Polynomial time-frequency distributions and time-varying higher order spectra: Application to the analysis of multicomponent FM signals and to the treatment of multiplicative noise

Abstract One of the fundamental problems of time–frequency analysis that remained unsolved until recently is the time–frequency representation of signals with an arbitrary non-linear frequency variation in time. A certain type of higher-order time–frequency distributions (TFDs), referred to as polynomial Wigner-Ville distributions (PWVDs), can achieve delta function concentration in the time–frequency plane for mono-component polynomial FM signals (Boashash and O’Shea, 1994). This paper is a sequel to Boashash and O’Shea (1994) dealing with a broader class of signals and presenting an application to the treatment of multiplicative noise. The first part of the paper presents a general design procedure for PWVDs and the main properties of PWVDs. A specific class of polynomial time–frequency distributions that deals effectively with multicomponent signals is then described. In the second part of the paper we deal with random signals and introduce time-varying higher-order spectra (TV-HOS) an ensemble averaged PWVDs. TV-HOS are shown to be natural tools for analysis of non-stationary random signals, and we demonstrate this in the context of FM signals affected by multiplicative noise. Both moment and cumulant fourth-order TV-HOS, referred to as the Wigner-Ville trispectra are shown to be superior to the second-order methods for this application.

[1]  R. F. Dwyer,et al.  Range and Doppler information from fourth-order spectra , 1991 .

[2]  Roger F. Dwyer,et al.  Fourth‐order spectra of Gaussian amplitude‐modulated sinusoids , 1991 .

[3]  F. Hlawatsch,et al.  Linear and quadratic time-frequency signal representations , 1992, IEEE Signal Processing Magazine.

[4]  Ananthram Swami,et al.  Cramer-Rao bounds for deterministic signals in additive and multiplicative noise , 1996, Signal Process..

[5]  Braham Barkat,et al.  A performance analysis of polynomial FM signals , 1997, Proceedings of ICICS, 1997 International Conference on Information, Communications and Signal Processing. Theme: Trends in Information Systems Engineering and Wireless Multimedia Communications (Cat..

[6]  Joseph M. Francos,et al.  Bounds for estimation of multicomponent signals with random amplitude and deterministic phase , 1995, IEEE Trans. Signal Process..

[7]  Antonia Papandreou-Suppappola,et al.  The hyperbolic class of quadratic time-frequency representations. I. Constant-Q warping, the hyperbolic paradigm, properties, and members , 1993, IEEE Trans. Signal Process..

[8]  Olivier Besson,et al.  On estimating the frequency of a sinusoid in autoregressive multiplicative noise , 1993, Signal Process..

[9]  Chrysostomos L. Nikias,et al.  Wigner Higher Order Moment Spectra: Definition, Properties, Computation and Application to Transient Signal Analysis , 1993, IEEE Trans. Signal Process..

[10]  Boualem Boashash,et al.  Relationship between the polynomial and the higher order Wigner-Ville distribution , 1995, IEEE Signal Processing Letters.

[11]  Benjamin Friedlander,et al.  Asymptotic statistical analysis of the high-order ambiguity function for parameter estimation of polynomial-phase signals , 1996, IEEE Trans. Inf. Theory.

[12]  Harry L. Van Trees,et al.  Detection, Estimation, and Modulation Theory, Part I , 1968 .

[13]  Patrick Flandrin,et al.  Wigner-Ville spectral analysis of nonstationary processes , 1985, IEEE Trans. Acoust. Speech Signal Process..

[14]  Georgios B. Giannakis,et al.  Estimating random amplitude polynomial phase signals: a cyclostationary approach , 1995, IEEE Trans. Signal Process..

[15]  Boualem Boashash,et al.  Analysis of FM signals affected by Gaussian AM using the reduced Wigner-Ville trispectrum , 1993, 1993 IEEE International Conference on Acoustics, Speech, and Signal Processing.

[16]  B. Porat,et al.  Comparative performance analysis of two algorithms for instantaneous frequency estimation , 1996, Proceedings of 8th Workshop on Statistical Signal and Array Processing.

[17]  Leon Cohen,et al.  The scale representation , 1993, IEEE Trans. Signal Process..

[18]  Chrysostomos L. Nikias,et al.  Analysis of finite-energy signals higher-order moments- and spectra-based time-frequency distributions , 1994, Signal Process..

[19]  B. Senadji,et al.  A mobile communications application of time-varying higher order spectra to FM signals affected by multiplicative noise , 1997, Proceedings of ICICS, 1997 International Conference on Information, Communications and Signal Processing. Theme: Trends in Information Systems Engineering and Wireless Multimedia Communications (Cat..

[20]  Boualem Boashash,et al.  Polynomial Wigner-Ville distributions and their relationship to time-varying higher order spectra , 1994, IEEE Trans. Signal Process..

[21]  B. Friedlander,et al.  Multicomponent signal analysis using the polynomial-phase transform , 1996, IEEE Transactions on Aerospace and Electronic Systems.

[22]  Ananthram Swami,et al.  Multiplicative noise models: parameter estimation using cumulants , 1994, Fifth ASSP Workshop on Spectrum Estimation and Modeling.

[23]  A. Swami Third-order Wigner distributions: definitions and properties , 1991, [Proceedings] ICASSP 91: 1991 International Conference on Acoustics, Speech, and Signal Processing.

[24]  B. Boashash,et al.  The Wigner-Ville trispectrum: definition and application , 1993, [1993 Proceedings] IEEE Signal Processing Workshop on Higher-Order Statistics.

[25]  Boaz Porat,et al.  Estimation and classification of polynomial-phase signals , 1991, IEEE Trans. Inf. Theory.

[26]  N. L. Gerr Introducing a third-order Wigner distribution , 1988, Proc. IEEE.

[27]  Braham Barkat,et al.  Polynomial Wigner-Ville distribution for the analysis of polynomial FM signals: a performance study , 1997, Proceedings of 13th International Conference on Digital Signal Processing.

[28]  Boaz Porat,et al.  The Cramer-Rao lower bound for signals with constant amplitude and polynomial phase , 1991, IEEE Trans. Signal Process..

[29]  Ljubisa Stankovic,et al.  A multitime definition of the Wigner higher order distribution: L-Wigner distribution , 1994, IEEE Signal Processing Letters.

[30]  Branko Ristic,et al.  Some aspects of signal dependent and higher-order time-frequency and time-scale analysis of non-stationary signals , 1995 .

[31]  Douglas L. Jones,et al.  Unitary equivalence: a new twist on signal processing , 1995, IEEE Trans. Signal Process..

[32]  C. L. Nikias,et al.  Signal processing with higher-order spectra , 1993, IEEE Signal Processing Magazine.

[33]  Boualem Boashash,et al.  Application of cumulant TVHOS to the analysis of composite FM signals in multiplicative and additive noise , 1993, Optics & Photonics.

[34]  Ljubisa Stankovic,et al.  On the realization of the polynomial Wigner-Ville distribution for multicomponent signals , 1998, IEEE Signal Processing Letters.