1. Master of science
in Electrical Engineering and Information technology
Master Thesis
Design of Coded Excitation and Filter Design for 3D
USCT II system
Submitted by:
Shreyank Gupta
Completion date: 16.03.2015
First Academic
Supervisor
Prof. Dr. Herbert Krauß Hochschule Darmstadt
University of applied sciences
Second Academic
Supervisor
Prof. Dr. Ulrich Schultheiß Hochschule Darmstadt
University of applied sciences
Industrial Supervisor Dipl.-Ing Michael Zapf Institute for Data
Processing and Electronics
2. Shreyank Gupta
Date of Birth: 27.11.1989, Uttar Pradesh, India
Matriculation No: 732226
First Academic Supervisor: Prof. Dr. Herbert Krauß
Second Academic Supervisor: Prof. Dr. Ulrich Schultheiß
Title: Design of Coded Excitation Techniques and Filter Design For 3D USCT II System
Abstract: 3D Ultrasound Computer Tomography (USCT) II aims at early detection
of breast cancer. The acquired data consists of multiple superimposed reflections which
are difficult to separate due to the system’s limited bandwidth. Non-separable reflections
limit the resolution of the resulting images. Side lobes are the artifacts which limit the
separability of the reflections. The aim of this work was to suppress the side lobes and
increase the separability. High side lobe suppression was achieved applying customized
Mismatched filters. As result the contrast of the images could improved by 143% in the
analyzed best case.
.
In partial fulfillment of the requirements of the University of Applied Sciences
Hochschule Darmstadt (h da) for the degree Master of Science in Electrical Engi-
neering carried out in collaboration with Industrial Enterprise
Company: Institute of Data Processing and Electronics, Karlsruhe Institute of
Technology, Germany.
Address: IPE, Geb¨aude 242, Hermann-von-Helmholtz-Platz 1, D- 76344,
Eggenstein-Leopoldshafen
This Master Thesis is subject to a non-disclosure agreement between the University of Ap-
plied Sciences Hochschule Darmstadt (h da) and the industrial partner.
(Signature)
First Academic Supervisor:
I
3. Shreyank Gupta
First Academic Supervisor: Prof. Dr. Herbert Krauß
Second Academic Supervisor: Prof. Dr. Ulrich Schultheiß
Declaration
I hereby declare that this thesis is a presentation of my original research work and that no
other sources were used other than what is cited. I furthermore declare that wherever contri-
butions of others are involved, this contribution is indicat-ed, clearly acknowledged and due
reference is given to the author and source. I also certify that all content without reference
or citation contained in this thesis is original work. I acknowledge that any misappropriation
of the previous declarations can be considered a case of academic fraud.
Darmstadt,
Signature:
II
4. Acknowledgements
I acknowledge my sincere thanks to Institute for Data Processing and Electronics (IPE),
for providing me an opportunity to carry out my master thesis work and exposing me to
challenging research.
I express my sincere thanks to Prof. Dr. Herbert Krauß and Prof. Dr. Schultheiß, my
academic supervisors, for their timely help and support throughout. Their suggestions and
care-taking stance are admirable.
I express my deep gratitude and sincere thanks to my supervisor Michael Zapf who is a
senior scientific staff at IPE, KIT Karlsruhe, for his valuable guidance and kind co-operation
throughout the thesis work. Without his guidelines and encouragement, this work may not
have been completed. It was a great pleasure to work under his guidance.
Finally I thank one and all who helped me directly or indirectly in completing the the-
sis work.
III
5. Abstract
3D Ultrasound computer tomography (USCT) II is a new imaging method aimed at early
detection of breast cancer. Synthetic aperture focusing technique (SAFT) is used for re-
constructing images which requires unfocused omnidirectional ultrasound emission and re-
ception. Thus the SNR received in the signals is low. Coded excitation (CE) and matched
filtering techniques are used to increase the SNR. The acquired data contain multiple super-
imposed reflections due to the unfocused and omnidirectional emission and reception. The
separability of the superimposed reflections is limited by the system’s bandwidth by caus-
ing side lobe artifacts in the signals. Non-separable reflections limit the resolution of the
resulting images. Currently applied CE is a sine damped linear chirp. In a previous work
evaluated alternative CEs (Human sequences (HS)), indicating the possibility of suppressing
the side lobes by trading-off on main lobe energy. To gain more control over side lobes, a
combination design of CEs and Mismatch filters is evaluated in this work. For this, several
customized Mismatch filters are derived using linear equation systems. Different objective
functions were designed and evaluated with metrics like MSR, ISL and self designed metrics
like SLD (Side Lobe Distribution). The designed filters were evaluated in the signal and
image domain. Signal based evaluation was carried out for localization error, detectability,
separability, and SNR metrics over the number of superimposed reflections. Evaluation with
SAFT imaging was done on base of simulated data analyzing the influence over varying
object complexity (number of superimposed reflections) and amount of data available. Used
image evaluation metric was a contrast-to-back-ratio metric which was derived in analogy to
the signal domain metric SNR. The results show that designed combinations of Mismatch
filters and CEs have better MSR and ISL than the currently used Matched filter and CE
combination. The best combination of filter and CE performs under the same system con-
straints 143% better in image contrast-to-back-ratio than the assumed USCT combination
of filter and CE.
IV
8. CONTENTS
About workplace
The Karlsruhe Institute of Technology, briefly referred to as KIT, was established by the
merger of the Forschungszentrum Karlsruhe GmbH and the University Karlsruhe (TH) on
October 1st, 2009. KIT combines the tasks of a university of the state of Baden-W¨urttemberg
with those of a research center of the Helmholtz Association in the areas of research, teaching,
and innovation. I worked at the Institute of Data Processing and Electronics (IPE) which
specializes in the development of customized detector, trigger and data acquisition systems
for the highest data rates as well as control and monitoring systems in the broadest sense.
The research and development of IPE are embedded in the superordinate program structure
of the Hermann von Helmholtz Association of German Research Centers (HGF). At the
IPE around 100 people are employed. In this institute I worked with Ultrasound Computer
Tomography (USCT) group which built a system aiming at detecting breast cancer. USCT
promises better image quality than conventional ultrasonic methods. It is a small group
comprising of 13 people and is lead by Dr. N. V. Ruiter. During the work I recognized there
are many research opportunities available in this group related to transducers building and
characterization, signal processing, image reconstruction methods etc.
VII
9. Chapter 1
Introduction
1.1 Motivation
Breast cancer is an uncontrolled growth of breast cells [1]. Signs of breast cancer may
include a lump in the breast, a change in the breast shape, dimpling of the skin, fluid coming
from the nipple, or a red scaly patch of skin [2]. According to World Health Organization
(WHO) ”Breast cancer is the most common cancer in women both in the developed and
the developing world.” [3]. The size of a breast cancer and how far it has spread are some
of the most important factors in predicting the outlook of a woman with this disease [3].
Fig 1.1 shows that number of new breast cancer cases are highest among Europe, Australia,
and North America. Research into the causes, prevention, and treatment of breast cancer is
being done in many medical centers throughout the world. Suggested by American Cancer
Society the best method to treat breast cancer is to find it early when it is small, and has
not spread [5]. Finding breast cancer early is called early detection. It should be the aim
any breast cancer imaging system to detect breast cancer early. Most doctors feel that early
detection for breast cancer save thousands of lives each year [4].
1.2 Concept and Principle of Ultrasound Computer
Tomography
Considering the importance of detecting breast cancer, the 3D USCT II system (Fig. 1.2)
was developed at KIT, Germany which aims at detecting early breast cancer. The goal
of developing USCT is to find cancers before they spread. Ultrasound is a pressure wave
which can not be heard to human beings ( >16 kHz). It not only complements the more
traditional approaches such as X-ray mammography, but possesses unique characteristics
that are advantageous in comparison to other modalities. One of the key advantage of using
ultrasound as the source of emission is that it is a non-ionizing radiation and is therefore
considered to be safe as compared to x-ray. Ultrasound frequencies which are used in USCT is
in MHz region. Water is the contact medium between ultrasound and object to be diagnosed.
Container comprises the aperture of piezoelectric transducers as shown in Fig. 1.4 which
are spatially positioned in 3D. Transducers convert electrical energy into ultrasound energy
and vice-versa using the piezoelectric effect. Transducers have the capability to become
emitter and receiver. Emitter and receiver are part of TAS (Transducer Array System),
which comprise backing layer, matching layer, all electrical connections, and controls and
amplification electronics. Fig.1.3 depicts a prototype of TAS.
1
10. Chapter 1. Concept and Principle of Ultrasound Computer Tomography
Figure 1.1: It depicts number of new breast cancer cases worldwide per 100,000 people
1.2.1 Technical details of 3D USCT II system
Fig. 1.5 shows the model of 3D USCT II system. An excitation signal is sent to the system
and then a so-called A-scan (pressure of time signal) is received. This process is repeated
several times before all the A-scan are collected and sent for imaging. 3D USCT II can be
divided in two parts one is measurement step where all the measurements are done and the
second part is the imaging is done.
Image reconstruction of the of the breast in 3D USCT II system is done off-line Therefore
the signals can be processed with sophisticated and - within limits- time consuming algo-
rithms. Therefore a signal processing step is included which aims for achieving high SNR,
removal of artifacts and noise etc. A brief description of USCT imaging method has been
shown in Fig 1.6. An excitation signal, i.e. the coded excitation (CE), is applied to excite
the emitters, the emitters are excited sequentially. The USCT container in which breast is
submerged has the capability to rotate and lift. After the complete scan of breast is done,
raw data is collected by the DDR (Data storage device) and is transfered to the SSD. The
image reconstruction takes place from the data received.
It includes a signal processing chain which involves coded excitation signal (CE), and
matchedfiltering (A filter designed for producing optimal SNR and the process using this filter
is called as Matched filtering). This whole chain can be modeled into a system description
as shown in Fig. 1.6.
2
11. Chapter 1. Concept and Principle of Ultrasound Computer Tomography
Figure 1.2: (a) Position of a patient while measurement. (b) container in which breast is submerged
for detection. (c) 3D USCT II system.
1.2.2 Signal Processing chain
Coded Excitation Techniques
Conventionally, a dirac pulse or step function is applied to excite the transducers since they
have all frequencies covered. Since USCT is limited by the maximum transmitted peak
power of transducers because of hardware limitation and medical safety , a high energy dirac
pulse can not be send out, so we do not achieve good penetration which may results into
poor image quality. An alternate method to improve is, to distribute its energy over the
time domain which increase the pulse duration while keeping its transmitted peak power to
a maximum permissible value as shown in Fig. 1.8
The methods which have the potential to improve the SNR without increasing the peak
transmitted power are called coded excitation methods as shown in Fig. 1.7
Coded excitation methods were initially developed in the radar research field in 1950’s.
Their application to ultrasound started in 1970’s when they were first used to improve the
performance of flow estimation [?] and tissue engineering [?] [?].
In general, a unit amplitude FM ( Frequency modulation) Chirp pulse whose instanta-
neous frequency varies linearly with time can be expressed as :
ft(t) = rect(t/T)cos(kt2
/2 + ωot) for | t |< T/2, (1.1)
where T is the duration of the chirp in seconds, ωo is the central frequency, and K is a
3
12. Chapter 1. Concept and Principle of Ultrasound Computer Tomography
Figure 1.3: Prototype of TAS which comprises of emitters, receivers and all electrical connections.
chirp rate parameter that controls the pulse bandwidth. Fig. 4.1 shows linear rectangular
chirp and it’s frequency domain according to the requirement of 3D USCT II system.
Match filtering Technique
To achieve a high SNR ”Pulse Compression Technique” needs to be applied. 3D USCT II
system is very fortunate to have the knowledge of CE which is being sent, because for pulse
compression technique we need this knowledge to apply. In the following paragraphs we will
discuss its properties and how it works.
Pulse compression technique is a signal processing technique mainly used to increase the
resolution and signal to noise ratio. This is achieved by correlating the received pulse with
the transmitted pulse as shown in Fig. 1.9
This pulse compression technique is also called as Match filtering technique.
As already described in 1.2.2 spreading the signal to long time for increasing the en-
ergy despreading is required to sharply construct the localized information, to satisfy this
requirement concept ofmatchedfiltering is done which leads into optimal performance of
USCT system regarding SNR.
The characteristics of matched filter in USCT system is designed by a frequency response
function. In the frequency domain the matched filter transfer function, H(ω), is the complex
conjugate function of the spectrum of the signal that is to be processed in an optimal fashion.
Thus in general terms,
H(ω) = S∗
(ω), where S(ω) is the frequency response of the input signal.
4
13. Chapter 1. Concept and Principle of Ultrasound Computer Tomography
Ultrasound
Emitters
Ultrasound
Receivers
Surface of
TAS
comprising
emitters and
receivers
Figure 1.4: Aperture of 3D USCT II which shows positioning of transducers. Blue dots show
position of transducers which consists of emitters and receivers. Black circles show geometrical
boundary of array system which contain transducers. The Combination of all transducers position
is the aperture.
System responsenUSCT Container
Collected system
response
Figure 1.5: A simple model of 3D USCT II system where ”System responsen” shows a system
responses for each iteration.
5
14. Chapter 1. Concept and Principle of Ultrasound Computer Tomography
0 50 100
−1
0
1
NormalizedUamplitude
ConvolutionU
ChirpUasUCE
TimeU2µsd
ExcitationUsignal
0 100 200 300
−2
0
2
A-scan
TimeU2µsd
3DUUSCTUIIUsystem
TimeU2µsd
PressureUp2td
0 100 200 300
−50
0
50
100
Reflections
MatchedUfilteredUoutput
PressureUp2td
3DUSAFT
ReflectivityUimage
MeasurementUprocessU SignalUprocessingUchain
A-scan
Imaging
Figure 1.6: Illustration of imaging method of 3D USCT II system including signal processing chain
Time(s) 10-5
0 1 2 3 4 5 6
Amplitudex(mv)
0
100
200
300
400
500
600
A dirac pulse
Energyx=x511
Energyx=x511
Codedxexcitationxmethod
Time(s) 10
-5
0 1 2 3 4 5 6
Amplitudex(mv)
Trasmittedxpeakxpower
Trasmittedxpeakxpower
{
Energyxspread
-1
-0.5
0
0.5
1
Coded excitation Energyx=x511
Figure 1.7: Shows the process of coded excitation methods which distribute the energy of a dirac
pulse over the domain resulting in a low peak power but still having the same energy. This process
is called as coded excitation .
6
15. Chapter 1. Motivation of this work: USCT constraints
Time (s) 10-5
0 1 2 3 4 5 6
Normalizedamplitude
-1
-0.5
0
0.5
1
Linear rectangular chirp as CE
Figure 1.8: Rectangular linear chirp with sampling frequency of 10 MHz and bandwidth of 5 MHz.
Chirp as CE USCT Container System response
Filtered output
Figure 1.9: Chirp and system response are the transmitted and received pulse respectively. This
diagram is specific to USCT and showsmatchedfilter process where the output is the cross-correlation
of the received signals and the transmitted pulse time-reversed conjugate.
Characteristics of Match filter
1. It compresses the signal to the location where it begun. This is possible because all
the phases of CE is collected at one point in time.
2. It shifts the energy of the pulse at the beginning.
3. Conservation of energy holds and there is a increase in the peak power of the compressed
pulse by which we have higher SNR in the filtered output.
All the three characteristics are shown in Fig.1.10
1.3 Motivation of this work: USCT constraints
Matched filter produces optimal SNR but it has one severe disadvantage for 3D USCT II.
Any real digital system has limited bandwidth which leads to longer pulses with side lobes
7
16. Chapter 1. Motivation of this work: USCT constraints
Time(s) 10-4
0 0.2 0.4 0.6 0.8 1 1.2
Amplitude
-1
-0.5
0
0.5
1
Match filtered output overlayed on chirp
Compresseda
Pulsealength
OriginalaaPulsealength
(a)
Time(s) 10
-5
0 2 4 6
Amplitude
-1
-0.5
0
0.5
1
Match filter output overlayed on Ascan
Ascan
Matchgfilteredgoutput
ggggggggEnergygspread
gggggggShiftsgallgthegenergygatgbegininnggofgPulse
(b)
Time(s) 10-4
0 0.5 1 1.5 2 2.5 3
Amplitude
-200
-100
0
100
200
Match filter output overlayed on Ascan
Ascan
Match:filtered:output
X::2.76e-05
Y::1
X::5e-06
Y::215.5
Gain:by:factor:of:215:regarding:amplitude
(c)
Figure 1.10: (a) Figure shows compression property ofmatchedfilter which compressed the pulse. (b)
Shows the shifting property ofmatchedfilter by which all the energy of pulse shifts at the beginning
(c) Shows by shifting all the energy, high energy is gained at the position where reflection occurred
thereby improving the SNR
as shown in Fig. 1.11. For any ideal system with maximum bandwidth the pulse after
Matched filter results in a short pulse, as short as possible for a given sample rate or Nyquist
frequency. In a real system with limited bandwidth the pulse is longer due to side lobes.
Fig. 1.12 shows how side lobes creating false results with reflections which are not
separable. In a real match filter output reflections are much smaller and difficult to separate
as shown in Fig. 1.13
Aim
1. We are interested in suppressing the side lobes while maintaining the main
lobe energy.
2. Second we are interested in separating the multiple overlaid small reflec-
tions. Contrast of the image can be improved by separating each overlaid
reflection.
8
18. Chapter 1. Motivation of this work: USCT constraints
Timev(s) × 10
-3
1.4 1.45 1.5 1.55 1.6 1.65
Normalizedvamplitude
-0.5
0
0.5
1
Two reflections apart by 30.8 mm
Filteredvsystemvresponse
Envelope
reflectionvpositionv1
reflectionvpositionv2
Speedvofvsound
=v1540vm/sec
(a)
Timen(s) × 10
-3
1.46 1.48 1.5 1.52 1.54
Normalizednamplitude
-1
-0.5
0
0.5
1
Two reflections apart by 1.08 mm
Filterednsystemnresponse
Envelope
reflectionnpositionn1
reflectionnpositionn2
Speednof
soundn=
1540nm/sec
(b)
Figure 1.12: (a) Shows that if two reflections are separated by a large distance then they are
separable. (b) Shows if two reflections are separated by a small distance they are not separable and
they give false results when reconstructed as two reflections are hidden in one pulse.
Time (s) 10-3
0 0.5 1 1.5 2 2.5 3
NormalizedPressurep(t)
-1
-0.5
0
0.5
1
Matched filter output of Real A-scan
Several overlaid small reflections
Figure 1.13: This shows real A-scan with match filtering applied to it that contains several overlaid
small reflections which are difficult to separate.
10
19. Chapter 2
State of Art
2.1 Literature
The concept of coded excitation (CE) was first introduced in mid 1950’s [10] [11] and became
an indispensable feature of modern radar systems. Since 1950s, several techniques have
been studied and published. In 1953 Barker [12] has shown that there are codes available
which have desirable auto-correlation and with uniform distribution of energy. Storer and
Turyn [13] have concluded that Barker sequence does not exist that have length exceeding
thirteen. Barker codes have been independently discovered by a number of authors a good
review of the literature of the subject has been given in [?]. In 1962 D.A. Huffman [15]
suggested a sequence which have zero side lobes in the auto-correlation except at the end.
But these waveform do not have high energy. In 1971 Ackroyd [16] has formulated Huffman
sequences with high energy using Schroeder’s rule [17] and shown these sequences can be
used as equivalent to linear chirp with higher side lobe suppression. Similar scheme have
also been made to design filters to suppresses the side lobes. Tapped delay-line techniques
for suppressing the side lobes seems to be first discussed by Key et al. [18] who studied the
particular case of 13-element Barker code. In 1971 Rihaczek and Golden has presented a
similar scheme in [19]. Most of the side lobes reducing schemes have been discussed depend
on cascading a matched filter with a further side lobe reducing filter. A filter can be designed
instead of match filter which takes care of both the features of optimizing the SNR and side
lobes reduction. The filter was designed first by Ackroyd’s and Ghani in [20] which they
have tested for 13-element barker codes. Thereafter much research has been done for side
lobe suppression [21] [22] [23] [24].
2.2 Previous Work
2.2.1 Signal Processing involved in USCT
At KIT USCT dates back in 2000 when first results were published using its first version
which was 2D by R. Stotzka et al [25]. Untill now there were many updates on USCT all of
which aimed at improving its imaging capability. In 2007 first version of 3D USCT system
was published by H. Gemeke et al [26]. Until 2009 a simple Impulse function was used
as excitation signal. In 2009 an attempt has been made by M. Zapf et al [27] to evaluate
different CE (Golay codes, Barker codes and linear chirps) for 3D USCT and results showed
that CE can be used in USCT to gain SNR while conserving the resolution of the images. It
was done by applying pulse compression technique at the receiver end. Until now USCT is
using sine damped linear chirps as CE and match filtering technique for gaining higher SNR.
Recently it has been discovered that sine damped linear chirps have several disadvantages.
11
20. Chapter 2. Previous Work
Time=s7 10S5
0 1 2 3 4 5 6
NormalizedcAmplitude
S1
S0b5
0
0b5
1
Sine damped linear chirp
Frequenciesc=Hz7 106
0 1 2 3 4 5
NormalizedcMagnitude
0
0b2
0b4
0b6
0b8
1
Frequency domain of Chirp
Bandwidthc=c1b67cMHz
Centercfrequencyc=c2b4cMHz
Time=s7 10S4
1b45 1b5 1b55
NormalizedcAmplitude
S0b5
0
0b5
1
Auto-correlation output
Sideclobes
Mainclobe
(a)
Time(s) 10
-5
0 1 2 3 4 5 6
NormalizedcAmplitude
-1
-0.5
0
0.5
1
Coded Excitation using Schroeder's rule
Frequencyc(Hz) 106
0 1 2 3 4 5
NormalizedcMagnitude
0
0.2
0.4
0.6
0.8
1
Frequency domain of CE
Time(s) 10-4
1.44 1.46 1.48 1.5 1.52 1.54 1.56
NormalizedcAmplitude
-1
-0.5
0
0.5
1
Auto correlation output
Sideclobes
Mainclobe
(b)
Figure 2.1: (a) Sine damped chirp. (b) Linear chirp designed using Schroeder’s rule with superior
properties.
Time8sW 10-4
0 0.5 1 1.5 2 2.5
NormalizednAmplitude
-1
-0.5
0
0.5
1
Huffman Sequence as CE
Paddednwithnzeros
Frequencyn8HzW 106
0 1 2 3 4 5
NormalizednMagnitude
0
0.2
0.4
0.6
0.8
1
Frequency domain
Wellndefinednripples
Time8sW 10-4
1.46 1.48 1.5 1.52 1.54
NormalizednAmplitude
-0.5
0
0.5
1
Auto correlation output
Sidenlobes
Mainnlobe
Figure 2.2: Designing of Huffman sequence with limited bandwidth, frequency domain consists of
well defined ripple which causes the side lobe distribution.
New kind of CE has been designed by M. Zapf and S. Gupta which uses Schroeder’s rule [17]
for designing linear chirps as it has superior properties than sine damped linear chirps as it
has more energy in the time domain which allows more use of bandwidth as shown in Fig.
2.1 .
2.2.2 Design of Huffman sequences
Huffman sequences have designed to suppress side lobes. Huffman sequences (HS) have been
designed for 3D USCT II system. Since out system has limited bandwidth HS needed to
designed under limited bandwidth. Simply filtering the HS results in very poor energy.In
literature there was no description of designing HS under limited bandwidth. It was shown
[29] that Huffman sequences can be constructed with the properties of distributing side lobes
compromising on main lobe energy under limited bandwidth Fig.2.2
A Comparison between Huffman sequences and Schroeder’s chirp was done and is shown
in Fig.2.3. The simulated results show that suppression of side lobes on the compromise on
main lobe energy. The suppression of side lobe is due to the aperiodic ripples in frequency
domain.
12
21. Chapter 2. Previous Work
Timezs6 10
-3
2.11 2.115 2.12 2.125 2.13 2.135 2.14 2.145
Amplitude
-40
-30
-20
-10
0
10
20
30
40
Auto correlation comparsion between Huffman and Schroeder chirp
Huffman
Schroedersbchirp
Sideblobes
suppression
Compromisebon
MainblobebEnergy
FrequencybzHz6 106
0 1 2 3 4 5
NormalizedbMagnitude
0
0.2
0.4
0.6
0.8
1
Frequency domain comparison
Huffman
Chirp
Figure 2.3: This figure compares auto-correlation Schroeder’s chirp and Huffman sequence It clearly
shows side lobe suppression peak side lobe suppression of around 26 %.
So finding a new trade-off between side lobes suppression and SNR is a hot topic in the
research field today and this idea can be extended to design filters which have such kind of
properties in presence of limited bandwidth and noise.
It was learnt from HS that side lobes can be suppressed by changing the magitudes and phases
of frequency domain. So new degrees of freedom which are available to us are magnitude and
phases in the frequency domain. Currently the filter is same as CE. i.e. Matched filter. For
side lobes suppression another kind of filter is suggested in literature which breaks bound
and is called as Mismatch filter.
2.2.3 Compensated Match filter - Ground truth
The basic idea of a compensated match filter is to make resultant frequency domain of
filtering process to be flat so that it uses more bandwidth. In contrast matched filter uses
only the bandwidth provided by CE as shown in Fig. 2.4
Design process
The first step is to compute the discrete fourier transform of the a sequence i.e.CE. Sup-
pose (A0, A1, A2, A3, A4, .., AN ) represents frequency bin where N is the length of sequence.
The sequence (1/A0, 1/A1, 1/A2, 1/A3, .., 1/AN ) represents the sample values of the transfer
function of the inverse filter. By computing its inverse discrete Fourier transform, an ideal
inverse filter weighting sequence is obtained. This is the simple form of an inverse filter which
is designed using a rectangular window. But instead of using rectangular window available
other windows can be used to construct it. Different inverse filters has been designed using
different windows like Tukey [30], Blackmanharris [30], Chebyshev [31], Bartlett [32] etc.
Fig. 2.6 shows the result of a compensated matched filter. Compensated match filter
design is referred to as ground truth since this is most modern filter being used in the USCT
but using rectangular window. Compensated match filter with the combination of CE will
be referred as Compensated match filter appended with the name of CE.
2.2.4 Conclusions from previous work
Distribution of side lobes
with limited bandwidth one idea is that, side lobes can be distributed far apart from the
main lobe. This is beneficial for the systems to separate close reflections in the A-scans.
13
22. Chapter 2. Previous Work
106
0 1 2 3 4 5
Normalizedpfrequencypmagnitude
0
0.2
0.4
0.6
0.8
1
Frequency domain of the resultant filtering process
Matchedpfiltering
Inputpcode
Compensated
matchpfiltering
Lossp1
Lossp2
Totalplosspforpapmatchedpfilter:pLossp1+pLossp2
Figure 2.4: Figure shows that resulting frequency domain of compensated matched filter is flat
which means it conserves bandwidth as compared to the matched filter which has greater loss to its
bandwidth (Loss 1 + Loss 2).
USCT has a limited specified time window out of which images are constructed so side lobes
which are distributed outside this window can be omitted. By appropriate selection of CE
and filter, our preferable filter output with limited bandwidth should look like as shown in
Fig. 2.7.
Suppression of side lobes
Complete side lobes removal is only possible if full bandwidth is available but still an attempt
has to be made to reach the desired output as close as we can. An ideal output is shown in
Fig. 2.8. We aim to approach this ideal as close as possible.
Compensated match filter
Applying inverse of the code in frequency
domain
Figure 2.5: Compensated match filter designing process
14
23. Chapter 2. Previous Work
Timepdomain Frequencypdomain
TimepIsL 10-3
1.34 1.36 1.38 1.4 1.42 1.44 1.46
NormalizedpAmplitude
-1
-0.5
0
0.5
1
Compensated match filter
MSRp=p2.27
ISLp=p14.74
FrequencypIHzL 106
0 1 2 3 4 5
NormalizedpMagnitude
0
0.2
0.4
0.6
0.8
1
Frequency magnitude domain of CE
Figure 2.6: Compensated matched filter output. Please refer to the metrics definition section for
MSR and ISL.
TimeC(s) 10-3
0 0.5 1 1.5 2 2.5 3
NormalizedCamplitude
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Distribution of side lobes of two near by reflections
SideClobes
USCTCTimeCwindow
Figure 2.7: Ideal output of filter for a limited bandwidth which shows the region where the side lobes
are positioned out of specified USCT time window in red.
15
24. Chapter 2. Metrics Definition
Time(s) × 10
-3
1.48 1.49 1.5 1.51 1.52 1.53
NormalizedfAmplitude
-1
-0.5
0
0.5
1
Supression of side lobes for two near by reflections
Preferedfoutput
Simulatedfoutput
Twofnearfbyfreflections
Reflections
Figure 2.8: Preferable output of filter vs simulated output for two near by reflections.
2.3 Metrics Definition
Before continuing further the metrics which are used for the evaluation purpose are intro-
duced and Fig. 2.9 illustrates the metrics.
1. Energy : Is defined as sum of squares of the absolute amplitude per sample. E =
N
1
x(n)2
where E is the energy of the sequence x(n) and N are the number of samples.
2. Main lobe : is the lobe which contains maximum energy.
3. Integrated side lobe energy(ISL) : Side lobes are the lobes which are not the main
lobe. The sum of squares of the absolute amplitude per sample of all these side lobes
is called Integrated Side Lobe Energy.
4. Main lobe to peak side lobe ratio (MSR) : MSR is defined as the ratio of main
lobe to the side lobe which has the highest amplitude per sample.
MSR = |x(n)|n=0
max(|x(n)|n=0)
where MSR is the defined ratio, x(n) is the function for which MSR is calculated.
5. Main lobe width: Is the minimum distance between the main lobe samples such that
the samples magnitude is 50% of the maximum amplitude of the main lobe.
16
25. Chapter 2. Metrics Definition
Peak side lobe
Second peak side lobe
Third peak side lobe
Area where side lobes get
integratred
Figure 2.9: Matched filter output with limited bandwidth. Figure depicts some metrics for evalua-
tion.
6. Side lobes distribution: It is defined as side lobe distribution in the time domain
until peak of side lobes diminished to 1% of the main lobe peak.
SLD = (.01 · |x(n)|n=0)−1
Fig. 2.9 shows side lobes. Since we have multiple reflections of different magnitudes it is
possible that if two targets are so close that second peak side lobe may also add together and
create a false target position. So it is important to analyze filters not just for suppressing
peak side lobes but also for second peak and third peak side lobe etc.
17
26. Chapter 3
Analysis of USCT Constraints and
Requirements
3.1 Comparison of Conventional Ultrasound Imaging
and USCT Imaging
Conventional ultrasound imaging is done as shown in Fig. 3.1 where transducers lie in an
linear phased array system which can be focused or unfocussed whereas 3D USCT II system
uses unfocussed transducers which are distributed on 3D aperture as shown in Fig. 3.2 . To
separate reflectors in the image it is important for the system to have good spatial resolution.
The spatial resolution of any imaging system is defined as its ability to distinguish two points
as separate in space. Spatial resolution is measured in units of distance such as mm. Spatial
resolution categorized into axial resolution and lateral resolution. Both resolution are shown
in Fig.3.1.
Axial resolution refers to the ability to distinguish two structures that lie along the axis
(i.e. parallel) of the ultrasound beam as separate. Axial resolution also corresponds to the
bandwidth of the transducers which means it is limited by the bandwidth of the ultrasound
pulse used i.e. CE. Lateral resolution is the ability to distinguished between two reflectors
situated side by side in a direction perpendicular to the ultrasound beam. Lateral resolution
in the context of USCT means higher number of emitter and receiver combination i.e. A-
scans, higher the lateral resolution. The reason why we need lateral resolution is needed is
explained in the Fig. 3.2.
3.2 Focus of this work
The resolution of the USCT system is dependent on the number and the bandwidth of
transducers. Both are fixed for the real system. The focus of this work therefore is to
manipulate the side lobes in order achieve a better separability of the pulses and with that
a higher resolution of the system.
18
27. Chapter 3. Focus of this work
Ultrasound device
Axial
resolution
Lateral
resolution
Lateral direction
Axialdirection
Transducers
Reflectors
Emissiondirection
Ultrasound
Figure 3.1: Axial and lateral resolution using an example of a conventional ultrasound imaging.
Axial resolution is the capability to separate the reflection along the axis of the emission whereas
lateral resolution is to separate the reflections which are parallel to the emission direction.
Emitter
Receiver
Reflectors
Spatial7domain Time7domain
Time7(s) 10-3
1.61 1.62 1.63 1.64 1.65 1.66 1.67
Normalized7Amplitude
-1
-0.5
0
0.5
1
1.5
System response
Reflection7position71
Reflection7position72
Reflection7position73
Reflection7position77
Ellipsoid7
Figure 3.2: The multicolored spheres represent reflectors on an ellipsoid, i.e. they have the same
time of arrival for the given emitter-receiver combination. Therefore they are received at the same
time in the A-scan (see right figure) and superimpose.
19
28. Chapter 4
Methods
4.1 Optimization of CE
Real systems has limited bandwidth, so does our USCT system. Transducers used in USCT
have limited bandwidth with a center frequency of 2.4 MHz and bandwidth of 1.67 MHz.
So there is a need to generate chirp within this bandwidth.
As we see from the Fig. 4.1 (c) time and frequencies are not well separated as exactly
at the beginning and at the end of spectrogram we have all of frequencies at one point in
the time domain. These frequencies can not be removed by bandpass filtering since it will
remove the signal also. Also while doing auto-correlation we have unwanted side lobes when
these frequencies auto-correlate with the starting frequencies and vice-versa. So there is a
need of damping function which suppresses such kind of side lobes. At first 3D USCT II
system used a sine damped linear chirp.
A sine damped chirp has been introduced as shown in Fig. 4.2.
While analyzing this sine damped chirp one interesting fact came out which leads to the
introduction of plateau in the time domain.
We understood that for a particular bandwidth we have a well defined slope of the
envelope of chirp. So slope of both the rising and falling edge of chirp kept constant and
plateau has been introduced between these slopes. An introduction of plateau has been done
to increase the energy of chirp keeping the slope constant. Much effort has been devoted
towards devising the schemes which has maximum energy and much smoother spectrum
(smoother spectrum is needed to avoid the artifacts which sharp cut-off spectrum has).
One useful way of shaping a chirp pulse is to apply a Tukey window which has a much
smoother spectrum, which in turn leads to a high energy and reduced side lobes as seen in
Fig. 4.4. There are many methods which can achieve this via different windowing functions
as suggested in [9]. They all have different properties so to choose an optimum is still a
debatable topic and research is still going on to find the optimum windowing function which
has optimal properties regarding energy and smoothing of frequency spectrum .
It was further found that Tukey windowed matched filter has lower side lobe energy than
the compensated match filter. This is because compensated match filter has the frequency
domain with a sharp cut-off in the required bandwidth as result of which in the time domain
higher side lobes occur as shown in Fig. 4.5.
Since different CE results into different matched filters so for each CE which is being
evaluated a different name is given to that combination of CE and filter. When CE used is
Tukey damped chirp then matched filter is named as ”Matched filter 1” and sine damped
chirp then filter is named ”Matched filter 2”.
20
29. Chapter 4. Optimization of filters
Timeh(s) 10-4
0 0.5 1 1.5 2 2.5 3
NormalizedhAmplitude
-1
-0.5
0
0.5
1
Chirp as CE
Paddinghwithhzeros
(a)
Frequencies8(Hz) 106
0 1 2 3 4 5
Normalized8Magnitude
0
0.2
0.4
0.6
0.8
1
Frequency domain of Chirp
Center8Frequency8=82.48MHz
Bandwidth8=81.678MHz
(b)
(c)
Timew(s) 10-4
0 0.5 1 1.5 2 2.5 3
NormalizedwAmplitude
-1
-0.5
0
0.5
1
Auto correlation
Sidewlobes
Mainwlobe
Sidewlobeswduewtow
unwantedwfrequencies
(d)
Figure 4.1: (a) Linear Chirp with padded zeros so that frequencies do not alias inside the bandwidth
(b) Frequency domain (c) Spectrogram shows linear phase per frequency with time and frequency
not well separated.(d) Auto-correlation of linear chirp with unwanted frequencies auto-correlating
with each other and producing the artifacts which are shown inside the rings.
4.2 Optimization of filters
In general matched filtering is only applied to maximize SNR of the received reflection side
lobe suppression is not considered. Matched filter is fixed which means we can not generate
other kind of filters. Now we are looking for the kind of filter which takes care of side lobe
suppression also. Currently matched filter is same as CE. Now the question is do we have
any degrees of freedom to break this hard connection between CE and filtering, the answer
is yes. Now the objective is to modify the impulse response of the filter such that the output
of this filter would have desirable side lobe suppression. During this modification we require
certain degrees which are discussed in the next section.
21
30. Chapter 4. New degrees of freedom in the filter design
Time (s) 10-4
0 0.5 1 1.5 2 2.5 3
Amplitude
-1
-0.5
0
0.5
1
Chirp as CE
(a)
Frequency8(Hz) 106
0 1 2 3 4 5
Normalized8Magnitude
0
0.2
0.4
0.6
0.8
1
Frequency domain of chirp
Bandwidth8=81.78MHz
2.58MHz
Center8frequency8=82.58MHz
(b)
Time (s)
0.5 1 1.5 2
Frequency()
0
1
2
3
4
5
Spectrogram
Bandwidth
MHz
(c)
Time(s) 10-4
0 0.5 1 1.5 2 2.5 3
Amplitude
-30
-20
-10
0
10
20
30
40
Auto correlation of Sine damped chirp
No artifacts
(d)
Figure 4.2: (a) Sine damped Chirp with padded zeros (b) Frequency domain (c) Spectrogram shows
linear phase per frequency with time frequency separation(d) Auto-correlation of sine damped chirp
which shows no artifacts inside the rings due to well defined phase spectrum with no unwanted
frequencies auto-correlating with each other
4.3 New degrees of freedom in the filter design
4.3.1 First degree of freedom
Huffman sequence as evident from Fig. 2.3 shown is making the uses magnitude in the
frequency domain and modulating it in a certain manner to suppress side lobes. Thus the
magnitude in the frequency domain, shown in Fig. 4.6, must be changed in a certain manner
to result in side lobes suppression.
4.3.2 Second degree of freedom
Another degree of freedom is phases as shown in Fig. 4.6. In the Matched filter process
they are just inverse of the phases of the CE. We believe by changing the phases per fre-
quency, a better distribution of side lobes as it was seen in Huffman sequence can be achieved.
22
31. Chapter 4. Mismatch filters
Time(s) 10-5
0 1 2 3 4 5 6
NormalizedAmplitude
0
0.2
0.4
0.6
0.8
1
1.2
Envelope of sine damped chirp
θ1
(a)
Frequency8(Hz) 106
0 1 2 3 4 5
Normalized8Magnitude
0
0.2
0.4
0.6
0.8
1
Frequency domain of chirp
2.58MHz
Bandwidth8=858MHz
Center8frequency8=828.58MHz
(b)
Time(s) 10-5
0 1 2 3 4 5 6
NormalizedAmplitude
0
0.2
0.4
0.6
0.8
1
Envelope of sine damped chirp
θ2
(c)
Frequency8(Hz) 106
0 1 2 3 4 5
Normalized8Magnitude
0
0.2
0.4
0.6
0.8
1
Frequency domain of chirp
Bandwidth8=81.78MHz
2.58MHz
Center8frequency8=82.58MHz
(d)
Figure 4.3: It shows the envelope of four sine damped chirps with having different slopes (θ1 < θ2).
Slope gets increased when bandwidth decreases .
4.4 Mismatch filters
Mismatched filters can differ in magnitude and phases from CE [20]. In USCT this is the
first time that these filters are applied to suppress the side lobes. To design Mismatch filters
there are two approaches one is applying a side lobe reduction aftermatchedfiltering or the
second one is use a single Mismatch filter resulting in side lobe reduction and acceptable
SNR. Unlike a matched filter, the mismatched filter can be of any length.
4.4.1 Creation of Mismatch filter based on Phase variations
There are many ways to design mismatch filters, one of the design which was tried earlier by
M. Zapf is to iteratively changing the phase per frequency of the auto-correlation function
to distribute the side lobes but that do not result in good properties as only some of the side
lobes are distributed which was considered negligible. Second design which was tried was
based on equation system solver in the frequency domain where I tried hard to formulate
a system which would solve for a desired output by changing the magnitude and phase per
frequency but was unable to do that.
So finally one of the new design which is used in this thesis is based on time domain
23
32. Chapter 4. Creation of Mismatch filter based on linear equation systems
Timer(s) 10-4
0 0.5 1 1.5 2 2.5 3
Amplitude
-1
-0.5
0
0.5
1
Tukey damped Chirp as CE
Wellrdefinedrsloper
withinrtherbandwidth
Introductionrofr
plateau
(a)
Frequencies (Hz) 10
6
0 1 2 3 4 5
Magnitude
0
0.2
0.4
0.6
0.8
1
Frequency domain of Tukey damped Chirp
(b)
Time (s)
0.5 1 1.5 2
Frequency()
0
1
2
3
4
5
Spectrogram of Tukey damped chirp
MHz
(c)
Time(s) 10-4
0 0.5 1 1.5 2 2.5 3
Amplitude
-60
-40
-20
0
20
40
60
Auto correlation of Tukey damped chirp
(d)
Figure 4.4: (a) Chirp with padded zeros with the introduction of plateau which pushes more energy in
the time domain and making frequency domain more flatter this uses more bandwidth (b) Smoother
Frequency domain compared to linear chirp within the bandwidth of the system (c) Spectrogram
shows linear phase per frequency (d) Auto-correlation function of Tukey damped chirp with side
lobes compared to rectangular linear chirp.
formulation which uses linear equations.
4.5 Creation of Mismatch filter based on linear equa-
tion systems
If the filter have the weighting sequence [b1, b2, b3, b4.......bn] and it is being convoluted with
the input sequence [a1, a2, a3, a4.......am] the resulting filtered sequence is [c1, c2, c3, c4.......cm+n].
Our aim is to minimize
E =
m+n
0
d2
t dt (4.1)
by proper choice of weighting sequence. The sum E represents the ”energy” of the
difference between the actual response and the ideal response sequences and ”d” represents
the difference in the amplitude between the actual response and ideal response. An ideal
24
33. Chapter 4. Creation of Mismatch filter based on linear equation systems
TimeC(s) 10-4
1.52 1.54 1.56 1.58 1.6 1.62 1.64
NormalizedCamplitude
0
0.2
0.4
0.6
0.8
1
MatchCfilteredCoutput
CompensatedCfilterCoutput
Comparison for Tukey damped chirp
Figure 4.5: Figure shows compensated matched filter has higher side lobe energy than the matched
filter.
response sequence can be called as an objective function in the following chapters. I have
chosen to minimize ”E” by solving a linear system of equations.
Lets suppose Filter coefficients are given by a vector ”B” and Toeplitz matrix is given
by ”A” and objective function is given by ”C”. So rearranging the equation 4.1 we get our
new system of linear equation given by eq 4.2 as :
A · B = C (4.2)
A11 A12 . . . A1n
A21 A22 . . . A2n
. . . . . .
. . . . . .
. . . . . .
An1 An2 . . . Ann
B1
B2
B3
.
.
.
Bn
=
C1
C2
C3
.
.
.
Cn
Solving linear system of equations is very wide domain and contains lot of open questions
[34]. Different objective functions were evaluated. An ideal objective function is a dirac but
others are also evaluated which have freedom to distribute the side lobes. For solving linear
system of equations we have different solvers which are available and needs to be evaluated.
25
34. Chapter 4. Creation of Mismatch filter based on linear equation systems
Frequency8PHzh 106
0 1 2 3 4 5
Normalized8Magnitude
0
0.2
0.4
0.6
0.8
1
Frequency magnitude domain of CE
Frequency8PHzh 106
0 1 2 3 4 5
Magnitude
-4
-2
0
2
4
Phases of CE
Frequency8PHzh 106
0 1 2 3 4 5
Magnitude
-4
-2
0
2
4
Phases of Matched filter
TimePsh 10-4
0 0.5 1 1.5 2 2.5 3
Normalized8Amplitude
-1
-0.5
0
0.5
1
A-Scan
TimePsh 10-4
0 0.5 1 1.5 2 2.5 3
Normalized8Amplitude
-1
-0.5
0
0.5
1
Matched filter output
Side8lobes
Main8lobe
Frequency8PHzh 106
0 1 2 3 4 5
Normalized8Magnitude
0
0.2
0.4
0.6
0.8
1
Frequency magnitude of Matched filter
MagnitudePhases
First8degree8of8freedomSecond8degree8of8freedom
Figure 4.6: This figure shows two degrees of freedom. First is magnitude, and the second is phases.
4.5.1 Objective function
Designing a objective function should be designed as per the requirements. An objective
function can be designed in several ways and some of them have been evaluated in this work.
There are four general ideas behind constructing an objective function:
1. The first idea is to have minimum integrated side lobe energy, i.e. a Dirac. This
objective function is named as objective function 1.
2. The second Idea is to distribute side lobes out of specified region for (USCT time
window). This objective function is made as the combination of sine pulses at the
end, an empty part with USCT time window and cosine at the start. (To see their
formulation please refer to the appendix section) This objective function is named as
objective function 2.
3. The third idea is to have a trade-off between the main lobe width and the integrated
side lobe energy. This objective function is made by increasing the width of an impulse
function. This objective function is named as objective function 3.
4. The fourth idea is to have a Sinc pulse which has very low side lobes. The basic idea
26
35. Chapter 4. Creation of Mismatch filter based on linear equation systems
Time(s) 10-4
0 1 2 3 4 5 6
NormalizedAmplitude
0
0.2
0.4
0.6
0.8
1
A simple dirac function
Figure 4.7: An objective which aims at achieving minimum integrated side lobe energy (objective
function 1)
behind constructing a Sinc pulse as an objective function is to relax the restrictions
during optimization. This objective function is named as objective function 4.
4.5.2 Modeling of multiple overlaying reflections as linear equa-
tions
Generally, Mismatch filters which are designed are for one or two reflectors to detect but in
USCT we have multiple reflectors.
There is a requirement to build a 2D matrix which covers all reflections and their posi-
tions. In linear algebra, a Toeplitz matrix is a matrix which seems to fulfill this requirement.
Toeplitz matrix is a matrix in which each descending diagonal from left to right is constant.
For instance, the following matrix is a Toeplitz matrix:
A =
a b c d e f g
h a b c d e f
i h a b c d e
j i h a b c d
k j i h a b c
Any square, symmetric matrix is a Toeplitz matrix. If ij element of Aij (A is a Toeplitz
matrix and aij are the elements inside it.) follows the form of equation
Aij = Ai+1,j+1 = ai−j (4.4)
27
36. Chapter 4. Methods to solve linear system of equations
TimeC(s) 10-3
0 0.5 1 1.5 2 2.5 3
NormalizedCAmplitude
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Shifting side lobes outside USCT Time window
USCTCTimeCwindow
Figure 4.8: An objective which aims at distributing the side lobes outside the specified USCT time
window (objective function 2)
Generally, a matrix equation is the general problem of n linear simultaneous equations
to solve. For USCT system the Toeplitz matrix has some special relevance. Toeplitz matrix
in our case consists of auto-correlation of CE.
For USCT system the y-axis of Toeplitz matrix represents all those reflections and all the
positions where they can lie is represented by x-axis. Toeplitz matrix in our case is a square,
symmetric and positive definitive matrix. In linear algebra, a symmetric n*n real matrix M
is said to be positive definite if zT
Mz is positive for every non-zero column vector z of n real
numbers. Here zT
denotes the transpose of z. Toeplitz matrix being a positive definitive
matrix is very important in solving linear system of equation as it will be explained later in
the next section.
4.6 Methods to solve linear system of equations
There are two class of methods which tries to solve equation 4.3 for n − equations and
n − variables
1. Direct Methods
2. Iterative methods
Direct methods theoretically give an exact solution in a finite number of steps. Unfortu-
nately, while doing these we encounter rounding errors which builds up in all the following
steps to come.
28
37. Chapter 4. Methods to solve linear system of equations
Timeb(s) 10-3
2.98 2.99 3 3.01 3.02 3.03
NormalizedbAmplitude
0
0.2
0.4
0.6
0.8
1
A simple impulse function with wider main lobe
Mainblobebwidthb=b.31bms
Figure 4.9: An objective which also aims for suppressing the side lobes by giving the solver, to solve
for very low side lobes. This width depicts 3dB width of the peak (objective function 3)
Iterative methods construct a series of solution approximations which converges to the
solution of system. Iterative methods are preferred over direct methods primarily due to two
reasons
1. Iterative methods are less sensitive to rounding errors.
2. Very large problems including hundreds or thousands of equations and unknown vari-
ables may be time demanding to solve by standard direct methods.
4.6.1 Gradient stabilization methods
Literature [35] suggests gradient stabilization methods to solve linear system of equations.
There are many methods which solve these equations and to pick any one of them must be
analyzed in our case. Choosing an optimal solver is a well known problem so an attempt
must be made still to choose a better one among all the available solvers.
Principle of Gradient stabilization method
The Toeplitz matrix A is poitive − definitive and symmetric. By looking into the matrix
formulation it is clear that we have n equations and n variables to solve for. Iterative
methods use a quadratic form which is represented by :
f(B) = .5 · (BT
AB − BT
C) + d (4.5)
29
38. Chapter 4. Methods to solve linear system of equations
Time (s) 10-3
1.49 1.495 1.5 1.505 1.51
NormalizedAmplitude
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
A Sinc Pulse
Figure 4.10: An objective which aims suppressing the side lobes by giving relaxation for the solvers
to optimize (objective function 4)
Where A is a matrix, B and C are vectors and d is a scalar constant. Because A is
positive definite, the surface defined by f(B) is shaped like a paraboloid bowl. It will be
shown that f(B)is minimized by the solution to A · B = C.
The gradient of a quadratic form is defined to be
f (B) =
dfB
dB1
dfB
dB2
dfB
dB3
.
.
dfB
dxn
.
This gradient is a vector whose elements point in the direction of greatest increase of
f(B). At the bottom of paraboloid bowl, the gradient will be zero. The vector which point to
this gradient can be derived by differentiating f(B) with respect to B and setting it to zero.
f (B) = .5 · (BT
AB + A(B)) − C. (4.7)
As we know A is symmetric so this equation reduces to
f (B) = A · B − C. (4.8)
30
39. Chapter 4. Mismatch filtering process
Toeplitz 2D matrix
�1(�)
�1(�)
�1(�)
�1(�)
�2(�) �3(�) �� (�)
��−1(�)
��−1(�)��−2(�)
�3(�)
�3(�)
�2(�)
�2(�)
�� (�)
�� (�)
�� (�)
Time
Time
Representsallpossiblereflections
Represents positions where they can lie
�� (x)= Auto-correlation values
�=1.........N, N=length of samples
Figure 4.11: Toeplitz matrix is a 2D matrix in which each in the time domain. Physically y-axis
shows all the possible reflections and x-axis shows all the positions of reflections where they can lie.
setting 3..9 to be zero gives AB=C. In our case the resulting Toepiltz matrix is complete,
which means it has n − equations and n − variablesand solution for each equation is unique
and stable i.e. (A−1
is continuous). In that case we do not need to regularize our matrix.
The starting point from where the algorithm should start its searching for minimum point
is taken as default to be a zero vector. The number of iterations for the iterative methods
must be analyzed.
4.7 Mismatch filtering process
Fig. 4.12 shows Mismatched filtering process. The steps in designing a mismatch filter
starts with defining a CE within the limited bandwidth. The auto-correlation function of
Chirp as CE USCT Container System response
Derived objective
function
Derived Mismatch
filter coefficients
Input auto-correlation function Filtered output
Figure 4.12: Mismatch filtering process showing decoupling from CE and filtering part. Green color
shows the difference between the design ofmatchedfilter and mismatch filter
31
40. Chapter 4. Mismatch filtering process
the CE used to form a Toeplitz matrix. Next, the Toeplitz matrix and derived objective
function are input parameters for the iterative solver. After deriving the filter coefficients,
derived A-scans can be convoluted with the filter coefficients to give filtered output.
4.7.1 Evaluation of different solvers
Conjugate gradient square methods, Bi-conjugate gradient stabilization methods, Gener-
alized minimum residual method etc. are one of the many methods which are available in
MATLAB. Every method available in MATLAB was evaluated in our case. Main aim of
this evaluation was which solver gives the best results. Objective function 2 has been used
randomly for evaluating the solver. The filter generated from this solver will be named as
Mismatch filter 1 which will be appended by the name of coded excitation it uses for e.g.
Mismatch filter sine 1 means combination of sine damped linear chirp and Mismatch filter 1
has been used.
Parametrization
A solver needs to be parametrized. There is no need of regularization as we have n equations
and n variables. Only the number of iterations has to be investigated. This analysis has
been done by varying the number of iterations for a solver. Biconjugate gradient stabilized
method has been chosen randomly for this parametrization to show the reader how number
of iterations influences the metrics.
Iteration Analysis
We chose a change of less than 1% or less as a saturation limit. Fig. 4.14 and Fig. 4.15
show the analysis for integrated side lobe energy and main lobe to peak side lobe ratio.
The conclusion can be made that solver needs around 1000 iterations before converging to
produce the acceptable result.
Conclusion
Based on the evaluation chart 4.13 it is clear that Biconjugate gradients stabilized method
produces superior results compared to all other solvers baring only on Integrated side lobe
energy where it lags behind conjugate gradient square method by very little margin. By
analyzing the solvers I propose to select the solver based on Biconjugate gradients stabilized
method for deigning mismatch filters with 1000 iterations.
4.7.2 Evaluation of different objective functions
For the evaluation Biconjugate gradient stabilization method has been chosen. Our main
aim of this evaluation is to pick an objective function which produces minimum Integrated
side lobe energy and all main lobe peak side lobes ratio to be as maximum as possible. The
ISL and MSR are not significantly different.
Conclusion from the designed objective functions
Designed objective functions have maximum bandwidth. Conventionally everything needs
to be designed in the limited bandwidth as we did in the case of CE. So next task was to
design an objective function which has limited bandwidth. This task can be achieved via
HS.
32
41. Chapter 4. Mismatch filter design based on Huffman sequence
Performance chart for different solvers
Solvers Integrated Side4lobe4
Energy
Main4lobe4to4peak4side4
lobe4ratio
Main4lobe4to4second4
peak4side4lobe4ratio
Main4lobe4to4third4peak4
side4lobe4ratio
Biconjugatelgradientsl
method
15.11 2.40 12.64 15.89
Biconjugate4gradients4
stabilized4(l)4method
14.64 2.48 14.64 19.20
Conjugatelgradientsl
squaredlmethod
14.57 2.40 13.80 18.19
Generalizedlminimuml
residuallmethod
15.25 2.21 11.51 15.15
LSQRlmethod 17.39 2.09 11.35 16.11
Minimumlresiduall
method
14.89 2.33 12.22 15.61
Preconditionedl
conjugatelgradientsl
method
15.11 2.40 12.64 15.89
Quasi-minimallresiduall
method
14.89 2.33 12.22 15.61
SymmetriclLQlmethod 15.05 2.43 13.00 16.21
Transpose-freelquasi-
minimallresiduall
method
14.59 2.35 13.27 17.58
Figure 4.13: Evaluation showing the performance of different solvers. It can be assumed that for the
real data also as compared to all other solvers Biconjugate gradients stabilized method will produce
superior results than the other one for the selected objective function as it does for the simulated
data.
4.8 Mismatch filter design based on Huffman sequence
D.A. Huffman investigated sequences, whose autocorrelation function side lobes except the
last one are zero. He showed, that roots of the Z-transformation of these sequences lie on
two circles [15]. Because of the sharp (means zero side lobes and lowest possible main lobe
width) auto-correlation these sequences are of particular interest for USCT. D.A. Huffman
and Ackroyd’s [16] both did not mentioned in their papers how to produce Huffman sequences
with limited bandwidth. In [29] it was shown that Huffman sequences can be designed for
a limited bandwidth conserving their superior properties of low side lobes, This paper has
explored one new degree of freedom i.e. the number of zeros are no bound anymore to the
length of the sequence whereas in the previous approaches number of zeros were same as
length of sequence. Making use of this new degree of freedom an objective function can be
designed with limited bandwidth which has very low side lobes.
In generating Huffman sequences we have two degrees of freedom which can be varied
independent of each other i.e. the number of zeros in the Z-domain, and their positions i.e.
radius of two circles. An analysis was carried out how to select number of zeros and their
position to determine the Huffman sequence which has lowest possible side lobes for the
given limited bandwidth. Both the parameters were varied which in a 2D analysis resulting
33
42. Chapter 4. Mismatch filter design based on Huffman sequence
No.%of%iterations
0 500 1000 1500 2000 2500
Main%lobe%to%peak%side%lobe%ratio
2
2.1
2.2
2.3
2.4
2.5
Performance of solver on number of iterations
Saturation%point%where%MSR%change%is%less%than%1%
Figure 4.14: It shows by increasing no. of iterations from 1 to 1000 MSR increases by almost 21
%
in the best combination of numbers of zeros and their positions to have the lowest possible
first peak side lobe level and minimum integrated side lobe energy.
4.8.1 Analysis
The used CE is of length 100 µs. The same length has been used to design a Huffman
sequence in the limited bandwidth of 1.67 MHz with center frequency of 2.4 MHz aiming
for lowest possible side lobes of its auto-correlation function. So combining the variation
of number of zeros and radius together at the same time makes it a 2D evaluation. Fig.
4.17 shows number of zeros on the x-axis and radius values on the y-axis. This evaluation
shows that both the zeros and the radius should be small for highest main lobe to peak side
lobe ratio.
2D analysis of mismatch filters for varying number of zeros and radius values was done
which calculates the MSR for each designed mismatch filters for 50 number of zeros and
radius combinations. The aim of this analysis was to see how many number of zeros are
actually required for producing maximum possible MSR.
Fig. 4.19 shows objective function and their frequency domain for our limited bandwidth.
If looked closely then its objective function has at the end very high side lobes, since we have
multiple overlaid reflections which are merged together so we ignore the area at the end of
this objective function and give solver the objective function with the ends ignored.
34
43. Chapter 4. Mismatch filter design based on Huffman sequence
No.cofciterations
0 500 1000 1500 2000 2500
Integratedcsideclobecenergy
14
14.5
15
15.5
16
16.5
17
17.5
Performance of solver on number of iterations
SaturationcpointcwherecISLcchangecisclesscthanc1%
Figure 4.15: Integrated side lobe energy decreases by around 18 % by increasing the number of
iterations from 1 to 1000.
4.8.2 Filtering process
After optimizing the objective function derived from HS, it serve as an objective function
for the design of the mismatch filter and the rest process remains the same. Fig. 4.21 shows
the output of mismatch filter and its frequency domain.
4.8.3 Qualitative comparison
Fig. 4.22 shows qualitative comparison of the output of different filters. This figure shows
a reflection which is simulated under real parameters i.e. in the limited bandwidth of 1.67
MHz which is strongly cut by using Tukey window. By looking into the figures it is clear
that among the best available filter i.e. compensatedmatchedfilter, designed mismatch filters
perform better in the context of MSR. Designed Mismatched filter has an MSR gain of 32%
compared to Compensated matched filter.
4.8.4 Quantitative comparison
Sine damped chirp is used as reference with a length of 100 µs, bandwidth of 1.67 MHz and
sampling frequency of 10 MHz. For initial testing the performance of all the filters are seen
for only one reflection.
35
44. Chapter 4. Mismatch filter design based on Huffman sequence
Timensc 1bU3
1C36 1C38 1C4 1C42 1C44
NormalizedRAmplitude
UbC5
b
bC5
1
Mismatchedvoutput
MSRR=R2C49
ISLR=R14C7b
TimeRnsc 1b
U3
1C36 1C38 1C4 1C42
NormalizedRAmplitude
U1
UbC5
b
bC5
1
Mismatchedvfiltervoutput
MSRR=R2C49
ISLR=R14C71
FrequencynHzc 1b6
b 1 2 3 4 5
NormalizedRMagnitude
b
bC2
bC4
bC6
bC8
1
Mismatchedvfiltervfrequencyvdomain
FrequencynHzc 1b6
b 1 2 3 4 5
NormalizedRMagnitude b
bC2
bC4
bC6
bC8
1
Mismatchedvfiltervfrequencyvdomain
TimeRnsc 1bU3
1C49 1C495 1C5 1C5b5 1C51
NormalizedRAmplitude
UbC4
UbC2
b
bC2
bC4
bC6
bC8
1
AvSincvPulse
TimeRnsc 1bU4
b 1 2 3 4 5 6
b
NormalizedRAmplitude
bC2
bC4
bC6
bC8
1
Avsimplevdiracvfunction
TimeRnsc 1bU3
b bC5 1 1C5 2 2C5 3
NormalizedRAmplitude
UbC4
UbC2
b
bC2
bC4
bC6
bC8
1
ShiftingvsidevlobesvoutsidevUSCTvTimevwindow
USCTRTimeRwindow
Timensc 1bU3
1C36 1C38 1C4 1C42
NormalizedRAmplitude
U1
UbC5
b
bC5
1
Mismatchedvoutput
MSRR=R2C49
ISLR=R14C7b
FrequencynHzc 1b6
b 1 2 3 4 5
NormalizedRMagnitude
b
bC2
bC4
bC6
bC8
1
Mismatchedvfiltervfrequencyvdomain
Objectivevfunction FrequencyvdomainResultingvfilteredvoutput
nac
nbc
ncc
Figure 4.16: Different objective functions results. Every objective function which was analyzed have
similar results.
2D analysis for varying number of zeros and radius values
Zeros
10 20 30 40 50
Radius
0
5
10
15
20
.2
.4
.6
.8
1
2D analysis of varying number of zeros and radius values
Zeros
10 20 30 40 50
Radius
.2
.4
.6
.8
1
Mainlobetopeaksideloberatiovalues
0
1
2
3
4
5
6
Energyvalues
Figure 4.17: 2D analysis of MSR performance of Huffman sequences with varying number of zeros
and radius values. This evaluation shows that for the best MSR possible there is a trade off on
the energy of Huffman sequence. As the energy increases MSR tends to decrease. So a Huffman
sequence has been selected by trading off on main lobe energy.
36
45. Chapter 4. Mismatch filter design based on Huffman sequence
2D analysis of MSR performance of Mismatch filter 2
Number of zeros
10 20 30 40 50
Radius
.200
.600
.800
1
.400
Mainlobetopeaksideloberatios
0
0.5
1
1.5
2
2.5
3
Figure 4.18: 2D analysis of the mismatch filters for each zero and radius combination. This figure
shows that only a small number of zeros are required for the best possible MSR and also the area
where maximum MSR lie is limited till radius values of .4.
The comparison table 4.1 shows several metrics comparison which are already discussed
before. Matched filter output would be considered as reference.
Compared to the current filter which is used in USCT i.e. compensatedmatchedfilter,
designed mismatch filter shows better performance in side lobes suppression. Mismatch filter
sine 1 compromise less on main lobe energy as compared to the compensatedmatchedfilter
which suggests better SNR for mismatch filters compared to compensated one.
Filters MSR ISL Main lobe energy SLD
Matched filter 2 1.38 26.75 19.05 65 ms
Compensated Match sine filter 2.27 14.64 17.75 291 ms
Mismatch filter sine 1 2.48 14.70 18.54 251 ms
Mismatch filter sine 2 3 15.65 17.16 495 ms
Table 4.1: A quantitative comparison of all the filters under one table. For this comparison Sine
damped chirp of 100µs has been used.
37
46. Chapter 4. Length Analysis of the Filter
4.9 Length Analysis of the Filter
We have a limited frequency space which means the side lobes will alias inside the bandwidth
because of this limit. To overcome this limit we need higher resolution in the frequency
domain. Making a longer filter will improve the resolution in frequency domain and the
aliasing frequency components causing the side lobes will be moved outside the signal’s
bandwidth which will ultimately lead to less side lobe energy. Fig.4.23 shows where longer
filter lengths results in to less side lobe energy.
38
47. Chapter 4. Length Analysis of the Filter
TimeHdomain FrequencyHdomain
TimeHbs) 10-4
0.9 1 1.1 1.2
NormalizedHamplitude
-0.5
0
0.5
1
Objective function with cutted ends
FrequencybHz) 106
0 1 2 3 4 5
NormalizedHMagnitude
0
0.2
0.4
0.6
0.8
1
Frequency domain
TimeHbs) 10-4
0.95 1 1.05 1.1
NormalizedHamplitude
-0.5
0
0.5
1
Objective function with bandpass filtered
FrequencybHz) 106
0 1 2 3 4 5
NormalizedHMagnitude
0
0.2
0.4
0.6
0.8
1
Frequency domain
TimeHbs) 10-4
0 0.5 1 1.5 2
NormalizedHamplitude
-0.5
0
0.5
1
Objective function derived from HS
Artifacts
FrequencybHz) 106
0 1 2 3 4 5
NormalizedHMagnitude
0
0.2
0.4
0.6
0.8
1
Frequency domain
ba)
bb)
bc)
Figure 4.19: (a) Design of objective function within the limited bandwidth with very low side lobes
in time domain. (b) As designed objective function contains artifacts at the end shown in the red
rings, so these are cut. (c) Cutted objective function has some frequency components outside the
desired bandwidth so it was band-passed using tukey window and the resulting objective function has
very low side lobes with in the desired bandwidth.
Chirp as CE USCT Container Derived A-Scans
Auto-correlation
function from HS
Derived Mismatch
filter coefficients
Input auto-correlation function Filtered output
Design of HS
Figure 4.20: Mismatch filtering process based on Huffman sequence. Pink color shows the difference
in the design from the conventional mismatch filtering approach.
39
48. Chapter 4. Length Analysis of the Filter
TimeL2sg 10-3
1.3 1.35 1.4 1.45
NormalizedLAmplitude
-0.5
0
0.5
1
Mismatched output
MSRL=L3.01
ISLL=L15.65
TimeLdomain FrequencyLdomain
Frequency2Hzg 106
0 1 2 3 4 5
Magnitude
0
5
10
15
Mismatched filter frequency domain
Figure 4.21: Mismatched filter output derived from HS. In the further evaluations this filter will be
named as Mismatch filter 2 which will be appended by the name of coded excitation it uses for e.g.
Mismatch filter sine 2 means combination of sine damped linear chirp and Mismatch filter 2 has
been used..
TimeF(s) 10-3
2.43 2.435 2.44 2.445 2.45 2.455 2.46
NormalizedFamplitude
-0.5
0
0.5
1
Qualitative Comparison of all the filters
MatchedFfilterF1
MismatchFFilterFtukeyF2
MismatchFFilterFtukeyF1
CompensatedFMatchFTukeyFfilter
Figure 4.22: A qualitative comparison of all the filters which Mismatch filter tukey 2 has the MSR
among the all designed filters.
40
49. Chapter 4. Length Analysis of the Filter
Lengthbofbfilterb(s) 10-3
0 1 2 3 4 5
Percentageb(Log)
1.8
1.85
1.9
Percentage of energy removed out (%)
Lengthbofbfilterb(s) 10-3
0 1 2 3 4 5
Energybofbsidelobesb(Log)
2.93
2.94
2.95
2.96
2.97
2.98
2.99
3
Sidelobe Energy cancellation
Saturationbpointbwherebthebchange
inbthebenergybofbsideblobesbisblessbthanb1p
Saturationbpointbwherebthebchangebinbthe
percentagebisblessbthanb1p
Figure 4.23: Figure showing that by increasing the length of filter approximately 6-7% of the side
lobe energy can be removed from the desired window. It is beneficial for the USCT project since
without much effort we are able to remove out some of the side lobe energy. It is recommended
to use longer filters since by doing this side lobe energy removes out but with some computational
effort.
41
50. Chapter 5
Simulation Model and Metrics Design
5.1 Preamble
This chapter focuses on the modeling of multiple reflections and their analysis. The analysis
of different filters has been done so far only for one reflection. In this chapter these filters will
be tested for several reflections which is more closer to reality. Metrics will be designed to
evaluate the performance of filters with several reflections and how side lobes behave when
number of reflection increases.
5.2 Modeling
The geometry of 3D USCT II system is simulated. Ideal point scatterers are defined within
the USCT and A-scans are created, see Fig. 5.1. For the following evaluation three point
scatterers are used.
Emitter
Receiver
Reflectors
Spatialudomain Timeudomain
Timeu(s) 10
-3
0 0.5 1 1.5 2 2.5 3
Normalizeduamplitude
-1
-0.5
0
0.5
1
Mapping of spatial domain to time domain
Transmissionusignal
Reflections
Figure 5.1: A simulated model depicting reflections which are placed at an unequal distance. For
this simulation the amplitude from each reflection is equal to 1. This simulation depicts the ability
of the system to separate these multiple reflections which will ultimately improve the axial resolution
of the system after designing mismatch filters.
42
51. Chapter 5. Metrics for evaluating several reflections
Timek(s) 10-3
2.3 2.4 2.5 2.6 2.7 2.8
Normalizedkamplitude
-0.5
0
0.5
1
Two reflections are far apart
Backgroundkartifacts
Reflections
Mainklobe
Sideklobes
Figure 5.2: Simulated reflections where the distance between two of them is large so that they are
very easy to detect and separate.
5.3 Metrics Design for Threshold Based Detection
In this section first we will discuss of side lobes when the distance between the two reflections
vary, and secondly several metrics definitions will be given for the analysis.
5.3.1 Superposition of side lobes
Depending on the distance of the reflectors three types of superposition patterns can be
identified.
1. Two reflections are far apart as shown in Fig.5.2 and thus very easy to separate.
2. Two reflections near to each other and could not be separated as seen in Fig. 5.3.
3. Two reflections near to each other but separable as shown in Fig.5.4
After realizing these three combinations, filters should be tested for the robustness against
such kind of combinations.
5.4 Metrics for evaluating several reflections
These metrics are designed to evaluate the performance of each filter to check their robustness
for several reflections. There are many metrics which are designed and can be divided in
three categories i.e. 2D, 1D.
43
52. Chapter 5. Metrics for evaluating several reflections
Timec(s) 10-3
1.38 1.39 1.4 1.41 1.42 1.43
NormalizedcAmplitude
-1
-0.5
0
0.5
1
Two reflections which are very near by
d
Reflections
Figure 5.3: Simulated reflections where the distance (d) between two of them is too small, that they
are not detectable .
5.4.1 Shifting
Superposition of side lobes can cause the main lobe to appear as shifted as shown in Fig.5.5
Shifts detection
Detecting shifts is difficult. For simulations this shift is known. In reality it is impossible to
detect such kind of phenomenon since the positions of the reflections are not known.
If the value of the time sample is larger than the time sample of the reflection position
is greater then it is referred to as lying on the negative slope of the main lobe i.e. shifting
happens to the right side of the reflection as shown in Fig. 5.6
If the value of the time sample is lower than the time sample of the reflection position
then it is referred to as lying on the negative slope of the main lobe i.e. shifting happens to
the right side of the reflection as shown in Fig. 5.6
After detecting the shifts a search is carried on the desired direction to find the nearest
maximum point which gives the total shift.
5.4.2 Separability
When several reflections overlap the side lobes can add up or cancel out. This cause the
difference in the ability of a filter to separate the reflection as shown in the example Fig.5.7
44
53. Chapter 5. Metrics for evaluating several reflections
Time (s) 10-3
1.36 1.38 1.4 1.42 1.44
NormalizedAmplitude
-1
-0.5
0
0.5
Two reflections in which side lobes help
d
Figure 5.4: Simulated reflections where the distance between two of them is too small but accidentally
side lobes helps in detecting the reflections by spreading out far away from them.
If ”h”, i.e. the distance between the 3dB amplitude to the minimum, is positive the
pulses are separable and the value ”1” is given to calculate the mean separability of the
filter. If ”h” is negative, the value ”0” is given.
5.4.3 Detectability
The detectability for each reflection is calculated as ratio of the height of its main lobe and
standard deviation of the side lobes surrounding it which ultimately means the SNR. If the
value of SNR comes less than 1 then this reflection is undetectable from others and will be
marked as ”0” for calculating the mean detectability of the filter. If the value of SNR comes
greater than 1 then this reflection is detectable from others and will be marked as ”1”.
5.4.4 SNR per reflection
The SNR for each reflection is calculated. In a time window as shown in Fig. 5.9 one
reflection is taken at a time and ratio of the height of its main lobe and side lobes or noise
surrounding results in the SNR.
45
54. Chapter 5. Metrics for evaluating several reflections
Timen(s) 10-4
7.62 7.64 7.66 7.68
Normalizednamplitude
-1
-0.5
0
0.5
1
Shifting effect
Filterednoutput
Originalnreflectionnposition
ShiftsShifts
Figure 5.5: A simulation showing the shifting effects which occur due to the superposition of several
reflections. Red line shows the original position where the reflections occur, Green line shows where
the reflection were detected
Timen(s) 10-4
7.62 7.64 7.66 7.68
Normalizednamplitude
-1
-0.5
0
0.5
1
Shifts detection
Filterednoutput
Originalnreflectionnposition
ShiftsShifts
(-venslope)
(+venslope)
Figure 5.6: Different kind of slopes for shifting.
46
55. Chapter 5. Metrics for evaluating several reflections
Timef(s) × 10
-4
4.33 4.34 4.35 4.36 4.37 4.38
Normalizedfamplitude
-0.1
0
0.1
0.2
0.3
Effect on separability between two reflections
Reflectionfpositions
Mismatchffilterfoutput
Matchffilterfoutput
d
h
Figure 5.7: Figure showing simulated compensatedmatchedand mismatch filter output. In mismatch
filtered output reflections are separable as level of the ”d” denotes the distance between the 3dB
amplitudes of the main lobe. ”h” is the distance of the 3dB amplitude to the minimum.
Timef(s) 10-3
2.7 2.71 2.72 2.73 2.74 2.75 2.76
Normalizedfamplitude
-0.5
0
0.5
1
Reflections which are detectable
Matchffilteredfoutput
Reflectionfpositions
Sideflobesforfartifacts
Timef(s) 10-3
1.32 1.325 1.33 1.335 1.34 1.345 1.35
Normalizedfamplitude
-1
-0.5
0
0.5
Reflections which are not detectable
Matchffilteredfoutput
Reflectionfpositions
Sideflobesforfartifacts
(a)f1 (b)f2
Figure 5.8: Simulated matched filter output. Two reflections are detectable if the ratio of the height
of the reflection and the standard deviation of side lobes is ≥ 1 . Two reflections are not detectable
if this ratio is <1 as shown in figure b
47
56. Chapter 5. Metrics for evaluating several reflections
Timeh(s) 10-3
0.5 1 1.5 2 2.5
Normalizedhamplitude
-1
-0.5
0
0.5
1
Process showing calculation of SNR per reflection
Filteredhsystemhresponse
Reflectionhpositions
h
a b
SNRh=hh/(std(a)+std(b))
Figure 5.9: Simulated filtered system response which shows the calculation of SNR per reflection.
In this figure a and b shows the noisy region, std is used as a term for standard deviation, h the
height of main lobe.
48
57. Chapter 6
Evaluation and Results
In literature these metrics has been designed for testing two reflections of equal magnitude,
in this thesis we have extended this by increasing the number of reflections. These all
simulations follows simulated USCT system with the limited bandwidth. There two kind of
simulations one is on the signal level and second part is on the image level. The first part
of the simulation test the robustness of the filters. Second part is dedicated to show how
reducing the side lobes by designing filters will improve the contrast of the images.
6.1 Signal level
For these simulations 1000 randomly generated reflections of equal magnitudes have been
used. Sine damped chirp of 100µs has been used with a bandwidth of 1.67 MHz for these
simulations with a sampling frequency of 10 MHz. For achieving higher resolution in the
time domain filtered output frequency domain is up-sampled by a factor of 10. In these
simulation no noise has been introduced. Reflections has been placed at the discrete sample
points in the time domain. Inter sample reflection points are not used.
6.1.1 Shifting comparison
This section evaluates how much shifts each filters produce per reflection. The comparison
was done between all the four filters i.e. matched filter 1, compensated matched filter,
mismatch filter sine 1, mismatch filter sine 2. The filter with the smallest shift per reflection
is the best since the images which are constructed out of these filters will have less probability
to contain reflections shifted from its original location.
Fig. 6.1 shows the mean shift per reflection for increasing number of reflections. Gen-
erally the mean shift increases with the number of reflections. Matched filter has worst
performance. The other three filters have similar shift probability.
To further analyze the shifting property in more intuitive way, 2D histogram for each
filter is being made. Ideally mean and standard deviation of the histograms should be zero.
But it is impossible to achieve for very high number reflections. So designed filters are
analyzed to see which filter reaches close to the ideal result. But before making a histogram
it is important to select the bin width of histogram.The shape of the histogram sometimes is
particularly sensitive to the number of bins. If the bins are too wide, important information
might get omitted. If the bins are too narrow, what may appear to be meaningful information
really may be due to random variations that show up because of the small number of data
points in a bin. The resolution of our data is 1µs. So the optimum bin width in which our
data do not get omitted and neither we have fuzzy variations is 1µs. After selecting the bin
width for each filter 2D histogram is being made as shown in Fig. 6.2, red color shows the
49
58. Chapter 6. Signal level
Number of reflections
0 200 400 600 800 1000
Meanshiftperreflection
0
1
2
3
4
5
6
7
Mean shift per reflector over number of reflectors
Matched filter 2
Compensated match filter sine
Mismatch filter sine1
Mismatch filter sine 2
Figure 6.1: Mean shift per number of reflectors. Matched filter seems to be produce maximum shifts,
followed by Mismatch filter 1 seems to be the second one which produces maximum shifts, the two
other filters do not differ significantly.
standard deviation for the distribution function of the histogram. Mismatch filter 2 produces
the least variation of the shifts as the number of reflection increases compared to the other
filters. After comparing both the analysis i.e. shifts per reflection and 2D histograms it is
highly recommended to use Mismatch filter 2 as it has smallest standard deviation.
6.1.2 Separability
This section evaluates the ability of the filters to separate the reflections. Fig. 6.3 shows
the mean variation of separability factor for increasing number of reflections using the four
filters.
Matched filter is worse for separability as it has lower values for the 1-600 number of
reflections but gets better for 800-1000 number of reflections. All the other filters seems to
perform similar but conclusive decision can not be made based on this figure.
6.1.3 Detectability
Filter which has the best detectability will have high probability that the images constructed
with it have high contrast. Fig. 6.4a shows the variation of the mean detectability factor
as the number of reflectors increases. Secondly the other important metrics is the analyze
the behavior of SNR per reflection. To analyze the behavior of SNR, mean of the SNR per
reflection has been taken to see how SNR changes as the number of reflection increases for
50
59. Chapter 6. Simulation: Image level
× 10-6Shifts / sec
-2 0 2 4
Numberofreflections
200
400
600
800
1000 0
0.2
0.4
0.6
0.8
1
Matched filter 2
× 10
-6Shifts / sec
-2 0 2
Numberofreflections
200
400
600
800
1000 0
0.2
0.4
0.6
0.8
1
Mismatch filter sine 2Mismatch filter sine 1
× 10
-6Shifts / sec
-4 -2 0 2 4
Numberofreflections
200
400
600
800
1000 0
0.2
0.4
0.6
0.8
1
Compensated Match filter sine
× 10
-6
Shifts / sec
-2 0 2
Numberofreflections
200
400
600
800
1000 0
0.2
0.4
0.6
0.8
1
Figure 6.2: 2D histogram of the shifts. Shifts represents a localization error. It is done for all the
designed filters. Figure shows the standard deviation ( in red ) of mean shifts as the number of
reflection increases. Mismatch filter 2 has smallest deviation for the shifts compared to other filters.
all the designed filters. Fig. 6.4b shows the variation of SNR against the increasing number
of reflections.
Table6.2 compares the performance of different filters. If overall comparison is done than
on detectability and shifting property matched filter does perform worse than the designed
filters.
6.2 Simulation: Image level
On signal level the Mismatched filters showed a better performance than the Matched filter.
Therefore we expect on image level a high contrast of the images using Mismatch filters.
USCT uses SAFT (Synthetic Aperture Focusing Technique) [36] for reflection tomogra-
phy. Point scatterers are simulated and imaged using SAFT technique. 30 Point scatterers
are distributed randomly in 2D and are imaged using SAFT approach. Each point scatter
Filters Matched filter 2 Compensated match sine filter Mismatch filter sine 1 Mismatch filter sine 2
Overall detectability 49% 57% 58% 59%
Overall separability 62% 65% 61% 66%
Overall SNR .978 1.102 1.091 1.103
Table 6.1: This table is made by taking the mean from the above graph and it shows quantitative
results for the performance of the filters for overall detectability, separability and SNR. From the
table it is clear that Mismatch filter 2 clearly has the advantage over the other designed filters.
51
60. Chapter 6. Simulation: Image level
Number of reflections
0 200 400 600 800 1000
Meanseaprabilityperreflection
0.2
0.4
0.6
0.8
1
Mean separability for the increasing number of reflectors
Matched filter 2
Compensated match filter sine
Mismatch filter sine 1
Mismatch filter sine 2
Figure 6.3: Performance of different filters for mean separability as number of reflection increases.
Number.of.reflections
0 200 400 600 800 1000
Mean.detectability.per.reflection.('1'.=.100C)
0.4
0.5
0.6
0.7
0.8
0.9
1
Mean detectability for increasing number of reflectors
Matched.filter.2
Compensated.match.filter.sine.
Mismatch.filter.sine.1
Mismatch.filter.sine.2
(a)
NumberCofCreflections
0 200 400 600 800 1000
MeanCSNRCcomparedCtoCMatchCfilter
1
1.05
1.1
1.15
1.2
1.25
1.3
CompensatedCmatchCCfilterCsine
MismatchCfilterCsineC1
MismatchCfilterCsineC2
SNR per reflector normalized to Matched filter for increasing
number of reflectors
(b)
Figure 6.4: (a) Figure shows detectability as the number of reflection increases for different filters.
Mismatch filter 2 has high probability of detecting the reflections compared to all other filters. (b)
Figure showing mean SNR per reflection and comparison has been done against Matched filter. It is
clearly seen that Mismatch filter 2 has higher mean SNR as the number of reflection increases which
also proves that probability of detecting reflections with Mismatch filter 2 is highest as compared to
other filters.
52
61. Chapter 6. Modeling the image SNR
Filters Compensated match filter sine Mismatch filter sine 1 Mismatch filter sine 2
MSR 64% 79% 117%
ISL 82% 80% 71%
Main lobe energy -7% -2% -11%
Table 6.2: Table shows quantitative results for the performance of the filters against matched filter.
Positive sign shows improvement in the metrics as compared to Matched filter and Negative sign
shows trade off against Matched filter.
Filters Compensated Matched filter Mismatch filter 1 Mismatch filter 2
Overall contrast improvement using sine damped chirp 15% 42% 56%
Overall contrast improvement using tukey damped chirp -0.5% 9.15% 22%
Table 6.3: Performance of the designed filters for the contrast improvement compared to Matched
filter for sine damped and Tukey damped chirp. As expected Mismatch filter 2 produced the best
results It achieved a contrast improvement of more than 20% in both the cases
depicts a reflection point. Fig. 6.5 shows a 30 simulated point scatterers.
Images are reconstructed using all the designed filters. Fig. 6.6 shows the images for all
filters. A sine and Tukey damped chirp of length 100µs has been used with a bandwidth of
1.7 MHz.
Comparison of contrast was done for the filters compared to the contrast values of the
Matched filter results. The aim of this comparison was to quantify the contrast, with in lim-
ited A-scans can have in the contrast of the images using different filters. Clearly Mismatch
filter 2 has highest contrast to back ratio compared to all other filters. as shown in Fig. 6.6
The next comparison was done with the Tukey damped chirp which gives better result
than the sine damped chirp as shown in the Fig. 4.4 that it uses more bandwidth and has
higher energy. Comparison of contrast for 1000 A-scans for different CE’s and is shown in
Fig. 6.8.
6.2.1 2D Analysis
The main aim of this analysis is to evaluate the behavior of filters over the varying number
of point scatterers. For this analysis number of A-scans are kept fixed to be 1000 to reduce
the computational effort. Number of reflections vary from 1 to 1000 with sampling of 30
reflections. Two analysis has been done, first Fig. 6.9 is the 2D representation of the contrast
to back ratio of the reflections over the number of A-scans. Second Fig. 6.10 is 1D evaluation
which is done by taking average of the contrast ratio of all the reflections per A-scan.
6.3 Modeling the image SNR
The existing model of Image SNR takes only grating lobes [36] from SAFT imaging method
assuming they are behaving similar like Gaussian noise. Generally expected contrast to back
ratio of an image in USCT changes proportional to
√
N where N is the number of A-scans.
SNRImagemodel1 =
√
N (6.1)
Hereby advice a new refined model which also takes image complexity into account,
modeled by the numbers of point scatterers. This model also assumes that the axial side
53
62. Chapter 6. Modeling the image SNR
Position of Point Scatters
m
0.115 0.12 0.125 0.13 0.135 0.14
m 0.115
0.12
0.125
0.13
0.135
0.14
Figure 6.5: 30 simulated point scatterers with resolution of .6 mm. Red circles indicate the area
where point scatterers are.
lobes behave like Gaussian noise and sums up Gaussian noise like.
SNRImagemodel2 = N/R (6.2)
Where N is the number of A-scan and R is the number of point scatterers.
Fig. 6.11 shows the corrected expected SNR for 2000 reflectors over 10000 A-scans.
Comparison between Matched filter 1 and Mismatched filter tukey 2 has been given to
further analyze which filter seems to come close to the expectation. As existing formulation
has much larger variation against the real SNR, on the contrary newly redefined formulation
seems to fit our expectation both for maximum and limited bandwidth. It has some offset
that is due to the limited bandwidth.
6.3.1 Conclusive Table
54
63. Chapter 6. Modeling the image SNR
m
0.115 0.12 0.125 0.13 0.135 0.14
m 0.115
0.12
0.125
0.13
0.135
0.14
,kMismatchkfilterksinek1,kBW:1.694[MHz],kContrast-to-backk5.51
m
0.115 0.12 0.125 0.13 0.135 0.14
m
0.115
0.12
0.125
0.13
0.135
0.14
Matchedkfilterk2,kBW:1.694[MHz],kContrast-to-backk3.87
m
0.115 0.12 0.125 0.13 0.135 0.14
m
0.115
0.12
0.125
0.13
0.135
0.14
Compensatedkmatchkfilterksine,kBW:1.694[MHz],kContrast-to-backk4.45k
m
0.115 0.12 0.125 0.13 0.135 0.14
m
0.115
0.12
0.125
0.13
0.135
0.14
Mismatchkfitlerksinek2,kBW:1.694[MHz],kContrast-to-backk6.04
Image reconstructions using sine damped chirp
Figure 6.6: Image reconstruction of 30 simulated point scatterers using SAFT for 1000 A-scans.
Contrast to back ratio shows the performance of the filter.
Filter + CE combinations contrast-to-back ratio improvement compared to Match filter 2
Compensated match filter sine 15%
Compensated match filter Tukey 99%
Mismatch filter sine 1 42%
Mismatch filter sine 2 56%
Mismatch filter Tukey 1 118%
Mismatch filter Tukey 2 143%
Match filter 1 100%
Table 6.4: Conclusive performance of the combination of the designed filters and CE for the contrast
improvement compared to Matched filter for sine damped chirp. As expected Mismatch filter 2
with the Tukey damped chirp produced the best results as it achieved a contrast improvement of
approximately 143% against the ground truth
55
64. Chapter 6. Modeling the image SNR
m
0.115 0.12 0.125 0.13 0.135 0.14
m
0.115
0.12
0.125
0.13
0.135
0.14
CompensatedMmatchMfilterMTukey,MBW:1.694[MHz],MContrast-to-backM7.72M
m
0.115 0.12 0.125 0.13 0.135 0.14
m
0.115
0.12
0.125
0.13
0.135
0.14
,MismatchMfilterMTukeyM1,MBW:1.694[MHz],MContrast-to-backM8.47
m
0.115 0.12 0.125 0.13 0.135 0.14
m
0.115
0.12
0.125
0.13
0.135
0.14
MatchedMfilterM1,MBW:1.694[MHz],MContrast-to-backM7.76
m
0.115 0.12 0.125 0.13 0.135 0.14
m
0.115
0.12
0.125
0.13
0.135
0.14
Image reconstructions using Tukey damped chirp
MismatchMfitlerMTukeyM2,MBW:1.694[MHz],MContrast-to-backM9.42
Figure 6.7: Image reconstruction of 30 simulated point scatterers using SAFT for 1000 A-scans.
Contrast to back ratio shows the performance of the filter. Higher contrast to back ratio means
higher suppression of artifacts or side lobes in an image.Figure shows compensatedmatchedfilter
gives worse results than the matched filter. Explanation for this phenomenon is given in Fig. 4.5
56
65. Chapter 6. Modeling the image SNR
NumberpofpA-scans
0 200 400 600 800 1000
Contrastptopbackpratio
-1
0
1
2
3
4
5
6
7
Comparison of different filters
Matchpfilterp2
Compensatedpmatchpfilterpsinep
Mismatchpfilterpsinep1
Mismatchpfilterpsinep2
(a)
NumberyofyA-scans
0 200 400 600 800 1000
0
2
4
6
8
10
Comparison of different filters
Matchedyfiltery1
CompensatedymatchyfilteryTukeyy
MismatchyfilteryTukeyy1
MismatchyfilteryTukeyy2
Contrastytoybackyratio
(b)
Figure 6.8: (a) Figure showing comparison of contrast to back ratio for different filters using sine
damped chirp for 30 Point scatterers. Mismatch filter 2 has the best contrast ratio which means
better suppression of side lobes compared to other filters. The expected curve of contrast to back
ratio varies as the square roots of the number of A-scans. (b) Figure showing comparison of contrast
to back ratio for different filters using tukey damped chirp. Interestingly compensatedmatchedfilter
is even worse than the matched filter, the reason is explained in Fig. 4.5
57
66. Chapter 6. Modeling the image SNR
NumbergofgA-Scans
200 400 600 800 1000
Numbergofgreflecions
100
200
300
400
500
600
700
800
900
-1.5
-1
-0.5
0
0.5
1
1.5
Mismatch filter Tukey 1
NumbergofgA-Scans
200 400 600 800 1000
Numbergofgreflecions
100
200
300
400
500
600
700
800
900
-1.5
-1
-0.5
0
0.5
1
1.5
Compensated match filter Tukey
NumbergofgA-Scans
100 200 300 400 500 600 700 800 900
Numbergofgreflections
0
100
200
300
400
500
600
700
800
900
1000
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
Matched filter 1
NumbergofgA-Scans
200 400 600 800 1000
Numbergofgreflecions
100
200
300
400
500
600
700
800
900 -2
-1
0
1
2
3
Log(Contrastgtogbackgratio)Log(Contrastgtogbackgratio)
Log(Contrastgtogbackgratio)Log(Contrastgtogbackgratio)
Mismatch filter Tukey 2
Figure 6.9: Figure shows 2D analysis of contrast to back ratio for varying number of reflection
over A-scans. This simulation was done using tukey window of length 100µs with a bandwidth of
1.7 MHz. It is a logarithmic curve which shows contrast to back ratio tends to decrease as number
of reflection or point scatterers increases. Designed mismatch filters seems to perform better than
Matched filters as they have larger values of contrast to back ratios.
58
