Compress-and-Forward Rates for the Gaussian Relay with ISI and Colored Noise

The achievable rate of a relay channel with inter-symbol interference and additive colored Gaussian noise is examined under an input power constraint and the compress-and-forward (CF) strategy at the relay. Using classical techniques, it is shown that the circular degraded relay channels is equivalent to that of the linear one in the limit of infinite block lengths. The achievable rate for CF for a circular Gaussian relay channel is equivalent to that of coupled collection of parallel, memoryless relay channels. Additionally, this parallel channel decomposition is shown to be information lossless for the evaluation of the CF rate.