Optimal Zak-OTFS receiver and its relation to the radar matched filter

This paper presents optimal receiver implementations for Zak-OTFS modulation. Zak-OTFS has a receiver structure that includes a twisted convolution filter followed by delay-Doppler domain sampling. We first show that this receiver is equivalent to a correlation demodulator where the receive pulses are determined by the choice of the receive delay-Doppler domain twisted convolution (TC) filter. We formulate the notion of a matched TC filter as the receive TC filter that maximizes the SNR in an additive white Gaussian noise (AWGN) channel, for a given transmit TC filter. We show that the matched TC filter formulation is crucial for understanding noise processes in the delay-Doppler domain. More generally, for a doubly dispersive channel, we define a receive TC filter that is matched to the twisted convolution of the channel with the transmit TC filter. We show that this receive TC filter, sampled at the delay-Doppler grid points is the optimal Zak-OTFS receiver that recovers sufficient statistics for maximum likelihood detection of the data symbols. We first present an implementation of this optimal receiver for a general sparse doubly dispersive channel which requires radar matched filter processing and involves computing ambiguity functions. We then present a second implementation that uses a receive TC filter that is matched to the transmit TC filter (not the channel) and only requires time and frequency windowing. We show that these two approaches converge when the window supports are large relative to the fundamental periods of the delay-Doppler grid. We also show that the second approach has an interpretation as a rake receiver operating in the delay-Doppler domain.