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

## TL;DR

This paper introduces a joint transmit and receive filter optimization framework for delay-Doppler estimation that exceeds traditional bandwidth constraints, improving parameter estimation accuracy through Bayesian Cramér-Rao bound optimization.

## Contribution

It proposes a novel transceiver design that violates the Nyquist-Shannon bandwidth limit by jointly optimizing filters for better delay-Doppler estimation accuracy.

## Key findings

- Optimized filters outperform traditional designs in estimation accuracy.
- The approach explores the Pareto-optimal filter region under weighted MSE.
- Monte-Carlo simulations confirm improved estimator performance.

## Abstract

In this article, 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 pre-filter $B$ to the rate of the analog-to-digital converter $f_s$ to comply with the well-known Nyquist-Shannon sampling theorem. In contrast, here we consider a transceiver that by design violates the common paradigm $B\leq f_s$. To this end, at the receiver, we allow for a higher pre-filter bandwidth $B>f_s$ 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 Cram\'{e}r-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 which 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.