THE WIGNER DISTRIBUTION - A TOOL FOR TIME-FREQUENCY SIGNAL ANALYSIS
THE WIGNER DISTRIBUTION - A TOOL FOR TIME-FREQUENCY SIGNAL ANALYSIS
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发表时间：
1980
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通讯作者：
T. Claasen;W. Mecklenbrauker
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作者：
T. Claasen;W. Mecklenbrauker
在本文的第二部分中，维格纳分布适用于离散时间信号的情况。结果表明，这种时频信号表示的大部分特性直接适用于离散时间情况，但有些特性会引起问题。这些问题与以下事实相关：一般来说，离散时间信号的维格纳分布包含混叠贡献。表明如果信号被过采样至少两倍或者是解析信号，则这些混叠分量将不会出现。 1. 简介 在本文的第一部分中，1）讨论了连续时间信号的维格纳分布（WD），并表明该函数具有一些非常有趣的性质。与频谱一样，该分布函数的确定需要评估傅里叶类型的积分。理想情况下，这需要所有人都知道该信号。时间，但实际上可以使用窗口技术来放宽这一要求。第 1 部分讨论了加窗对 WD 的影响。一般来说，可以区分两种不同的方法来计算这些傅立叶型积分。第一种是通过模拟信号处理，并且最近已经提出了光信号处理方法来确定WD 2)的合适近似值。第二种方法基于数字信号处理。这为应用计算有效的方法来评估离散傅里叶变换开辟了道路，但需要将维格纳分布的概念转移到离散时间信号的情况。这就是本文这一部分的目的。可以预见，离散时间信号的 WD 与连续时间信号的 WD 非常相似，但在某些方面存在特征差异。为了强调相似性并指出差异，我们将尝试尽可能遵循与第一部分相同的路线，并且仅对那些与连续时间对应物的结果不同的结果进行评论。 276 Philps Journalof Research Vol.35 Nos.4/5 1980 Philips Journalof Research Vol.35 Nos.4/5 1980 277 维格纳分布 方程的编号也使得相应的方程具有相同的编号。这样做的结果是，如果方程被删除，有时方程编号不连续，并且在第 I 部分中未出现的方程具有特殊编号。如果参考第 I 部分中的方程，则该方程将被赋予前缀 I。除第 1 节外的所有部分。第 7 节与第一部分具有相同的主题和标题。第 7 节在第一部分中处理带限信号的 WD，现在处理有限持续时间序列的 WD。本节中的方程一般不对应于第 I 部分的方程。 2. 离散时间信号的维格纳分布 2.1 预备知识 在本文中，我们考虑一般复数值的离散时间信号 f(n), feC, n e Z，其（傅里叶）频谱由 3) 00 F(e) = (fJdf) (e) = L f(n) e -jnlJ 定义。 (2.l.a) n=-c:o 逆变换由下式给出 11 f(n) = (fJd-l F) (n) = _1_ J F(e) ejnlJ de, 21t (2.l.b) -11 信号和频谱的内积由 00 (I, g) = L f(n) g*(n) (2.2.a) n=-oo 和 11 定义(F, G) = _1 J F(e) G*(e) de 21t (2.2.b) -11 分别。范数和帕塞瓦尔的关系分别与等式（1.2.3）和（1.2.4）中的相同。将使用以下运算符。信号 (9{f) (n) = f(n k)、kEZ (2.5.a) 和频谱 (.9'cF) (e) = F(e C)、CE R、时域 T. A. C. M. Claasen 和 W. F. G. Mecklenbrauker ( .At!!,!) (n) = f (n) ein!! 以及频域中的（复）调制的移位算子。 c;«; F) (0) = F(O) einD ceR (2.6.a) neZ, (2.6.b) 频谱微分 1 (fi))F) (0) = -;-F 1(0), J (2.7) 乘以运行变量 (Rlf) (n) = nf(n), (2.8) 时间反转 (f!JlJ)(n) = f( -nl· (2.9) 有有几个不同的 连接模拟和的方法。数字信号和系统，因此有多种方法来定义维格纳分布的离散时间版本。通过这样的定义，人们希望得到的是（1）获得一个简单的概念； (2)尽可能多地保留连续时间信号的WD特性； (3) 找到通过模拟采样获得的离散时间信号的离散时间和连续时间 WD 之间的简单关系 信号。我们认为最符合这些要求的定义是等式1 所建议的定义。 (1.7.10)。 2.2.维格纳分布的定义 两个离散时间信号 f(n) 和 g(n) 的交维格纳分布定义为 00 u-j,g(n, 0) = 2 L e-i2kDf(n + k)g*(n kl。 (2.10) k=-oo a 的 autoWigner 分布 信号由 00 u-j(n, 0) = Wj,f(n, 0) = 2 L e-i2kDf(n+k)f*(n kl. (2.11) k=-oo 两个函数称为维格纳分布 (WD)。为了获得类似于 (1.2.13) 的关系，光谱的 WD 必须由 278 Philips Journalor Research Vol.35 定义1980年第4/5号 维格纳分布使得 11 wF,G(e, n)= ~ J et; F(e + C) G*(e C) dc (2.12)
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)