Multiple window spectrogram and time-frequency distributions ...

MULTIPLE WINDOW SPECTROGRAM AND TIME-FREQUENCY DISTRIBUTIONS Gordon Frazer and Boualem Boashash Signal Processing Research Centre, QUT, P. 0. Box 2434, Brisbane 4001, AUSTRALIA

ABSTRACT

where zn in this case extends for { n : n E 2 ) ;

We extend the spectrum estimation method of Thomson to non-stationary signals by formulating a multiple window spectrogram. The traditional spectrogram can be represented as a member of Cohen’s class of time-frequency distributions (TFDs), where the smoothing kernel is the Wigner distribution of the signal temporal window. We show the unusual shape of the Cohen’s class smoothing kernels corresponding to the Thomson method multiple windows. These are a class of smoothing kernels not hitherto used in time-frequency (t-f) analysis. Examples of the mnltiple window spectrogram applied to a noisy dual linear FM test signal and t o actual underwater acoustic d a t a demonstrate the merit of the method.

1.

INTRODUCTION

Non-stationary signals are traditionally analysed using the spectrogram [l],or more recently any member of Cohen’s see equaclass of time-frequency distributions ( T F D tion 8), e.g. the Wigner-Ville distribution ( a V 6 ) , the reduced interference distribution (RID), the cone-kernel distribution (CKD) [2]. Typical non-stationary signals include active and passive sonar signals, radar return signals, biological acoustic signals, mechanical vibration signals and many others. When the signal to be analysed is modelled as a random process then issues of bias and statistical stability must be considered in the design of the estimator of the T F D [2-41. Smoothing kernels used in Cohen’s class T F D s are typically designed to achieve a variety of localisation, positivity, cross-term removal and other goals in time and frequency (t-f). Consequently, such kernel designs do not address the bias and variance control issues central to statistical estimator design. We present an extension for non-stationary signals of a recent spectrum analysis procedure, due to Thomson [571, and demonstrate that time-varying spectrum estimates with low bias and variance can be computed as compared with the spectrogram and existing TFDs.

and the power spectral density, the quantity of interest, is obtained from;

1: _-

v N ( f

- u)dX(y)

Where the Dirichlet kernel, V N ,dependent on N , is defined o as;

One particular solution to equation 3, Thomson’s multiple window method computes an approximate least squares solution using a local eigen-expansion, where the eigenfunctions of V N are discrete prolate spheroidal sequences (DPSS), or Slepian sequences (see figure 1). Thomson’s method comprises several iterations of the following three steps, with the initial =

2.1. Multiple W i n d o w Spectrum E s t i m a t i o n Consider N samples of a signal, 20,X I , . . . , Z N - ~ from a stationary stochastic process z, with discrete Fourier transzne-12ffnf and inverse discrete Fourier form y(f) =

Cfzt

J-++ y(f)e”““fdf

Y(f) =

(3)

2. BACKGROUND In this section we briefly review Thomson’s multiple window spectrum estimator and provide some relevant background on TFDs.

transfoim zn =

E{ldX(f)121 = S W f

(2)

The problem addressed by Thomson [5-7 is; given the data z,, compute the second moment of d X ( f ) . I t is possible t o relate the random orthogonal increment process, dX(f), in the Cramer representation to the discrete time Fourier transform of the data, ~ ( ,fthrough a Fredholm integral equation of the first kind [8{;

s(f) i(S,(f)+S~(f));

1. Given N observations, choose an analysis bandwidth W . Typically W = j$; if W is too small then the estimate is statistically unstable, while if W is too large then the resolution is poor. Choice of W dictates the number of windows, K (approx. 2 N W - 3). 2. Compute fi’ functions; N-1

(5)

n=0

where the vk(n;N, W) are Slepian sequences (DPSS) [91. 3. Compute the power spectrum estimate S(f)iteratively from;

n = 0,. . . , N - 1. A

stationary stochastic process has a Cramer representation,

zntJk(n;N,W)e-””’”

yk(f) =

i k ( f ) =dk(f )Yk(f) 1.

(6)

=[xi s ( . f ) / ( A k s ( f ) + (1 - xk)bZ)lYk(f)

IV-293 0-7803-1775-0/94 $3.00 Q 1994 IEEE

and IC-1

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