CHIRPLET SIGANL DECOMPOSITION OF ULTRASONIC SIGNAL: ANALYSIS, ALGORITHMS AND APPLICATIONS BY YUFENG LU Submitted in partial fulfillment of the requirements for the degree of Doctor in Philosophy in Electrical Engineering in the Graduate College of the Illinois Institute of Technology Approved _________________________ Adviser Chicago, Illinois May 2007
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CHIRPLET SIGANL DECOMPOSITION OF ULTRASONIC SIGNAL:
ANALYSIS, ALGORITHMS AND APPLICATIONS
BY
YUFENG LU
Submitted in partial fulfillment of the requirements for the degree of
Doctor in Philosophy in Electrical Engineering in the Graduate College of the Illinois Institute of Technology
Approved _________________________ Adviser
Chicago, Illinois May 2007
iii
ACKNOWLEDGEMENT
I would like to express my sincere gratitude and appreciation to my advisor, Dr.
Jafar Saniie, for his encouragement, motivation, inspiration, guidance and friendship
throughout all phases of my Ph.D study at Illinois Institute of Technology. I am very
grateful to my defense committee members: Dr. Guillermo E. Atkin, Dr. Erdal Oruklu,
and Dr. Xiangyang Li, for their valuable comments and suggestion on this work. I am
also thankful to my colleagues and friends: Dr. Ramazan Demirli, Dr. Guillerme
Cardoso, Dr. Fernando Martinez Vallina, and Mr. Logan Sorenson, in particular, to
Ramazan and Guillerme for their valuable discussion to enhance the work, to Logan for
the collaboration in the hardware implementation chapter.
I would like to dedicate the work to my family: my wife, my parents, and my
sister. This work would not be possible without their years of constant support,
encouragement and love. The special thanks to my wife, Jie Jiao for the endless patience
and understanding. The work witnesses the days from China to United States, from
Syracuse to Chicago.
iv
TABLE OF CONTENTS
Page
ACKNOWLEDGEMENT ....................................................................................... iii
LIST OF TABLES ................................................................................................... vii
LIST OF FIGURES ................................................................................................. viii
ABSTRACT ............................................................................................................. xi
5. COMPARITIVE STUDY OF CTSD AND MPSD ALGORITHMS ...... 54
5.1 Introduction ................................................................................. 54 5.2 Derivation of Cramer-Rao Lower Bounds .................................. 55 5.3 Monte Carlo Simulation .............................................................. 59 5.4 Observation and Analysis ........................................................... 60 5.5 Summary ..................................................................................... 61
6. TARGET DETECTION OF ULTRASONIC BACKSCATTERED
SIGNAL .................................................................................................. 64
6.1 Introduction ................................................................................. 64 6.2 Real Time Ultrasonic Measurement System .............................. 64 6.3 Target Detection in Ultrasonic Backscattered Signal ................. 67 6.4 Bat Chirp Signal Analysis ........................................................... 75 6.5 Summary ..................................................................................... 76
7. STATISTICAL EVALUATION USING ULTRASONIC GRAIN
SIGNAL ................................................................................................... 83
9.1 Introduction ................................................................................. 109 9.2 Embedded DSP System Based on Xilinx Virtex II Pro FPGA. . 110 9.3 Summary ..................................................................................... 114
3.1 Parameters of Decomposed Echoes (CTSD Method) ..................................... 32 3.2 Parameters of Decomposed Echoes (Gabor Decomposition Method) ............ 33 4.1 Parameters of Decomposed Echoes for the Simulated Chirp Signal (MPSD-MLE Algorithm) ................................................................................ 45 4.2 Parameters of Decomposed Echoes for the Simulated Chirp Signal (MPSD-MAP Algorithm) ................................................................................ 51 5.1 Comparison of the CRLB’s with the Variances of CTSD and MPSD for Different SNR. ................................................................................................ 62 6.1 Parameter Estimation Results for Ultrasonic Signal (CTSD Algorithm). ...... 71 6.2 Parameter Estimation Results for Ultrasonic Backscattered Signal (MPSD Algorithm). ......................................................................................... 74 6.3 Parameter Estimation Results for Bat Chirp Signal (CTSD Algorithm). ....... 79 6.4 Parameter Estimation Results for Bat Chirp Signal (MPSD Algorithm). ....... 82 7.1 Scattering Coefficients as a Function of Mean Grain Diameter and Frequency. ....................................................................................................... 86 7.2 Upward Frequency Observed for Grain Signal from Steel Specimens. .......... 93 8.1 Parameter Estimation Results for Multilayered Echoes .................................. 104 8.2 Estimated Coefficients of Reverberant Echoes ............................................... 105 8.3 Thickness Estimation of Multilayered Structure ( 31 ≤≤ k ) ................... 107
viii
LIST OF FIGURES
Figure Page
2.1 Comparisons of Time Frequency Techniques. a) a Simulated Signal. b) WVD of the Signal. c) STFT of the Signal (Using Hamming Window). d) CWT of the Signal (Using Morlet Wavelet). ............................................ 14
3.1 The Flowchart of CTSD Algorithm ................................................................ 28 3.2 Basic Illustration of Dominant Echo Windowing Method. a) CT of Three Interfering Chirp Echoes. b) Projection in Frequency Domain and the Frequency Window Boundary Points (Dashed Lines). c) Projection in Time Domain and the Time Window Boundary Points (Dashed Lines) ................. 29 3.3 Simulated Ultrasonic Highly Overlapping Echoes(Solid Line), Superimposed
with the Reconstructed Signals by CTSD Algorithm and Gabor Decomposition Method. ........................................................................................................... 34
3.4 Comparisons of CTSD Method and Gabor Decomposition Method. a) Simulated
Highly Overlapping Echoes. b) WVD of the Original Simulated Signal. c) Reconstructed Signal by CTSD Method. d) WVD of the Reconstructed Signal (Using CTSD). e) Reconstructed Signal by Gabor Method. f) WVD of the Reconstructed Signal (Using Gabor)... ........................................................... 35
4.1 The Flowchart of MPSD Algorithm. .............................................................. 42 4.2 Overlapping Chirp Signal Superimposed with the Reconstructed Signal Using
MPSD-MLE Algorithm. ................................................................................. 43 4.3 a) Overlapping Chirp Signal. b) WVD of the Overlapping Chirp Signal. c) the Estimated Signal Using MPSD-MLE Algorithm. d) WVD of the Estimated Signal. ............................................................................................................. 44 4.4 Overlapping Chirp Signal Superimposed with the Reconstructed Signal Using
MPSD-MAP Algorithm. ................................................................................. 49 4.5 a) Overlapping Chirp Signal. b) WVD of the Overlapping Chirp Signal. c) the Estimated Signal Using MPSD-MAP Algorithm. d) WVD of the Estimated Signal. ............................................................................................................. 50 5.1 a) Input SNR vs. Output SNR for a Single Noisy Chirp Echo Using CTSD Algorithm. b) Input SNR vs. Output SNR for a Single Noisy Chirp Echo Using MPSD Algorithm. ............................................................................................ 62 6.1 Real Time Ultrasonic Measurement System. .................................................. 66
ix
6.2 Ultrasonic Backscattering Signal Superimposed with the Reconstructed Signal (CTSD Algorithm) ............................................................................... 67 6.3 a) Ultrasonic Backscattering Signal. b) TF representation of the Ultrasonic Backscattering Signal. c) Estimated Signal Using CTSD Algorithm. d) TF Representation of the Estimated Signal. ......................................................... 70 6.4 Ultrasonic Backscattering Signal Superimposed with the Reconstructed Signal (MPSD Algorithm) .............................................................................. 72 6.5 a) Ultrasonic Backscattering Signal. b) WVD of Ultrasonic Backscattering Signal. c) the Reconstructed Signal. d) WVD of the Reconstructed Signal Using MPSD Algorithm. ................................................................................ 73 6.6 Experimental Bat Chirp Signal Superimposed with the Estimated Result (CTSD Algorithm). ......................................................................................... 77 6.7 a) Experimental Bat Chirp Signal. b) TF Representation of the Experimental Bat Chirp Signal. c) Estimated Signal. d) TF Representation of the Estimated Signal. .............................................................................................................. 78 6.8 Experimental Bat Chirp Signal Superimposed with the Estimated Result (MPSD Algorithm). ......................................................................................... 80 6.9 a) Experimental Bat Chirp Signal. b) WVD of Bat Chirp Signal. c) the
Reconstructed Signal. d) WVD of the Reconstructed Signal Using MPSD Algorithm. ...................................................................................................... . 81
7.1 Microscopes of Specimens. a) Steel-ref. b) Steel-1600. c) Steel-1900. .......... 89 7.2 Grain Signals of Steel Specimens. (I) Shows Grain Signal. (II) Shows Magnitude Spectrum. a) Steel-ref. b) Steel-1600. c) Stell-1900. .................... 91 8.1 Reverberation Path in Signal Thin Layer. ....................................................... 96 8.2 Multilayered Structures Consisting of Four Different Regions. ..................... 97 8.3 Variation of Wave Paths with Equivalent Traveling Time for Case Where k=2 and L=2. ................................................................................................... 98 8.4 The Reconstructed Reverberant Echoes Superimposed with the Experimental Reverberant Echoes of Multilayered Structure. .............................................. 103 8.5 Comparison of Envelope of Class “a” Echoes, “b” Echoes and “c” Echoes. . 106
x
9.1 Architecture Overview of Embedded FPGA-Based System........................... 113 9.2 Process Experimental Ultrasonic Echoes on FPGA-Based DSP System ....... 115
xi
ABSTRACT
A major and challenging problem in ultrasonic nondestructive evaluation (NDE)
is the ultrasonic backscattered signal analysis in presence of high scattering noise. The
pattern of Ultrasonic backscattered signal represents the shape, size and orientation of
ultrasonic reflectors and the physical property of propagation path. The signal loss by the
effect of scattering and absorption imposes a limit on the detection capability of
ultrasonic NDE systems. Therefore, signal modeling and parameter estimation of the
nonstationary ultrasonic signal is critical for precise evaluation of objects.
Joint time-frequency signal representation is an important method to evaluate the
nonstationary characteristic of ultrasonic backscattered signal. It can be shown that the
conventional time frequency transform such as Wigner Ville Distribution and Short time
Fourier transform introduce cross-terms , offer poor resolution, and are sensitive to noise
level. On the other hand, the continuous wavelet transform shows higher time resolution
in smaller scale and higher frequency resolution in high scale. This is a preferable
property for tracking the time-varying frequency of nonstationary signal, especially in
ultrasonic model based algorithm design.
In this study, we introduced chirplet transform (CT) as a means not only to obtain
time frequency representation of signal, but also to be utilized for chirplet signal
decomposition and successive parameter estimation. Based on the assumption that the
signal to be processed, no matter how complex, can be decomposed into superimposition
of multiple chirplet echoes, the chirplet signal decomposition based on chirplet transform
(CTSD) algorithm is developed. It utilizes the chirplet transform of signal to locate the
most dominant chirplet component and successively estimate its parameters, such as
xii
time-of-arrival, center frequency, chirp rate, phase and intensity. Compared with signal
decomposition based on Gabor function, the chirplet signal decomposition algorithm is
very effective in representing dispersive ultrasonic echoes due to the parameter diversity
of chirplets. Analysis and simulation results show that the performance of chirplet signal
decomposition overwhelms that of the Gabor decomposition with less number of
components to reconstruct the same high overlapping signal.
As an alternative, we developed matching pursuit signal decomposition(MPSD)
algorithm through incorporating statistical methods such as Maximum Likelihood
Estimation (MLE) and Maximum a Posteriori (MAP) into a general nonstationary signal
analysis frame work (i.e., matching pursuit algorithm). The MPSD algorithm iteratively
optimizes the parameters of a chirplet function to match the signal and achieve high
resolution decomposition. This approach avoids the exhaustive search of a large number
of dictionary functions and leads to a more efficient implementation.
Furthermore, we derived analytical Cramer Rao Lower Bound (CRLB) of chriplet
estimator. The performance of CTSD and MPSD algorithm are evaluated against the
CRLB bounds. Computer simulation indicates noise is better suppressed in CTSD
algorithm than it is in MPSD algorithm. Monte Carlo analysis shows that both algorithms
are minimum variance unbiased (MVU) estimators, hence they provide optimal
parameter estimation and robust chirplet signal decomposition.
We also explored different applications of the chirplet signal decomposition
approaches. The estimated parameters from the experimental signals have been
successfully used to locate the target echo in ultrasonic reverberant signal, evaluate grain
size of materials, and classify ultrasonic multilayered reverberant echoes. Moreover, an
xiii
embedded hardware system is implemented on Xilinx Virtex II Pro FPGA platform to
accelerate the chirplet signal decomposition algorithm. Through computer simulation and
analysis of experimental signals, this type of study addresses a broad range applications
including target detection, deconvolution, object classification, velocity measurement,
and ranging system.
1
CHAPTER 1
INTRODUCTION
1.1 Brief Introduction to Research
Ultrasonic waves have been applied in testing and imaging of material for a long
time. In the ultrasonic pulse-echo testing, ultrasonic signal travels through medium
without changing their physical states. The signal undergoes an energy loss due to
absorption and scattering of the internal microstructure on the propagation path. Hence,
the information of microstructure is inherent to the measured backscattered ultrasonic
signal. It can be utilized to characterize the propagation path which determines the
physical properties of reflectors, in terms of their location, geometric shape, size,
orientation and microstructure. Through the signal analysis, the useful feature of the
medium can be extracted. This is the property that supports the broad applications of
ultrasound in non-destructive evaluation (NDE) of material, and medical diagnosis.
The extraction of the desired information related to the properties of the medium
requires models to simulate the formation of echoes. From system point of view, the
measured backscattered signal can be simplified as the convolution result of input signal
(i.e., the transducer excitation pulse) and system response. The parameters of the
backscattered echoes such as time-of-arrival, center frequency, amplitude, bandwidth,
phase, and chirp rate are of important for their significance to dissolve the system
response. For example, the time-of-arrival and amplitude of the echo can be attributed to
the target response in term of target location, size and orientation. The variation of time-
of-arrival and amplitude can be attributed to the energy loss and the traversed time. The
center frequency, bandwidth and the phase of the echo can be attributed to the frequency
2
modification of the propagation path (i.e. characterization of media impedance). The
chirp rate can be attributed to the dispersion phenomenon in the traveling of ultrasonic
wave.
In this research, to form an efficient way to model the ultrasonic backscattered
echoes, we propose chirplet signal decomposition algorithm based on the chirplet
transform. The mathematical foundation of the algorithm is discussed. Another
decomposition implementation scheme which is based on the matching pursuit
framework is compared and discussed. The analytical Cramer-Rao bounds of the
algorithms are explored and compared with the simulated results. Furthermore, the
proposed algorithm is tested and verified in the different applications such as target
detection, bat chirp signal analysis, material grain size evaluation, and multilayered
structure inspection. Furthermore, an embedded FPGA-based DSP system for signal
decomposition is analyzed.
1.2 Thesis Outline
Chapter 2 presents a brief review concerning time frequency representation. Three
notably used time frequency representations such as short time Fourier transform,
Wigner-Ville distribution, and continuous wavelet transform are outlined. The time
resolution and frequency resolution of the three time frequency representations are
discussed.
Chapter 3 lays out the mathematical foundation of chirplet signal decomposition.
The basic idea behind the chirplet signal decomposition is to decompose any complex
signal into a linear combination of chirplet model and estimate all the parameters of the
3
model precisely. First, the chirp signal and its application background are presented.
Then, the successive parameter estimation algorithm based on chirplet transform is
elaborately derived with mathematical details. Furthermore, a windowing strategy is
applied in both time domain and frequency domain to generalize the successive
parameter estimation algorithm to decompose multiple high overlapping signals. In order
to demonstrate the robustness of chirplet model and the efficiency of chirplet signal
decomposition algorithm, we simulate a signal with multiple highly-overlapping echoes.
The simulated signal is examined by the chirplet signal decomposition algorithm and
another decomposition algorithm from the literature, which is based on Gabor function.
The performances of these two algorithms are compared with each other and discussed
with details.
Alternatively, Chapter 4 introduces signal decomposition based on matching
pursuit (MPSD) framework. The matching pursuit framework was proposed by Mallat et.
al for non-stationary signal analysis. In the original matching pursuit algorithm, it uses
correlation criteria to search the best matching function in dictionaries. It has been
reported that this criterion obtains decompositions adaptive to global signal
characteristics. Since in some applications, it is preferable to be best adapted to the local
structures of signal, we incorporate the statistical analysis tools such as Maximum
Likelihood Estimation and Maximum a Posteriori into the implementation of
decomposition. The implementation details of the algorithms and simulation results are
discussed in Chapter 4. To benchmark the proposed signal decomposition algorithms,
Chapter 5 explores the analytical lower bound, i.e., the Cramer-Rao lower bound (CRLB).
4
We evaluate the performance of the signal decomposition and parameter estimation
algorithms against the analytical CRLB bounds through Monte Carlo simulation.
Chapter 6 presents the applications of the chirplet signal decomposition algorithm
and the signal decomposition based on matching pursuit in ultrasonic target detection and
bat chirp signal analysis. Chapter 7 introduces the application of material grain size
evaluation. The chirplet signal decomposition algorithm is applied to estimate the grain
size of materials which are processed under different heat treatment condition. As another
important aspect of ultrasonic nondestructive evaluation, Chapter 8 lay out the discussion
of the multilayered reverberant structures. The proposed algorithm is evaluated by
ultrasonic multilayered reverberant echoes. To verify the feasibility of hardware
implementation and acceleration of the algorithm, In Chapter 9, an embedded hardware
design of signal decomposition algorithm is analyzed and implemented on Xilinx Virtex
II Pro Field Programmable Gate Array (FPGA) Platform. Finally, Chapter 10 summaries
the research of chirplet signal decomposition algorithm and its applications.
5
CHAPTER 2
REVIEW OF TIME-FREQUENCY REPRESENTATION
2.1 Introduction
In this chapter, the background of time-frequency representation is reviewed.
Then three commonly used methods of time-frequency signal representation such as short
time Fourier transform, Wigner Ville distribution and continuous wavelet transform are
introduced. The time resolution and frequency resolution are discussed and compared
among the three time frequency representations.
The need for time-frequency representation is from the nonstationary nature of
most signals in real world. Usually it is inadequate to fully describe the signal using
either time domain or frequency domain analysis. Time-frequency representation is a
useful tool for simultaneous characterization of a signal in time and frequency domain. It
provides information about how the spectrum of the signal changes with time, thus
leading to accurately describe, analyze and interpret the nonstationary signal. The time-
frequency process is performed by mapping the signal from time domain, where the
signal is one-dimensional, into a two dimensional expression (i.e., time frequency
domain). A variety of methods for obtaining time frequency representation have been
devised, most notably the short time Fourier transform (STFT), the Wigner-Ville
distribution (WVD) and the continuous wavelet transform.
6
2.2 Short Time Fourier Transform (STFT)
During the 1940s, the motivation to analyze the human speech, which is
nonstationary and rapidly varying spectral components, led to the invention of sound
spectrogram (i.e., STFT). In order to analyze such a non-stationary signal, it is
reasonable to apply a small window along time axis in order to examine the frequency
content of the signal in the given time window. The STFT aims to obtain the short time
Fourier transform of a signal by sliding a time window and then taking the Fourier
transform of the windowed signal. In doing so, it is assumed that the signal is stationary
during the duration of the time window. The STFT of a signal can be expressed as:
( ) ∫+∞
∞−
−−= dtettgtftSTFT tif
0)()(, 000ωω (2.1)
Here, )( tg is a normalized real and symmetric window
)()( tgtg −= , 1)( =tg (2.2)
Using different type windows result in different TF representations. Since there
already have extensive research efforts in the classic signal processing field, such as
efficient implementation of Fourier transform, correlation and filter design theory in past
years, they can be imported into the implementation of STFT.
The downfall of STFT is from the windowing process, which leads to inherent
trade off between time resolution and frequency resolution. The resolution problem of
STFT can be revealed by the following expression of time spread tσ and frequency
spread ωσ of the window function )( tg .
Let )(ˆ ωg denote ))(( tgFT , then from properties of Fourier transform,
7
( ) ))((ˆ 0)(
000 ttgFTeg it −=− −− ωωωω (2.3)
Hence, the time spread tσ and frequency spread ωσ are
( ) ( )
( )∫∫
∞+
∞−
+∞
∞−
−
=
−−=
dttgt
dtettgtt tit
22
2
02
02 0ωσ
(2.4)
( ) ( )
( )∫
∫∞+
∞−
+∞
∞−
=
−−=
ωωωπ
ωωωωωπ
σ ωω
dg
deg ti
22
2
02
02
ˆ21
ˆ21
0
(2.5)
From Equation 2.4 and Equation 2.5, it can be seen that the spreads are independent of
the time shift, 0t , and the frequency shift, 0ω . Therefore, STFT has the same time
resolution and the same frequency resolution across time frequency plane. Can the time
resolution and the frequency resolution of STFT both be arbitrarily small to reveal the
non-stationary property of signal? Unfortunately, Heisenberg uncertain principle limits
the scheme.
Heisenberg uncertainty principle [Mal99] expresses a fundamental relationship
between the time spread and the frequency spread of the windowed signal. It states the
mathematical fact that a narrow waveform yields a wide spectrum and a wide waveform
yields a narrow spectrum. Both the time waveform and the frequency spectrum can not
be made arbitrarily small simultaneously.
The Heisenberg uncertain principle can be derived as following. Given a
signal ( ) ( )RLtf 2∈ , the mean and variance of signal in time domain and frequency
domain can be expressed as following.
8
Mean in time domain ( )∫+∞
∞−= dttft
fu 2
2
1
Mean in frequency domain: ( )∫+∞
∞−= ωωω
πξ df
f
2
2ˆ
21
Variance in time domain: ( ) ( )∫+∞
∞−−= dttfut
ft
222
2 1σ
Variance in frequency domain: ( ) ( )∫+∞
∞−−= ωωξω
πσω df
f
222
2 ˆ2
1
Hence,
( ) ( )
( ) ( )
( ) ( )
( ) ( ) ( ) ( )[ ]
( )( ) 2'2
4
2'**'
4
2'
4
2'2
4
22
422
41
21
*1
1
ˆ2
1
⎥⎦⎤
⎢⎣⎡≥
⎥⎦⎤
⎢⎣⎡ +≥
⎥⎦⎤
⎢⎣⎡≥
=
=
∫
∫
∫
∫∫
∫∫
∞+
∞−
∞+
∞−
∞+
∞−
∞+
∞−
∞+
∞−
+∞
∞−
+∞
∞−
dttftf
dttftftftftf
dttfttff
dttfdtttff
dfdtttff
t ωωωπ
σσ ω
(2.6)
Since ( ) 0lim =∞→ tftt
( )( )
( )
414
1
41
22
4
2'24
22
≥
⎥⎦⎤
⎢⎣⎡≥
⎥⎦⎤
⎢⎣⎡≥
∫
∫∞+
∞−
∞+
∞−
dttff
dttftf
t ωσσ
(2.7)
9
The principle shows that there is a lower bound of ωσσ t .
Through the above discussion of time resolution and frequency resolution, it can
be seen that in STFT, the resolutions solely depend on the resolution property of the short
time window. The inherent lower bound of Heisenberg principle determines the tradeoff
between time resolution and frequency resolution of STFT. For a non-stationary signal,
it is always problematic to find an appropriate type and size of the window to fit the
specific signal analysis in STFT of signal. To demonstrate the STFT of a signal, Figure
2.1a shows a simulated ultrasonic signal consisting of two chirp echoes. Figure 2.1c
shows the STFT of the signal in Figure 2.1a using Hamming window.
2.3 Wigner-Ville Distribution (WVD)
Another well-known time frequency representation, Wigner-Ville Distribution
(WVD), has been received research attention for many years. In 1932, Wigner presented
a joint probability function for the coordinates and moment in the study of statistical
quantum mechanics [Wig32]. Ville derived the Wigner distribution for analytic signals
in 1948, which is known as Wigner-Ville distribution (WVD) [Vil48]. In 1946, Gabor
presented the method to expand the given signal into a sum of elementary signals of
“minimum” spread in time and frequency [Gab46]. In 1966, Cohen generalized time-
frequency representation into different distribution functions [Coh89].
A great interest was shown in time-frequency analysis in the 1980’s when a large
number of researchers started exploring the field of time frequency representation in
signal processing area [Coh89]. In the implementation of discrete Wigner-Ville
distribution, Classsen discussed the sampling rate to avoid aliasing [Cla80]. Boualem
10
Boashash et. al made a significant contribution towards Wigner-Ville analysis of time
varying signals, non-stationary random signals, cross spectral analysis, estimation and
interpretation of instantaneous frequency[Boa03].
The WVD of signal can be expressed as
∫∞+
∞−
−⎟⎠⎞
⎜⎝⎛ −⎟
⎠⎞
⎜⎝⎛ += τττω τω detftftWVD i
f0
22),( 0
*000 (2.8)
From the analysis of STFT in Section 2.2, it can be seen that the time and
frequency resolution is limited by the resolution of correlated window ( )tg in STFT. But
in WVD representation of signal, it is calculated by correlating the signal with a time and
frequency translation of the signal. From Equation 2.8, it can be seen that the time
resolution and frequency resolution are solely determined by the signal ( )tf itself.
Hence the WVD representation does not have the resolution loss from windowing.
Although WVD has excellent time and frequency resolution, the quadratic
property of WVD is that the cross terms (i.e., artifacts) are introduced when dealing with
multi-component signals. The artifacts lead to an erroneous interpretation of the time
frequency representation of the signal. The cross terms indicate that the time-frequency
energy is distributed to the place where the signal doest not really exist on the joint time-
frequency domain. To demonstrate the cross terms problem of WVD representation in
the case of multi-component signal, an example is demonstrated in the Figure 2.1. Figure
2.1a is the simulated multi-component signal. Figure 2.1b clearly shows the cross term
between these two components of the signal.
Many researchers worked on the problem of cross-terms in WVD by smoothing,
windowing, interpolating, filtering in time domain, frequency domain, or joint time-
11
frequency domain so that to attenuate the cross terms [And87, Gre96, and Oeh97].
Usually, the suppression and elimination of the cross-terms is achieved at the cost of
marginal properties and computation.
2.4 Continuous Wavelet Transform (CWT)
In the STFT implementation, a window is designed to slide along the time axis.
Once the window is chosen, the time resolution and the frequency resolution are fixed. In
certain applications, it is more desirable to have better time resolution at higher
frequencies than that at lower frequency. As a result of this characteristic, wavelet
transform have become a useful tool for non-stationary signal analysis. Since wavelet
theory were developed independently in multiple fields such as mathematics, quantum
physics, and electrical engineering, it is difficult to track a unique origin of wavelet
theory.
In 1984, Grossman and Morlet broadly defined wavelets in the context of
quantum physics. They discussed decomposition of hardy functions into square
integrable wavelets of constant shape [Gro84]. In 1985, Stephane Mallet gave wavelets
an ice-break jump through his work in digital signal processing [Mal89, Mal99]. For the
first time, he discovered some relationship between quadrature mirror filters, pyramidal
algorithm, and orthonormal wavelet bases. After that, many researchers such as Meyer,
Ingrid Daubechies worked out many sets of wavelets [Dau92, Mey93].The continuous
wavelet transform, discrete wavelet transform, and the fast implementation of wavelet
transform have been extensively explored by researchers. The wavelets have been
applied to a broad range of applications such as denoising, compression, spectral
12
estimation, pattern recognition, human vision, radar and sonar etc [Dau90, Mal91, Ant92,
Rod98, Zen01, and Cha06]. Wavelet becomes a general mathematical tool in the similar
way as the Fourier transform does. Nevertheless, we are not going to discuss the discrete
wavelet transform and the details of different wavelet base functions. We focus on the
similar resolution argument in the introduction of continuous wavelet transform as the
discussion in the STFT and WVD section. Unlike STFT and WVD, continuous wavelet
transform (CWT), through the correlation of the signal with a scaling and translating
function of wavelet ( )tψ , has varying resolution at different scale. The role of scale acts
as the role of frequency in WVD and STFT.
The CWT of signal ( )tf can be expressed as
( )
( ) ( )∫
∫∞+
∞−
∞+
∞−
=
⎟⎠⎞
⎜⎝⎛ −
=
ωωψπ
ψ
ω desstf
dtstt
stfstCWT
ti 0*
0*0
ˆˆ21
1),(
(2.9)
Here ( )tψ satisfies ( )∫+∞
∞−=0dttψ and ( )ωψ̂ denotes ( )( )tFT ψ .
0t denotes the center time of ( )tψ
0ω denotes the center frequency of ( )ωψ̂
tσ denotes the time spread of ( )tψ
ωσ denotes the frequency spread of ( )ωψ̂
Then the time spread of ⎟⎠⎞
⎜⎝⎛ −
stt
s01 ψ is
13
( ) ( ) 222222
0*20
1tsdtttsdt
stt
stt σψψ ==⎟
⎠⎞
⎜⎝⎛ −
− ∫∫∞+
∞−
∞+
∞− (2.10)
and the frequency spread of ( ) 0*ˆ tiess ωωψ is
( )( ) ( )
2
2
20
220
0
2*2
0ˆ
21
ˆ21
ss
ddss
sωσ
ωωψωωπωωψωω
π=
−=⎟
⎠⎞
⎜⎝⎛ −
∫∫
+∞
∞+
(2.11)
Hence the wavelet window ⎟⎠⎞
⎜⎝⎛ −
stt
s01 ψ centered at ⎟
⎠⎞
⎜⎝⎛
st 0
0 , ω in time frequency domain
and the time spread is tsσ , frequency spread issωσ . And the product ωσσ t still keeps
unchanged, which is the inherent property of Heisenberg uncertain principle. It is worth
to point out that the time resolution and frequency resolution depend on the scale s . This
shows higher time resolution in smaller scale and higher frequency resolution in higher
scale. As a comparison, the CWT using morlet wavelet is shown in Figure 2.1d.
2.5 Summary
In this chapter, we reviewed the time frequency representation of signal and
introduced three conventional time frequency representations such as short time Fourier
transform, Wigner-Ville distribution, and continuous wavelet transform. From all the
preliminary analysis, it can be seen that the conventional time frequency representation
such as WVD and STFT introduce cross terms, have poor resolution and are sensitive to
noise level. On the other hand, CWT shows higher time resolution in smaller scale and
higher frequency resolution in higher scale. Hence, it is a preferable property for tracking
14
the time-varying frequency of non-stationary signals, especially in our ultrasonic model
base algorithm design.
Figure 2.1 Comparisons of Time Frequency Techniques. a) a Simulated Signal. b) WVD of the Signal. c) STFT of the Signal (using Hamming Window) d) CWT of the Signal (Using Morlet Wavelet).
15
CHAPTER 3
CHIRPLET SIGNAL DECOMPOSITION
3.1 Introduction
It has been reported [San89, Wan91, and San94] that the broadband ultrasonic
backscattered signal depicts a downward shift in frequency due to signal attenuation. It
means that the higher frequencies are experienced more attenuation than the lower
frequencies. On the other hand, in the Rayleigh region of scattering, an upward trend in
frequency due to scattering is experienced. This implies that the high frequency
components are backscattered with more intensity than the low frequency components.
The echo reflected from a discontinuity (flaw) has lower frequency due to attenuation
effect compared with that of the echoes backscattered from internal microstructure of
materials. Furthermore, dispersion is a phenomenon in which the velocity of sound
depends on its frequency and consequently different frequency components arrive at
different time. Hence, the shift in frequency with depth and the random arrival of
different frequency components with random amplitude in backscattered ultrasonic signal
make it a non-stationary signal. By Fourier analysis, we can decompose signal into
individual different frequency components. However, the spectrum of signal does not
shows how the frequencies evolve with time. Therefore, joint time-frequency (TF)
representation is required by the non-stationary property of ultrasonic backscattered
signal.
Chirp signal is a type of signal that is often encountered in seismic signal, radar,
sonar, speech and ultrasound [Ma98, Fan02, Wan02, Wan03, Zan03, Lu05, and Lu06a].
The chirplet transformation has been applied as a useful and practical method for time-
16
frequency analysis of radar signals [Man92, Man95, Nei99, Qia98, Xia00, and Yin02].
Further implementations and applications of the adaptive chirplet transform for sonar,
speech, CFAR detection, medical signal and seismic signal analysis have been presented
in [Wan00a, Wan00b, Lij03, Lop02, Lop03, and Cui06]. The chirp signal parameters are
very important in analysis the physical interpretation of the signal in these applications.
More recently, a modified continuous wavelet transform (MCWT), which is based on the
Gabor-Helstorm transformation, has been introduced as a means to estimate parameters
of ultrasonic echoes [Car05a, Car05b]. The MCWT decomposition has not been found
effective in representing ultrasonic echoes with chirp characteristics.
Compared with Gaussian Gabor function, chirplet has one more parameter
freedom and thereby can better match chirp signal. Moreover, Gaussian Gabor function is
the special chirplet with zero chirp rates. We introduce a chirplet signal decomposition
algorithm to represent chirp-type signals in terms of Gaussian chirplet, which is sparse
and energy preserving. The sparseness property aims for a compact representation of the
complex signal by decomposing it into a limited number of chirp components. The
energy preservation property, by coherently distributing the signal energy into composing
functions, enables the linear addition of the time-frequency distributions of composing
functions to represent the TF of the signal. Furthermore, once the signal is decomposed
by a family of chirplet echoes, these echoes, individually or collectively can be used to
describe the nonstationary behavior of the signal.
The chirplet signal decomposition method utilizes the chirplet transform and a
successive parameter estimation algorithm. Based on the chirplet transform of the signal,
the algorithm identifies the location and duration of the most dominant chirp component
17
in time frequency domain. Then, a successive parameter estimation algorithm is used to
estimate the parameters of this dominant chirp component. The algorithm can recover
the parameters of a noise-free chirp signal without requiring any initial guess for
parameters. It accounts for a variety of differently shaped echoes, including narrow-
band, broad-band, symmetric, skewed, dispersive or nondispersive.
In this chapter, we first introduce the successive parameter estimate algorithm
and address the details of its mathematical derivation. Moreover, an efficient windowing
method is designed to iteratively handle the echo estimation process of more complex
signals. To compare with the performance of MCWT algorithm, the proposed signal
decomposition based on chirplet transform (CTSD) algorithm is utilized to process the
same high overlapping signal as the MCWT algorithm does.
3.2 Successive Parameter Estimation Algorithm
Under the assumption that the signal to be processed, no matter how complex, it
can be decomposed into the superposition of multiple chirplet echoes. The objective of
the successive parameter estimation algorithm is to efficiently estimate the parameters of
the individual chirp echoes.
In most application case, a single chirp echo can be modeled as
( ) ( ) ( ) ( )( )22
21 2exp ταφτπταβ −++−+−−=Θ tiitfittf c (3.1)
Where ],,,,,[ 21 βφαατ cf=Θ denotes the parameter vector of the chirp echo
τ denotes the time-of-arrival
cf denotes the center frequency
18
1α denotes the bandwidth factor
2α denotes the chirp-rate
φ denotes the phase
β denotes the amplitude
These parameters can be estimated successively using the chirplet transform (CT).
The successive parameter estimation algorithm is a recursive method that starts with a
time-frequency (TF) representation of the superimposed chirp signal based on the CT.
The CT of ( )tf Θ with respect to a chirplet kernel ( )tΘ
Ψ ˆ is defined as
( ) ( ) dtttfCT ∫
+∞
∞− ΘΘ Ψ=Θ )(ˆ *ˆ
(3.2)
Where ⎥⎦⎤
⎢⎣⎡=Θ ηθγγ
πω ,,,,2
,ˆ21
0
ab denotes the parameter vector of chirplet kernel. The
chirplet kernel ( )tΘ
Ψ ˆ is
( ) ( ) ( ) ⎟⎟
⎠
⎞⎜⎜⎝
⎛−++⎟
⎠⎞
⎜⎝⎛ −
+−−=ΨΘ
220
21ˆ exp btii
abtibtt γθωγη
(3.3)
Where ( )tΘΨ ˆ* denotes the conjugate of ( )tΘ
Ψ ˆ . In order to normalize the energy of the
chirplet kernel, the term 41
12⎟⎠⎞
⎜⎝⎛=πγη . Hence, the CT of a signal chirp echo ( )tfΘ given by
Equation 3.2 can be expressed as
19
( ) ( )
( ) ( )
( ) ( ) ( )
( ) ( ) ( )
⎥⎥⎥⎥
⎦
⎤
+−+
−⎟⎠⎞
⎜⎝⎛ ++−
+
+−+−+−
−
−+
⎢⎢⎢⎢⎢
⎣
⎡
+−+
⎟⎠⎞
⎜⎝⎛ −
−
+−+=Θ
2211
21210
2211
22121
2211
20
2211
41
1
4exp
12ˆ
γαγα
τγγωααω
γαγατγγαα
θφγαγα
ωω
γαγαπγβ
ii
biiia
i
iibii
iii
a
iiCT
c
c
(3.4)
Where cc fπω 2= . The maximum similarity between the input signal, ( )tfΘ , and the
chirplet kernel, ( )tΘΨ , leads to correct estimation of echo parameters, Θ̂ . It can be
shown that the peaks of TF representation )ˆ(ΘCT of the superimposed signal ( )tfΘ can
be used to estimate the center frequency, cf , and time-of-arrival, τ . To accomplish
this goal, the magnitude of )ˆ(ΘCT is used for estimation of the signal parameters, which
is given by
20
( ) ( ) ( ) ( )[ ]
( )
( ) ( )( )
( )( )
( ) ( )( )( )
( ) ( ) ⎥⎥⎦
⎤
−++
−+++−
−++
−+⎟⎠
⎞⎜⎝
⎛ −−
⎢⎢⎢⎢⎢
⎣
⎡
−++
+⎟⎠
⎞⎜⎝
⎛ −−
−++=Θ−
222
211
21
221
211
221
21
222
211
12210
222
211
211
20
412
222
114
11
4exp
2ˆ
γαγαταγαγγαγα
γαγα
τγαγαω
ω
γαγα
γαω
ω
γαγαπγβ
b
ba
a
CT
c
c
(3.5)
The maximum of the above equation can be obtained by taking partial derivatives
of )ˆ(ΘCT in respect to a (which corresponds to the center frequency, cf ) and b
(which corresponds to the time-of-arrival, τ ).
( ) ( ) ( )( )( ) ( )( )
( )
( ) ( )( ) 02
2ˆ
ˆ
222
211
110
20
222
211
12212
0
=
⎪⎪⎭
⎪⎪⎬
⎫
−++
+⎟⎠
⎞⎜⎝
⎛ −+
⎪⎩
⎪⎨⎧
−++
−+−Θ=
∂
Θ∂ −
γαγα
γαωωω
γαγατγαγαω
caa
baCT
a
CT
(3.6)
21
( ) ( ) ( )( )( ) ( )( )
( )
( ) ( )( ) 04
2ˆ
ˆ
222
211
12210
222
211
221
2111
221
21
=
⎪⎪⎭
⎪⎪⎬
⎫
−++
+⎟⎠
⎞⎜⎝
⎛−
+
⎪⎩
⎪⎨⎧
−++
−+++−Θ=
∂
Θ∂
γαγα
γαγαωω
γαγατγαγαγαγα
ca
bCT
b
CT
(3.7)
The solutions of Equation 3.6 and Equation 3.7 are
τ=b caωω
=0 (3.8)
It is important to point out that under the condition of Equation 3.8, the estimation
of the peak position of )ˆ(ΘCT in TF domain is not a function of the bandwidth factor,
1γ ,chirp-rate, 2γ , and phase, θ of the echo. Furthermore, the peak value of )ˆ(ΘCT is
proportional to the amplitude of the actual echo and leads to the estimation of β .
Based on the above estimations of a and b , the estimation of the chirp-rate, 2γ ,
becomes a one-dimensional estimation problem. This can be achieved by taking the
derivative of )ˆ(ΘCT in respect to 2γ and setting it to 0,
22
( ) ( )( ) ( )( )
( ) ( )
( ) ( )
( ) ( )
( ) ( )( )( ) ( )( )
( ) ( )( )( ) ( ) ( )
( ) ( )( ) ⎥⎥⎦
⎤
−++
−+++−−
−++
−+⎟⎠
⎞⎜⎝
⎛−−
−
−++
+⎟⎠
⎞⎜⎝
⎛−−
−
−++
−+−⎟⎠
⎞⎜⎝
⎛−
−
⎢⎢⎣
⎡
−++
−Θ=
∂
Θ∂
2222
211
21
221
211
221
2122
2222
211
12210
22
2222
211
211
20
22
222
211
212
01
222
211
22
2
2
2
2
2
2ˆ
ˆ
γαγα
ταγαγγαγαγα
γαγα
τγαγαω
ωγα
γαγα
γαω
ωγα
γαγα
ταγτω
ωα
γαγαγα
γ
b
ba
a
bba
CTCT
c
c
c
(3.9)
Hence, the maximum of )ˆ(ΘCT yields the optimal solution of 2γ
( ) ( )( ) ( )
( ) 0ˆ2ˆ
0
0
,222
211
22
,2
=Θ−++
−=
∂
Θ∂
==
==c
ca
b
ab
CTCT
ωω
τ
ωω
τγαγα
γαγ
(3.10)
The solution to Equation 3.10 is
22 αγ = (3.11)
23
Similarly, the estimation of the bandwidth factor, 1γ , is carried out by taking the
partial derivative of )ˆ(ΘCT in respect to the bandwidth factor, 1γ , and setting it to 0.
( ) ( ) ( ) ( )( ) ( )( )
( )
( ) ( )( )( ) ( )( )
( ) ( )
( )
( ) ( )( )( ) ( )( )
( ) ( )( )( )( )( )
( ) ( )( ) ⎥⎥⎦
⎤
−++
−++++−
−++
−+⎟⎠⎞
⎜⎝⎛ −+
−
−++
+⎟⎠⎞
⎜⎝⎛ −
−
−++
−++−⎟⎠⎞
⎜⎝⎛ −
−
−++
+⎟⎠⎞
⎜⎝⎛ −
−
⎢⎣
⎡
−++−+−
Θ=∂
Θ∂
2222
211
21
221
211
221
2111
2222
211
12210
11
2222
211
311
20
222
211
222
21
02
2222
211
11
20
222
2111
222
21
21
1
2
2
2
2
4ˆ
ˆ
γαγα
ταγαγγαγαγα
γαγα
τγαγαωωγα
γαγα
γαωω
γαγα
ταατωωα
γαγα
γαωω
γαγαγγαγα
γ
b
ba
a
bba
a
CTCT
c
c
c
c
(3.12)
Hence,
( )( )
( ) 0ˆ4
ˆ
22
0
22
0
,
,2
111
21
21
,
,1
=Θ⎟⎟⎠
⎞⎜⎜⎝
⎛
+−
=∂
Θ∂
=
=
=
=
=
=
αγω
ωτ
αγω
ωτ γαγ
γαγ c
ca
b
a
bCT
CT
(3.13)
The solution to Equation 3.13 yields
24
11 αγ = (3.14)
Since there is no information about signal phase in the magnitude representation
of the CT, the real part of the CT is used to estimate the phase of the echo, θ .
( )( ) ( ) ( )
( )
( ) ( )( )( ) ( )( ) ( )
( )
( ) ( )( ) ( )
( )
( )
( ) ( )( )
⎥⎥⎥⎥
⎦
⎤
−−++
−⎟⎠
⎞⎜⎝
⎛ ++
−−++
+−++
−−++
+−++
−++
−⎟⎠
⎞⎜⎝
⎛ −−
−+⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛+−
Θ=Θ −
τγαγα
γαγαω
ω
τγαγα
ωααγγω
τγαγα
γααγγα
γαγα
γαω
ω
θϕγαγα
ba
ba
b
a
CTCT
c
c
c
222
211
22110
222
211
022
21
22
21
22
222
11
222
21
22
212
222
211
22
20
11
221
4
tan21cosˆˆRe
(3.15)
Based on the above estimation of a , b and 2γ , the estimation of phase, θ ,
becomes a one-dimensional estimation problem. The maximum of ( )( )Θ̂Re CT yields the
optimal solution for θ . This can be obtained by taking the partial derivative of
( )( )Θ̂Re CT with respect to θ and setting it to 0,
( )( ) ( ) ( )( ) 0ˆResinˆRe
22
0
22
0,,
,,=Θ−=
∂Θ∂
=
==
=
== αγω
ωτ
αγω
ωτ
φθθ c
ca
b
ab
CTCT
(3.16)
The solution of Equation 3.16 yields
25
πφθ k2±= , ,...3,2,1=k (3.17)
In summary, the mathematical steps present above show that the chirplet
transform leads to an exact estimation of the time-of-arrival, center frequency, phase,
bandwidth factor, and chirp-rate of the chirp echo signal. The parameter estimation based
on these equations can be implemented successively using signal correlation (see
Equation 3.2). A grid search is performed of these parameters are refined with a fast
Gauss-Newton algorithm [Dem00, Dem01a, Dem01b]. The refinement improves the
parameter estimation beyond the resolution of the search grid. The successive parameter
estimation based on CTSD method can recover the exact value of the parameters of a
noise-free Gaussian chirp echo. It does not require any initial guess for the parameters
before estimation. Furthermore, it can also estimate the parameters of a noise corrupted
echo with high accuracy.
3.3 Windowing Algorithm
We utilize the successive parameter estimation technique to decompose a
complex signal into a small number of Gaussian chirplets. The complex signal is
presented by the linear addition of a number of chirplets:
( )∑
−
=Θ=
1
0)(
N
jtfts
j
(3.18)
where ( )tfjΘ is the chirplet model and jΘ is the parameter vector of ( )tf
jΘ , (refer to
Equation 3.1).
26
The goal of signal decomposition is to express the signal, )(ts , as a linear
combination of chirp components. The decomposition is performed as follows. First,
based on the CT of the signal (i.e.,TF representation), the most dominant chirp echo is
windowed and estimated using the successive parameter estimation algorithm presented
in Section 3.2. Then, the estimated echo is subtracted from the original signal. Next, the
second echo is estimated from the remaining signal. This process is repeated until the
reconstruction error, Er, is below an acceptable value Emin. The value of Emin is
determined based on the requirements of the reconstruction quality of the signal. This
iterative decomposition method ensures energy preservation by coherently distributing
the signal energy into composing function. Energy preservation allows us to add the TF
distribution of composing function ( )tfjΘ to estimate the TF distribution of the
signal )( ts . Meanwhile, the sparseness of decomposition is ensured by searching for
the most dominant chirp echo per iteration. A block diagram summarizing the chirplet
signal decomposition algorithm is shown in Figure 3.1.
The procedure used to design the window is based on the determination of the
peaks and valleys of the CT of the signal. Figure 3.2 illustrates the windowing method
with simulated data containing 3 interfering echoes. First, the maximum peak of the CT
of the signal (Figure. 3.2a) is identified. Next, the CT of the signal is projected onto the
time domain (Figure. 3.2c) and frequency domain (Figure. 3.2b). The windowing
algorithm uses these projections to isolate the dominant echo by tracing the nearest
valleys around the peak. The closest two valleys confining the time-projection peak are
defined as the boundaries of the time-window (i.e., Tbegin and Tend in Figure. 3.2c).
Similarly, the closest two valleys confining the frequency projection peak are defined as
27
the boundaries of the frequency-window (i.e., Fbegin and Fend in Figure 3.2b). The time-
of-arrival τ and center frequency cf parameters are in fact the peak locations of the
projections (see Equation 3.2). The dominant signal along with the time window and
frequency window is used to estimate the remaining chirplet parameters (i.e.,
amplitude β , bandwidth 1γ , chirp rate 2γ , and phaseθ ) using signal correlation (see
Equation 3.2).
When there are heavily overlapping echoes and high noise levels, the performance
of the automatic windowing method may be compromised as the peak separation process
becomes more difficult. The distance between peaks becomes shorter and artificial
valley points may be created due to the noise. In these cases, a time window and
frequency window with predetermined size can be used to separate out the time and
frequency projection peaks. The windows are centered at the peaks. The sizes of the
windows can be determined by inspecting the CT of the measured signal for given noise
levels. A good window size selection strategy is to keep as much of the signal energy as
possible while suppressing the contribution of noise energy in the window. For the
simulated and experimental signals presented in this study, the automatic windowing
method performed adequately in extracting the individual echoes. However, one can
apply the predetermined windowing method for signals with very poor SNRs (2 dB and
below).
28
E stim ate α 2 ,α 1 and φ
E r < E m in
Y es
N o Subtract the estim ated
echo from the signal
S tore the estim ated param eters
G enerate ( )Θ̂C T and
localize dom inant echo
by w indow ing m ethod
M ultip le E choes
C alculate reconstruction error E r
E stim ate β , fc and τ
Figure 3.1 The Flowchart of the CTSD Algorithm.
29
Figure 3.2 Basic illustration of dominant echo windowing method: a) CT of three interfering chirp echoes. The most dominant echo is emphasized after time and frequency windowing b) Projection in frequency domain and the frequency-window boundary points (dashed lines) c) Projection in time domain and the time-window boundary points (dashed lines)
30
3.4 Comparison with Gabor Decomposition Algorithm
The CTSD algorithm is very effective in representing dispersive ultrasonic
echoes. An alternative decomposition algorithm [Car05a] uses a Gabor kernel to analyze
ultrasonic echoes. However, if the ultrasonic signal has a dispersive or frequency shift
property, Gabor decomposition requires many components. The chirplet model is
expected to have better decomposition efficiency with extra parameter diversity. To
demonstrate chirplet decomposition efficiency, a noisy chirp signal containing highly
overlapping echoes is simulated, and then the algorithm presented in [Car05a] and the
CTSD algorithm are both applied to reconstruct the signal. Figure 3.3 shows the noisy
chirp signal and the two reconstruction results from these two different decomposition
strategies, under the same output SNR criteria. More specifically, the parameters of the
decomposed echoes are listed in Table 3.1 and Table 3.2. Furthermore, Figure 3.4 shows
the time frequency difference of the reconstructed signal using CTSD method (see Figure
3.4c and Figure 3.4d) and using Gabor method (see Figure 3.4e and Figure 3.4f). It can
be seen that, under the same quality of reconstructed signal (i.e., the same output SNR
criteria), the chirplet decomposition algorithm requires significantly a less number of
components than Gabor decomposition [Lu06a].
The compact representation achieved by the chirplet decomposition is more
powerful in revealing the physical properties of chirp-type signals (e.g., the Doppler shift
in a radar system, the dispersive echoes in an ultrasonic nondestructive testing system).
31
3.5 Summary
In this chapter, we introduce a successive and efficient chirplet decomposition
algorithm that employs an adaptive chirplet kernel as the general model for the parameter
estimation of the superimposed chirp signal. This algorithm adaptively tracks and locates
the individual echoes for efficient and precise estimation of all echo parameters. Analysis
results showed that the performance of chirplet signal decomposition overwhelmed that
of the Gabor decomposition algorithm with less number of components to reconstruct the
same high overlapping signal. Hence, the chirplet signal decomposition and parameter
estimation algorithm allows for high fidelity signal reconstruction.
32
Table 3.1. Parameters of Decomposed Echoes (CTSD Method)
Table3.2. Parameters of Decomposed Echoes (Gabor Decomposition Method)
Echo # τ [μs] ƒc [MHz] α1 [MHz]2 φ [rad] β
1 1.91 4.86 19.70 3.91 1.01
2 2.54 6.92 16.18 9.09 0.97
3 3.02 4.65 27.00 0.73 0.88
4 1.10 4.06 7.13 2.69 0.65
5 2.69 3.95 41.11 5.67 0.39
6 1.97 7.31 3.44 -2.57 0.34
7 3.31 5.73 65.66 4.47 0.27
8 1.58 3.66 36.10 -2.85 0.26
9 0.86 2.68 4.22 -3.10 0.23
10 1.82 5.87 1.84 6.63 0.18
11 2.72 8.82 10.08 -1.84 0.11
34
Figure 3.3 Simulated Ultrasonic Highly Overlapping Echoes (Solid Line), Superimposed with the Reconstructed Signals by CTSD Method and Gabor Decomposition Method.
35
Figure 3.4. Comparisons of CTSD Method and Gabor Decomposition Method. a) Simulated Highly Overlapping Echoes. b) WVD of the Original Simulated Signal. c) Reconstructed Signal by CTSD Method. d) WVD of the Reconstructed Signal (Using CTSD). e) Reconstructed Signal by Gabor Method. f) WVD of the Reconstructed Signal (Using Gabor).
36
CHAPTER 4
SIGNAL DECOMPOSITION BASED ON MATCHING PURSUIT
4.1 Introduction
The matching pursuit (MP) algorithm has been initially introduced by Mallat and
Zhang [Mal89, Mal93]. It aims to provide a signal analysis framework for non-stationary
signal under energy conservation signal decomposition condition. Hence, a high
resolution TF representation can be achieved by decomposing ultrasonic backscattering
signal into a limited number of elementary functions with known TF distribution such as
WVD.
The real challenge of matching pursuit algorithm is that different matching
criteria can get different decomposition results [Adl96, Che98, and Cot98]. The original
matching pursuit algorithm uses correlation criteria (the inner product between signal
residue and a pre-defined dictionary function) to determine the best matching function.
This matching criterion obtains decompositions adaptive to global signal characteristics,
but is not best adapted to its local structures.
Recently, an enhanced version of MP algorithm, called high resolution matching
pursuit (HRMP) algorithm, is proposed by Grilbonval et. al [Gri96]. The HRMP uses a
different correlation function, which allows the pursuit to emphasize local fit over global
fit at each step. The new correlation function avoids creating energy at time location
where there are none. Compared with MP algorithm, HRMP algorithm performs higher
time resolution decomposition but the frequency resolution is decreased [Gri96]. This
limits the use of HRMP algorithm in the case for ultrasonic signal where local signal
structure change in frequency.
37
In this chapter, we first introduce matching pursuing signal decomposition
algorithm based on Maximum Likelihood Estimation (MPSD-MLE). The principle of
MPSD-MLE algorithm is discussed. Moreover, another implementation scheme, which is
the matching pursuit signal decomposition based on Maximum a Posteriori (MPSD-
MAP), is presented. Furthermore, the performance of these two algorithms is
demonstrated by applying both algorithms to simulated overlapping signal.
4.2 MPSD-MLE Algorithm
In the implementation of the original MP algorithm, the best match criterion is
based on the projection coefficient obtained by projecting the signal residue of current
stage onto a dictionary function. The signal residue of next stage is the remaining signal
after the best matching function has been subtracted from the signal residue of current
stage. When the energy summation of signal residue at all stages is a fraction of the
energy of the original signal, the decomposition is said to be completed. The final
decomposition is a linear expansion of all chosen matching functions.
In our MP algorithm, by incorporating the statistical strategies such as Maximum
Likelihood Estimation (MLE) and Maximum a Posteriori (MAP) method, we adaptively
optimize the parameters of the chirplet function to achieve high resolution
decompositions. This approach avoids the exhaustive search of a larger number of
dictionary functions and leads to a more efficient implementation.
At any stage of the MP algorithm, the signal residue is represented by a chirplet
function and a remaining signal (i.e., next residue),
38
sRtgsR nn 1);( ++Θ= (4.1)
Here, sR n is the current residue of signal )(ts , sR n 1+ is the next signal residue and
);( Θtg is a chirplet echo defined by the model,
])()(2cos[);( 22
2)( 21 φτατπβ τα +−+−=Θ −− ttfetg c
t
(4.2)
Where ],,,,,[ 21 τφβαα cf=Θ denotes the parameter vector of );( Θtg .
If we assume sR n 1+ has white Gaussian noise characteristics, the maximum likelihood
estimation of the parameter vector Θ can be obtained by minimizing:
2);(minargˆ Θ−=Θ Θ tgsR n
MLE (4.3)
Therefore, the parameter vector of the best matching function at stage n is chosen
by minimizing the least-square error. By assuming the remaining signal residue sR n 1+
is white Gaussian, Maximum Likelihood Estimation is simplified to Least Square
estimation [Kay93, Dem01a]. Hence, the optimization problem in Equation 4.3 replaces
the search for the best matching function. The MLE parameter vector, MLEΘ̂ , maximizes
the inner product between signal residue and normalized chirplet function, );(, ΘtgsR n .
In summary, for the signal )(ts , the MPSD-MLE algorithm can be outlined in
the following computation steps:
1. Set iteration index 0=n and first signal residue )(0 tssR = .
2. Find the best parameter vector of the chirplet function such that
2
);(minargˆ Θ−=Θ Θ tgsRnn (4.4)
39
3. Computer the next residue )ˆ;(1n
nn tgsRsR Θ−=+.
4. Check convergence: If Thresholdts
sR n
≤+
2
21
)(, STOP;
OTHERWISE, set 1+→ nn , and go to Step 2.
Step 1 of the algorithm initializes current signal residue as the original signal.
Step 2 finds the best matching function for the current signal residue by optimizing the
parameters of the chirplet function. Step 3 computes the next signal residue by
subtracting the best matching chirplet function. Step 4 checks for convergence: if the
residue energy is some fraction of the original signal energy, the algorithm stops,
otherwise a new chirplet function is matched to current signal residue. The flow chart of
MPSD algorithm is shown in Figure 4.1.
In the decomposition algorithm, Step 2 is essentially the most important step. An
optimal solution is critical in achieving the best decomposition. Since the model );( Θtg
is a nonlinear function ofΘ , there is no closed form solution available for Equation 4.4.
An iterative estimator can be obtained by successive linearizing the objective function.
i.e., by taking Taylor series expansion of );( Θtg at ( )nΘ
))(();();( )()()( nnn Htgtg Θ−ΘΘ+Θ≈Θ (4.5)
Where )(
)()( )(
n
gH n
Θ=ΘΘ∂Θ∂
=Θ
Then Equation 4.5 can be expressed as
WHX n +ΘΘ= )(~ )( (4.6)
Where ( ) )()()( )(;~ nnnn HtgsRX ΘΘ+Θ−= , and sRW n 1+=
Lemma 1: Optimality of the MLE for the linear model [Kay98]
40
For linear model WHX +Θ= , where ),0(~ WCNW .Then the minimum variance
unbiased MVU estimator is XHHH TT 1][ˆ −=Θ . Therefore, assuming that sR n 1+ has
white Gaussian noise (WGN) characteristics in Equation 4.6, the MLE estimation of the
4. Check convergence: If Thresholdkk ≤Θ−Θ + )()1( , then STOP;
OTHERWISE, set 1+→ kk , and go to Step 2.
The MPSD-MLE method described above yields a greedy approximation of the
signal. As long as a function matches the signal residue, it is included in the
decomposition. We demonstrate the performance of MPSD-MLE with a simulation
41
example. This example simulates two overlapping ultrasonic echoes sampled at 100 MHz
sampling frequency. The parameter vectors used to generate these functions are
[ ]0.10.0][0.4][0.80.40.5 221 radMHzMHzMHzsμ=Θ
[ ]0.10.1][0.3][0.60.65.5 222 radMHzMHzMHzsμ=Θ
These two echoes are very close in terms of center frequency and bandwidths.
Figure 4.2 shows the overlapping signal superimposed with the reconstructed result.
Figure 4.3a and Figure 4.3c display the original simulated signal and the reconstructed
signal using MPSD-MLE algorithm. It can be seen that the MPSD-MLE algorithm
successfully reconstructs the original signal. When the MPSD-MLE algorithm is applied
to this signal, the decomposition consists of 4 chirplets is obtained. The estimated
parameters are listed in the Table 4.1. Figure 4.2b shows the WVD representation of the
signal in Figure 4.3a. As a comparison, the WVD representation of estimated chirplets is
shown in Figure 4.3d.
From the estimation results of simulation example, it can be seen that in MPSD-
MLE algorithm, the decomposition is globally adaptive to signal structures. However, the
globally decomposition may smear out fine local structures in the signal.
42
Figure 4.1. The Flowchart of MPSD Algorithm.
43
Figure 4.2. Overlapping Chirp Signal Superposed with the Reconstructed Signal Using MPSD-MLE Algorithm.
44
Figure 4.3. a) Overlapping Chirp Signal. b) WVD of the Overlapping Chirp Signal. c) the Estimated Signal Using MPSD-MLE Algorithm. d) WVD of the Estimated Signal.
45
Table 4.1. Parameters of Decomposed Echoes for the Simulated Chirp Signal (MPSD- MLE algorithm)
Figure 5.1. a) Input SNR vs. Output SNR for a Single Noisy Chirp Echo Using CTSD Algorithm. b) Input SNR vs. Output SNR for a Single Noisy Chirp Echo Using MPSD Algorithm.
64
CHAPTER 6
TARGET DETECTION OF ULTRASONIC BACKSCATTERED SIGNAL
6.1 Introduction
In this chapter, the CTSD and MPSD algorithms are applied to estimate a target
embedded in a ultrasonic experimental backscattered signal which is acquired from a
ultrasonic nondestructive testing system. First, a real time ultrasonic pulse-echo
measurement system, which is used to acquire the ultrasonic backscattered data for signal
analysis, is reviewed. Then the CTSD and MPSD algorithms are used to process the
ultrasonic experimental data. Moreover, we evaluate the proposed algorithms by using an
experimental bat chirp signal, which is a benchmark signal from literatures for time
frequency signal analysis.
6.2 Read Time Ultrasonic Measurement System
The ultrasonic pulse echo method has been one of efficient non destructive
evaluation methods in the past decades. In general, a ultrasonic pulse echo system
requires one ultrasonic transducer (a device, usually refers to piezoelectric transducer,
that can convert electrical energy to acoustic pressure and generate electrical voltage
when a proper amount of acoustic pressure is forced on it) as the measuring probe, an
electrical pulse generation unit (transmitter) for transducer excitation and a display unit
for inspection of the received echoes. The principle of ultrasonic pulse echo method is to
launch acoustic waves into a medium and inspect the returning echoes. The incident
acoustic waves propagate through the medium and partially reflect from the impedance-
65
mismatched boundaries. The reflected acoustic waves excite the piezoelectric transducer
and form the returning ultrasonic signal.
The objective of ultrasonic pulse echo test is to evaluate the functionality and
characterization of specimen, which could be material, vegetation, or tissue texture.
Although there are a lot of information embedded in the return ultrasonic echoes, one of
the most common applications is ultrasonic target detection, which usually only addresses
the detection and positioning of defects in materials. The proposed signal decomposition
algorithms are used not only to detect and locate the targets but also enable us to
determine the characteristics of sound propagation and reflection as well as quantitatively
evaluate physical properties of targets.
The ultrasonic pulse-echo system used in this study for data acquisition is a real
time ultrasonic measurement system, which is depicted in Figure 6.1. It can be seen that
the basic elements of the system are transducer, stepper motor and controller, pulse
transmitter/receiver unit, oscilloscope with digitizer unit. The pulse transmitter/receiver
unit launches an impulse train to excite the transducer and generates a triggering signal to
control the timing of events in the system. A computer with virtual instrument
programming (i.e., LabVIEW programming) is used to control two stepper motors to
moving in both X and Y directions, and configure the sampling and digitizing parameters
for data acquisition. Due to the difficulties to reproduce the same conditions of coupling
between transducer and the specimen in transducer-contact method, we use water as
couplant in this experiment and immerse the specimen and transducer in a water tank.
66
Figure 6.1. Real Time Ultrasonic Measurement System.
67
According to the transducer scan mode, there exist different testing procedures
(i.e., A-scan, B-scan and C-scan). When the transmitted signal scans the specimen along
the transducer axis through one fixed point, the acquired data is called A-scan (Amplitude
scan). When the transducer is moved along X or Y direction, it yields 2-D image, which
is called B-scan (Brightness scan). When the transducer is moved both in X and Y
directions, it would yields 3-D image. The image slice perpendicular to the transducer
axis is called C-scan (Constant depth scan). It can be seen that the A-scan is the base of
B-scan and C-scan. The quality of B-scan and C-scan in turn depend on the quality of A-
scan data in certain extent. The accurate analysis and enhancement of A-scan can
improve B-scan and C-scan for ultrasonic imaging and further process. The signal
decomposition algorithms aim to efficiently analysis an A-scan data.
6.3 Target Detection in Ultrasonic Backscattered Signal
The CTSD algorithm is utilized to evaluate an ultrasonic experimental
backscattered signal consisting of many interfering echoes and detect a embedded target.
The experimental signal is acquired from a steel block with a flat-bottom hole (i.e., target)
using a nominal center frequency 5MHz transducer and sampling rate of 100 MHz.
Figure 6.2 shows the reconstructed signal using CTSD algorithm (dash line)
superimposed the experimental ultrasonic backscattered echoes (solid line). The
experimental signal has poor SNR and the target echo shows interference from
microstructure scattering and measurement noise. The reconstructed signal and its
chirplet transform representation are shown in Figure 6.3c and Figure 6.3d. The
parameters of each decomposed chirplet using CTSD algorithm are listed in the Table 6.1.
68
Furthermore, the comparison between the experimental signal and the reconstructed
signal using CTSD algorithm (see Figure 6.2 and Figure 6.3) clearly demonstrates that
the chirplet signal decomposition has been successful in estimating echoes and filtering
out the noise.
Similarly, the MPSD algorithm is evaluated using the same ultrasonic
experimental backscattered signal consisting of many interfering echoes to detect the
embedded target. Figure 6.4 shows the reconstructed signal (dash line) using MPSD
algorithm superimposed the experimental ultrasonic backscattered echoes (solid line).
The experimental signal has poor SNR and the target echo shows interference from
microstructure scattering and measurement noise. The WVD representation of the
experimental signal (see Figure 6.5b) clearly shows that the experimental signal has poor
SNR and the target echo is completely embedded in the interference from microstructure
scattering, measurement noise. The cross-term effect of WVD also smears the target
information in the time frequency representation. After the process of decomposition, the
reconstructed signal and its WVD representation are shown in Figure 6.5c and Figure
6.5d. The parameters of each decomposed chirplet using MPSD algorithm are listed in
the Table 6.2. The comparison between the experimental signal and the reconstructed
signal using MPSD algorithm (see Figure 6.4 and Figure 6.5) also clearly demonstrates
that the decomposition has been successful in detecting the target echo and filtering out
the noise.
From the discussion of ultrasonic target detection, it can be seen that the CTSD
and MPSD algorithm can decompose and reconstruct the heavily overlapped ultrasonic
backscattered signal with high accuracy. The time frequency representations show that
69
the target echo can be successfully detected and the parameters of targets can be used to
further locate, evaluate, and analyze its physical properties.
Figure 6.2. Ultrasonic Backscattering Signal Superimposed with the Reconstructed Signal (CTSD Method).
70
Figure 6.3. a) Ultrasonic Backscattering Signal. b) TF Representation of the Ultrasonic Backscattered Signal. c) Estimated Signal.d) TF representation of the Estimated Signal.
71
Table 6.1 Parameter Estimation Results for Ultrasonic signal (CTSD)
Figure 6.4. Ultrasonic Backscattering Signal Superimposed with the Reconstructed Signal (MPSD Method).
73
Figure 6.5. a) Ultrasonic Backscattering Signal. b) WVD of Ultrasonic Backscattering Signal. c) the Reconstructed Signal d) WVD of the Reconstructed Signal Using MPSD Method.
74
Table 6.2 Parameter Estimation Results for Ultrasonic Backscattered Signal (MPSD)
Bat is one of species that use ultrasound for echolocation. The research of its
sound is important in scientific research, which providing insights into the biology of
hibernation and sonar mechanisms. There is an experimental chirp data which is emitted
by a large brown bat in the signal analysis literatures [Qia98, Fen01, Wan01, Cap03,
Rub05, and Don06]. It has been used as a benchmark signal for time frequency signal
analysis. Thanks to Beckman Institute, University of Illinois for offering the data, we can
evaluate the proposed signal decomposition algorithms using the bat chirp signal.
The CTSD algorithm is applied to process the bat chirp signal emitted by the large
brown bat, which is digitized within 2.2 ms duration with sampling period of 7 us. Figure
6.6 shows the reconstructed signal using CTSD algorithm (dash line) superimposed the
experimental bat chirp signal (solid line). The parameters of each decomposed chirplet
using CTSD algorithm are listed in the Table 6.3. The bat chirp signal has poor SNR and
contains heavily overlapping chirp components. From the reconstructed signal and its
chirplet transform representation (shown in Figure 6.7c and Figure 6.7d), it can be seen
that the bat chirp signal includes three main stripes. These stripes are highly overlapped
with each other in both time domain and frequency domain, which add the difficulties for
signal analysis. The process results (shown in Figure 6.6, Figure 6.7 and Table 6.3)
clearly demonstrate that the chirplet signal decomposition not only successful analyzes
the contents of bat echoes as the other literatures did, but offer the details of parameters
for better scientific analysis of the species.
Similarly, the MPSD algorithm is evaluated using the same experimental bat chirp
echoes. Figure 6.8 shows the reconstructed signal (dash line) using MPSD algorithm
76
superimposed the experimental bat chirp echoes (solid line). . The parameters of each
decomposed chirplet using MPSD algorithm are listed in the Table 6.4. The WVD
representation of the experimental bat chirp echoes (see Figure 6.9b) shows that the cross
term effect of WVD conceals the characteristics of bat signal. After the process of
MPSD algorithm, the WVD representation of the reconstructed signal (see Figure 6.9d)
suppresses the cross terms and reveals the similar three main stripes in time frequency
domain. The process results (shown in Figure 6.8, Figure 6.9 and Table 6.4) show the
decomposition with high efficiency in the bat chirp signal analysis.
6.5 Summary
In this chapter, the CTSD and MPSD algorithm has been evaluated in the
ultrasonic target detection and bat chirp signal analysis. Experimental results and
performance analysis indicate the robustness of the proposed algorithms in these
applications.
77
Figure 6.6. Experimental Bat Chirp Signal Superposed with Estimated Result (CTSD).
78
Figure 6.7. a) Experimental Bat Chirp Signal. b) TF Representation of the Experimental Bat Chirp Signal. c) Estimated Signal. d) TF representation of the Estimated Signal.
79
Table 6.3. Parameter Estimation Results for Bat Chirp (CTSD)
Table 8.2 Estimated Coefficients of Reverberant Echoes
Echo Time of arrival [μs] Amplitude
a1 0.1831 1.1380
a2 0.5852 0.6306
a3 0.9873 0.2951
a4 1.3858 0.1701
b1 3.0557 0.3284
b2 3.4572 0.3928
b3 3.8557 0.3537
b4 4.2519 0.2863
c1 5.9069 0.3186
c2 6.3088 0.2896
c3 6.7044 0.1826
c4 7.0937 0.0866
106
1 2 3 40
0.2
0.4
0.6
0.8
1
1.2
Reverberation number
Am
plitu
de
a echob echoc echo
Figure 8.5. Comparison of Envelope of Class “a” Echoes, “b” Echoes and “c” Echoes.
107
Table 8.3 Thickness Estimation of Multilayered Structure ( 31 ≤≤ k )
Difference of TOA Mean [μs] Variance
kk aa −+1 0.4009 4.3200e-6
kk bb −+1 0.3987 7.0633e-6
kk cc −+1 0.3956 1.3363e-5
kk ab −
2.8698 9.4425e-6
kk bc −
2.8483 2.0569e-5
108
8.4 Summary
In this chapter, we have analyzed theoretical model of multilayer structure. An
echo classification model of multilayered structure has been developed to reveal the
physical nature of reverberant path and re-grouped the general expression of reverberant
signal into different type of sequential echoes based on the traveling distance in the media.
The chirplet signal decomposition algorithm has been utilized to reconstruct the
experimental ultrasonic multilayered reverberant echoes with high accuracy. The
expected echo patterns, based on the theoretical model, not only have been shown in the
acquired experimental ultrasonic data, but also shown by the parameter estimation results
of chirplet signal decomposition algorithm. Through extensive experimental studies we
have shown that the reverberation model of thin layers coupled with chirplet signal
decomposition allows for a very accurate estimation of transmission/reflection properties
of each layer and also leads to an accurate estimation of the thicknesses of the layers by
an order of magnitude beyond the resolution of the ultrasonic measuring system.
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CHAPTER 9
EMBEDDED FPGA-BASED DSP SYSTEM FOR SIGNAL DECOMPOSITION
9.1 Introduction
Field programmable gate arrays (FPGAs) are digital integrated circuits that
contain configurable logic blocks (CLBs) along with programmable interconnects
between these blocks. The Virtex series FPGAs are intended as system integration
platform which offer a combinations of performance, capability, and low system cost.
The Virtex integrates high level of system functions such as processors, delay lock loops,
clock managers, memory, and serial transceivers on a single FPGA chip [Xil06a].
Due to the flexibility of the FPGA to add custom hardware to accelerate software
bottlenecks and its quick development time, speeding the prototype process by allowing
in-platform testing and debugging of the system, we choose the Xilinx University
Program Virtex II Pro (XUPV2P) development board to verify the CTSD algorithm. The
XUPV2P board provides an advance hardware platform that has a 100 MHz system clock
and consists of a high performance Virtex-II Pro platform FPGA (i.e., XC2VP30)
surrounded by a comprehensive collection of peripheral components such as RS-232, on-
board 10/100 Ethernet device, up to 2GB of Double Data Rate(DDR) SDRAM, AC-97
audio CODEC, and on-board video port[Xil05].
Moreover, Xilinx provides its own implementation of a 32 bit RISC processor
soft core (i.e., MicroBlaze), which is tailored and optimized for implementation in Xilinx
FPGAs with minimum configurable logic resource. It features a 5-stage pipeline, with
most instructions completing in a single cycle. Both instruction and data words are 32
bits. Many aspects of the MicroBlaze can be configured at compile time due to the
110
configurable nature of FPGAs[Xil06a]. The XC2VP30 FPGA can be configured to
contain multiple MicroBlaze cores for multiprocessor system design. One of the most
useful features of the MicroBlaze is the fast simplex link (FSL) bus, which provides a
simple and high-throughput point-to-point communications between MicroBlaze and
custom hardware cores. The MicroBlaze has special assembly instructions to place and
retrieve data on and from the FSL bus.
In this chapter, the CTSD algorithm is implemented as a System-on-Chip (SoC)
based on Xilinx Virtex-II Pro FPGA to rapid prototype and further probes its suitability
for embedded hardware [Sor06]. The CTSD algorithm is implemented in software and
profiled with standard software tools to identify the parts of the algorithm which consume
the most execution time. The dedicated hardware accelerator is designed to increase the
performance of the embedded system. Simulated and experimental ultrasonic signals are
used to verify the functionality of the system design.
9.2 Embedded DSP System Based on Xilinx Virtex II Pro FPGA
From a computational complexity standpoint, the complexity of the chirplet
transform, a correlation operation between the signals and the scaled chirplet kernels (see
Equation 3.2), is )( mnO , where m is the number of scaled chirplets, and n is the
number of samples in the signal. The windowing process of chirplet transform is a linear
search with computational complexity )( nO . In other words, the time-frequency
representation of the signal depends on the sampling frequency and the scale size of the
chirplet kernel. Meanwhile, in the successive parameter estimation stage, each parameter
111
is estimated through maximization of the correlation between windowed signal and
chirplet kernel. The accuracy is dependent on the step size used to estimate the parameter.
The C implementation of the CTSD algorithm is profiled using GNU tools to
isolate which parts of the algorithm consumed the most execution time. The chirplet
transform and successive parameter estimation process occupied most of execution time.
The chirplet transform and windowing algorithm consumes on average 45.3% of the total
processing time. The successive parameter estimation consumes on average 40.3%
processing time. Further analysis shows that forward and inverse Fourier Transform
consume the majority of the windowing algorithm execution time (72.1%). Moreover, in
the successive parameter estimation stage of the algorithm, the trigonometric functions
and exponent functions are found to be heavily used in calculating the time frequency
representation and reconstructing the signal, which are major contributors to execution
time. They are most promising candidates for hardware acceleration.
An architectural overview of the developed embedded DSP system is shown in
Figure 9.1[Sor07]. From Figure 9.1, it can be seen that this system consists of an analog
sensor (ultrasonic transmitter/receiver), an A/D converter devices to sample and digitize
the ultrasonic data, and two FPGAs (i.e., an interface FPGA and an application FPGA).
The interface FPGA pre-processes the data from A/D devices and manages queuing the
data for the application FPGA. In the application FPGA, two MicroBlaze are
implemented in the FPGA fabric. One MicroBlaze is referred to as the algorithm
processor, while the other is refereed to as the communication processor.
112
The algorithm MicroBlaze is connected to the hardware cores through
unidirectional FSL buses. The FSL buses are implemented internally as FIFO queues,
along with some control logic. Since the FSLs are unidirectional, the FSL master places
data on the bus and store in the internal buffer of FSL core. The FSL slaver is in charge
of reading the data out of FSL. The transmission occurs asynchronously. This allows the
accelerators to run with a higher clock frequency than the MicroBlaze to achieve better
performance. A dedicated cache connects the algorithm MicroBlaze and the system
memory. It speeds execution on the algorithm processor since most of the data is kept in
on-chip memory. In the design of hardware accelerator cores, based on the profile
results, those time-consuming software functions are transferred to custom hardware
accelerator cores. For the Fourier transform hardware acceleration, the FFT core, a
pipelined architecture is chosen based on decimation in frequency Radix-2 butterfly units
for maximum throughput [Sek99], offered by Xilinx is capsulated with FSL interface. A
CORDIC-based core is selected for the sine, cosine acceleration, which improves
performance by calculating both sine and cosine in hardware simultaneously [Men98].
The communication MicroBlaze is mainly to provide interfacing to fetch and send
processed results from the system. It is supplemented with hardware cores to handle
RS232, video, audio and Ethernet interfacing.
113
Figure 9.1. Embedded System Architecture. [Sor06]
114
Figure 9.2 shows the process results of processing actual experimental ultrasonic
measurements through the system. These results show that the reconstructed signal
demonstrates high fidelity to the original signal. Also, the result of estimated parameters
for each echoes from FPGA system matched the results obtained from software
implementation of the CTSD algorithm, proving the feasibility of constructing an
embedded implementation of the CTSD algorithm.
9.3 Summary
In this Chapter, A Xilinx Virtex II Pro FPGA-Based DSP system is designed to
verify the feasibility of hardware implementation of CTSD algorithm. Embedded
MicroBlaze processors and FSL buses are utilized to manage the hardware system. Based
on the profile results of CTSD algorithm, hardware acceleration cores such as FFT cores
and CORDIC-based core are used to accelerate the computation of the algorithm. The
simulation and experimental results functionally verified the system design. This work
demonstrates an embedded FPGA-based DSP system for ultrasonic detection and
estimation using the CTSD algorithm. Further algorithm analysis and hardware
acceleration strategies are expected to be done for the future real time ultrasonic signal
processing.
115
Figure 9.2. Process Experimental Ultrasonic Echoes on FPGA-Based DSP System.
116
CHAPTER 10
CONCLUSION AND FUTURE WORK
In ultrasonic applications, the patterns of detected echoes correspond to the shape,
size and orientation of the reflectors and the physical properties of the propagation path.
However, these echoes are often overlapped due to closely spaced reflectors and/or
microstructure scattering. Therefore, signal model and parameter estimation is critical for
these applications. In this research, we have developed chirplet signal decomposition
algorithms for signal analysis. Two different implementation strategies of decomposition
have been discussed. One is based on chirplet transform. Another one is based on the
matching pursuit framework. We developed the decomposition algorithms and
demonstrated them in different ultrasonic applications such as ultrasonic target detection,
bat chirp signal evaluation, grain size estimation, and backscattered reverberant analysis.
The chirplet signal decomposition algorithm aims to decompose the signal to be
processed into a linear combination of chirplets. In the signal decomposition algorithm
based on chirplet transform (CTSD) algorithm, from the point view of time frequency
resolution, the chirplet transform has similar resolution advantage as wavelet transform
does. The chirplet transform is used not only used as a mean for time frequency
representation, but also to estimate the echo parameters including the amplitude, time of
arrival, center frequency, bandwidth factor, phase, and chirp rate. Once these parameters
are estimated, one can achieve a quantitative representation leading to the identification
of echoes and physical property analysis of specimen. The successive parameter
estimation algorithm coupled with windowing strategy in time frequency representation
domain showed robustness in chirp signal decomposition, compared with the Gabor
117
decomposition algorithm [Car05b]. This comparison revealed one important fact about
the CTSD algorithm, that is, it uses fewer components to reconstruct the chirp type signal
and the parameters reveal the chirp nature of original signals.
Another algorithm is matching pursuit signal decomposition (MPSD) algorithm.
We incorporated statistical signal processing methods such as Maximum Likelihood
Estimation (MLE) and Maximum a Posteriori (MAP) into matching pursuit framework.
The signal analysis results show that, if proper prior information is offered, MPSD-MAP
can be more matched to the local physical properties of signals than MPSD-MLE. In both
implementations of the MPSD algorithm, the parameters of chirplet are adaptively
optimized to best match the signal residues. It avoids the exhaustive search of a large
number of dictionary function and leads to a more efficient implementation.
Furthermore, in order to determine the effect of noise level in parameter
estimation, we derived the analytical Cramer Rao Lower Bounds (CRLB) for chirplet
signal decomposition. The CRLB provides the bounds on the variance of parameter
estimators. Through Monte Carlo simulation, we demonstrated that the chirplet parameter
estimation of both algorithms is unbiased with minimum variance, i.e., it attains
analytical derived CRLB bounds. When applied to simulated ultrasonic signals, both
algorithms perform robustly, yield accurate echo estimations and result in considerable
SNR enhancements. Moreover, the MPSD algorithm outperforms the CTSD in moderate
noise levels whereas the CTSD performs better than MPSD in severe noise levels. This
can be explained by the different nature of algorithms. First, the processing domain is
different. The CTSD algorithm is to process signal and estimate the parameters in time
frequency domain whereas the MPSD algorithm performs only in time domain. Hence,
118
the noise is better suppressed in CTSD algorithm than it is in MPSD algorithm. Secondly,
the iterative optimization of MPSD algorithm may become more dependent on the initial
guess in severe noise levels.
One immediate application of the chirplet signal decomposition algorithm is
ultrasonic target detection. The CTSD algorithm has been evaluated using an ultrasonic
experimental backscattered signal consisting of many interfering echoes to detect a target
embedded in it. The reconstructed signal and time frequency representation showed that
the target echo was successfully detected and the parameters can be used to further
evaluate and analyze the physical properties.
We studied the performance of our algorithm in an experimental bat chirp echoes,
which is emitted by a large brown bat and used as the benchmark signal in literatures for
time frequency analysis. The time frequency representation shows that the bat chirp
signal is highly overlapped in both time and frequency domain, which add the difficulties
in the signal analysis. The bat chirp signal has poor SNR and contains heavily
overlapping chirp components. The chirplet signal decomposition not only successful
analyzes the contents of bat echoes as the other literatures did, but offer the details of
parameters for better scientific analysis of the species.
Another application of our algorithms is grain size estimation, which is critical to
determine the mechanical properties of materials. We reviewed a model for the ultrasonic
grain backscattered signal and discussed the effect of frequency dependent scattering and
attenuation. By estimating the expected frequency, our algorithm verified the spectral
shift trend in different specimens which were processed under different heat treatment
119
and have different grain size. Our algorithm exhibits a new angle to extract the mean
grain size information from ultrasonic backscattering echoes.
One can also use the chirp signal decomposition algorithms in the classification of
ultrasonic multilayered reverberant echoes. An echo classification model of multilayered
structure has been developed to reveal the physical nature of reverberant. Our algorithm
has been utilized to successfully classify different type of echoes in ultrasonic
experimental reverberant signal and estimate the physical parameters of multilayered
structure.
Furthermore, an embedded FPGA-based DSP system is successfully designed to
verify the feasibility of hardware implementation and acceleration for the CTSD
algorithm.
Overall, it has been shown through computer simulation and analysis of
experimental data that the chirplet decomposition algorithms can efficiently decompose
the nonstationary signal and estimate the parameters of the chirplets. The estimated
parameters have been successfully used to locate the target echo in ultrasonic
backscattered signal, evaluate grain size of material, and classify ultrasonic multilayered
reverberant echoes. This type of study addresses a broad range of applications such as
target detection, data compression, deconvlution, object classification, velocity
measurement, and material characterization.
120
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