A joint time-invariant filtering approach to the linear Gaussian relay problem

TL;DR

This paper addresses the linear Gaussian relay problem under LTI assumptions, formulating it in the frequency domain and proposing a practical joint filter design method that outperforms traditional schemes in ISI channels.

Contribution

It introduces a joint source and relay filter design approach for the LTI Gaussian relay problem using the projected subgradient method, with proven optimality in flat-fading channels.

Findings

Proposed method outperforms amplify-and-forward in ISI channels.

Joint filter design improves relay communication performance.

Optimality of AF scheme established in flat-fading channels.

Abstract

In this paper, the linear Gaussian relay problem is considered. Under the linear time-invariant (LTI) model the problem is formulated in the frequency domain based on the Toeplitz distribution theorem. Under the further assumption of realizable input spectra, the LTI Gaussian relay problem is converted to a joint design problem of source and relay filters under two power constraints, one at the source and the other at the relay, and a practical solution to this problem is proposed based on the projected subgradient method. Numerical results show that the proposed method yields a noticeable gain over the instantaneous amplify-and-forward (AF) scheme in inter-symbol interference (ISI) channels. Also, the optimality of the AF scheme within the class of one-tap relay filters is established in flat-fading channels.