A measure of some time–frequency distributions concentration
Introduction
Efficient time–frequency distribution concentration measurement can provide a quantitative criteria to evaluate performances of different distributions and can be used for adaptive and automatic parameters selection in time–frequency analysis, without interference by a user. Measures for distribution concentration of monocomponent signals date back to [4], [9], [10], [17]. For more complex signals, some quantities in the statistics were the inspiration for defining measures for time–frequency distributions concentration in the form of the ratio of distribution norms by Jones and Parks [12], and the Rényi entropy by Williams et al. [8], [15], [18]. Distribution energy was also used, first by Baraniuk and Jones, for optimal kernel distributions design [2], [3], [6], [7], [11]. Common for all of these measures is that they are based on the distribution norms, i.e., sums over the distribution values raised to a power greater than one. They provided good quantitative measure of the auto-terms concentration. Norms themselves failed to behave in the desired way when the cross terms appeared. Various and efficient modifications were used in order to take into account the appearance of nondesirable oscillatory zero-mean distribution values. The distribution norm has been divided by a lower-order norm in [12], [15], while some strict constraints were imposed on the kernel form in [2]. However, even the normalized forms of the norm-based measures are not quite appropriate for the cases where there are two or more components (or regions in time–frequency plane of a single component) of approximately equal energies (importance) whose concentrations are very different. The norm-based measures, due to raising of distribution values to a high power (fourth, in [12], third, in [8], [15]) will favor distributions with “peaky” components [12]. It means that if one component (region) is “extremely highly” concentrated, and all the others are “very poorly” concentrated, then they will not look for a compromise, for example, when all components are “very well” concentrated. In order to deal with this kind of problem, that are present in time–frequency analysis, Jones and Parks [12] introduced local concentration measure, that locally measures concentration, and increases the calculation complexity.
Here, we will present a simple measure for distributions concentration, that can overcome some of the mentioned drawbacks. It behaves well with respect to the auto-terms, cross-terms and does not discriminate low concentrated components against very concentrated ones during the optimization procedure. Its application will be demonstrated for automatic determination of the “best window length” for the spectrogram or “the optimal number of terms” in the S-method (SM) [16]. It could be used in other similar problems in time–frequency analysis.
Section snippets
A concentration measure
Consider time–frequency representation of a signal x(t) denoted by . Assume that satisfies the unbiased energy condition
Let us, just for the beginning, assume that Px(t,ω)≠0 only for (t,ω)∈Dx(t,ω). For a large p we have thatwhere Sx is the area of Dx(t,ω).
As a criterion for the distributions-concentration measure we will assume: Among several given unbiased energy distributions, the best concentrated is the
Applications and examples
1. Consider the spectrogramwhere STFT(n,k)=DFTm{w(m)x(n+m)} is the short-time Fourier transform; E is the energy of the lag window w(m). Among several spectrograms, calculated with different window lengths or forms, the best one according to the proposed measure with p=2, will be that which minimizesLet us illustrate this by an example. Consider the signalsampled at . The
Optimization
Parameters optimization may be done by a straightforward computation of a distribution measure for different parameter values. The best choice according to this criterion (optimal distribution with respect to this measure) is the distribution that produces the minimal value of . In the cases, when one has to consider a wide region of possible parameter values for the distribution calculation (like for example window lengths in spectrogram), this approach can be numerically
Review of the existing measures and their comparison with the proposed measure
Here we will briefly review the existing measures of time–frequency distributions concentration. For each of them, we will point out a drawback that lead us to consider and propose another form of measure for time–frequency distributions concentration.
1. Ratio of norms based measures: Jones and Parks proposed [12] the fourth power of the L4 norm of time–frequency distribution divided by the L2 norm:
This norm is similar to “kurtosis” in statistics. They have also
Conclusion
A very simple criterion that can provide an objective measure for time–frequency distributions concentration is presented. It has been used in automatic determination of some time–frequency distributions parameters. A review of the existing measures of time–frequency distributions concentration, and their comparison with the proposed measure is given.
Acknowledgments
The author is thankful to the reviewers for the comments, that helped to make the paper more complete. Author also thanks Prof. Johann F. Böhme, ARAL Research Bochum, and the Signal Theory Group at the Ruhr University Bochum, for the car engine data. A part of this research has been supported by the Alexander von Humboldt foundation, and the Volkswagen Stiftung, Federal Republic of Germany.
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