On conditional spectral moments of Gaussian and damped sinusoidal atoms in adaptive signal decomposition

It is well known that the convergence (with different speeds) of the matching pursuit (MP) signal decomposition algorithm for any dense dictionary is guaranteed. In this paper, we have analysed (in theory and through simulations) the performance and properties of both Gaussian and damped sinusoidal atoms for the MP signal decomposition. We have examined the decomposed signal in ambiguity space (to look for auto-terms concentrated around the origin), and investigated the requirement to have a positive time-frequency representation. We are thus able to propose what kind of dictionary might be more suitable for MP signal decomposition. We have also derived general formulae for the first and second conditional spectral moments, which are useful generalizations of the concept of instantaneous frequency and instantaneous bandwidth, respectively. While the second conditional moment is not positive for many bilinear time-frequency distributions, thus making useless its interpretation as instantaneous bandwidth, we have proved that for MP decomposition based on Gaussian or damped sinusoidal atoms, it is always guaranteed positive.

[1]  M. Goodwin,et al.  Atomic decompositions of audio signals , 1997, Proceedings of 1997 Workshop on Applications of Signal Processing to Audio and Acoustics.

[2]  Eric J. Diethorn,et al.  The generalized exponential time-frequency distribution , 1994, IEEE Trans. Signal Process..

[3]  Stéphane Mallat,et al.  Adaptive time-frequency transform , 1993, 1993 IEEE International Conference on Acoustics, Speech, and Signal Processing.

[4]  Piotr J. Durka,et al.  Stochastic time-frequency dictionaries for matching pursuit , 2001, IEEE Trans. Signal Process..

[5]  Qinye Yin,et al.  A fast refinement for adaptive Gaussian chirplet decomposition , 2002, IEEE Trans. Signal Process..

[6]  Richard S. Orr,et al.  The Order of Computation for Finite Discrete Gabor Transforms , 1993, IEEE Trans. Signal Process..

[7]  Natasa Kovacevic,et al.  Algorithm 820: A flexible implementation of matching pursuit for Gabor functions on the interval , 2002, TOMS.

[8]  P. Loughlin,et al.  Instantaneous kurtosis , 2000, IEEE Signal Processing Letters.

[9]  Stéphane Mallat,et al.  Matching pursuits with time-frequency dictionaries , 1993, IEEE Trans. Signal Process..

[10]  Robert J. Marks,et al.  The use of cone-shaped kernels for generalized time-frequency representations of nonstationary signals , 1990, IEEE Trans. Acoust. Speech Signal Process..

[11]  Boualem Boashash,et al.  Estimating and interpreting the instantaneous frequency of a signal. I. Fundamentals , 1992, Proc. IEEE.

[12]  Zhenyu Guo,et al.  The time-frequency distributions of nonstationary signals based on a Bessel kernel , 1994, IEEE Trans. Signal Process..

[13]  Patrick Flandrin,et al.  Some features of time-frequency representations of multicomponent signals , 1984, ICASSP.

[14]  Jechang Jeong,et al.  Kernel design for reduced interference distributions , 1992, IEEE Trans. Signal Process..

[15]  Douglas L. Jones,et al.  A signal-dependent time-frequency representation: optimal kernel design , 1993, IEEE Trans. Signal Process..

[16]  Patrick J. Loughlin,et al.  Instantaneous spectral skew and kurtosis , 2000, Proceedings of the Tenth IEEE Workshop on Statistical Signal and Array Processing (Cat. No.00TH8496).

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

[18]  William J. Williams,et al.  Improved time-frequency representation of multicomponent signals using exponential kernels , 1989, IEEE Trans. Acoust. Speech Signal Process..

[19]  Braham Barkat,et al.  A high-resolution quadratic time-frequency distribution for multicomponent signals analysis , 2001, IEEE Trans. Signal Process..

[20]  L. Cohen,et al.  Time-frequency distributions-a review , 1989, Proc. IEEE.