Correlative time-frequency analysis and classification of nonstationary random processes

Abstract

The ezpected ambiguity function (EAF) is shown to provide a generalization of stationary correlation analysis to nonstationary random processes. Important properties of the EAF are discussed, and the EAFs of special processes are considered. Based on the EAF, a fundamental classification (underspread/overspread) of nonstationary processes is introduced and shown to be relevant to timevarying spectral analysis. I N T R O D U C T I O NThe correlative analysis of stationar processes using theis of fundamental importance [l]. In particular, the power spectrum S,(f) is the Fourier transform of the ACF ( Wiener-Khintchine relation)',For a nonstationary process x ( t ) , the ACF r z ( t i , t z ) = E { x ( t i ) x*(t2)} is a 2-D function [l], and the power spectrum S,(f) is replaced by a time-varying power spectrum T,(t, f ) such as the physical spectrum, the (generalized) Wagner-Ville spect m m , or the evolutionary spectrum [2, 3, 41. Nonstationary processes exhibit spectral correlation [5] as measured by the spectral ACF R Z ( f 1 , f 2 )= E{X(f')X*(fi)} (assuming existence of the Fourier transform X ( f ) of z(t)).We now ask if there exists a joint time-frequency (TF) correlation function which combines the temporal ACF r , ( t i , t 2 ) and the spectral ACF R , ( f l , f i in a meaningful way, and which is related to a meanin d ul time-varying spectrum by a Fourier transform (generalization of the Wiener-Khintchine relation (1)). In this paper, we show that a satisfactory answer to this question is provided by the expected ambiguity function (EAF) recently proposed in [S, 7 . We demonstrate that the effective support region nonstationarity inherent in a rocess. We then introduce a fundamental classification &nderspread/overspread) of nonstationary processes. For underspread processes, various timevarying spectra (such as the generdized WignerVille spectra and the evolutionary spectra) are shown to be effectively equivalent. Furthermore, the underspread property is relevant to timevarying spectrum estimation, and finally, the physical spectrum of an underspread process is a complete second-order description of the process. 'Funding by grant 4913 of the Jubilaumsfondf der Osterrei-'Integrals go from -cu to 00 unless specified otherwise. of the k A F provides useful indications about the type of chischen Nationalbank and FWF grant The (generalized) ambiguity function (AF) [SI of a signal x ( t ) is where a is a realvalued parameter. We define the ezpected (generalized) ambiguity function (EAF) of a nonstationary random process s(t) as the expectation of the AF, EA?)(^, e' E{A?)(~, VI} .It follows that the EAF is the Fourier transform of the a-parameterized ACF with respect to t , J tSince the ACF can be recovered from the EAF by inversion of the Fourier transform (2) followed by a simple substitution to obtain r , ( t l , t z ) from r p ) ( t , T ) , the EAF is a complete second-order description of the process.Interpretation as TF Correlation. An intuitively reasonable measure for the statistical correlation between ...

Kernels and Multiple Windows for Estimation of the Wigner-Ville Spectrum of Gaussian Locally Stationary Processes

“…The WVS of a process with covariance function is defined by [1], [31], [36], [37] The expected ambiguity function (EAF) is defined by (10) The WVS of an LSP is , and the EAF is . Both and can be said to be concentrated around the origin, since (it does not oscillate), and .…”

Nonstationary spectral analysis based on time-frequency operator symbols and underspread approximations

“…In this section, we explain the underspread property and introduce a concept of underspread processes that extends the original definition given by Kozek [25]- [28], [60]. We first review a TF correlation function on which the underspread concept is based.…”

“…A joint description of the temporal and spectral correlations is provided by the generalized expected ambiguity function (GEAF) defined as [25]- [27], [30] Comparing this definition with the GSF in (13), it is seen that the GEAF is the GSF of the correlation operator , i.e., . The GEAF is a global measure of the correlation of all components of that are separated by in time and by in frequency [25]- [27], [30]. For example, the GEAF of a stationary white process with PSD is .…”

“…For a nonstationary white process with correlation function , we have (where is the Fourier transform of ), which indicates the absence of temporal correlations. The relation allows us to use the shorthand notation Further properties of the GEAF are described in [25], [27], [30].…”

“…We consider the practically important class of underspread processes, which are nonstationary processes with negligible high-lag TF correlations [25]- [32]. Our principal goal is to develop a simple and powerful TF calculus for the analysis and filtering of underspread processes.…”