THE WIGNER DISTRIBUTION - A TOOL FOR TIME-FREQUENCY SIGNAL ANALYSIS

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. 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 Vol.35 Nos.4/5 1980 Philips Journalof Research Vol.35 Nos.4/5 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 2.1 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. (2.l.a) n=-c:o The inverse transform is given by 11 f(n) = (fJd-l F) (n) = _1_ J F(e) ejnlJ de, 21t (2.l.b) -11 Inner products are defined for the signals and spectra by 00 (I, g) = L f(n) g*(n) (2.2.a) n=-oo and 11 (F, G) = _1 J F(e) G*(e) de 21t (2.2.b) -11 respectively. Norms and Parseval's relation are then the same as in eqs (1.2.3) and (1.2.4) respectively. The following operators will be used. The shift operator for the signals (9{f) (n) = f(n k), kEZ (2.5.a) 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 (2.6.a) neZ, (2.6.b) differentiation of the spectrum 1 (fi))F) (0) = -;-F 1(0), J (2.7) multiplication by the running variable (Rlf) (n) = nf(n), (2.8) time reversal (f! JlJ)(n) = f( -nl· (2.9) 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. (1.7.10). 2.2. 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. (2.10) 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. (2.11) k=-oo Both functions will be called a Wigner distribution (WD). Aiming at obtaining a relation similar to (1.2.13) the WD for the spectra must be defined by 278 Philips Journalor Research Vol.35 Nos.4/5 1980 The Wigner distribution so that 11 wF,G(e, n)= ~ J et; F(e + C) G*(e C) dc (2.12)