Estimating time-frequency distributions and scattering functions using the Rihaczek distribution

In this paper we study two problems: estimation of time-frequency distributions (TFDs) and estimation of wireless or radar scattering functions (SFs). We show that the most general quadratic, delay- and modulation-invariant estimator of each may be written as a convolution of a Rihaczek TF density with a TF kernel. This representation complements other equivalent representations and establishes a fundamental connection between the analysis of the two problems. However, the representation also illustrates a fundamental difference in design for the two problems. For TFD estimation, the Rihaczek TF density is the raw Rihaczek for the time series and the kernel is designed to convolve in time and frequency for smoothing. For SF estimation, the kernel is the true SF and the Rihaczek TF density is that for a transmitter signal designed to deconvolve in time and frequency for inversion. In each case, the Fourier transform identities in a four-corners diagram allow for kernel or Rihaczek design in the transformed space of ambiguity functions. Design in this space then produces spectrograms and interferograms for TFD estimation, and ideal transmitter signals for SF estimation.