Parameters Selection for Optimising Time-Frequency Distributions and Measurements of Time-Frequency Characteristics of Nonstationary Signals

Transcription

1 Parameters Selection for Optimising Time-Frequency Distributions and Measurements of Time-Frequency Characteristics of Nonstationary Signals Victor Sucic Bachelor of Engineering (Electrical and Computer Engineering) First Class Honours School of Electrical and Electronic Systems Engineering Queensland University of Technology Brisbane, Australia Submitted as a requirement for the degree of Doctor of Philosophy, Queensland University of Technology. March, 2004.

2 Not everything that can be counted counts, and not everything that counts can be counted. Albert Einstein

3 Keywords Time- distribution, Quadratic Class of time- distributions, nonstationary signal, monocomponent signal, multicomponent signal, modulated (FM) signal, linear FM signal, non-linear FM signal, kernel filter, separable kernel, Fourier transform, crossterms, autoterms, concentration, resolution, mainlobe amplitude, sidelobe amplitude, instantaneous, instantaneous bandwidth, crossterms amplitude, comparison criteria, performance measure, parameter optimisation, optimal time- distribution, design requirements, additive white Gaussian noise, real-life signal, Modified B distribution, Smoothed windowed Wigner-Ville distribution, separable kernel TFD. i

4 Abstract The quadratic class of time- distributions (TFDs) forms a set of tools which allow to effectively extract important information from a nonstationary signal. To determine which TFD best represents the given signal, it is a common practice to visually compare different TFDs time- plots, and select as best the TFD with the most appealing plot. This visual comparison is not only subjective, but also difficult and unreliable especially when signal components are closely-spaced in the time- plane. To objectively compare TFDs, a quantitative performance measure should be used. Several measures of concentration/ complexity have been proposed in the literature. However, those measures by being derived with certain theoretical assumptions about TFDs are generally not suitable for the TFD selection problem encountered in practical applications. The non-existence of practically-valuable measures for TFDs resolution comparison, and hence the non-existence of methodologies for the signal optimal TFD selection, has significantly limited the use of time- tools in practice. In this thesis, by extending and complementing the concept of spectral resolution to the case of nonstationary signals, and by redefining the set of TFDs properties desirable for practical applications, we define an objective measure to quantify the quality of TFDs. This local measure of TFDs resolution performance combines all important signal time-varying parameters, along with TFDs characteristics that influence their resolution. Methodologies for automatically selecting a TFD which best suits a given signal, including real-life signals, are also developed. The optimisation of the resolution performances of TFDs, by modifying their kernel filter parameters to enhance the TFDs resolution capabilities, is an important prerequisite in satisfying any additional application-specific requirements by the TFDs. The resolution performance measure and the accompanying TFDs comparison criteria allow to improve procedures for designing high-resolution quadratic TFDs for practical time- analysis. The separable kernel TFDs, designed in this way, are shown to best resolve closely-spaced components for various classes of synthetic and real-life signals that we have analysed. ii

5 Contents Keywords Abstract Acronyms and Symbols Author s Publications Authorship Acknowledgements Preface i ii xiii xiv xvi xvii xviii 1 Introduction Objectives of the Thesis Contributions Thesis Organisation Time, Frequency, and Time-Frequency Signal Representations Classical Signal Analysis Tools Time domain and domain representations Limitations of the classical representations Joint Time-Frequency Representations Time- domain representation the need and advantages Short-time Fourier transform and the spectrogram The Quadratic Class of TFDs The Wigner-Ville distribution Asymptotic signals and smoothing of the WVD Dual domain representations and the ambiguity function Time-Frequency Analysis of Multicomponent Signals Monocomponent and multicomponent signal Multicomponent signal and crossterms Crossterms reduction techniques Reduced interference TFDs iii

6 Signal dependent TFDs Summary and Conclusions Resolution Performance Measure for Quadratic TFDs Introduction Performance Measures for TFDs Resolution Performance Criteria for TFDs Monocomponent signal Multicomponent signal Concentration Resolution Resolution Performance Measure for TFDs Using the measure P to compare the resolution performance of TFDs Summary and Conclusions TFD Properties and Kernel Design for High-Resolution TFSAP Classical Properties of Quadratic TFDs Desirable Properties for Practical High Resolution TFSAP Classical properties and TFDs practical applications Revised set of desirable properties Desirable properties, resolution performance measure, and TFD design Optimal Separable Kernel TFD Design The design methodology Examples of designing signals optimal separable kernel TFDs Three-component synthetic signal Real-life signal in noise Summary and Conclusions Methodology for Selecting Best-Performing TFD The TFD Optimisation/Selection Methodology The TFD Optimisation/Selection Algorithm Examples of Selecting Signal Best-Performing TFD Two-LFM-component signal in additive white noise Signal with a linear and a non-linear FM component in noise Signal with different inter-component separations Signal with components of different amplitudes Summary and Conclusions Modifications of the Resolution Performance Measure Normalized Resolution Performance Measure Limiting the range of the measure values Example of selecting signal best-performing TFD using the measure P i

7 6.2 Improving the Definition of the Measure P i More discriminative resolution performance measure Examples of selecting signals best-performing TFDs using the measure P im Two closely-spaced sinusoids Two closely-spaced parallel LFMs Two LFMs with the opposite IF laws Statistical Performance of TFD Resolution Measures Summary and Conclusions Time-Frequency Analysis of Real-Life Signals Introduction Selecting Real-Life Signal Best TFD in Time-Frequency Regions of Interest The methodology Real-life signal example Automatic Selection of Real-Life Signal Best-Performing TFD The methodology Test signal: A multicomponent, synthetic signal Real-life signals examples Bird song signal Bat echolocation signal Summary and Conclusions Conclusions and Future Directions 113 A Algorithm for Comparing TFDs Resolution Performances 116 A.1 Closest Components Identification A.2 TFDs Optimisation and Comparison Bibliography 125

8 List of Figures 2.1 Time and representations of the whale signal The whale signal domain representation The whale signal time- representation Time- representation (Wigner-Ville distribution) of a linear FM signal. The signal instantaneous power is given on the left, and its magnitude spectrum on the bottom Time- representations of two three-component signals Dual domain representations of signal z(t) obtained from the signal kernel K z (t, τ) Monocomponent signal. For a given value of time there is a single value of Multicomponent signals. For a given value of time there are two values of Instantaneous and instantaneous bandwidth of a monocomponent signal Instantaneous and instantaneous bandwidth of a multicomponent signal Location of autoterms and crossterms in the time- domain (left), and the Doppler-lag domain (right) The Choi-Williams distribution Doppler-lag kernel filter The time-lag kernel filter (g(τ) = 1, and a = 2) of the Zhao-Atlas- Marks time- distribution TFDs of a multicomponent bird song signal Slice of a TFD ρ z (t, f) of a monocomponent FM signal z(t) taken at the time instant t = t 0. The dominant peak is the signal component with magnitude A M and instantaneous bandwidth V i, centred about the signal IF f i (t). The smaller peaks are the sidelobes with magnitude A S Resolution in a PSD estimate of the signal consisting of two sinusoids f 1 and f 2, with the corresponding bandwidths V 1 and V 2. The two lobes are clearly distinguishable from each other, so the components are said to be resolved vi

9 3.4 WVD of a two-lfm-component signal (a), and the slices of the WVD at the middle of the signal duration interval of the two contributing terms (b and c), and of the multicomponent signal itself (d) Slice of a TFD ρ z (t, f) of a two-component FM signal z(t) taken at the time instant t = t 0. The two dominant peaks are the signal resolved components with the corresponding magnitudes A M1 and A M2, and the instantaneous bandwidths V i1 and V i2, centred about the components IFs f i1 (t) and f i2 (t). The middle peak is the crossterms of magnitude A X, while other smaller peaks are the two components sidelobes with magnitudes A S1 and A S Slice of a TFD ρ z (t, f) of a two-component FM signal z(t) taken at the time instant t = t 0. The signal components and the crossterm have merged in a single (broader) peak. In such cases we say that the components are unresolved TFDs of a signal consisting of two LFMs whose frequencies increase from 0.15 to 0.25 Hz and from 0.2 to 0.3 Hz, respectively, over the time interval t [1, 128]. The sampling is f s = 1 Hz Resolution parameters and the performance measure P at t = 64 for TFDs of the signal consisting of two LFMs whose frequencies increase from 0.15 to 0.25 Hz and from 0.2 to 0.3 Hz, respectively, over the time interval t [1, 128]. The sampling is f s = 1 Hz. In each of the four subplots, the TFDs are numbered as follows: 1 = Born-Jordan, 2 = Choi-Williams (σ = 2), 3 = Modified B (β = 0.01), 4 = Spectrogram (Hann, L = 35), 5 = WVD, and 6 = Zhao-Atlas- Marks (a = 2) time- distribution Normalized slices (taken at t = 64) of TFDs plotted in Fig The smoothed WVD and the closest components time- location of the three-component synthetic signal embedded in 10 db additive white noise The optimal separable kernel TFD of the three-component synthetic signal in 10 db additive white noise. The optimal time and lag smoothing windows (both corresponding to the overall closest time t = 72) are respectively the Kaiser window of length L t = 61 and with the parameter β t = 16, and the Kaiser window with L τ = 47 and β τ = The smoothed WVD and the closest components time- location of the Large Brown bat signal embedded in 10 db additive white noise The optimised TFDs of the Large Brown bat signal embedded in 10 db additive white noise

10 5.1 Optimized TFDs (over the time interval of interest t [33, 96]) of a signal consisting of two LFMs with frequencies increasing from 0.15 to 0.25 Hz and 0.2 to 0.3 Hz, respectively. The signal is embedded in 10 db additive white noise Comparison of the measured (dashed) and true (solid) IF laws of a signal with two LFM components embedded in 10 db additive white noise: (a) The mean-square-errors (MSEs) of the IF estimates obtained from the two dominant peaks of the signal optimized Modified B distribution, (b) the MSEs of the IF estimates obtained from the peaks of the signal optimized spectrogram Signal s 1 (t) TFDs comparison using different measures of concentration/complexity/resolution. The plots correspond to the measures values given in Table 5.2. In each of the four subplots, the TFDs are numbered as follows: 1 = Born-Jordan, 2 = Choi-Williams (σ opt = 0.5), 3 = Modified B (β opt = 0.002), 4 = Spectrogram (Bartlett, L opt = 47), 5 = WVD, and 6 = Zhao-Atlas-Marks (a opt = 8) time distribution Diagram illustrating the IF laws of the signal s 2 (t) Optimised TFDs of the signal s 2 (t), whose components IF laws are defined in Fig 5.4, for the case when the signal is embedded in 0 db additive white noise The time slices (t = 50) of the Modified B distribution with β = 10 4 (thick solid line), and the spectrogram with the rectangular window (thin solid line) and with the Hanning window (thin dashed line) for the two crossed LFMs The TFDs average performance measure P a, t {[40, 50], [80, 90]}, for the two crossed LFMs The mean and the standard deviation of the measure P (34) for TFDs of the signal with two LFM components embedded in noise with different SNRs The mean and the standard deviation of the measure P i (34) for TFDs of the signal with two LFM components embedded in noise with different SNRs The mean and the standard deviation of the measure P im (34) for TFDs of the signal with two LFM components embedded in noise with different SNRs Optimised Modified B distributions of multicomponent signals, which are used for evaluating statistical performances of different resolution measures for signals TFDs The mean and the standard deviation of the measure P (33) for TFDs of a signal with two parallel quadratic FM components in noise with different SNRs

11 6.8 The mean and the standard deviation of the measure P i (33) for TFDs of a signal with two parallel quadratic FM components in noise with different SNRs The mean and the standard deviation of the measure P im (33) for TFDs of a signal with two parallel quadratic FM components in noise with different SNRs The mean and the standard deviation of the measure P (77) for TFDs of a signal with two quadratic FM components and a linear FM component in noise with different SNRs The mean and the standard deviation of the measure P i (77) for TFDs of a signal with two quadratic FM components and a linear FM component in noise with different SNRs The mean and the standard deviation of the measure P im (77) for TFDs of a signal with two quadratic FM components and a linear FM component in noise with different SNRs The mean and the standard deviation of the measure P (299) for TFDs of the large brown bat echolocation signal in noise with different SNRs The mean and the standard deviation of the measure P i (299) for TFDs of the large brown bat echolocation signal in noise with different SNRs The mean and the standard deviation of the measure P im (299) for TFDs of the large brown bat echolocation signal in noise with different SNRs The Noisy Minor song signal represented in (a) time, (b) Selection of the time- ROIs for the Noisy Minor song signal Regions of Interest (ROIs) for the Noisy Minor song signal s time distributions ROI 1 of the optimised TFDs of the Noisy Minor song signal. The TFDs optimal kernel parameters are given in Table 7.1, column two ROI 2 of the optimised TFDs of the Noisy Minor song signal. The TFDs optimal kernel parameters are given in Table 7.1, column three ROI 3 of the optimised TFDs of the Noisy Minor song signal. The TFDs optimal kernel parameters are given in Table 7.1, column four ROI 4 of the optimised TFDs of the Noisy Minor song signal. The TFDs optimal kernel parameters are given in Table 7.1, column five The synthetic signal components IF laws and the signal s Smoothed Windowed WVD Pairs of closest components of the synthetic signal Time- plots of the optimised TFDs of the synthetic multicomponent signal The synthetic signal s three best-performing optimised TFDs at t = The bird song signal s SWWVD and its pairs of closest components The bat signal s SWWVD and its pairs of closest components

12 7.14 Time- plots of the optimised TFDs of the bird song signal Time- plots of the optimised TFDs of the bat signal in noise.111

13 List of Tables 3.1 Parameters and the resolution performance measure P of TFDs in Fig. 3.7 for the time instant t = Optimization and comparison results for the TFDs of the Large Brown bat signal in 10 db additive white noise. A P i value close to 1 indicates a TFD s good resolution performance Optimization results for TFDs of a signal consisting of two LFMs whose frequencies increase from 0.15 to 0.25 Hz and from 0.2 to 0.3 Hz, respectively (the time interval of interest t [33, 96]). The signal is embedded in 10 db additive white noise Signal s 1 (t) TFDs comparison using: M = concentration measure defined by Jones and Parks (Eq. (3.3)), R 3 = Renyi complexity measure for α = 3 (Eq. (3.5)), M2 2 = concentration measure defined by Stankovic (Eq. (3.6)), and P opt = resolution performance measure (defined using Eq. (6.4)) Optimisation results for TFDs of the noiseless signal s 2 (t) Optimisation results for TFDs of the signal s 2 (t) embedded in 5 db additive white noise Optimisation results for TFDs of the signal s 2 (t) embedded in 0 db additive white noise Instantaneous (t = 64) resolution performance measure P i, defined by Eq. (6.4), for TFDs of the three-component signal s 3 (t) for the pair of in closest components (P i{f1,f 2 ), and the pair of } more separated signal components (P i{f2,f 3 ) } Resolution performance measure P i (64) of the spectrogram of a twosinusoid signal for different ratios of the components mainlobe amplitudes A 2 /A 1 and different components separations f. The spectrograms are calculated using the rectangular and the Hamming windows of length L Resolution performance measure P i (64) of the Modified B distribution of a two-sinusoid signal for different ratios of the components mainlobe amplitudes A 2 /A 1 and different components separations f. The MBDs are calculated for four different values of the kernel filter parameter β xi

14 6.1 Resolution performance measure P i for TFDs of a signal with two parallel linear FM components, and the measure P i values corresponding to the case when the separation between the two components decreases (P is ) and when it increases (P il ) Normalised sidelobe-to-mainlobe ratio SM R, crossterm-to-mainlobe ratio XM R, components separation measure CSM, and instantaneous (t = 64) resolution performance measure P im of TFDs for the two sinusoids Normalised sidelobe-to-mainlobe ratio SM R, crossterm-to-mainlobe ratio XM R, components separation measure CSM, and instantaneous (t = 64) resolution performance measure P im of TFDs for the two parallel LFMs The resolution performance measure P a, for t {[40, 50], [80, 90]}, of the three (in the noiseless case) best-performing TFDs of the two crossed LFMs embedded in additive white noise for different values of signal-to-noise ratio Optimal kernel filter parameters for TFDs of the considered signals The performance measure P opt and the optimal parameter values of TFDs of the Noisy Minor song signal for four different regions-ofinterest (ROIs) as defined in Fig The performance measure M and the optimal value of the kernel parameter for TFDs of the signals considered in Section

15 Acronyms and Symbols TFD TFSAP FT IF IB FM WVD AF CWD MBD SWWVD PSD SNR s(t) z(t) ρ z (t, f) γ(t, f) g(ν, τ) A z (ν, τ) f i (t) V i (t) A M (t) A S (t) A X (t) E z E zr p(t) P (t) M j time- distribution time- signal analysis and processing Fourier transform instantaneous instantaneous bandwidth modulated (signal) Wigner-Ville distribution Ambiguity function Choi-Williams distribution Modified B-distribution Smoothed windowed WVD Power spectral density signal-to-noise ratio real signal analytic associate of s(t) time- distribution of z(t) kernel filter in the time- domain kernel filter in the Ambiguity domain Ambiguity function of z(t) instantaneous instantaneous component mainlobe bandwidth component mainlobe amplitude component sidelobe amplitude crossterm amplitude energy of signal z(t) signal z(t) energy in the time- region R resolution performance measure for monocomponent FM signals resolution performance measure for multicomponent FM signals resolution performance measure of the j th TFD in a set of different TFDs xiii

16 Author s Publications In chronological order: 1. V. Sucic, B. Barkat, and B. Boashash, Performance evaluation of the B- distribution, in Proc. Fifth Internat. Symp. on Signal Processing and its Applications (ISSPA 99), vol. 1, pp , Brisbane, Australia, August B. Boashash and V. Sucic, A resolution performance measure for quadratic time- distributions, in Proc. Tenth IEEE Workshop on Statistical Signal and Array Processing (SSAP 00), pp , Pocono Manor, PA, August V. Sucic and B. Boashash, On the selection of quadratic time- distributions and optimisation of their parameters, in Proc. Third Australasian Workshop on Signal Processing Applications (WoSPA 00), Brisbane, Australia, December 2000, On CD-ROM. 4. V. Sucic and B. Boashash, Parameter selection for optimising time- distributions and measurements of time- characteristics of nonstationary signals, in Proc. IEEE Internat. Conf. on Acoustics, Speech and Signal Processing (ICASSP 01), vol. 6, pp , Salt Lake City, UT, 7 11 May V. Sucic and B. Boashash, Optimisation algorithm for selecting quadratic time- distributions: Performance results and calibration, in Proc. Sixth Internat. Symp. on Signal Processing and its Applications (ISSPA 01), vol. 1, pp , Kuala Lumpur, Malaysia, August V. Sucic and B. Boashash, Selecting the optimal time- distribution for real-life multicomponent signals under given constraints, in Proc. Eleventh European Signal Processing Conf. (EUSIPCO 02), vol. 1, pp , Toulouse, France, 3 6 September V. Sucic, B. Boashash, and K. Abed-Meraim, A normalised performance measure for quadratic time- distributions, in Proc. Second IEEE Internat. Symp. on Signal Processing and Information Technology (ISSPIT 02), pp , Marrakech, Morocco, December xiv

17 8. B. Boashash and V. Sucic, Resolution measure criteria for the objective assessment of the performance of quadratic time- distributions, IEEE Trans. Signal Processing, vol. 51, pp , May V. Sucic and B. Boashash, An approach for selecting a real-life signal bestperforming time- distribution, in Proc. Seventh Internat. Symp. on Signal Processing and its Applications (ISSPA 03), pp , Paris, France, 1 4 July V. Sucic, H. MacGillivray, and B. Boashash, Improving the time- distribution approach in investigating non-stationary signals, in Proc. Fifth Internat. Congress on Industrial and Applied Mathematics (ICIAM 03), pp. 234, Sydney, Australia, 7 11 July K. Abed-Meraim, N. Linh-Trung, V. Sucic, F. Tupin, and B. Boashash, An image processing approach for underdetermined blind separation of nonstationary sources, in Proc. Third Internat. Symp. on Image and Signal Processing and Analysis (ISPA 03), pp , Rome, Italy, September B. Boashash and V. Sucic. High performance time- distributions for practical applications. In L. Debnath, editor, Wavelets and Signal Processing, chapter 6, pp , Birkhäuser, Boston, MA, B. Boashash and V. Sucic. Resolution performance assessment for quadratic time- distributions. In B. Boashash, editor, Time-Frequency Signal Analysis and Processing, chapter 7.4, pp , Elsevier, V. Sucic and B. Boashash, The optimal smoothing of the Wigner-Ville distribution for real-life signals time- analysis, in Proc. Tenth Asia- Pacific Vibration Conf. (APVC 03), pp , Gold Coast, Australia, November 2003.

18 Authorship The work contained in this thesis has not been previously submitted for a degree or diploma at this or any other higher education institution. To the best of my knowledge and belief, the thesis contains no material previously published or written by another person except where due reference is made. Signed:... Date:... xvi

19 Acknowledgements First of all, I would like to thank my principal supervisor Prof. Boualem Boashash for his assistance and support throughout the entire course of my PhD research. Also, I would like to thank my Associate supervisor A/Prof. Helen MacGillivray for her guidance and help in improving my work, as well as to thank all the former and current students and staff of the Signal Processing Research lab for their help. A special thank goes to my parents and my sister for their generous support and encouragement through all these years of my undergraduate and postgraduate studies at Queensland University of Technology. xvii

20 Preface Over the last several decades, the field of signal processing has been a major driving force in electrical engineering. The developed techniques and methods have found their applications not only in engineering, but also in science, medicine, military, industry, just to name few. Signal processing has grown to become an independent field of research, seeing many of its original components maturing and starting to follow independent paths themselves too. To this fast-developing and vibrant area of research, I was attracted at the early stage of my undergraduate studies. The introductory subjects on signal processing opened a new chapter in my engineering education. In those I found an excellent ground to use and further develop my skills in mathematics, which has always been both enjoyment and challenge for me. My first exposure to time- signal analysis and processing (TFSAP) techniques, and indeed the first hands-on experience with signal processing, dates back to the time when I was doing my final year undergraduate project. While exploring how the time- methods can be used in newborn babies electroencephalogram (EEG) signals analysis, I became aware of an important fact. The TFSAP, while currently being one of the most powerful tools for signal analysis, lacks a set of guidelines (methodologies) that would make it more user-friendly and therefore more attractive for practical applications. This finding served as the main motivation for my PhD studies, the results of which are presented in this thesis. The major challenge that a signal analyst (both specialist and non-specialist) faces when using time- tools to represent and analyse a given signal in the joint time- domain is to select the time- distribution (TFD) which is best suited for that signal. In this thesis we limit our studies to the quadratic class of time- distributions, since this is the most-studied, and in practice the most-frequently used class of TFDs. The presented results, however, are applicable to other classes of TFDs, such as higher-order TFDs, for example. If the TFD selection is inadequate, important signal characteristics may be corrupted or become lost in the signal time- representation (e.g. the signal components instantaneous laws, the components duration intervals and bands, or the components amplitudes). How to optimise and compare different TFDs in an objective and automatic way, in order to select the most suitable one among them for representing a given signal in the time- plane, is the topic of this thesis. xviii

21 It is my hope that the thesis will not only serve as an inspiration to the fellow colleagues in the field, but also be of use to the engineers who have been applying the time- techniques in practice. By providing the necessary guidelines on how to use the TFSAP techniques in practical situations, I hope that this thesis will path the way for the time- tools to become one of the primary tools for signal analysis. Author, Brisbane, March 2004.

22 Chapter 1 Introduction The majority of signals we encounter in our everyday life are nonstationary. The main characteristic of these signals is the time-varying nature of their information. The standard practice for analysing a given signal is to represent the signal in either the time domain or the domain. When representing the signal exclusively as a function of time or a function of, the obtained information about the signal is incomplete. Therefore, any conclusion/decision we consequently make based on this information may not be best, if not inappropriate. To completely define a nonstationary signal, it needs to be represented in time and simultaneously. Using time- distributions (TFDs), which are the most popular time- analysis tools, the distribution of signal energy is obtained in the joint time- plane. The difficulty of interpreting the signal phase spectrum in and the signal autocorrelation function in time, made the magnitude spectrum a primary signal analysis tool in practice. However, its major drawback is that the time information which characterises the spectra of nonstationary signals is discarded. Time- distributions, on the other hand, preserve this time information and provide the signal analyst with a complete picture of the signal spectrum variations over time. In addition, TFDs also provide information on the number of components that exist in the signal, the components individual time durations and bands, as well as the components FM laws and relative amplitudes. Depending on the analysed signal, some time- distributions perform better than others. By optimising TFDs and comparing their performances for a given signal in an objective way, the time- distribution which is the best for that signal can be found. The signal optimal TFD provides a complete, interference least corrupted, accurate description of the signal, with the signal information well preserved/enhanced.

23 2 INTRODUCTION 1.1 Objectives of the Thesis The main objective of this thesis is to establish a set of guidelines that would allow the existing time- analysis tools be more easily applied in practice. By building on the classical spectral analysis concepts, and extended those to the nonstationary signals case, we aim to provide signal analysts, both specialists and non-specialists like, with: 1. The comparison criteria and objective measures for quantifying the resolution performance of time- distributions, 2. The methodologies for optimising and comparing the resolution performance of time- distributions in an automatic way, and selecting the best TFD to represent a given signal in the time- plane, 3. The set of properties that a time- distribution should satisfy in order to be a useful tool for practical applications, and 4. The procedures for designing quadratic TFDs for high-resolution practical time- analysis. 1.2 Contributions The work presented in this thesis contains the following original contributions to the field of time- signal analysis and processing (for the references refer to the list of publications on page xiv): 1. Identification of the nonstationary signals parameters and the quadratic TFDs characteristics that affect the TFDs resolution performance (Chapter 3 and [2, 4, 5, 8, 12, 13]), 2. Definition of the criteria for comparison of the resolution performance of quadratic time- distributions for both monocomponent and multicomponent modulated signals (Chapter 3 and [2, 8, 12, 13]), 3. Definition of the measure for an objective evaluation of the resolution performance of quadratic time- distributions (Chapter 3 and [2 4, 7, 8, 10, 12, 13]), 4. Introduction of the methodologies for automatic optimisation of quadratic time- distributions and selection of the best-performing TFD for a given signal (Chapters 4, 5, and 7, and [2, 4 6, 8 14]), 5. Outline of the properties of the quadratic class of time- distributions which are desirable for TFDs used in practical applications (Chapter 4 and [2, 8, 12]),

24 1.3 Thesis Organisation 3 6. Outline of the TFD kernel filter design procedure for high-resolution time signal analysis (Chapter 4 and [1, 2, 8, 12, 14]), 7. Improvements to the resolution performance measure for quadratic TFDs to make it more suitable for the analysis of real-life signals in noise (Chapters 6 and 7, and [7, 9 11, 13, 14]), 8. Introduction of the methodologies that allow an automatic selection of the optimal time- distribution for a given real-life signal (Chapters 4 and 7, and [6, 9 14]). 1.3 Thesis Organisation The thesis is organised in eight chapters, whose contents are briefly described below. Chapter 1 describes the motivation behind this work, and outlines the main research objectives and contributions. It also summarises the contents of each chapter in the thesis. Chapter 2 provides a review of the classical signal analysis tools, and explains the need for the joint time- representations of signals. The advantages of time- analysis are illustrated on examples of synthetic and real-life nonstationary, multicomponent signals. The chapter also defines important time- concepts, such as the analytic signal, monocomponent and multicomponent signals, the instantaneous and bandwidth, and the Quadratic Class of TFDs. In this chapter we review the nature of crossterms that are found in multicomponent signals quadratic TFDs, and discuss the commonly-used techniques for their reduction. Chapter 3 defines an objective measure for evaluating the resolution performance of quadratic TFDs, and illustrates its use on an example of a two-component, nonstationary signal. From the point of view of TFDs practical usage, the resolutionrelated nonstationary signals parameters and TFDs characteristics needed to quantify the TFD quality are identified. This has led to define the criteria for comparing the resolution performance of TFDs for both monocomponent and multicomponent modulated signals. The shortcomings of the commonly-used visual inspection of TFDs time- plots, and the practical limitations of the existing measures of TFDs concentration/complexity are also discussed. Chapter 4 defines the properties of quadratic TFDs which are desirable for their practical applications. The relationship between those properties and the resolution performance criteria is also established. Based on the revised set of TFDs desirable properties and the introduced resolution performance measure for TFDs, we define in this chapter a methodology for designing high-resolution quadratic TFDs with

25 4 INTRODUCTION separable kernel filters. This methodology is then used to design optimal separable kernel TFDs for both synthetic and real-life signals in noise. Chapter 5 presents the methodology for optimising TFDs kernel filter parameters in an automatic way, and selecting a given signal s best-performing TFD. Its use is illustrated on an example of a multicomponent, nonstationary signal embedded in additive white noise. From the signal s best-performing TFD, important signal and TFD parameters are automatically measured. The algorithm which implements this methodology is outlined in details, and its robustness is tested on an extensive range of signals, including signals with components of different FM laws and amplitudes. In Chapter 5 we also provide an example of comparing TFDs performances using the proposed resolution measure and several existing measures of TFD concentration. Chapter 6 discusses the normalisation of the resolution performance measure introduced in Chapter 3. A normalised version of the measure, in which the contributing terms are combined in a sum, rather than a product, is defined. However, it is found that such a measure fails to discriminate good enough different TFDs performances, especially for real-life signals. By normalising over the considered TFDs each of the contributing terms of the performance measure, more discriminative comparison results are obtained. The statistics of the original resolution measure and its two modified versions confirms those findings for different multicomponent signals in noise analysed using the same set of TFDs. Chapter 7 defines the methodologies for selecting the best TFD to represent a reallife signal in the time- domain. Two approaches are presented. The first requires the analyst to identify the regions-of-interest before applying the methodology described in Chapter 5 to identify the best-performing TFD for each of those regions. In the second approach, the resolution performances of TFDs are compared for the pairs of in the direction closest components of the signal at each of the observed time instants. With this approach, TFDs are automatically optimised and compared using the normalised resolution performance measure defined in Chapter 6. Examples of real-life signals are used to show how the methodologies can be applied in practice. Chapter 8 summarises the main conclusions of the thesis. It also provides the recommendations and guidelines on the future research directions to be followed based on and by extending the methods and techniques presented in this thesis.

26 Chapter 2 Time, Frequency, and Time-Frequency Signal Representations In this chapter we review the classical signal analysis tools, and explain the need for signals joint time- representations. We introduce the Wigner-Ville distribution and the Quadratic Class of TFDs as the most commonly-used time tools. The concepts of monocomponent and multicomponent signals are explained. Several solutions for designing TFDs that reduce the interfering crossterms, which are a result of the bilinear nature of multicomponent signals WVD, are presented. The chapter also reviews other important time- concepts, including the instantaneous and bandwidth, asymptotic and analytic signals, TFDs kernel filters, and the ambiguity function. 2.1 Classical Signal Analysis Tools Time domain and domain representations The two most common signal representations in the classical signal analysis are the time representation and the representation. The time representation, s(t), tells us how the amplitude of the signal varies with time, and in certain cases provides indications of the variation in the signal content. The representation, S(f), on the other hand, indicates the presence (absence) of different components in the signal, as well as information on the relative magnitudes of those components. The two representations are related by the Fourier transform (FT) as: S(f) = F {s(t)} = s(t)e j2πft dt (2.1) t f

27 6 TIME, FREQUENCY, AND TIME-FREQUENCY SIGNAL REPRESENTATIONS 1000 Time representation of the whale signal 6 x 105 Frequency representation of the whale signal Amplitude Amplitude time (a) Signal in time (b) Signal in Figure 2.1: Time and representations of the whale signal. or by the inverse Fourier transform (IFT) as: s(t) = F 1 {S(f)} = S(f)e j2πft df (2.2) f t Eq. (2.2) indicates that signal s(t) can be expressed as the sum of complex exponentials of different frequencies, whose amplitudes are the complex quantities S(f) defined by Eq. (2.1). The signal representation, in practice this is often the magnitude squared of the signal Fourier transform S(f) 2, allows in general more easier interpretation of the signal nature than its time representation. Consider, for example, a whale signal s time and representations in Fig Fig. 2.1(a) shows how the amplitude of the whale signal varies with time, while Fig. 2.1(b) indicates that three major components exist in the signal. Although the magnitude spectrum of the whale signal is more informative than its time representation, neither the time nor the representation of the signal is capable of providing any information about the nature of the signal components, e.g. their durations in time, their modulation laws, etc. These information are considered to be crucial when dealing with modulated (FM) signals, as well as other signal with time-varying spectrum nonstationary signals Limitations of the classical representations The whale signal example shows that the existing tools for classical signal analysis are insufficient when used to represent and analyse nonstationary signals. This

28 2.1 Classical Signal Analysis Tools 7 served as a motivation to search for new tools that would allow representation of signals in a more appropriate (complete) way. A solution was found in the introduction of time- distributions (TFDs), which are joint time and representations of signals, capable of preserving all signal information and displaying those in a meaningful way. By distributing the signal energy over the time- plane, TFDs provide the analyst with information usually unavailable from the signal time-domain representation or its -domain representation. This includes the number of components present in the signal, the time durations and bands over which these components are defined, the components relative amplitudes, phase information, and the laws which components follow in the time- plane. But before we further introduce TFDs and important concepts in the field of TFSAP, we will first explain what makes the magnitude spectrum fail for the class of nonstationary signals. It is known that the signal Fourier transform is a complex quantity [1, 2] which can be defined in terms of its magnitude and phase. The magnitude spectrum tells us what components are present in the signal and what their respective amplitudes are. However, it does not tell anything about the modulation laws of the components, nor what their time duration are. Unless the components are disjoint in for the entire duration of the signal, the spectrum also gives no information about the bands signal components occupy. In the definition of the Fourier transform, the stationary nature of the signal is assumed, i.e. the spectrum is assumed to be independent of time and all signal components are assumed to exist for the entire time duration of the signal. All important information about signals are contained in the signal phase spectrum, which although useful, usually is omitted from the analysis due to the difficulty of its interpretation. Consider the magnitude and phase spectra of the whale signal in Fig We can see that the phase spectrum provides little information about the signal, at least in a way that would be easy to interpret. Therefore, in order to be useful for nonstationary signals analysis, any new signal analysis tool need to preserve the phase information, and display this information in an easy-to-interpret way. The indication as to how the content of a signal changes with time, information that is important when one deals with nonstationary signals, is what makes TFDs a powerful tool for representing and analysing such signals. In Fig. 2.3 a time- representation of the whale signal is given. It is evident that such a representation more completely characterises the analysed signal than any of the classical analysis tools. In addition, there may exist signals that are fundamentally different, but their magnitude spectra are identical. The information which allows one to discriminate between them is contained in their phase spectra, which is lost when we form the square modulus of the Fourier transform [3 5]. It follows then that the magnitude spectrum by itself is insufficient (and hence inadequate) for representing signals in a way useful for precise characterisation and

29 8 TIME, FREQUENCY, AND TIME-FREQUENCY SIGNAL REPRESENTATIONS 16 x 104 Magnitude spectrum of the whale signal Phase spectrum of the whale signal Amplitude Amplitude (a) Magnitude spectrum (b) Phase spectrum Figure 2.2: The whale signal domain representation. Fs=1Hz N=7001 Time res= B distribution; beta= Time (secs) Frequency (Hz) Figure 2.3: The whale signal time- representation. identification. This serves as a motivation for devising a more sophisticated signal analysis tool, which preserves all the information of the signal and discriminates signals in a better way, using one single complete representation instead of attempting to interpret signals magnitude and phase spectra separately. 2.2 Joint Time-Frequency Representations Time- domain representation the need and advantages Revealing the time and dependence of a signal is achieved by using a joint time- representation. In Fig. 2.4, the Wigner-Ville distribution is used to represent a chirp (linear FM) signal. This time- representation is

30 2.2 Joint Time-Frequency Representations 9 Wigner Ville distribution of s a (t) Fs=1Hz N=128 Time res=2 Wigner Ville distribution of s a (t) Time (sec) Fequency (Hz) Figure 2.4: Time- representation (Wigner-Ville distribution) of a linear FM signal. The signal instantaneous power is given on the left, and its magnitude spectrum on the bottom. optimal for linear FM signals [3,5 7]. In Fig. 2.4, the signal start and stop times are clearly identifiable from its time- representation, as is the time variation of the content of the signal. This information cannot be retrieved solely from either the signal instantaneous power s(t) 2 or its spectrum S(f) 2. The ease of interpreting plots such as the plot in Fig. 2.4 makes the concept of a joint time- signal representation attractive. Another advantage of using time- representations of signals is that they reveals whether the signal is monocomponent or multicomponent, a fact that cannot be easily obtained from the signal time domain or its domain representation. Let us consider a case of two different signals having almost identical magnitude spectra (see Fig. 2.5). Both signals contain 3 linear FMs. The differences in the time intervals, bands, and the FM laws that characterise the signals components are not shown clearly in the time domain and the domain, but do appear clearly in the signals joint time- representation Short-time Fourier transform and the spectrogram The time information, in addition to the information, plays an important role when describing the nature and content of nonstationary signals. Attempts to localise the spectrum information in time, can therefore be regarded as a natural, logical and simplest transition from stationary to time-varying spectrum analysis. One such attempt was to multiply a signal in time with a window function w(t) centred about time of interest t, and then take the Fourier transform of the windowed signal. To completely describe signal spectrum variation with time, one would have to repeat this procedure for all time instants. This method is known as the Short-time Fourier transform (STFT) [3, 6], and

31 10 TIME, FREQUENCY, AND TIME-FREQUENCY SIGNAL REPRESENTATIONS MBD MBD Fs=1Hz N=512 Time res=10 MBD Fs=1Hz N=512 Time res=10 MBD Time (secs) Time (secs) Frequency (Hz) Frequency (Hz) (a) Time- representation (Modified B distribution) of the first signal (b) Time- representation (Modified B distribution) of the second signal Figure 2.5: Time- representations of two three-component signals. its magnitude squared as the spectrogram, which is defined as: 2 S(t, f) = s(τ)w(t τ)e j2πfτ dτ (2.3) From Eq. (2.3) it is evident that in order to achieve best time resolution in the spectrogram, the window function w(t) should be as short as possible. This in return causes the resolution to degrade significantly. The trade-off in time and resolution is a consequence of the uncertainty principle which puts the lower limit of 1/4π on the time- resolution product t f [3,4,8]. The time- resolution trade-off is a major drawback when using the spectrogram as a tool for time- analysis The Quadratic Class of TFDs The Wigner-Ville distribution The introduction of the Wigner-Ville distribution (WVD) was fundamental for development of the whole field of TFSAP. The WVD is a result of independent works of Wigner in quantum mechanics, and Ville in probability theory. The Wigner-Ville distribution of real signal s(t) is defined as [3, 6, 9 11]: W z (t, f) = z(t + τ 2 )z (t τ 2 )e j2πfτ dτ (2.4) where z(t) is the analytic associate of s(t) [3, 4]. The analytic signal z(t) is a complex signal that contains only positive frequencies. The real signal s(t) can not have only positive frequencies, as the FT of s(t) has a Hermitian symmetry.