Novel time-frequency analysis techniques for deterministic signals

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(1)NOVEL TIME-FREQUENCY ANALYSIS TECHNIQUES FOR DETERMINISTIC SIGNALS a dissertation submitted to the department of electrical and electronics engineering and the institute of engineering and sciences of bilkent university in partial fulfillment of the requirements for the degree of doctor of philosophy. By Lutye Durak December 2003.

(2) I certify that I have read this thesis and that in my opinion it is fully adequate, in scope and in quality, as a thesis for the degree of Doctor of Philosophy.. Assoc. Prof. Dr. Orhan Arkan (Supervisor). I certify that I have read this thesis and that in my opinion it is fully adequate, in scope and in quality, as a thesis for the degree of Doctor of Philosophy.. Prof. Dr. Haldun Ozakta s. I certify that I have read this thesis and that in my opinion it is fully adequate, in scope and in quality, as a thesis for the degree of Doctor of Philosophy.. Prof. Dr. Enis Cetin. ii.

(3) I certify that I have read this thesis and that in my opinion it is fully adequate, in scope and in quality, as a thesis for the degree of Doctor of Philosophy.. Prof. Dr. U lku Gurler. I certify that I have read this thesis and that in my opinion it is fully adequate, in scope and in quality, as a thesis for the degree of Doctor of Philosophy.. Prof. Dr. Salim Kayhan. Approved for the Institute of Engineering and Sciences:. Prof. Dr. Mehmet Baray Director of Institute of Engineering and Sciences iii.

(4) ABSTRACT NOVEL TIME-FREQUENCY ANALYSIS TECHNIQUES FOR DETERMINISTIC SIGNALS Lutye Durak Ph.D. in Department of Electrical and Electronics Engineering Supervisor: Assoc. Prof. Dr. Orhan Arkan December 2003 In this thesis, novel time-frequency analysis techniques are proposed for deterministic signals. It is shown that among all linear time-frequency representations only the short-time Fourier transformation (STFT) family satis es both the shift-invariance and rotation-invariance properties in both time, frequency and all fractional Fourier domains. The time-frequency domain localization by STFT is then characterized by introducing a novel generalized time-bandwidth product (GTBP) de nition which is an extension of the time-bandwidth product (TBP) on the fractional Fourier domains. For monocomponent signals, it is shown that GTBP provides a rotation-independent measure of compactness. The GTBP optimal STFT, which is a well-localized and high resolution time-frequency representation is introduced and its computationally e cient form is presented. The GTBP optimal STFT provides optimal results for chirp-like signals which can be encountered in a variety of application areas including radar, sonar, seismic and biological signal processing. Also, a linear canonical decomposition of the GTBP optimal STFT analysis is presented to identify its relation to the rotationally-invariant STFT. iv.

(5) Furthermore, for signals with non-convex time-frequency support, an improved GTBP optimal STFT analysis is obtained through chirp multiplication or equivalently shearing in the 2-D time-frequency domain. Finally, we extend these ideas from time-frequency distributions to joint fractional Fourier domain representations.. Keywords: time-frequency analysis, time-frequency distributions, linear timefrequency representations, short-time Fourier transform, fractional Fourier transform, generalized time-bandwidth product, rotation invariance, joint fractional domain signal representation.. v.

(6) O ZET GEREKI_RCI_ SI_NYALLER I_CI_N YENI_ ZAMAN-FREKANS ANALI_Z TEKNI_KLERI_ Lutye Durak Elektrik ve Elektronik Muhendisli gi Doktora Tez Yoneticisi: Do c. Dr. Orhan Arkan Aralk 2003. Bu tezde, gerekirci sinyaller icin yeni zaman-frekans sinyal analiz teknikleri onerilmektedir. Tum do grusal zaman-frekans gosterimleri icinde yalnzca ksa-sureli Fourier donusumu (KSFD) ailesinin zaman, frekans ve tum kesirli Fourier bolgelerinde kaydrma ve dongusel de gismezli gi sa glad g gosterilmistir. Yeni bir genellestirilmis zaman-bant genisli gi carpm (GZBC) tanm ortaya konularak, KSFD icin zaman-frekans duzleminde sinyal desteklerinin yerlerinin snrlans karakterize edilmistir. Zaman-bant genisli gi carpmnn kesirli Fourier bolgelerini kapsayacak sekilde genellestirilmis hali olan GZBC'nin tek bilesenli sinyaller icin bir dongusel-ba gmsz yo gunluk olcutu oldu gu gosterilmistir. Sinyal desteklerininin yerlerini iyi snrlayan ve yuksek cozunurluklu bir zaman-frekans gosterimi olarak GZBC optimal KSFD tantlms ve hzl bir hesaplama algoritmas verilmistir. GZBC optimal KSFD, radar, sonar, sismik ve biyolojik sinyal islemeyi de iceren de gisik uygulama alanlarnda karslaslan corp-tipi sinyaller icin en iyi sonuclar uretir. Ayn zamanda, GZBC optimal KSFD'nin dongusel de gismez KSFD ile ilgisini ortaya koymak icin da glmn bir do grusal do gal bicim ayrsm da vi.

(7) gosterilmistir. Ayrca, dsbukey olmayan zaman-frekans deste gine sahip sinyaller icin yeni bir GZBC optimal KSFD, corp carpm ya da buna esde ger bir islem olarak zaman-frekans duzleminde makaslanma yontemiyle gelistirilmistir. Son olarak, zamanfrekans gosterimlerinde uzerinde durulan kirler bilesik kesirli Fourier bolgelerindeki sinyal gosterimlerine genisletilmistir.. Anahtar Kelimeler: zaman-frekans analizi, zaman-frekans da glmlar, do grusal zaman-frekans gosterimleri, ksa-sureli Fourier donusumu, kesirli Fourier donusumu, genellestirilmis zaman-bant genisli gi carpm, dongusel de gismezlik, bilesik kesirli bolge sinyal gosterimi.. vii.

(8) ACKNOWLEDGMENTS. I would like to express my sincere gratitude to Dr. Orhan Arkan for his supervision, guidance, suggestions, and encouragement throughout the development of this thesis. I would also like to thank the members of my committee, Dr. Enis Cetin, Dr. Haldun Ozakta s, Dr. U lku Gurler and Dr. Salim Kayhan for their valuable comments on the thesis. I wish to thank Curtis Condon, Ken White and Al Feng of the Beckham Institute of the University of Illinois for the bat data and for permissions to use it in this thesis. Special thanks to all my friends for their lasting friendship and support. Finally, I would like to thank my family for the endless love, support and encouragement they have provided throughout my life.. viii.

(9) To my parents, Sevim and Tark Durak for their endless love and support.. ix.

(10) Contents 1 Introduction. 1. 2 Time-Frequency Distributions and the Fractional Fourier Transform 11 2.1 Linear Time-Frequency Representations . . . . . . . . . . . . . . . . . . . 12 2.1.1 De nition of the Short-Time Fourier Transformation . . . . . . . 13 2.1.2 Time and Frequency Domain Shift-Invariance of the STFT . . . . 14 2.1.3 Time and Frequency Localization by STFT . . . . . . . . . . . . 14 2.1.4 Alternative Algorithms on Improving the Time-Frequency Localization of the STFT . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.1.5 Other Linear Time-Frequency Distributions . . . . . . . . . . . . 23 2.2 Quadratic Time-Frequency Representations . . . . . . . . . . . . . . . . 24 2.2.1 Cohen's Class of Time-Frequency Distributions . . . . . . . . . . 25 2.2.2 Relation Between the Cross-WD and the STFT . . . . . . . . . . 27 2.3 The Fractional Fourier Transform . . . . . . . . . . . . . . . . . . . . . . 27 2.3.1 Fractional Fourier Domain Shift-Invariance of the STFT . . . . . 28 x.

(11) 2.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32. 3 Linear Shift-Invariant Time-Frequency Distributions and the Rotation Property 33 3.1 Linear Shift-Invariant Time-Frequency Distributions . . . . . . . . . . . . 34 3.2 A Generalized Shift-Invariance Property of Linear Time-Frequency Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.2.1 The Fractional Fourier Domain Shift Operator and FractionalDomain Shift-Invariance of Distribution Functions . . . . . . . . . 40 3.2.2 Linear Shift-Invariant Systems in Fractional Fourier Domains . . . 42 3.3 Linear Time-Frequency Distributions and the Rotation Property . . . . . 46 3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54. 4 Time-Frequency Domain Localization by STFT and Optimal STFT Implementations 55 4.1 Time-Bandwidth Product Optimal STFT . . . . . . . . . . . . . . . . . . 57 4.2 Generalized Time-Bandwidth Product . . . . . . . . . . . . . . . . . . . 61 4.3 GTBP Optimal STFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.3.1 Determination of Signal Adaptive Parameters in Evaluating GTBP optimal STFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.3.2 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.4 GTBP Optimal STFT with Rotation Property . . . . . . . . . . . . . . . 72 4.5 GTBP Optimal STFT Analysis of Multi-component Signals . . . . . . . 77 xi.

(12) 4.5.1 Analysis of a Recorded Bat Echolocation Signal . . . . . . . . . . 84 4.6 Discussions and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . 87. 5 An Optimal STFT Implementation Through Shearing. 89. 5.1 Improving Time-frequency Localization by Shearing . . . . . . . . . . . . 90 5.2 GTBP of Signals in Sheared Domains . . . . . . . . . . . . . . . . . . . . 95 5.3 SGTBP Optimal STFT and Its Canonical Decomposition . . . . . . . . . 95 5.3.1 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 5.3.2 Linear Canonical Decomposition of the SGTBP Optimal STFT . 101 5.4 Discussions and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . 102. 6 Generalization of Time-Frequency Signal Representations to Joint Fractional Fourier Domains 104 6.1 Distribution of Signal Energy on Joint Fractional Fourier Domains . . . . 106 6.2 Properties of the JFSR . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 6.3 Fast Computation of the JFSR . . . . . . . . . . . . . . . . . . . . . . . 115 6.4 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117. 7 Conclusions. 120. Vita. 139. xii.

(13) List of Figures 2.1 The time-frequency domain support of the STFT representation for x(t) with kernel g(t) can be zoned into a rectangular region of respective timefrequency dimensions of (Tx2 + Tg2 )1=2 and (Bx2 + Bg2 )1=2 . . . . . . . . . . . 16 2.2 The STFT localization depends on the choice of the kernel function. . . . 17 2.3 To illustrate the eect of the choice of the STFT kernel function, the two-component chirp signal shown in Figure 2 is used. . . . . . . . . . . . 18 2.4 STFT with the zeroth-order Hermite-Gaussian kernel shown in (a) still suers from the problem of limited resolution compared to the WD shown in (b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.5 The combined multi-resolution algorithm in 1] is presented using the example multi-component chirp signal expressed in Equation 2.12. . . . . 22 2.6 Fractional Fourier domains interpolate between time and frequency domains with the fractional Fourier order parameter a and the ath0 order fractional Fourier domain is illustrated on a time-frequency plane. . . . . 30. xiii.

(14) 3.1 The STFT satis es the rotation property only with Hermite-Gaussian kernels. To illustrate this, a two-component chirp signal is used. In (a) real part of the signal and in (b) real part of the a = 0:5th order FrFT of the signal are presented. . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.2 The STFT satis es the rotation property only with Hermite-Gaussian kernels. To illustrate this, a two-component chirp signal is used. In (c) the STFT of the signal and in (d) the STFT of its a = 0:5th order FrFT with zeroth-order Hermite-Gaussian kernel are shown. The STFT of the transformed signal is the rotated version of the original STFT shown in (c). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.3 The STFT satis es the rotation property only with Hermite-Gaussian kernels. To illustrate this, a two-component chirp signal is used. However, as shown in (e) and (f), the STFT fails to satisfy the rotation property if its kernel is not a Hermite-Gaussian function. In (e) the STFT of the signal, and in (f) the STFT of the fractionally Fourier transformed signal with the kernel h(t) = e;t2 =3 are shown. . . . . . . . . . . . . . . . . . . 51 3.4 The magnitude of the numerically computed kernel function (t f 0) of the constructed linear time-frequency distribution satisfying the rotation property. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.5 Using the constructed linear distribution kernel satisfying the rotation property, the magnitude of the resultant time-frequency representation (a) of a chirp signal centered at the origin and (b) of a shifted chirp signal are shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54. xiv.

(15) 4.1 The time-frequency domain support of the STFT representation for x(t) with kernel g(t) can be zoned into a rectangular region of respective timefrequency dimensions of (Tx2 + Tg2 )1=2 and (Bx2 + Bg2 )1=2 . . . . . . . . . . . 58 4.2 Time-domain representation of the chirp signal whose time-frequency domain localization by the STFT is compared in Fig. 4.3. . . . . . . . . 59 4.3 Time-frequency domain localization of the STFT by (a) the zeroth-order window function and (b) TBP optimal window function is compared. . . 60 4.4 The bounding rectangle of a chirp signal support on the time-frequency plane may be much larger than the signal support itself. . . . . . . . . . 61 4.5 TBP which is the area of the dashed rectangle is not usually a tight measure for the time-frequency support of signals. . . . . . . . . . . . . . 62 4.6 Even though the rotation operation does not change the area of the support of a signal, the TBP changes. . . . . . . . . . . . . . . . . . . . . 63 4.7 GTBP is the area of the tightest bounding rectangle to the support of the signal in the time-frequency domain, providing a better measure for the time-frequency domain signal supports. . . . . . . . . . . . . . . . . . . . 64 4.8 By searching peaks of the jxa (r)j2 signals computed at various a values, the orientation of the signal supports can be estimated robustly. This is illustrated for a chirp signal embedded in additive Gaussian noise with an SNR level of 5 dB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 4.9 Time-frequency domain localization by the TBP and GTBP optimal STFT of a quadratic FM signal. . . . . . . . . . . . . . . . . . . . . . . . 70. xv.

(16) 4.10 To illustrate the eect of noise, the noisy quadratic FM signal shown in (a) is analyzed by both the TBP optimal STFT and the GTBP optimal STFT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.11 Block diagram of GTBP optimal STFT. . . . . . . . . . . . . . . . . . . 72 4.12 Block diagram of the linear canonical decomposition of GTBP optimal STFT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.13 For a chirp-like signal shown in (a), the FrFT is computed so that the chirp is converted to a sinusoidal as in (c). The corresponding STFTs are shown in (b) and (d), respectively. . . . . . . . . . . . . . . . . . . . . . . 75 4.14 Through appropriate scaling xa (t) is converted to a zeroth-order HermiteGaussian enveloped sinusoidal as illustrated in (e) and its STFT is computed with the Gaussian window as shown in (f). This is followed by 2-D scaling which inverts the scaling on the signal as shown in (g), and. nally the distribution is rotated back to its original orientation removing the FrFT eect as illustrated in (h). . . . . . . . . . . . . . . . . . . . . . 76 4.15 By searching peaks of the jxa (r)j2 signals computed at various a values, the orientation of the signal supports can be estimated robustly. This is illustrated for a two-component signal embedded in additive Gaussian noise with an SNR level of 5 dB. . . . . . . . . . . . . . . . . . . . . . . 79 4.16 GTBP optimal STFTs associated with each signal component of the signal illustrated in Fig. 4.15. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81. xvi.

(17) 4.17 The supports of the signal components can be determined through thresholding the individual distributions by choosing a threshold value as 10% of the maximum value of the corresponding GTBP optimal STFTs illustrated in Fig. 4.16. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.18 Fusion of the individual GTBP optimal STFTs of multi-component signals. 83 4.19 A digitized 2.5 ms duration bat echolocation pulse emitted by a bat, Eptesicus Fuscus 2]. (a) The time-domain signal and (b) its STFT with zeroth-order Hermite-Gaussian kernel. . . . . . . . . . . . . . . . . . . . 85 4.20 The TBP optimal STFT analysis of the bat sound is performed as illustrated in (a). We determine the orientation of the signal components through computing the FrFT of the signal at various orders. The algorithm gives one peak at ;16 , indicating the orientation angle of all components as illustrated in (b). The GTBP optimal STFT is presented in (c) which introduces a distinctive result compared to other STFTs. . . 86 5.1 The real part of the quadratic FM signal x(t). . . . . . . . . . . . . . . . 92 5.2 (a) The WD of the quadratic FM signal x(t). (b) The GTBP optimal STFT x(t). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 5.3 The GTBP as a function of the rate of the multiplying chirp, q. . . . . . 94 5.4 The WD of sx(t) = x(t)e|2:2t2 . . . . . . . . . . . . . . . . . . . . . . . . . 94 5.5 The block diagram of the SGTBP optimal signal representation. . . . . . 96 5.6 The block diagram of the SGTBP optimal STFT. . . . . . . . . . . . . . 97 5.7 The SGTBP optimal STFT of the quadratic FM signal. . . . . . . . . . . 98 xvii.

(18) 5.8 Time-frequency localization comparison of the GTBP optimal STFT illustrated in the above plot and the SGTBP optimal STFT illustrated in the below plot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 5.9 The real part of the quadratic FM signal. . . . . . . . . . . . . . . . . . . 99 5.10 (a) The WD of the noisy quadratic FM signal x(t). (b) The GTBP optimal STFT of the noisy signal. The WD has inner interference terms which drastically clutter the distribution and the GTBP optimal STFT is crossterm free. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 5.11 The GTBP as a function of the rate of the multiplying chirp, q in the noisy signal case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 5.12 Time-frequency localization comparison of the GTBP optimal STFT illustrated in the above plot and the SGTBP optimal STFT illustrated in the below plot for the noisy signal case. . . . . . . . . . . . . . . . . . . . 101 5.13 The block diagram of the linear canonical decomposition of the SGTBP optimal STFT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 6.1 The value of the time-frequency distribution at a point P contributes to the energy densities of the fractional Fourier domains u and v at points. u = u0 and v = v0, respectively. . . . . . . . . . . . . . . . . . . . . . . . 109 6.2 The JFSR of x(t) = rect(t=10)e|2(0:5 t2 +t) at joint fractional Fourier domains.118 6.3 The JFSR of x(t) = e;((t=3)2 +0:3|t3 ) at joint fractional Fourier domains. . 119. xviii.

(19) List of Abbreviations EEG. electroencephalogram. ECG. electrocardiogram. FM. frequency modulated. FFT. fast Fourier transform. FrFT. fractional Fourier transform. GTBP. generalized time-bandwidth product. JFSR. joint fractional domain signal representation. RWT. radon-Wigner transform. SGTBP generalized time-bandwidth product of signals in sheared domains STFT. short-time Fourier transformation. TBP. time-bandwidth product. WD. Wigner distribution. xix.

(20) Chapter 1 Introduction Spectral content of natural signals typically shows a variation as a function of time. A large collection of signals including speech, music, biomedical signals as EEG (electroencephalogram) and ECG (electrocardiogram), biological signals as bat or whale sounds, impulse responses of wireless communication channels, vibrational signals, acoustic sounds from structure failure, and also most of the sonar, radar and seismic signals convey time-varying frequency behavior 3{8]. Hence, time-frequency signal processing has been one of the fundamental research areas covering a rich variety of applications 9,10]. Some of these application areas can be summarized as follows. In biomedical signal processing, ECG waveforms and heart sound 11, 12], vibroartrographic signals emitted by human knee joints 13], EEG signals 14{16] and various other biomedical waveforms 17, 18] are analyzed by time-frequency analysis techniques.. Besides. biomedical signals, many time-varying biological signals such as bat sonar signals 19] and marine mammal sound 20] are processed using time-frequency techniques. In radar signal processing, time-frequency analysis methods are used for target feature 1.

(21) extraction 21,22], ISAR imaging and motion compensation 23], SAR imaging of moving targets 24], target classi cation 25]. Similarly, time-frequency analysis techniques have been extensively used in sonar signal processing 26]. Time-varying signal processing in wireless communications is mainly focused on investigating time-frequency receivers, time-varying channel characterization 27{32], channel diversity 33{37], time-varying modulation schemes 38], suppressing nonstationary interference as chirp jammers 39,40]. In most of these applications, time-frequency signal processing includes signal detection 41{43], ltering 44], characterization, decomposition, estimation 27, 45{47], classi cation 48, 49], and signal synthesis 50, 51]. In literature, detectors designed by time-frequency analysis techniques are employed for a wide variety of applications including speech processing 52], industrial applications 53] and communications 37]. In all of these applications, time-frequency distributions can be used to analyze both deterministic and random signals. In statistical time-frequency signal processing, a random time-frequency distribution is de ned as the expected value of the distribution of the random signal. However, as ergodicity is not satis ed by non-stationary random signals, the expected value may be approximated by local time-frequency averaging. Alternatively, ensemble averaging can be used if multiple realizations are available. In 54,55], \Wigner-Ville spectrum" is developed through ensemble averaging the Wigner distribution which is one of the widely used time-frequency distributions. Also in 56{58] time-frequency spectral analysis and estimation for non-stationary random processes are investigated and speci c applications are addressed. In this thesis, we focus on obtaining readable time-frequency representations and consider only the deterministic signals, however the introduced ideas and developed techniques can be extended to random signals as well. 2.

(22) In time-frequency signal processing, much of the eort has been invested on obtaining clear and sharp time-frequency description of signals. Because there exist inherent problems such as the uncertainty relations between time and frequency domains which result poor signal energy localization on the time-frequency plane. Uncertainty principle dictates that the spreads of the energy of a signal in time and frequency domains, cannot be simultaneously small. The time-width and the bandwidth of a signal x(t) which are represented by Tx and Bx, respectively, satisfy the following relationship1:. Tx Bx =. R. 1=2 R. (t ; t )2 jx(t)j2dt. kxk. (f ; f )2jX (f )j2df. 2. 1=2. 41. (1.1). where jx(t)j2 and jX (f )j2 are the energies of signals in time and frequency domains and. t and f are expressed as t = f =. Z Z. . t jx(t)j dt =kxk2 . (1.2). f jX (f )j df =kxk2 :. (1.3). 2. . 2. Signals satisfying the minimum uncertainty product are all shifted and scaled Gaussian functions. Another problem on the readability of time-frequency representations is the unavoidable spurious structures which are called as the \cross-terms" of quadratic or higher order time-frequency representations. Such undesirable terms may even lead to unreadable time-frequency distributions. The most widely used quadratic distribution is the Wigner Distribution (WD) with nice theoretical properties. The WD of a signal. x(t) is represented by Wx(t f ) and de ned as 59]: Wx(t f ) =. Z. x(t + =2)x (t ; =2)e;|2 d. (1.4). WD describes the energy density of signals in the joint time-frequency plane. WD is always real and shift-invariant in time and frequency. Time or frequency domain 1 All integrals are from ;1 to +1 unless otherwise stated. 3.

(23) projections of the Wigner distribution give the energy distribution of the signal with respect to time or frequency. However, the bilinear form of WD gives rise to interaction both between signal components and single component signals with non-convex timefrequency supports. The interference structure of quadratic distributions has been extensively analyzed in 60]. The absence of undesirable cross-terms 61,62] and computational simplicity 63{65] are the major factors in the wide-spread use of the linear distributions in practice. With the advance of faster processors, the e ciency of these techniques have become less important. However, the ability of representing time-frequency content of signals free of cross-terms is still the major advantage of the linear time-frequency distributions over the quadratic or higher-order time-frequency distributions. However linear time-frequency distributions usually suer from the poor localization problem. Besides linear, quadratic and higher-order time-frequency distributions, an alternative approach is developed by linearly decomposing a signal using a redundant set of waveforms which forms a dictionary. After linear expansion of a signal, the timefrequency distribution of it is obtained by summing up the individual WDs of these waveforms. The set of waveforms, which are also called as atoms are chosen from a dictionary which contains either shifted and scaled Gaussian atoms 66], chirplets 67] or windowed exponential frequency modulated functions 68]. This way, interference between the individual signal components may be prevented, however these methods provide satisfactory results only if the signal components resemble to the atoms in the dictionary. Also suboptimal matching pursuit algorithms used in these techniques can be computationally ine cient if the number of atoms in the dictionary increases. As a remedy for poor localization and cross-term problems of the time-frequency. 4.

(24) representations either signal independent or signal dependent time-frequency representations have been developed 1,60,69{77]. Among these distributions, research on timefrequency domain characterization of signals has been focused on the variants of shorttime Fourier transform (STFT) 1,3,4,41,62,76,78] and Wigner distribution 43,79]. Among the linear time-frequency representations, most commonly used distributions are the STFT and the wavelet transform. STFT of a signal x(t) is de ned as: STFTx(t f ) =. Z. x(t0 ) h(t0 ; t) e;|2ft dt0 0. (1.5). where h(t0) is a low-pass, unit-energy window function centered around the origin and the spectrum of the windowed signal x(t0 )h (t0 ; t) is computed for all time. STFT is a shift-invariant distribution in time and frequency and complex-valued in general. Although the wavelet transform is originally introduced as a time-scale representation, by using the analysis scale as f0 =f where f0 is the center frequency of the analysis window, the time-frequency version of the wavelet transform is de ned as: WTx(t f ) =. Z. x(t0 ). p. (f=f0. ) h. . . f (t0 ; t) dt0 f0. (1.6). where the analyzing wavelet h(t) is a bandpass function centered around the frequency. f0 62]. Both the STFT and the wavelet transform can be interpreted as ltering the signal x(t0 ) with a bandpass lter, therefore they both suer from poor time-frequency localization. For the STFT case, the bandwidth of the analysis window is independent of the analysis frequency and time-frequency resolution is the same for each frequency value. Whereas for the wavelet transform case, the bandwidth of the lter is proportional to the analysis frequency. This way, the quality factor of the lter which is de ned by. Q = fB0 5. (1.7).

(25) where f0 is the center frequency and B is the bandwidth of the window function, is kept constant. As the frequency increases, such a \constant-Q" analysis of the wavelet transform results in poorer frequency resolution but higher time resolution 80]. Also, the wavelet transform preserves time shifts and scalings, however does not preserve frequency shifts. Therefore, the wavelet transform diers from the STFT based on both localization and shift-invariance properties. In this thesis, we mostly restrict our attention to shift-invariant representations in both time and frequency domains. Starting from the most general form of the linear time-frequency distributions, we show that STFT is the only linear distribution that is magnitude-wise shift-invariant in both time and frequency domains. We extend this idea to any fractional Fourier domains and show that STFT is the only linear distribution that is magnitude-wise shift-invariant in all fractional Fourier domains. The fractional Fourier domains are the set of all domains interpolating between time and frequency with an order parameter a and the fractional Fourier domains corresponding to order parameters a = 0 and a = 1 are the time and the frequency domains respectively. The fractional Fourier transformation (FrFT) with order parameter a transforms a signal into the ath fractional domain. Also, in the literature uncertainty relationship in the fractional Fourier domains can be found in 67,81{83]. Besides the shift-invariance property, we investigate the rotational eect of the FrFT on the STFT. The rotation property of time-frequency distributions is de ned as follows. A time-frequency distribution Dx(t f ) of a signal x(t) satis es the rotation property if. jDxa (t f )j = jR;fDx(t f )gj. (1.8). where is equal to a=2 and xa is the ath order fractional Fourier transformation (FrFT) of x(t) and R denotes for the 2-D counter-clockwise rotation operator. We also show that STFT with Hermite-Gaussian kernel functions is the only linear shift-invariant 6.

(26) time-frequency distribution satisfying the rotation property. Among the Hermite-Gaussian function family, since it has the minimum timebandwidth product (TBP), the Gaussian function is the most commonly used kernel function 77]. However, STFT with the Gaussian kernel still suers from the problem of limited resolution. In the literature, to overcome the inherent trade-o between the time and the frequency localization of the STFT, several alternatives have been investigated. In 1], using two kernel functions of dierent supports, a wide-band and a narrow-band spectrogram are obtained. In order to preserve the localization characteristics of both, a combined spectrogram is formed by computing the geometric mean of the corresponding STFT magnitudes. Whereas in 76], the STFT is evaluated by using a kernel function with an adaptive width in order to analyze the transient response of radar targets. In 75], a kernel matching algorithm is developed by locally adapting the Gaussian kernel functions to the analyzed signal. Although these investigations provide signi cant improvements in the time-frequency localization of signal components, in the presence of chirp-like signals, they still provide descriptions whose localization properties depends on the chirp rate of the components. Recently, 4] introduced an improved instantaneous frequency estimation technique using an adaptive STFT where the kernel functions are chosen from a set of functions through adaptation rules and compute the STFT with varying kernel functions at each time instance. Also, apart from the analysis of deterministic signals, there has been studies where time-varying spectra of random processes are investigated 84]. In this thesis, we characterize the time-frequency domain localization by STFT and investigate the eect of the STFT kernel on the obtained time-frequency representation 7.

(27) of signals. Using the fractional Fourier domains, we introduce the generalized timebandwidth product (GTBP) de nition to provide a rotation-invariant measure of signal support in the time-frequency domain. The GTBP is de ned as the minimum of the TBPs among all fractional Fourier domains. Then we obtain the optimal STFT kernel which provides the most compact representation considering the GTBP of a signal component. The proposed time-frequency analysis is shown to be equivalent to an ordinary STFT analysis conducted in a scaled fractional Fourier transform domain. The obtained GTBP optimal STFT representation yields optimally compact time-frequency supports for chirp-like signals on the STFT plane. In general, the GTBP optimal STFT representation does not satisfy the rotation property. However, as shown in detail, there exists a linear canonical decomposition of the GTBP optimal STFT that provides the link between the GTBP optimal STFT and the rotation-invariance property. As the GTBP is bounded by the TBP, the optimal STFT based on the GTBP provides more localized time-frequency representations compared to any scaled Gaussian kernels. Furthermore, since the support of chirp-like signals or signals with convex time-frequency support quite ts the bounding rectangle of the GTBP on the time-frequency plane, the GTBP optimal STFT analysis generates the best STFT results for such signals. However, it is shown that for signals with a non-convex time-frequency support, the GTBP of signals can be decreased through chirp-multiplication in time-domain. As chirp multiplication corresponds to shearing in the Wigner distribution case, we de ne a novel time-frequency support criterion as: GTBP of signals in sheared domains (SGTBP). The SGTBP is bounded by the GTBP and similar to the GTBP optimal STFT analysis, an SGTBP optimal STFT analysis is introduced and analyzed. In the literature, similar to the joint time-frequency representations, alternative joint densities of dierent domains depending on the application interest are also investigated 85{87]. These domains include time-scale, scale-hyperbolic time and 8.

(28) warped time-frequency or scale. Recently in 88], a joint fractional signal representation is derived by associating Hermitian fractional operators to FrFT variables that constitute the joint distribution. In this thesis, instead of using the operator theory and detailed mathematical derivations, we made use of the basic idea of time-frequency distributions that is: Time-frequency representations designate the energy content of signals in the time-frequency plane. This idea is generalized to a joint distribution de ned by two fractional Fourier domain variables, namely the joint fractional signal representation (JFSR). We show some properties of the JFSR including its relation to quadratic time-frequency representations. Actually, this representation corresponds to shearing operation followed by a frequency domain scaling. In this thesis we also present a fast computation algorithm of the radial slices of the distribution using the method developed in 89] with simulation examples. The thesis is organized as follows. In Chapter 2, a general overview on the linear and quadratic time-frequency distributions is given indicating the shift-invariance and timefrequency localization properties of the STFT in detail. Various alternative algorithms on the improvement of STFT resolution is discussed and STFT is compared to two other linear time-frequency distributions namely the Gabor expansion and the wavelet transform. Cohen's class of quadratic time-frequency representations are analyzed with a special emphasis given to the Wigner distribution. Since the introduced time-frequency representations and novel concepts in this thesis require the use of the fractional Fourier transformation (FrFT), we de ne and give the computation algorithm of the FrFT. Finally, the shift-invariance properties of the STFT is revisited and investigated on arbitrary fractional Fourier domains. In Chapter 3, we investigate shift-invariant linear time-frequency distributions in time and frequency domains, and show that STFT is the only linear time-frequency representation that satis es the magnitude-wise shift invariance property. We generalize 9.

(29) the shift-invariance property to fractional Fourier domains and investigate the generalized magnitude-wise shift-invariant linear time-frequency distributions in arbitrary fractional Fourier domains. We conclude that STFT satis es the shift-invariance property in all of these domains. Finally, the rotation property of the STFT is examined. In Chapter 4, the optimal STFT which minimizes the usual TBP of the windowed signal is determined. As an extension to the TBP de nition, the GTBP is de ned and the GTBP optimal STFT is introduced by giving the details of the new approach on simulated examples. A canonical decomposition of the GTBP optimal STFT analysis is also presented. This decomposition provided the introduction of a novel \natural domain" idea for the input signals. The extension of the algorithm to multi-component. signals is proposed and results are illustrated on real data which is a bat echolocation signal. In Chapter 5, the chirp-multiplication operator as a special case of the general class of linear canonical transforms (LCTs) is examined and the eect of chirp-multiplication on the GTBP of a signal is investigated. It is shown that the chirp multiplication may decrease the GTBP of the transformed signal, providing a new variant of the TBP: The GTBP of signals in sheared domains (SGTBP). The SGTBP is de ned and the SGTBP optimal STFT is derived and simulated. In Chapter 6, the JFSR is rede ned providing an equivalent alternative derivation given in 88]. The properties of the JFSR is examined and proved by providing a fast computation algorithm and applying the algorithm on simulation examples. Finally, the thesis is concluded in Chapter 7 with future research directions.. 10.

(30) Chapter 2 Time-Frequency Distributions and the Fractional Fourier Transform Time-frequency distributions are designed to characterize the frequency content of signals as a function of time and are usually applied to analyze, process or synthesize signals with time-varying spectral content. Many widely used time-frequency distributions are shift-invariant in both time and frequency, including the STFT which is a member of linear time-frequency representations and Cohen's class of quadratic distributions. Time and frequency shiftinvariance of these distributions implies a constant resolution time-frequency analysis on the entire time-frequency plane. As an alternative to the constant-bandwidth analysis of such distributions, wavelet transform and a ne time-frequency representations are proposed where analysis bandwidth is proportional to the analysis frequency resulting in a \constant-Q" analysis 80]. In this chapter, we de ne and explore linear and quadratic representations and concentrate on the shift-invariant time-frequency representations. In Section 2.1, starting 11.

(31) with the general de nition of linear time-frequency representations, STFT and its shiftinvariance and time-frequency localization properties are examined. Various alternative algorithms on the improvement of STFT resolution is discussed and STFT is also compared to two other linear time-frequency distributions namely the Gabor expansion and the wavelet transform. In Section 2.2, Cohen's class of quadratic time-frequency representations are analyzed with a special emphasis given to the Wigner distribution. Since the introduced time-frequency representations and novel concepts in this thesis require the use of the fractional Fourier transformation (FrFT), we de ne and give the computation algorithm of the FrFT in Section 2.3. Finally, the shift-invariance properties of the STFT is revisited and investigated on arbitrary fractional Fourier domains.. 2.1 Linear Time-Frequency Representations The general kernel based form of a linear time-frequency distribution Dx(t f ) is given by:. Dx(t f ) =. Z. (t f t0 ) x(t0 )dt0. (2.1). where (t f t0 ) is the kernel of the distribution 90]. For any linear combination of signals as x(t) =. P. i ci xi (t), all linear distributions are free of cross-terms and satisfy the. superposition principle. Dx(t f ) =. X. i. ciDxi (t f ). (2.2). which is a desirable property for multicomponent signals. Among the linear timefrequency distributions, STFT and wavelet transform are the most widely used ones. In this section, STFT is de ned and its time-frequency domain shift-invariance and localization properties are analyzed by comparing various alternative algorithms 12.

(32) proposed in the literature. Finally, two other linear time-frequency distributions are introduced.. 2.1.1 Denition of the Short-Time Fourier Transformation STFT can be explicitly de ned as: STFTx(t f ) =. Z. x(t0 ) h(t0 ; t) e;|2ft dt0. (2.3). 0. where h(t0) is a low-pass, unit-energy window function centered around the origin and the spectrum of the windowed signal x(t0 )h(t0 ; t) is computed for all time. STFT is a complex-valued distribution. The basic idea of the STFT is to Fourier analyze the time-varying signals during appropriate short time intervals and obtain the entire time-frequency behavior of the signal by concatenating these consecutive analyses. As an alternative de nition, STFT can be expressed in the Fourier domain as STFTx. (t f ) = e;|2ft. Z. X (f 0) H (f 0 ; f ) e|2f t df 0. (2.4). 0. where X (f ) and H (f ) are the Fourier transforms of the signal x(t) and the window function h(t). The discrete STFT can be de ned and computed as: STFTx(nT mF ) =. Z. ' T. x(t0 )h (t0 ; nT )e;|2mFt dt0. (2.5). 0. X. n. x(n0 T )h(n0T ; nT )e;|2mFn T 0. (2.6). 0. where n, m and n0 are integers, T and F are the sampling intervals of time and frequency. The STFT computation can be e ciently implemented by using FFT techniques 62].. 13.

(33) 2.1.2 Time and Frequency Domain Shift-Invariance of the STFT STFT is magnitude-wise shift-invariant in time and frequency. The STFT of a timeshifted signal xs(t) = x(t ; ts) is: STFTxs (t f ) = =. Z. x(t0 ; ts) h (t0 ; t) e;|2ft dt0 0. Z. x(t00) h(t00 ; (t ; ts)) e;|2f (t +ts) dt00 00. = e;|2fts STFTx(t ; ts f ) :. (2.7). The magnitude of the STFT of the time-shifted signal xs (t) equals to the magnitude of the shifted STFT of the original signal x(t) by ts in time domain. Similarly, STFT preserves frequency shifts and the STFT of the modulated signal. xm (t) = x(t) e;|2fm t is: STFTxm (t f ) =. Z. x(t0 ) h (t0 ; t) e;|2(f ;fm )t dt0 0. = STFTx(t f ; fm ) : Shift-invariance in time and frequency provides easy interpretation of a timefrequency distribution and also enables a uniform analysis on the whole time-frequency plane.. 2.1.3 Time and Frequency Localization by STFT The choice of the window function or equivalently the STFT kernel determines the timefrequency signal localization properties of the distribution. Time resolution improves by choosing more localized h(t) in the STFT form as expressed by Equation (2.3), whereas the frequency resolution increases as H (f ) is chosen more localized in frequency as 14.

(34) expressed in Equation (2.4). Hence there exists a trade-o between time and frequency resolution for all window functions. In the de nition of STFT, as the original signal is multiplied by a window function and a short-duration signal is extracted for each time t, the time-frequency support should be determined using the windowed signal x(t0 )g(t0 ; t). Let the time-width and bandwidth of a function x(t) be de ned as: Z. Tx =. Z. Bx =. (t ; t ) jx(t)j dt 2. 2. 1=2. (f ; f ) jX (f )j df 2. 2. =kxk2 . 1=2. (2.8). =kxk2. (2.9). where Z. t =. Z. f =. . t jx(t)j dt =kxk2 2 . (2.10). f jX (f )j df =kxk22. (2.11). 2. . 2. for a Fourier transform pair x(t) and X (f ). Then, for an STFT kernel g(t), the ;. time-width and the bandwidth of the windowed signal is obtained as Tx2 + Tg2 ;. Bx2 + Bg2. 1=2. 1=2. and. , respectively 3]. Thus, on the STFT plane, the time-frequency domain. support of the representation for x(t) can be zoned into a rectangular region of respective ;. time-frequency dimensions of Tx2 + Tg2. 1=2. ;. and Bx2 + Bg2. 1=2. as shown in Fig. 2.1 .. Therefore it is essential to optimize the kernel parameters Tg and Bg to reach welllocalized STFT pictures. To compare the choice of windows on the time-frequency localization of signals, various STFTs of a multi-component signal. x(t) =. 2 X. k=1. p1 e(t;tk ) =18 ej(ak (t;tk ) +2bk (t;tk )) 3. 2. 2. (2.12). where ak = arctan(k ) with 1 = =12 and 2 = =6, b1 = ;2, b2 = 2, and t1 = t2 = 1 are shown on Fig. 2.2 and Fig. 2.3. Each STFT is associated with the window function 15.

(35) f. (Bx2+Bg2)1/2/2. t - (Tx. 2. (Tx2+Tg2)1/2/2. +Tg2)1/2/2. - (Bx. 2. +Bg2)1/2/2. Figure 2.1: The time-frequency domain support of the STFT representation for x(t) with kernel g(t) can be zoned into a rectangular region of respective time-frequency dimensions of (Tx2 + Tg2)1=2 and (Bx2 + Bg2 )1=2 . put on the left. As the window function is less localized in time, ner frequency resolution is achieved as illustrated in Fig. 2.2-(d), Fig. 2.3-(b) and Fig. 2.3-(d), however what we pay for it is the time localization. Thus, there is an inherent trade-o between time and frequency localization in STFT while choosing the window function. Among the Hermite-Gaussian function family, since it has the minimum timebandwidth product (TBP), the zeroth-order Hermite-Gaussian function is the most commonly used kernel function. However, STFT with the zeroth-order Hermite-Gaussian kernel still suers from the problem of limited resolution compared to the WD as illustrated in Fig. 2.4 . In the literature, to obtain well-localized STFTs, either signal dependent or signal independent algorithms are developed and some of these approaches will be summarized and discussed in the following section.. 16.

(36) (b) 2. 1. 1 X(f). x(t). (a) 2. 0. −1. 0. −1. −2 −8. −6. −4. −2. 0 time. 2. 4. 6. −2 −8. 8. −6. −4. −2 0 2 frequency. (c). 4. 6. (d). 0.5. 8 6. 0.4. 1.5. 4 frequency. h1(t). 0.3 0.2. 2 1. 0. −2. 0.1. 0.5. −4 0 −0.1 −8. 8. −6 −6. −4. −2. 0 time. 2. 4. 6. −8 −8 −6 −4 −2. 8. 0 2 time. 4. 6. 8. Figure 2.2: The STFT localization depends on the choice of the kernel function. To illustrate this, a two-component chirp signal is used. In (a) real part of the signal and in (b) real part 2of Fourier transform of the signal are presented. In (c) the kernel function h1 (t) = e;4t and in (d) the corresponding STFT are shown.. 17.

(37) (a). (a). 0.5. 8. 2.5. 6. 0.4. 2. 4 frequency. h2(t). 0.3 0.2. 2. 1.5. 0 1. −2. 0.1. −4 0 −0.1 −8. 0.5. −6 −6. −4. −2. 0 time. 2. 4. 6. −8 −8 −6 −4 −2. 8. (c). 0 2 time. 4. 6. 8. (d). 0.5. 8 6. 0.4. 2.5. 4 frequency. x(t). 0.3 0.2. 2. 2. 1.5. 0. −2. 0.1. 1. −4 0 −0.1 −8. 0.5. −6 −6. −4. −2. 0 time. 2. 4. 6. −8 −8 −6 −4 −2. 8. 0 2 time. 4. 6. 8. Figure 2.3: To illustrate the eect of the choice of the STFT kernel function, the twocomponent chirp signal shown in Figure 2 is used. In (a) the kernel function h2 (t) = 2 ; t e and in2 (b) the corresponding STFT are shown, whereas in (c) the kernel function h3 (t) = e;t =4 and in (d) the corresponding STFT are presented. As the window is less localized in time, the STFT of the signal is more localized in frequency, however less localized in time.. 18.

(38) (a). (b). 8. 8. 2.5. 2. 6. 6. 1.5. 2. 2. 4 frequency. frequency. 4. 1.5. 0 1. −2 −4. 0.5. 0. 0. −1.5. −6. −8 −8 −6 −4 −2. −8 −8 −6 −4 −2. 6. −1. −4. 0.5. 4. −0.5. −2. −6 0 2 time. 1. 2. 8. −2. 0 2 time. 4. 6. 8. Figure 2.4: STFT with the zeroth-order Hermite-Gaussian kernel shown in (a) still suers from the problem of limited resolution compared to the WD shown in (b).. 2.1.4 Alternative Algorithms on Improving the TimeFrequency Localization of the STFT To overcome the inherent trade-o between the time and the frequency localization of the STFT, several alternatives have been investigated. In this section, four of them will be summarized by indicating their advantages and disadvantages. In the STFT computations in 1], Cheung and Lim used two kernel functions of dierent supports to obtain a wide-band and a narrow-band STFT simultaneously. They used Hamming windows of two dierent widths, speci cally, one window has a width eight times longer than the other. Then, they combined the STFT magnitudes through computing the geometric mean of the individual STFTs by viewing them as images. In other words, for each (n k) coordinate on the time-frequency plane, the resultant STFTr (n k) is computed by STFTr (n k) = jSTFTwb(n k)j jSTFTnb (n k)j]1=2 19. (2.13).

(39) where STFTwb(n k) and STFTnb(n k) indicates the wideband and narrowband STFTs, respectively. The result of the combined multi-resolution algorithm is presented in Fig. 2.5 using the example multi-component chirp signal expressed in Equation (2.12). A narrowband STFT and a wideband STFT illustrated in Fig. 2.5 -(a) and (b) are merged and the resultant distribution is obtained as in Fig. 2.5-(c). Although the resultant distribution has an improved time-frequency localization, such a nonlinear combination of two STFTs results in loss of information. Moreover, the localization limitations of the individual STFTs still persists in the resultant distribution. Alternatively in 76], an adaptive STFT is evaluated by using a kernel function with an adaptive width in order to analyze the transient response of radar targets. The adaptation of the STFT kernel is performed as follows. The STFT window width is determined by making use of the time-domain signal and changed as the signal changes its structure. The algorithm starts with a window of width Tw . The left edge of the window is kept xed and the right edge is expanded up to a specular re!ection observed in the time-domain signal. The successive analysis windows of the STFT is determined by this procedure repeatedly through dividing the signal into consecutive parts up to the nal specular re!ection encountered in the signal. As this method only changes the window width throughout the signal, the algorithm provides partial improvement in the frequency localization and the inherent time-frequency localization problem still remains. Although these investigations in 1] and 76] provide improvements in the timefrequency localization of signal components, in the presence of chirp-like signals, they still provide descriptions whose localization properties depend on the chirp rate of the components. 20.

(40) In 75], Jones and Boashash proposed a kernel matching algorithm which is developed by locally adapting the kernel functions to the analyzed signal by introducing \generalized instantaneous parameters" as functions of both time and frequency and associated these parameters to time-frequency distributions. They employed WD-derived measures to determine the generalized instantaneous parameters and nally produced a window-matched spectrogram. The window-matching is achieved by using the ideas of matched ltering which gives sharp time-frequency representation of signals. However, the computational cost of the full algorithm is too high, requiring O(KN 3) computations where N is the signal length and K is typically less than 100. Also, WD-based parameter estimates are noise-sensitive and therefore may not lead robust results while dealing with noisy signals. Recently, 4] introduced an improved instantaneous frequency estimation technique using an adaptive STFT where the kernel functions are chosen from a set of functions and compute the STFT with a time varying kernel function. As the mismatch of the window function and the signal is the main reason of the time-frequency localization problem of the STFT, Kwok and Jones used dierent windows at dierent time instances and the window function is determined among a set of windows by using two adaptation criteria: One based on maximum correlation and the other on a concentration measurement (minimum entropy). The success of the method mostly depends on the choice of window functions since the window set should depend on the nature of the signals to be analyzed. Among some alternatives, they have chosen a set of windows with dierent chirp rates. The overall computational complexity of their instantaneous frequency estimation algorithm based on the adaptive STFT is on the average 2N 2 +8N log N +40N FLOPS.. 21.

(41) (a) 8. 1.6. 6. 1.4 1.2. 2. 1. 0. 0.8. −2. 0.6. −4. 0.4. −6. 0.2. frequency. 4. −8 −8 −6 −4 −2. 0 2 time (b). 4. 6. 8. 8 6. 2.5. frequency. 4 2. 2. 1.5. 0. −2. 1. −4 0.5. −6 −8 −8 −6 −4 −2. 0 2 time. 4. 6. 8. (c) 8 2. 6. frequency. 4 1.5. 2 0. 1. −2 −4. 0.5. −6 −8 −8 −6 −4 −2. 0 2 time. 4. 6. 8. Figure 2.5: The combined multi-resolution algorithm in 1] is presented using the example multi-component chirp signal expressed in Equation 2.12. A narrowband STFT illustrated in (a) and wideband STFT illustrated in (b) are merged and the resultant distribution is obtained as in (c). 22.

(42) 2.1.5 Other Linear Time-Frequency Distributions Two other linear time-frequency representations widely used in various applications are the Gabor expansion coe cient functions 91,92] and the wavelet transform 93]. The Gabor expansion coe cient functions, Gx(n k) can be implicitly de ned by 92, 94]. x(t) =. XX. n. k. Gx(n k)g(t ; nT )ej2kFt. (2.14). where the signal x(t) is a linear combination of time and frequency shifted versions of an elementary function g(t). The basis signals g(t ; nT )ej2kFt must be chosen such that they are well-localized with respect to time and frequency, then the expansion coe cients Gx(n k) describe the energy content of the signal around (nT kF ). The Gabor coe cients can be chosen as the STFT samples, STFTx(nT kF ), however they are not in general uniquely de ned 62]. On the other hand, wavelet transform is originally introduced as a time-scale representation. By using the analysis scale as f0=f where f0 is the center frequency of the analysis window, the time-frequency version of the wavelet transform can be de ned as: WTx(t f ) =. Z. p x(t0 ) (f=f0) h( ff (t0 ; t)) dt0 0. (2.15). where the analyzing wavelet h(t) is a bandpass function centered around the frequency. f0 62]. Similar to the STFT, the wavelet transform can be interpreted as ltering the signal. x(t0 ) with a bandpass lter. However, the bandwidth of the window function changes proportionally with the analysis frequency. Therefore, as the frequency decreases, the wavelet transform provides higher frequency resolution but poorer time resolution. 23.

(43) Also, another discriminating feature of this transform is the shift-invariance property in time and frequency. The wavelet transform is not shift-invariant in frequency. Although the wavelet transform of a time-shifted signal xs(t) = x(t ; ts) equals to the shifted wavelet transform of the original signal x(t) by ts in time WTxs (t f ) = WTx(t ; ts f ) the wavelet transform does not preserve frequency shifts. In this thesis, our main consideration is on the shift-invariant distributions in time and frequency.. 2.2 Quadratic Time-Frequency Representations The main motivation of the quadratic representations is to construct a joint signal representation that describes the energy density of a signal both in time and frequency. Because of the energetic interpretation, quadratic time-frequency representations and especially the Wigner distribution (WD) has a central role in the time-frequency analysis. In general for a time-frequency distribution, it is desirable to have an energy density function D(t f ) so that the total energy in t + "t f + "f ] interval be equal to. D(t f ) "t "f . However the uncertainty relationship between time and frequency domains prohibits the determination of such an energy density at every point on the time-frequency plane as "t, "f approaches to 0. However, as a looser condition, it is possible to satisfy the marginal densities as: Z Z. D(t f )df = jx(t)j2 . (2.16). D(t f )dt = jX (f )j2 :. (2.17). A direct consequence of Equation (2.16) and Equation (2.17) is obtained by integrating 24.

(44) D(t f ) over the entire time-frequency plane which gives the signal energy: Z Z. D(t f )dt df = jjxjj22:. (2.18). The most widely used quadratic representation is WD which satis es Equation (2.16)Equation (2.18) and de ned as 59]:. Wx(t f ) =. Z. x(t + =2)x (t ; =2)e;|2 d :. (2.19). WD is a quadratic distribution and carries nice theoretical properties presented in detail in 3, 59, 78, 95, 96]. However since it is quadratic in signal, the WD of the sum of two signals is not equal to the sum of the individual WDs resulting in undesirable cross-terms and even there exists negative values over the time-frequency plane. WD is a member of a relatively broad class of energetic time-frequency distributions, the Cohen's class, which will be presented in the following section.. 2.2.1 Cohen's Class of Time-Frequency Distributions The class of all quadratic time-frequency representations satisfying the shift-invariance property both in time and frequency domain is named as the Cohen's class. In this class, the time-frequency distributions of a signal x(t) is represented by. TFx(t f ) =. Z Z Z. ( )x(u + =2)x (u ; =2) ej2(u;t;f ) dud d. (2.20). where the function ( ) is called the kernel of the distribution 3, 78]. Each member of the Cohen's class is associated with a unique kernel and the most commonly used distribution of this class, the WD is obtained by choosing the kernel ( ) = 1. The WD de nition expressed in Equation (2.19) can also be generalized to de ne a cross-Wigner distribution of two signals x(t) and y(t) as. Wxy (t f ) =. Z. x(t + =2)y(t ; =2)e;|2f d : 25. (2.21).

(45) WD is a real valued distribution. It is shift-invariant in both time and frequency domains and produces well-localized time-frequency supports for signal components, however the uncertainty relationship prohibits the distribution to be an exact energy density. Additionally, practical applications of the WD is often restricted by the occurrence of undesirable interference terms which may disturb the actual supports of the signals and clutters the time-frequency representations. All other members of the Cohen's class can be interpreted as 2-D ltered WDs and thus the kernel can also be taken as a smoothing function. In the literature, there has been extensive research on developing signal-dependent kernel functions which helps time-frequency localization of signal components 69,70,97]. The 2-D inverse Fourier transform of the WD is called as the ambiguity function and expressed as Z Z. Ax( ) =. Z. =. Wx(t f ) ej2(t+f ). x(t + =2)x (t ; =2)e|2t dt. (2.22) (2.23). which is commonly used in radar and sonar signal processing 5,8]. Similar to the de nition of the cross-WD, the cross-ambiguity function of two signals. x(t) and y(t) can be expressed as Axy (t f ) =. Z. x(t + =2)y(t ; =2)e|2t dt :. (2.24). In the following section, we will derive the relationship between the cross-WD of a signal x(t) and a kernel function h(t) and the STFT of the signal.. 26.

(46) 2.2.2 Relation Between the Cross-WD and the STFT The STFT of a signal x(t) using a kernel function h(t) is STFTx(t f ) =. Z. x(t0 ) h(t0 ; t) e;|2ft dt0 : 0. (2.25). By changing the variable of integration t0 to + 2t , Equation (2.25) can be re-expressed as Z. x( + 2t ) h ( ; 2t ) e;|2f d 1 t t ; |ft = 2 e Wxh 2 2. STFTx(t f ) =. e;|ft. (2.26) (2.27). yielding the relationship between the STFT and the cross-WD of the signal x(t) and the kernel function h(t). Also, the spectrogram of a signal which is de ned as jSTFTx(t f )j2, is one of the members of the Cohen's class of quadratic distributions and related to the WD of the signal, Wx(t f ) and the WD of the kernel function Wh(t f ) as 62]:. jSTFTx(t f )j = 2. Z Z. Wh(t0 ; t f 0 ; f ) Wx(t0 f 0)dt0df 0 :. (2.28). The smoothing function of the distribution in this case is the WD of the analysis window of the STFT, Wh(t f ), providing a substantial cross-term suppression but poorer timefrequency localization.. 2.3 The Fractional Fourier Transform The fractional Fourier transform is a generalization of the ordinary Fourier transform where the fractional order parameter a corresponds to the ath power of the Fourier transform operator, F . The ath order fractional Fourier transformation (FrFT) of x(t) 27.

(47) is given by 98,99]:. xa. (t) fF axg(t) =. Z. Ba (t t0)x(t0 )dt0 a 2 < 0 < jaj < 2. where the transformation kernel Ba(t t0 ) is: ;j ( sgn(sin )=4+=2) 2 ej(t2 cot ;2tt csc +t cot ) : Ba (t t0 ) = e j sin j1=2 0. 0. (2.29). (2.30). The rst-order FrFT is the ordinary Fourier transform and the zeroth-order FrFT is the function itself. The ath order FrFT interpolates between a function x(t) and its Fourier transform X (f ). The de nition in Equation (2.29) is easily extended to outside the interval of the order parameter a of ;2 2] since FrFT is additive in index, i.e.. F a F a = F a +a and F n is the identity operator for any integer n. 1. 2. 1. 2. The continuous FrFT given by Fig. 2.29 can be approximated closely from discrete samples of x(t) by using the fast computation algorithm proposed in 100] in O(NlogN ) operations. The steps of the algorithm is given in Table 2.1. In the following section, the shift-invariance property of the STFT is revisited and this property is investigated in arbitrary fractional Fourier domains.. 2.3.1 Fractional Fourier Domain Shift-Invariance of the STFT Fractional Fourier domains are the set of all domains which interpolate between time and frequency domains with the fractional Fourier order parameter a. When the order parameter a is 0, the fractional Fourier domain corresponds to time domain, and when the order parameter a is 1, the fractional Fourier domain corresponds to frequency domain. In Fig. 2.6, ath0 order fractional Fourier domain is illustrated on a time-frequency plane. In this section, we de ne the fractional Fourier domain shift operator, state the fractional domain shift-invariance property of any time-frequency distribution functions, and prove that STFT is also shift-invariant in fractional Fourier domains. 28.

(48) Object of the algorithm: Given x(n=dx), ;N=2 n N=2 ; 1, to compute xa (m=2dx), ;N m N ; 1, underpthe assumption that the WD of x(t) is con ned into a circle with diameter dx N . Steps of the algorithm: Interpolate the input samples by 2: x(n=dx) ! x(n=2dx) a0 := (a + 2 mod 4) ; 2 % After the modulo operation, a0 2 ;2 2) 0 % The cases of a 2 0:5 1:5] and a0 2 f;2 ; 0:5)U (;0:5 2)g have to be treated separately. if ja0j 2 0:5 1:5] then a00 := a0. else. a00 := (a0 + 1 mod 4) ; 2. end if 00. % After the modulo operation, a00 2 (0:5 1:5). := 2 a00 := cot 00

(49) := csc 00 (sin )=4+j=2) A := exp(;jsgn j sin j1=2 % Compute the following sequences: 1 (=dx2 ; =N )m2 | c1m] := e 4 p for ;N m N ; 1 | (m=2 N )2 for ;2N m 2N ; 1 c2m] := e 2 dx (=N ; =dx2 )m2 | c3m] := e 4N for ;N m N ; 1 gm] := c1m]x(m=2dx) for ;N m N ; 1 A ha (m=2dx) := 2dx c3m](c2 g)m] for ;N m N ; 1 %In the last step FFT is used to compute the convolution in O(N log N ) !ops. if jaj 2 0:5 1:5] then xa (m=2dx) := ha (m=2dx) 0. else. 00. % Compute samples of the ordinary FT using FFT. xa (m=2dx) := fF 1ha g(m=2dx). end if .. 00. Table 2.1: The Fast Fractional Fourier Transform Algorithm proposed in 100]. 29.

(50) f th. a 0 order fractional Fourier domain. φo. t. Figure 2.6: Fractional Fourier domains interpolate between time and frequency domains with the fractional Fourier order parameter a and the ath0 order fractional Fourier domain is illustrated on a time-frequency plane.. De nition 2.1 The fractional Fourier domain shift operator S is dened as: S = F ;a S0 F a. (2.31). where = a 2 and F a is the fractional Fourier transformation (FrFT) operator and S0. is the translation operator which can be dened as:. fS0 xg(t) = (t ; ) x(t) = x(t ; ). (2.32). as in 101].. When S is applied to a signal x(t), we obtain. fS xg(t) = F ;a S0 fF axg(t) = F ;a f (t ; ) xa (t)g :. (2.33). Using the FrFT de nition it can be shown that 101],. F a fx(t ; )g = ej 2 sin cos e;j2t sin xa (t ; cos ). (2.34). therefore,. fS xg(t) = e;|(. 2 sin. cos ;2t sin ). 30. x(t ; cos ) :. (2.35).

(51) In this thesis, we de ne the fractional domain shift-invariance of a time-frequency distribution function, Dx(t f ) as follows.. De nition 2.2 If the distribution function Dx(t f ) satises jDxsa (t f )j = jDx(t ; cos f ; sin )j 8 x(t) . (2.36). where xsa (t) is the fractionally shifted signal x(t) at an arbitrary ath order fractional Fourier domain by an amount of , then Dx (t f ) is shift-invariant in all fractional Fourier domains.. To show the magnitude-wise shift-invariance of STFT in any arbitrary fractional Fourier domain, we start with the STFT of the fractionally shifted signal xsa (t) with an arbitrary kernel function h(t): STFTxsa (t f ) =. Z. xsa ( )h( ; t)e;|2f d :. (2.37). where xsa (t) is. xsa (t) = fS xg(t) = fF ;aS F a xg(t) = e;| 2 sin cos e|2 sin x(t ; cos ) :. (2.38). Then STFT in Equation (2.37) becomes: STFTxsa. (t f ) = e;| 2 sin cos . Z. x( ; cos )h( ; t)e;|2

(52) f ; sin ]d :. (2.39). By changing variable t0 = ; cos , the right hand side of Equation (2.39) can be rewritten as: STFTxsa (t f ) =. e;| 2 sin cos . Z. x(t0 )h(t0 ; t ; cos ])e;|2

(53) f ; sin ]d (2.40). = e;| 2 sin cos STFTx(t ; cos f ; sin ) 31. (2.41).

(54) Thus, the magnitude of STFTxsa (t f ) equals to,. jSTFTxsa (t f )j = jSTFTx(t ; cos f ; sin )j. (2.42). indicating that STFT of the fractional-domain shifted form of a signal is equal to the shifted form of the STFT of the signal itself along the fractional Fourier domain making an angle of = a=2 with respect to the time axis. The equality in Equation (2.42) proves that STFT satis es the magnitude-wise shiftinvariance property in all fractional Fourier domains.. 2.4 Conclusions In this chapter an overview of linear and quadratic time-frequency representations are given indicating shift-invariance and localization properties of the STFT in greater depth. It is shown that, STFT is shift-invariant not only in time and frequency domains, but also in all fractional Fourier domains. Also, time-frequency localization of the STFT is investigated by illustrating the eect of the kernel functions on the resolution of the existing time-frequency signal components. As the STFT kernel is more localized in time, the resultant distribution is less localized in frequency and vice versa. To overcome the localization problem, several researchers proposed alternative algorithms four of which are explained, compared and discussed in this chapter.. 32.

(55) Chapter 3 Linear Shift-Invariant Time-Frequency Distributions and the Rotation Property Among its many important properties, STFT has a fundamental property that simpli es the interpretation of the resultant distribution: Magnitude-wise shift invariance in both time and frequency. In this chapter, within the class of all linear time-frequency representations, it is proven that STFT is the only linear distribution that satis es the magnitude-wise shift invariance property in both time and frequency domains. Shift invariance of STFT is also extended to fractional Fourier domains and it is also shown that STFT is the only linear distribution that is magnitude-wise shift invariant in any arbitrary fractional Fourier domains. Time-frequency domain rotation property within the general class of linear distributions is also investigated. This lesser known property, which is satis ed by the WD, is de ned as follows: A time-frequency distribution satis es the rotation property if the distribution of an arbitrary signal and the distribution of its ath order fractional 33.

(56) Fourier transformation are. a 2. radians rotated version of each other 102, 103]. In this. chapter, we will prove that STFT satis es the rotation property only if the kernel function is one of the Hermite-Gaussian functions. Thus, we reach to the conclusion that linear time-frequency distributions, which satisfy both the rotation property and the magnitude-wise shift invariance property is the STFT with Hermite-Gaussian kernels. The outline of the chapter is as follows. In Section 3.1, we investigate shift-invariant linear time-frequency distributions in time and frequency domains. In Section 3.2, we generalize the shift-invariance property to fractional Fourier domains and investigate the generalized magnitude-wise shift-invariant linear time-frequency distributions in arbitrary fractional Fourier domains. We reach to the conclusion that STFT satis es the shift-invariance property in all these domains. Then, the rotation property of STFT is examined in Section 3.3. Finally, in Section 3.4 conclusions are drawn.. 3.1 Linear Shift-Invariant Time-Frequency Distributions Time-frequency distributions are designed to characterize the time-frequency content of signals. Since time or frequency shifts do not change the time-frequency content of a signal, except relocating it correspondingly, it is important that time-frequency representations satisfy the magnitude-wise shift invariance property. A precise statement of this property is given as follows: A time frequency representation Dx(t f ) is magnitude-wise shift invariant, if for any x(t), xs (t) = x(t ; ts) e|2fst. jDxs (t f )j = jDx(t ; ts f ; fs)j 8 ts fs :. (3.1). The shift-invariance property of the Cohen class of representations have been investigated by Mustard in 104, 105] and he concluded that WD is the only representation which 34.

(57) satis es the shift-invariance property in time, frequency and any fractional Fourier domains. In this section, we investigate the magnitude-wise shift invariance property within the class of linear time-frequency representations. The magnitude-wise shift invariance of linear time-frequency distributions can be characterized fully as follows. The general kernel based form of a linear time-frequency distribution Dx(t f ) is given by:. Dx(t f ) =. Z. K (t f t0 ) x(t0 )dt0. (3.2). where K (t f t0 ) is the kernel of the distribution 90]. To proceed with the time-frequency distributions, we will state and prove a more general result on linear systems.. Theorem 3.1 If a linear system T satises magnitude-wise shift invariance in time, then there exist an h(t) and b(t) such that the output of T for any arbitrary input x(t) can be written as:. T fx(t)g = e|b(t) h(t) x(t)]:. (3.3). Proof: Since T is a linear system, by using the Riesz theorem, T can be represented as: Z. T fx(t)g = K (t t0 ) x(t0 ) dt0. (3.4). where K (t t0 ) is the kernel of the linear system. If T satis es magnitude-wise shift invariance in time, the outputs to impulses (t) and (t ; ts), y(t) and ys(t), respectively, should satisfy:. jys(t)j = jy(t ; ts)j. (3.5). jK (t ; ts 0)j = jK (t ts)j 8 t ts :. (3.6). which implies that. 35.

(58) In general the kernel function can be decomposed as: K (t t0 ) = %(t t0 ) e|(t t ) where 0. %(t t0 ) and (t t0) are the magnitude and phase functions, respectively. The condition in (3.6) requires that %(t t0) = %(t ; t0 ), therefore the kernel function can be decomposed as:. K (t t0 ) = %(t ; t0 ) e|(t t ) :. (3.7). 0. Next, it will be shown that the phase function satis es:. (t t0 ) = ;#(t ; t0 ) + b(t) :. (3.8). To prove (3.8), the input can be chosen as a linear combination of two weighted impulses, x(t) = 1 (t)+ 2 (t ; ), then the output is y(t) = 1K (t 0)+ 2K (t ). For the shifted input xs(t) = x(t ; ts ), the output becomes ys(t) = 1K (t ts)+ 2 K (t ts + ). The magnitude-wise shift invariance implies jys(t)j = jy(t ; ts)j. Thus, the kernel should satisfy:. j 1K (t ts) + 2K (t ts + )j = j 1K (t ; ts 0) + 2K (t ; ts )j. (3.9). for all t ts 1 2. Using the de nition in (3.7), (3.9) can be re-expressed as:. j 1%(t ; ts) e|(t ts ) + 2%(t ; ts ; ) e|(t ts + ) j = j 1%(t ; ts) e|(t;ts 0) + 2%(t ; ts ; ) e|(t;ts ) j : (3.10) Assuming that %(t) is not identically zero, (3.10) can only be satis ed if:. (t ts) ; (t ; ts 0) = (t ts + ) ; (t ; ts ) 8 t ts :. (3.11). After re-arranging the terms in (3.11), we obtain the following condition:. (t ts + ) ; (t ts) = (t ; ts ) ; (t ; ts 0) 8 t ts :. (3.12). It can be shown that if (t t0) satis es (3.12), then (t t0 ) = @t@ (t t0) exists. Thus, in 0. the limit approaching 0, (3.12) implies that:. (t ts) = (t ; ts 0) = (t ; ts) 8 t ts: 36. (3.13).

(59) Therefore, (t t0) satis es the following partial dierential equation:. @ (t t0) = (t ; t0) @t0. (3.14). (t t0) = ;#(t ; t0) + b(t). (3.15). which is solved by:. where (t) = dtd #(t) and b(t) is an arbitrary phase function. Thus, the kernel has the following form:. K (t t0 ) = %(t ; t0 ) e;|(t;t ) e|b(t) : 0. (3.16). Hence, the input-output relationship of the linear system can be written as:. y(t) = =. Z Z. K (t t0) x(t0 ) dt0 %(t ; t0) e;|(t;t ) e|b(t) x(t0 ) dt0 0. = e|b(t) h(t) x(t)]. (3.17). where h(t) = %(t) e;|(t) , can be called as the linear shift-invariant part of the response and b(t) can be called as the modulation part of the response. We can extend the result of Theorem 3.1 to any 2-D distribution functions Dx(t f ) as in the following Proposition 3.1:. Proposition 3.1 The magnitude-wise shift-invariance in one dimension of a 2-D linear distribution function Dx(t f ) yields:. Dx. (t f ) = e|b(t f ). Z. (t ; t0 f ) x(t0 )dt0. (3.18). where Dx(t f ) is the 2-D linear distribution function dened in Equation (3.2) and t is the shift-invariant variable.. 37.

(60) Proof: The kernel function of Dx(t f ) in Equation (3.2) can be decomposed as K (t f t0 ) = K(t f t0 )ej (t f t ) : 0. (3.19). 0. As stated in Theorem 3.1, shift invariance condition in a 1-D linear system produces an output of:. y(t) = ejb. Z. h(t ; t0) x(t0 ) dt0. (3.20). where h(t) is the 1-D kernel function as h(t) = %(t) e;|(t) . In Equation (3.19), there exists an additional independent variable f , besides the shift-invariant variable t and the variable of integration t0. Then, Dx(t f ) can be re-expressed as. Dx(t f ) = =. e|b(t). Z. K(t ; t0 f ) ej (t;t f ) x(t0)dt0 0. e|b(t f ). Z. 0. (t ; t0 f ) x(t0 )dt0. (3.21). where (t f ) = K(t f ) ej (t f ) . 0. Since the magnitude of time-frequency distributions are related to the energy distribution of the signals in the time-frequency plane, e|b(t f ) will be ignored in the rest of the derivations. Next, by making use of the relation between parameters time, t and frequency, f , we will show that STFT is the only shift-invariant distribution in time and frequency.. Theorem 3.2 Among the linear distributions, STFT is the only shift-invariant distribution in both time and frequency.. Proof: The shift-invariance in time requires the form stated in Equation (3.18), then the implications of magnitude-wise shift invariance in frequency can be investigated in the Fourier domain as:. Dx(t f ) = =. Z. (t ; t0 f ). e|2ft. Z. Z. X (f 0) e|2f t df 0 dt0 0 0. ;(f f 0) e;|2(f ;f )t X (f 0) df 0 0. 38. (3.22).