Characterization of Relay Channels Using the Bhattacharyya Parameter

Relay systems have large and complex parameter spaces, which makes it difficult to determine the parameter region where the system achieves a given performance criterion, such as probability of frame error. In this paper, we show that the union bound (UB) and the Bhattacharyya parameter (BP) can be used for fast analysis of the parameter space when error-control coding is used. This is applicable when amplify-and-forward (AF) or demodulate-and-forward (DemF) are used. For a given code ensemble, the associated UB threshold is found and can be used to define the signal-to-noise region where a given frame error rate can be achieved. Using asymptotic results, the UB threshold can be used to specify the signal-to-noise ratio region where successful decoding can be achieved for large blocklength. In addition, the UB with BP can be used when fractional cooperation is used, where each relay only relays a fraction of the source codeword. This makes the UB with BP a valuable tool in the system design of relay networks.