averaged

arrival time differences between the two reconstructed echoes were 0.8 µs for TEI and 0.2 µs for

DBPR. The time difference estimate of TEI was much closer to the geometrically calculated

value of 0.71 µs which suggests that the echoes identified by TEI are more accurate.

Overall, TEI performed much better than DBPR for the separation of the SDH and backwall

echoes. The reason why such a larger performance discrepancy existed is that both the SDH and

backwall echoes have relatively complex shapes. Looking at Figure 5.51, we see that the two

echoes have nonlinear modulation frequencies and irregular amplitude envelopes. Such complex

echo shapes cannot be accurately described by the selected DBPR model. This experimental test

suggests that TEI can sometimes provide better time difference estimates than DBPR because it

can adapt the shapes of its reconstructed echoes to the input signal.

95

Chapter 6 Conclusions

6.1 Thesis Summary

The goal of this research project was to develop a novel frequency-domain post-processing

algorithm for the separation of two overlapping ultrasonic echoes encountered in NDT

applications. The ability to separate overlapping echoes can improve the axial-resolution in

ultrasound imaging and provide more accurate flaw size estimates in time-of-flight based

ultrasonic tests.

The proposed echo separation method called Trigonometric Echo Identification (TEI) was

explicitly designed to address the disadvantages of echo separation algorithms reported in the

literature. Compared to Dictionary-Based Parametric Representation (DBPR), TEI can represent

ultrasonic echoes with more complex shapes using the same number of fitting parameters. In

addition, the shapes of the reconstructed echoes can adapt to the input signal data. These two

advantages of TEI were achieved by solving for the phase profiles of the two echoes instead of

explicitly defining their mathematical forms.

The echo separation performance of TEI was evaluated and compared to DBPR for both

simulation and experimental tests. For the simulation tests, we varied the frequency difference,

phase shift and amplitude ratio between the two echoes to obtain statistically relevant

comparisons of the two methods. We also repeated the amplitude ratio test at different SNR

levels to evaluate the noise sensitivity of the algorithms. The percentage timing error and

percentage reconstruction error were the two metrics used to quantitatively compare the

performances of the two echo separation methods.

Two different experimental tests were conducted to evaluate the applicability of the TEI

algorithm for processing of actual ultrasound testing data. For the first experiment, a TOFD scan

was conducted on a test sample containing multiple vertical notches. The TEI and DBPR

methods were applied to the collected experimental data to separate the lateral wave and notch

tip diffracted echoes. For the second experiment, a phased array pitch catch scan was conducted

on a test sample containing multiple side-drilled holes. For this experiment the echo separation

algorithms were applied to separate the side-drilled hole scattered echo and the back wall

reflection echo. The echo separation performance of each algorithm was assessed by comparing

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its extracted arrival time difference between the two echoes with the arrival time difference

estimated using geometric calculations.

6.2 Research Findings

The simulation tests conducted in this research project were divided into two major sets. For the

first set, each of the two overlapping echoes in the simulated signal was designed to have a

symmetric envelope and a constant modulation frequency. This set represented the ideal case for

both TEI and DBPR because the echoes were perfectly described by the mathematical models of

the two echo separation algorithms. For the second set, each echo in the simulated signal had an

asymmetric envelope and a time-varying modulation frequency. This set of simulation tests was

more challenging as the echoes did not perfectly comply with the mathematical models and

assumptions of the two echo separation algorithms. However, the second set of simulation tests

also better represented real-world NDT applications where the ultrasonic echoes cannot be fully

described by simple parametric expressions.

For the set of simulation tests using symmetric echoes, we found that the echo separation

performance of TEI was much more consistent than DBPR. The standard deviations of timing

errors were approximately 40% lower for TEI in the noise-free tests. The reason for this

consistency is that the solution space for TEI is relatively convex. It is in general difficult to

reconstruct near-linear phase profiles which have phase slope values that are drastically different

from the correct ones. For this reason, TEI is less likely than DBPR to produce outliers in the

estimate of echo arrival time difference.

For the set of simulation tests using asymmetric echoes, it was observed that TEI outperformed

DBPR significantly in the noise-free tests according to all performance metrics. The means of

the timing errors (13-15% vs. 21-31%) and the reconstruction errors (~1% vs. ~4%) were both

significantly lower for TEI. The standard deviations of the timing errors (14-15% vs 27-37%)

and the reconstruction errors (~0.5% vs 2.5%) were also much lower for TEI. This difference in

performance is due to the ability of TEI to adapt its echo shapes to the input signal. This result

suggests that TEI may perform more reliably than DBPR in applications where the ultrasonic

echoes have complex shapes.

97

From the simulation tests, we also observed that the performance of TEI was nearly independent

of the SNR level for symmetric echoes but its performance deteriorated with decreasing SNR

level for separation of asymmetric echoes. We postulated that this difference in noise sensitivity

is due to TEI’s difficulty of differentiating between phase profile fluctuations introduced by

noise and the inherent phase non-linearity in asymmetric echoes. In comparison, the performance

of DBPR deteriorated with decreasing SNR level for both symmetric and asymmetric echoes.

Despite the difference in noise sensitivity for symmetric and asymmetric echoes, the mean

timing errors of TEI were still lower than DBPR by a minimum margin of 10% across all SNR

levels tested.

From the results of the experimental tests, we saw that the estimates of time difference between

two echoes of TEI were in line with the theoretical time differences calculated based on

trigonometry. This suggests that the frequency domain assumptions of TEI were applicable for

the NDT applications demonstrated. In particular, we tested two different echo optimality

metrics for the notched sample and both sets of solutions provided accurate time difference

estimates. This result suggests that the TEI method is sufficiently flexible to allow the user to

tailor the formulation of the optimality metric based on the prior information available.

6.3 Future Work

In this section, we outline additional research work that can be pursued in four major areas to

further develop the TEI echo separation algorithm.

Algorithm Refinement

• Currently the value of ∆𝑡𝑚𝑖𝑛 used in the phase slope inequality constraint is chosen

heuristically for the simulation and experimental tests. It would be ideal to develop an

automatic procedure to calculate an optimal value of ∆𝑡𝑚𝑖𝑛 using the estimated center

frequency and bandwidth of the input signal. A high bandwidth (and hence short

duration) signal should allow for a smaller time separation between the two echoes.

• In Section 5.2.3, we postulated that the echo separation accuracy of the TEI algorithm

deteriorates when the total signal resembles a single ultrasonic echo. Consequently, one

can develop a metric to measure the similarity of the total signal to our assumed profile of

98

a single echo. This metric can then be used to predict the difficulty of resolving the two

overlapping echoes and provide us with an estimate of the reliability of the echo

separation results.

Simulation Tests

• In this thesis, we have only examined the noise sensitivity of TEI to white Gaussian

noise. It would be useful to examine the effects of correlated noise on the echo separation

algorithm. Correlated noise is encountered in ultrasonic testing of materials with large

grain structures such as stainless steel and some grades of aluminum.

• To assess the influence of the randomness of the CMAES solver on the quality of the

solutions, one could rerun the optimization solver multiple times for a fixed test

configuration and examine the variance of the converged solutions. However, it might be

difficult to extrapolate the solution variance from one test configuration to another.

Experimental Tests

• In this thesis, we have only tested the TEI algorithm on two specific NDT applications. It

would be illustrative to evaluate the performance of TEI for other NDT applications

containing overlapping echoes. Candidate applications include separation of corner-

trapped echo separation in angled pulse-echo testing and normal incidence thickness

measurement of thin layers.

Novel Applications of the TEI Algorithm

• Although TEI is designed for the separation of two echoes encountered in ultrasound

non-destructive testing, the method is sufficiently flexible that it can be adapted for other

applications. The amplitude profile models and the echo optimality conditions can both

be altered to suit the needs of the target application. Possible applications include

magnetic and radar testing used in geophysical imaging.

• A dual method of the TEI algorithm can be implemented in the time-domain for

separation of two long duration low-bandwidth signals that overlap completely in time

and overlap partially in frequency. Instead of reconstruction of the frequency phase

99

profiles, the dual algorithm would reconstruct the phase profiles of the Hilbert transforms

of the two signals using time-domain trial amplitude profiles. Difference in the Hilbert

transform phase slopes can be used to calculate the frequency difference of the two

signals.

100

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105

Appendix 1: Two-way Impulse Response of Van-Dyke Model

One-Way Transfer Function

The acoustic path of the Van Dyke approximate model is modelled as an equivalent series RLC

circuit. In this equivalent circuit, the “voltage” is used to represent the force experienced at the

face of the transducer and the “current” is used to represent particle velocity located at the face

of the transducer. For a series RLC circuit, using Kirchhoff’s voltage law, the voltage drop

across the RLC components is equal to the voltage source:

𝑉𝑠(𝑡) = 𝑅 ∙ 𝐼(𝑡) + 𝐿𝑑𝐼(𝑡)

𝑑𝑡+

1

𝐶∫ 𝐼(𝜏)𝑑𝜏

𝑡

−∞

(A1.1)

In Eq. (A1.1), 𝐼(𝑡) is the “current” passed through the acoustic path and represents the face

particle velocity of the transducer. By differentiating Eq. (A1.1), we obtain:

𝑑𝑉𝑠(𝑡)

𝑑𝑡= 𝑅

𝑑𝐼(𝑡)

𝑑𝑡+ 𝐿

𝑑2𝐼(𝑡)

𝑑𝑡2+

1

𝐶𝐼(𝑡)

(A1.2)

Equation (A1.2) is a constant coefficient second-order differential equation which has a well-

known solution. If the forcing term 𝑉𝑠(𝑡) is an impulse input, the solution of Eq. (A1.2) would

be of the form:

𝐼(𝑡) = 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦(𝑡) = 𝑐𝑜𝑛𝑠𝑡 ∙ 𝑒−𝑎𝑡 cos(𝜔𝑑𝑡 + 𝜙) (A1.3)

Where 𝛼 is the decay rate and 𝜔𝑑 is the damped frequency and are defined as follows:

𝑎 =𝑅

2𝐿; 𝜔𝑜 =

1

√𝐿𝐶; 𝜔𝑑 = √𝜔𝑜

2 − 𝑎2 (A1.4)

Equation (A1.3) is the under-damped solution to Eq. (A1.2). It is appropriate to select the under-

damped solution because the Van Dyke approximate model is only applicable for lightly-loaded

piezoelectric transducers. If we take the Fourier transform of Eq. (A1.3), we would obtain the

one-way voltage-to-velocity transfer function of transducer.

𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦(𝜔)

𝑉𝑖𝑛(𝜔)= 𝑒(𝑗𝜙) [


𝛼 + 𝑗(𝜔 − 𝜔𝑑)+


𝑎 + 𝑗(𝜔 + 𝜔𝑑)]

(A1.5)

Taking the absolute value of the positive frequency content, we obtain:

106

|𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦(𝜔)



√1 +1

𝑎2 (𝜔 − 𝜔𝑑)2

(A1.6)

Two-Way Transfer Function

To find the two-way voltage-to-voltage transfer function, we need to first obtain an expression

for the force transmitted from the transducer. The force transmitted from the transducer can be

related to its face particle velocity through the acoustic impedance of the contacting medium:

𝑓𝑜𝑟𝑐𝑒 = 𝑍𝑡 ∙ 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 (A1.7)

Consequently, the frequency voltage-to-force transfer function would be equal to Eq. (A1.5)

except for multiplication with the acoustic impedance 𝑍𝑡. In addition, since the transducer is

modelled as a linear reciprocal device, the force-to-voltage transfer function should have the

same frequency dependence as the voltage-to-force transfer function. Using these assumptions,

the two-way voltage-to-voltage transfer function of the transducer would have the form:

𝑉𝑜𝑢𝑡(𝜔)

𝑉𝑖𝑛(𝜔)= 𝑐𝑜𝑛𝑠𝑡 [

𝑓𝑜𝑟𝑐𝑒(𝜔)

𝑉𝑖𝑛(𝜔)] [

𝑉𝑜𝑢𝑡(𝜔)

𝑓𝑜𝑟𝑐𝑒(𝜔)] = 𝑐𝑜𝑛𝑠𝑡 [

𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦(𝜔)

𝑉𝑖𝑛(𝜔)]

2

(A1.8)

Using Eq. (A1.8) and (A1.6), the amplitude response of the two-way transfer function can be

finally obtained:




1 +1

𝑎2 (𝜔 − 𝜔𝑑)2

(A1.9)

Appendix 2: KLM Model of Broadband Transducer

For the KLM model employed in Section 3.1.3, there is one finite matching layer at the front and

a perfectly matching layer at the back of the transducer. The perfectly matching layer can be

modelled by setting the acoustic impedance of the backing medium to be identical to the acoustic

impedance of the piezoelectric material. In addition, in this KLM model we do not include the

effects of electrical matching. For this configuration, the transmission path can be modelled by

the following transmission matrices:

Figure A-1: Transmission path in KLM model

For the transmission path shown in Figure A-1, the overall transmission matrix and the

frequency domain transfer function can be calculated as follows:

[𝑇𝑡𝑟] = [1 𝑍𝑠

0 1] [𝑇𝐶𝑜][𝑇𝐶′

][𝑇𝑥𝑓][𝑇𝑃][𝑇𝑇][𝑇𝑀] [1 0

1/𝑍𝑇 1]

𝐻𝑡𝑟(𝜔) = 𝐹2(𝜔)/𝑉𝑆(𝜔) = 1/𝑇11𝑡𝑟

(A2.1)

Similarly following the procedure outlined above, the reception path of the transducer can be

modelled using the following transmission matrices:

108

Figure A-2: Reception path in KLM model

For the reception path shown in Figure A-2, the overall transmission matrix and the frequency

domain transfer function can be calculated as follows:

[𝑇𝑟𝑒] = [1 𝑍𝑇

0 1] [𝑇𝑀][𝑇𝑇][𝑇𝑃][𝑇𝐶′

][𝑇𝐶𝑜][𝑇𝑥𝑓] [1 0

1/𝑍𝑅 1]

𝐻𝑟𝑒(𝜔) = 𝑉𝑜𝑢𝑡(𝜔)/𝐹2𝑅(𝜔) = 1/𝑇11𝑟𝑒

(A2.2)

Assuming the transducer is coupled to a perfect wave reflector, the force detected by the receiver

𝐹2𝑅(𝜔) would be equal to twice the force transmitted from the transducer 𝐹2(𝜔). Using this

relationship, the two-way transducer transfer function can be calculated as:

𝐻𝑡𝑤𝑜−𝑤𝑎𝑦(𝜔) = 𝑉𝑜𝑢𝑡(𝜔)/𝑉𝑆(𝜔) = 2𝐻𝑡𝑟(𝜔)𝐻𝑟𝑒(𝜔) (A2.3)

Using the model parameters provided in Table 3.1, a MATLAB implementation of this model

was created. The source code of this implementation is shown in Appendix 3.

109

Appendix 3: Source Code of KLM model

clear variables

close all

%properties for PZT-5H

es = 1.3e-8; %clamped permittivity [F/m]

c = 4620; %speed of sound [m/s]

rho = 7500; %density [kg/m^3]

kt = 0.49; % material coupling factor

Zo = 34.6e6; % acoustic impedance [Rayl]

%geometry of the transducer

A = 10e-3*10e-3; %area [m^3]

l = 1e-3; %thickness [m]

%frequency-independent properties

Co = A*es/l; % frequency-independent capacitance

wo = pi/l*c; % unloaded anti-resonant frequency [rad/s]

fo = wo/(2*pi); % unloaded anti-resonant frequency [Hz]

%properties of backing medium

Zb = 1*Zo; %characteristic acoustic impedance of backing layer

%properties of transmitting medium

Zt = 46e6; % steel characteristic acoustic impedance

%properties of matching material

Zm = 40e6; %acoustic impedance

rhom = 9000; %density

cm = Zm/rhom; %speed of sound of backing layer

tm = 550e-6; %thickness

%Multiply all acoustic impedances by area of transducer

Za = Zo*A;

Zt = Zt*A;

Zm = Zm*A;

Zb = Zb*A;

%Frequency Analysis Parameters

fc = 2e6; % design center frequency

Np = 20; %number of points per period at fc

N = 2048; %number of points in time domain

Nf = N/2; %number of frequency points to analyze

df = Np*fc/N; %frequency resolution

dt = 1/(N*df); %sampling time

f = 0:df:(Nf-1)*df; %frequency analysis axis

%Initialize vectors for one-way transmission and reception responses

H_t = zeros(1,Nf);

H_r = H_t;

for count = 1:Nf %loop through all analysis frequencies

w = 2*pi*f(count); %angular frequency

cp = -Co/kt^2/sinc(w/wo); %frequency-dependent capacitance

phi = kt*sqrt(pi/(wo*Co*Za))*sinc(w/(2*wo)); %frequency-dependent

transformer ratio

110

beta = w/c; %wave-number for piezo material

beta_match = w/cm; %wave-number for matching

% creation of transmission matrices

Tco = [1 1/(1i*w*Co);0 1];

Tcp = [1 1/(1i*w*cp);0 1];

Txf = [phi 0;0 1/phi];

Tt = [cos(l*beta/2) 1i*Za*sin(l*beta/2); 1i*sin(l*beta/2)/Za

cos(l*beta/2)];

Tq = At*[1 0; 1/Zb 1];

Tp = [1 0; Tq(2,1)/Tq(1,1) 1];

Tm = [cos(tm*beta_match/2) 1i*Zm*sin(tm*beta_match/2);

1i*sin(tm*beta_match/2)/Zm cos(tm*beta_match/2)];

Tpulser = [1 50;0 1];

Treceiver = [1 0;1/50 1];

% calculation of overall transmission matrix

Ttt = Tpulser*Tco*Tcp*Txf*Tp*Tt*Tm*[1 0;1/Zt 1];

% calculation of transmission transfer function

H_t(count) = 1/Ttt(1,1);

% calculation of overall reception matrix

Trr = [1 Zt;0 1]*Tm*Tt*Tp*inv(Txf)*Tcp*Tco*Treceiver;

% calculation of reception transfer function

H_r(count) = 1/Trr(1,1);

end

% calculation of two-way transfer function

H_tr = 2*H_t.*H_r;

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Frequency Domain Method for Resolution of Two Overlapping ... · Frequency Domain Method for Resolution of Two Overlapping Ultrasonic Echoes Chi-Hang Kwan Doctor of Philosophy Department