Computationally Efficient Scale Covariant Time-Frequency Distributions
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Research Article Computationally Efficient Scale Covariant Time-Frequency Distributions Selin Aviyente Department of Electrical and Computer Engineering, Michigan State University, East Lansing, MI 48824, USA Correspondence should be addressed to Selin Aviyente, [email protected] Received 3 October 2008; Revised 14 January 2009; Accepted 3 February 2009 Recommended by Ulrich Heute Scale is a physical attribute of a signal which occurs in many natural settings. Time-frequency distributions (TFDs) belonging to Cohen’s class are invariant to time and frequency shifts, but are not necessarily covariant to the time scalings of the signal. Conditions on the time-frequency kernel for yielding a scale covariant distribution have been previously derived (Cohen, 1995) . In this paper, a new class of computationally efficient scale covariant distributions is introduced. These distributions are constructed using the eigendecomposition of time-frequency kernels (Burrus et al., 1997). The performance of this new class of distributions is illustrated with examples and is compared to conventional scale covariant distributions. Copyright © 2009 Selin Aviyente. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction Scale is a physical attribute of signals just like frequency. Self-scaling of signals is a phenomenon that is observed in different natural settings including biological and acoustic signals. Time-frequency distributions are designed to represent the energy distribution of nonstationary signals simultaneously in time and frequency but do not necessarily reflect the changes in scale. A bilinear continuous timefrequency distribution (TFD) belonging to Cohen’s class is represented as follows [1] (all integrals are from −∞ to ∞ unless otherwise specified):
C(t, ω) =
φ(θ, τ)s u + ×e
j θu−θt −τω
τ τ ∗ s u− 2 2
(1)
du dθ dτ,
where s is the signal, and φ(θ, τ) is the kernel function in the ambiguity domain. For a bilinear time-frequency distribution, scale covariance implies that when the signal is scaled in time, the TFD scales accordingly both in time √ and frequency, that is, if s(t) → C(t, ω), then as(at) → C(at, ω/a). For bilinear distributions belonging to Cohen’s class, it has been shown that scale covariance is satisfied when the kernel is a product kernel, that is, φ(θ, τ) = φ(θτ).
There have been various time-frequency representations designed to satisfy scale covariance such as the wavelet transform, the affine class, and the hyperbolic class of time-frequency distributions (e.g., [2–4]). These transforms achieve scale covariance at the expense of losing some desired properties such as time-frequency shift invariance and constant-bandwidth resolution. In this paper, the focus is on the scale covariance of Cohen’s class of distributions. Recent research results in the decomposition of time-frequency kernels for fast com
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