Joint Transmit and Receive Filter Optimization for Sub-Nyquist Delay-Doppler Estimation

In this paper, a framework is presented for the joint optimization of the analog transmit and receive filter with respect to a parameter estimation problem. At the receiver, conventional signal processing systems restrict the two-sided bandwidth of the analog prefilter <inline-formula><tex-math notation="LaTeX">$B$</tex-math></inline-formula> to the rate of the analog-to-digital converter <inline-formula><tex-math notation="LaTeX">$f_s$</tex-math> </inline-formula> to comply with the well-known Nyquist–Shannon sampling theorem. In contrast, here we consider a transceiver that by design violates the common paradigm <inline-formula><tex-math notation="LaTeX">$B\leq f_s$ </tex-math></inline-formula>. To this end, at the receiver, we allow for a higher prefilter bandwidth <inline-formula><tex-math notation="LaTeX">$B>f_s$</tex-math></inline-formula> and study the achievable parameter estimation accuracy under a fixed sampling rate when the transmit and receive filter are jointly optimized with respect to the Bayesian Cramér–Rao lower bound. For the case of delay-Doppler estimation, we propose to approximate the required Fisher information matrix and solve the transceiver design problem by an alternating optimization algorithm. The presented approach allows us to explore the Pareto-optimal region spanned by transmit and receive filters that are favorable under a weighted mean squared error criterion. We also discuss the computational complexity of the obtained transceiver design by visualizing the resulting ambiguity function. Finally, we verify the performance of the optimized designs by Monte Carlo simulations of a likelihood-based estimator.