Study of statistical properties of random signals in multirate filter banks

摘要

Abstract: Previous works on subband-related signal processing were mainly dedicated to the applications of subband systems and to the formulation of multirate filter banks. Only very limited results can be found that treat statistical properties of random signals inside a multirate filter bank. In this paper, such a theoretical study is performed from the statistical viewpoint. Our main interest lies in how a multirate structure interacts with a random signal. The key statistical properties examined are stationarity, autocorrelation, cross-correlation, power spectral density, and spectral flatness measure. Exact explicit expressions are obtained. These results have their counterparts in a fullband system; however, inside a multirate structure or a subband system, the aliasing effect caused by decimation should be taken into account. In a multirate system, stationarity is not preserved when an upsampling (or expanding) operation is encountered. Furthermore the equivalent filtering operation is nonlinear. A test example of an AR-1 process is included for demonstration. From this example, an interesting phenomenon is observed. When the correlation coefficient of the AR-1 process is close to 1, the lowpassed signal is not, in any sense, a rough replica of the source. This example justifies the significance and necessity of a theoretical analysis of subband systems from a statistical viewpoint. We believe that stochastic signal processing applications of a subband structure such as estimation, detection, recognition, etc. will benefit from study of this nature.!14