T. Claasen

Bio: T. Claasen is an academic researcher from Philips. The author has contributed to research in topics: Digital filter & Adaptive filter. The author has an hindex of 17, co-authored 26 publications receiving 2659 citations.

The wigner distribution - a tool for time-frequency signal analysis

Abstract: In this second part of the paper the Wigner distribution is adapted to the case of discrete-time signals It is shown that most of the properties of this time-frequency signal representation carry over directly to the discrete-time case, but someothers cause problems These problems are associated with the fact that in general the Wigner distribution of a discrete-time signal contains aliasing contributions It is indicated that these aliasing components will not be present if the signal is either oversampled by a factor of at least two, or is analytic 1 Introduetion In part I of this paper 1) the Wigner distribution (WD) of continuous-time signals was discussed, and it was shown that this function has some very interesting properties The determination of this distribution function requires, like the spectrum, an integral of the Fourier type to be evaluated Ideally this requires the signal to be known for all time, but in practice windowing techniques can be used to relax this requirement The effects of windowing on the WD were discussed in part 1 In general two different approaches can be distinguished to compute these Fourier-type integrals The first is by means of analogue signal processing, and recently optical signal processing methods have been proposed for determining suitable approximations to the WD 2) The second approach is based on digital signal processing This opens the way to apply computationally efficient methods for evaluating the discrete Fourier transform, but requires the concept of the Wigner distribution to be transferred to the case of discretetime signals This is the aim of this part of the paper As can be expected, the WD for discrete-time signals shows much similarity with that for continuous-time signals, but in some respects it has characteristic differences To emphasize the similarities and point out the differences we will try to follow as closely as possible the same lines as in part I, and give comments only on those results that differ from that of the continuous-time counterpart 276 Phillps Journalof Research Vol35 Nos4/5 1980 Philips Journalof Research Vol35 Nos4/5 1980 277 The Wigner distribution Also the numbering of the equations is made such that corresponding equations have the same number This has the consequence that sometimes equation numbers are not successive if equations have been deleted, and that equations that do not occur in part I have a special numbering If reference is made to an equation in part I the equation is given the prefix I All sections, except sec 7, have the same topic and heading as in part I Section 7, which in part I deals with the WD of band-limited signals, now deals with the WD of finite duration sequences Equations in this section do not correspond in general with an equation of part I 2 The Wigner distribution for discrete-time signals 21 Preliminaries In this paper Weconsider in general complex valued, discrete-time signals f(n), feC, n e Z for which the (Fourier) spectrum is defined by 3) 00 F(e) = (fJdf) (e) = L f(n) e -jnlJ (2la) n=-c:o The inverse transform is given by 11 f(n) = (fJd-l F) (n) = _1_ J F(e) ejnlJ de, 21t (2lb) -11 Inner products are defined for the signals and spectra by 00 (I, g) = L f(n) g*(n) (22a) n=-oo and 11 (F, G) = _1 J F(e) G*(e) de 21t (22b) -11 respectively Norms and Parseval's relation are then the same as in eqs (123) and (124) respectively The following operators will be used The shift operator for the signals (9{f) (n) = f(n k), kEZ (25a) and for the spectrum (9'cF) (e) = F(e C), CE R, (complex) modulation in the time domain T A C M Claasen and W F G Mecklenbrauker ( At!!,!) (n) = f (n) ein!!, and in the frequency domain c;«; F) (0) = F(O) einD ceR (26a) neZ, (26b) differentiation of the spectrum 1 (fi))F) (0) = -;-F 1(0), J (27) multiplication by the running variable (Rlf) (n) = nf(n), (28) time reversal (f!JlJ)(n) = f( -nl· (29) There are several different ways to link analogue and digital signals and systems, and hence a variety of ways to define a discrete-time version of the Wigner distribution What one would like with such a definition is (1) to obtain a simple concept; (2) to retain as many as possible of the properties of the WD of continuoustime signals; (3) to find a simple relation between the discrete-time and continuous-time WD's for discrete-time signals that are obtained by sampling of analogue signals The definition which, in our opinion, best matches these requirements is the one suggested by eq (1710) 22 Definition of the Wigner distribution The cross-Wigner distribution of two discrete-time signals f(n) and g(n) is defined by 00 u-j,g(n, 0) = 2 L e-i2kDf(n + k)g*(n kl (210) k=-oo The autoWigner distribution of a signal is then given by 00 u-j(n, 0) = Wj,f(n, 0) = 2 L e-i2kDf(n+k)f*(n kl (211) k=-oo Both functions will be called a Wigner distribution (WD) Aiming at obtaining a relation similar to (1213) the WD for the spectra must be defined by 278 Philips Journalor Research Vol35 Nos4/5 1980 The Wigner distribution so that 11 wF,G(e, n)= ~ J et; F(e + C) G*(e C) dc (212)

The wigner distribution - a tool for time-frequency signal analysis part ii: discrete-time signals

Abstract: In this second part of the paper the Wigner distribution is adapted to the case of discrete-time signals. It is shown that most of the properties of this time-frequency signal representation carry over directly to the discrete-time case, but some.others cause problems. These problems are associated with the fact that in general the Wigner distribution of a discrete-time signal contains aliasing contributions. It is indicated that these aliasing components will not be present if the signal is either oversampled by a factor of at least two, or is analytic. 1. Introduetion In part I of this paper 1) the Wigner distribution (WD) of continuous-time signals was discussed, and it was shown that this function has some very interesting properties. The determination of this distribution function requires, like the spectrum, an integral of the Fourier type to be evaluated. Ideally this requires the signal to be known for all. time, but in practice windowing techniques can be used to relax this requirement. The effects of windowing on the WD were discussed in part 1. In general two different approaches can be distinguished to compute these Fourier-type integrals. The first is by means of analogue signal processing, and recently optical signal processing methods have been proposed for determining suitable approximations to the WD 2). The second approach is based on digital signal processing. This opens the way to apply computationally efficient methods for evaluating the discrete Fourier transform, but requires the concept of the Wigner distribution to be transferred to the case of discretetime signals. This is the aim of this part of the paper. As can be expected, the WD for discrete-time signals shows much similarity with that for continuous-time signals, but in some respects it has characteristic differences.

Effects of quantization and overflow in recursive digital filters

Abstract: A classification is given of the various possible nonlinear effects that can occur in recursive digital filters due to signal quantization and adder overflow. The effects include limit cycles, overflow oscillations, and quantization noise. A review is given of recent literature on this subject. Alternative methods of avoiding some of these nonlinear phenomena are discussed.

Comparison of the convergence of two algorithms for adaptive FIR digital filters

Abstract: The convergence properties of two different algorithms for the updating of the coefficients of an adaptive FIR digital filter are investigated and compared with one another. These algorithms are the stochastic iteration algorithm and the sign algorithm. In this latter algorithm a one-bit gradient estimation is used which makes its implementation very simple. The convergence is characterized by the residual echo variance after convergence, and a parameter that indicates the speed of the convergence. It is shown that the convergence of the sign algorithm can always be assured but is much slower than that of the stochastic iteration algorithm if the same variance of the residual echo is to be obtained.

On stationary linear time-varying systems

Abstract: A comprehensive review of representations of linear timevarying systems is given, both in the time and in frequency domains. Subsequently a definition is given of a stationary deterministic signal. Based on this definition the notion of stationary systems is introduced. These systems have the useful property that the spectral relation between input and output has a simpler form than the corresponding relation for arbitrary time-varying systems. It is shown that causality puts rather severe constraints on the frequency mappings that can be realized by stationary linear systems. An extension of the theory of linear time-varying systems to the case of discrete-time and hybrid systems (analog input, digital output, or vice versa) is discussed. Examples of stationary systems are given, such as a decimator, a periodic sampler, and a bilinear A/D converter. Also, a recently proposed generalized sampling method is analyzed by means of the concepts discussed in this paper.

The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis

Abstract: A new method for analysing nonlinear and non-stationary data has been developed. The key part of the method is the empirical mode decomposition method with which any complicated data set can be dec...

A wavelet tour of signal processing

Abstract: Introduction to a Transient World. Fourier Kingdom. Discrete Revolution. Time Meets Frequency. Frames. Wavelet Zoom. Wavelet Bases. Wavelet Packet and Local Cosine Bases. An Approximation Tour. Estimations are Approximations. Transform Coding. Appendix A: Mathematical Complements. Appendix B: Software Toolboxes.

The wavelet transform, time-frequency localization and signal analysis

Abstract: Two different procedures for effecting a frequency analysis of a time-dependent signal locally in time are studied. The first procedure is the short-time or windowed Fourier transform; the second is the wavelet transform, in which high-frequency components are studied with sharper time resolution than low-frequency components. The similarities and the differences between these two methods are discussed. For both schemes a detailed study is made of the reconstruction method and its stability as a function of the chosen time-frequency density. Finally, the notion of time-frequency localization is made precise, within this framework, by two localization theorems. >

Time-frequency distributions-a review

Abstract: A review and tutorial of the fundamental ideas and methods of joint time-frequency distributions is presented. The objective of the field is to describe how the spectral content of a signal changes in time and to develop the physical and mathematical ideas needed to understand what a time-varying spectrum is. The basic gal is to devise a distribution that represents the energy or intensity of a signal simultaneously in time and frequency. Although the basic notions have been developing steadily over the last 40 years, there have recently been significant advances. This review is intended to be understandable to the nonspecialist with emphasis on the diversity of concepts and motivations that have gone into the formation of the field. >

Wavelets and signal processing

Abstract: A simple, nonrigorous, synthetic view of wavelet theory is presented for both review and tutorial purposes. The discussion includes nonstationary signal analysis, scale versus frequency, wavelet analysis and synthesis, scalograms, wavelet frames and orthonormal bases, the discrete-time case, and applications of wavelets in signal processing. The main definitions and properties of wavelet transforms are covered, and connections among the various fields where results have been developed are shown. >