Quadratic-Inverse Expansion of the Rihaczek Distribution

This paper describes some unexpected relationships between multitaper estimates of the spectrum and time–frequency distributions. In particular, there is an orthogonal decomposition of a localized Rihaczek distribution in terms of quadratic–inverse expansions. The frequency marginal of this estimate is the standard multitaper estimate and thus much more accurate than conventional forms that have the periodogram as the frequency marginal. The time marginal is a smoothed version of the instantaneous power. The first three terms of the quadratic–inverse expansion are approximately the multitaper estimate and the quadratic–inverse estimates of the time and frequency derivatives of the spectrum.

[1]  D. Thomson.  Quadratic-inverse spectrum estimates: applications to palaeoclimatology , 1990, Philosophical Transactions of the Royal Society of London. Series A: Physical and Engineering Sciences.

[2]  H. Margenau,et al.  Correlation between Measurements in Quantum Theory , 1961 .

[3]  Charles A. Stutt,et al.  Some results on real-part/imaginary-part and magnitude-phase relations in ambiguity functions , 1964, IEEE Trans. Inf. Theory.

[4]  August W. Rihaczek,et al.  Signal energy distribution in time and frequency , 1968, IEEE Trans. Inf. Theory.

[5]  D. Slepian.  Prolate spheroidal wave functions, fourier analysis, and uncertainty — V: the discrete case , 1978, The Bell System Technical Journal.

[6]  David J. Thomson,et al.  Nonstationary fluctuations in stationary time-series , 1993, Optics & Photonics.

[7]  Thomas P. Bronez,et al.  On the performance advantage of multitaper spectral analysis , 1992, IEEE Trans. Signal Process..

[8]  Petre Stoica,et al.  On nonparametric spectral estimation , 1998, 9th European Signal Processing Conference (EUSIPCO 1998).

[9]  T. Claasen,et al.  THE WIGNER DISTRIBUTION - A TOOL FOR TIME-FREQUENCY SIGNAL ANALYSIS , 1980 .

[10]  B. Roy Frieden,et al.  VIII Evaluation, Design and Extrapolation Methods for Optical Signals, Based on Use of the Prolate Functions , 1971 .

[11]  Khaled H. Hamed,et al.  Time-frequency analysis , 2003 .

[12]  B. Friedlander,et al.  Covariant estimators of time-frequency descriptors for nonstationary random processes , 1998, Conference Record of Thirty-Second Asilomar Conference on Signals, Systems and Computers (Cat. No.98CH36284).

[13]  D. Thomson,et al.  Spectrum estimation and harmonic analysis , 1982, Proceedings of the IEEE.

[14]  Steven M. Sussman,et al.  Least-square synthesis of radar ambiguity functions , 1962, IRE Trans. Inf. Theory.