SCAF estimation of time delay and doppler stretch

The conventional cross-ambiguity function (CAF) process assumes that the transmitted signal is a sinusoid having slowly varying complex modulation, and models a received signal as a delayed version of the transmitted signal, doppler shifted by the dominant frequency. For wide-band transmitted signals, it is more accurate to model a received signal as a time-scaled version of the transmitted signal, combined with a time delay, and wide-band cross-ambiguity models are well-known. We provide derivations of time-dependent wide-band cross-ambiguity functions appropriate for estimating radar target range and velocity, and time-difference of arrival (TDOA) and differential receiver velocity (DV) for geolocation. We demonstrate through simulations that for wide-band transmission, these scale CAF (SCAF) models are signficantly more accurate than CAF for estimating target range and velocity, TDOA and DV. In these applications, it is critical that the SCAF surface be evaluated in real-time, and we provide a method for fast computation of the scale correlation in SCAF, using only the discrete Fourier transform (DFT). SCAF estimates of delay and scale are computed on a discrete lattice, which may not provide sufficient resolution. To address this issue we further demonstrate simple methods, based on the DFT and phase differentiation of the time-dependent SCAF surface, by which super resolution of delay and scale, may be achieved.