Time-frequency Based Distance and Divergence Measures

A study of the phase and amplitude sensitivity of the recently proposed RCnyi 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 definite 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 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 [l, 2, 31. 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): MEASURES J cs(t, f) df = ls(t)I2, c s ( t , f) dt = I W ) I ~ , J H ( C s ) = J J Cs(t, r) log, Cs(t, f) dt df f J ~ . ( t , f ) d t d f = 1. The probabilistic interpretation of a TFD suggests the Shannon entropy 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 E n y i entropies [4] parameterized by a > 0 [l, 2, 31. The Shannon entropy belongs to this class, in the limit as a + 1. Unlike the Shannon entropy, however, the RCnyi measures are defined for virtually all Cohen’s class TFDs for all integer CY 2 2 [2, 31. The primary property of the RCnyi entropies studied thus far has been the component counting property: as two identical, overlapping components are separated in the time-frequency plane, the measure Ha(CS) increases by 1 bit (for a odd). As an example of this property [l, 2, 31, consider the H3(W,) information of the signal s ( t ) = g(t)+g(t+T), with g a lowpass Gaussian pulse and W, the Wigner distribution. This information is plotted in Fig. l(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 normalization, have the same information content as a solitary pulse.) The time-bandwidth product (TBP) of the signal is also plotted with a dashed line. It is clear from the figure that, unlike the TBP, which grows without bound with T , the information measure saturates exactly one bit above the value H 3 ( W g ) = -0.208.’ Similar results hold for three separated copies of g ( t ) (log,3 bits information gain), four copies (2 bits information gain), and so on. 2. PHASE SENSITIVITY The results of Fig. l(a) are very appealing, but also incomplete and unrealistic, because no phase differences were introduced between the two signal components. Figure l(b) illustrates a more complete set of curves of the H3(Ws) information for the signal g(t) cos(xt/6)+g(t+T) cos(x(t+T)/6+p). Each curve corresponds to a Merent relative phase angle p between 0 and x rad. It is apparent from the curves that while phase changes do not affect the saturation levels of the information measure, they allow many possible trajectories between the two levels, including even trajectories where an “overestimation” [I] of information content occurs. The phase sensitivity of the RCnyi information measure stems from cross-terms in the Wigner distribution and can be studied by developing an expansion for H,(W,) with s a simple twecomponent signal. Our signal model is s ( t ) = g ( t ) + (Dd,pg)(t), where g(t) is a Gaussian logon of the form 1 -2 g(t) = e 2w2 ( r 4 and def ej(2rtAf+p) g( t ~ t ) (Dd,lpg)(t) = denotes the same logon phase-shifted by angle p and translated in time-frequency by the normalized distance (2) At2 dZ = ( 2 r A f ~ ) ~ + 7, . , *This work was supported by URA 1325 ems, the National Science Foundation, gmt no. MP-9457438, and the Texas Advanced Technology Program, grant no. TX-ATP 003604-002. ‘&&em should not be alarmed by negative Rknyi entropy values. Even the Shannonentropy takes on negative values for certain distributions in the continuousvariable case. 0-7803-2127-8/94 $4.00 01994 IEEE