Time - Frequency Baseddistance and Divergence

A study of the phase and amplitude sensitivity of the recently proposed R enyi time-frequency information measure leads to the introduction of a new \Jensen-like" divergence measure. While this quantity promises to be a useful indicator of the distance between two time-frequency distributions, it is limited to the analysis of positive deenite TFDs. In spite of this rather severe limitation, this measure could prove useful for time-frequency based detection. We illustrate with an example of detecting a signal in additive noise. 1. TIME-FREQUENCY INFORMATION MEASURES Information and entropy functionals have shed new light on the question \what is a signal component" by providing quantitative measures of signal complexity in the time-frequency plane 1, 2, 3]. Their theoretical basis relies on the formal analogy between the time-frequency distributions (TFDs) of Cohen's class and bidimensional probability density functions. In particular, a large class of TFDs possesses the following marginal characteristics (for unit energy signals): Z Cs(t; f) df = js(t)j 2 ; Z Cs(t; f) dt = jS(f)j 2 ; Z Z Cs(t; f) dt df = 1: The probabilistic interpretation of a TFD suggests the Shan-non entropy H(Cs) = ? Z Z Cs(t; f) log 2 Cs(t; f) dt df as a natural candidate for a time-frequency information measure. Unfortunately, however, most Cohen's class TFDs take on negative values, prohibiting its application. Recent research has concentrated on alternative information measures, in particular the class of R enyi entropies 4] H(Cs) = 1 1 ? log 2 Z Z C s (t; f) dt df (1) parameterized by > 0 1, 2, 3]. The Shannon entropy belongs to this class, in the limit as ! 1. Unlike the Shan-non entropy, however, the R enyi measures are deened for virtually all Cohen's class TFDs for all integer 2 2, 3]. The primary property of the R enyi entropies studied thus far has been the component counting property: as two identical, overlapping components are separated in the time-frequency plane, the measure H(Cs) increases by 1 bit (for odd). As an example of this property 1, 2, 3], consider the H3(Ws) information of the signal s(t) = g(t)+g(t+T), with g a lowpass Gaussian pulse and Ws the Wigner distribution. This information is plotted in Fig. 1(a) versus the separation distance T (in units of the RMS time duration of g(t)). (At T = 0, the two pulses coincide and therefore, because of the assumed energy …