The wide-band ambiguity function and Altes´ Q-distribution: constrained synthesis and time-scale filtering

Two problems of interest in wideband signal processing are addressed. The first problem concerns the synthesis of wideband radar/sonar signal waveforms given the desired shape (as a function of time delay and time scale) of the cross correlator (matched filter) output; this function is termed the wideband ambiguity function (WBAF). Constraints on the transmitted waveforms that are manifest as closed convex subsets in the delay-scale space can also be incorporated into the synthesis procedure. The alternating convex projection theorem is used to generate a (weakly) convergent iterative algorithm for waveform synthesis. The second problem addressed concerns the closely related problem of signal recovery from R.A. Altes´ (IEEE Trans. Acoust., Speech, Signal Proc., vol.38, no.6, p.1005-12, June 1990) Q-distribution and modified versions thereof. The Q-distribution is intimately related to the WBAF via a unitary transformation. Admissible modifications include affine convolutions and other operators with closed convex range. The application in terms of filtering in a time-scale domain is considered.<>