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In this paper, the linear Gaussian relay problem is considered. Under the linear time-invariant (LTI) model the rate maximization problem in the linear Gaussian relay channel is formulated in the frequency domain based on the Toeplitz distribution theorem. Under the further assumption of realizable input spectra, the rate maximization problem is converted to the problem of joint source and relay filter design with 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 (adaptive) projected (sub)gradient method. Numerical results show that the proposed method yields a considerable 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.

A Joint Time-Invariant Filtering Approach to the Linear Gaussian Relay Problem