Wide-band, proportional-bandwidth Wigner-Ville analysis

The Wigner-Ville (W-V) distribution is a time-frequency representation that yields a highly accurate estimate of instantaneous frequency. It is related to the narrow-band ambiguity function by an internal transform, and it can be used in a variety of detection and estimation problems. A spectrogram constructed with constant bandwidth filters can be obtained by convolving two W-V distributions or by forming a magnitude-squared narrow-band cross-ambiguity function. The wide-band ambiguity function represents the Doppler effect with dilation or compression rather than with frequency shift as in the narrow-band approximation. The Q distributions, a modified W-V representation that is related to the wide-band ambiguity function by an integral transform, is defined. A spectrogram constructed with proportional-bandwidth or constant-Q filters can be obtained by a convolution-like operation involving two Q distributions or by forming a magnitude-squared wide-band cross-ambiguity function. The Q distribution is thus a wide-band version of the W-V distribution. Properties of the Q distribution indicate that it may prove useful for detection and parameter estimation as well as for tomographic measurement of wideband scattering functions with relatively few transmitted waveforms