On the Diversity Behavior for Fixed-Gain Amplify-and-Forward Relaying Over Frequency-Selective Channels Based on Linearly Precoded OFDM

Recent research has shown that the diversity gain of cooperation among fixed-gain amplify-and-forward (FG-AF) relays is not a simple function of power of signal-to-noise ratio (SNR), where a factor of the logarithm of SNR might be involved. It is more suitable to use the diversity gain function (DGF) of ρ^-y(log ρ)^x, with ρ denoting the SNR and nonnegative integers x and y, to describe such diversity performance. In this letter, we consider the use of FG-AF relaying over frequency-selective channel based on the orthogonal frequency-division multiplexing (OFDM) transmission employing linear precoding (LP) and present comprehensive analysis of the resulting DGF. Suppose that the number of multipath components for the channel from source to relay is L₁ and that from relay to destination is L₂. Theoretical analysis shows that, if L₁ ≠ L₂, the optimal DGF is precisely ρ^-L, with L = min{L₁, L₂}, which can be achieved as long as the precoding group size J ≥ L + 1, whereas only a DGF of ρ^-J(log ρ)^J is obtained by precoding with size J ≤ L. In the case of L₁ = L₂, we show that the best achievable DGF is ρ^-Llog ρ, which can be guaranteed by choosing J ≥ 2L - 1. We also demonstrate that these results can be easily extended to the multirelay case.