The capacity for the linear time-invariant Gaussian relay channel

In this paper, the Gaussian relay channel with linear time-invariant relay filtering is considered. Based on spectral theory for stationary processes, the maximum achievable rate for this subclass of linear Gaussian relay operation is obtained in finite-letter characterization. The maximum rate can be achieved by dividing the overall frequency band into at most eight subbands and by making the relay behave as an instantaneous amplify-and-forward relay at each subband. Numerical results are provided to evaluate the performance of LTI relaying.