We analyze the diversity-multiplexing trade-off (DMT) of fixed-gain amplify-and-forward (AF) protocols for the dual hop relay channel. Contrary to most prior work, we characterize the finite signal-to-noise ratio (SNR) DMT using a non-asymptotic framework. Our results are based on a new exact closed-form expression which we derive for the distribution of the received SNR, which takes into account the direct transmission link. We also extract the conventional asymptotic DMT as a special case of the finite-SNR result, and demonstrate that the asymptotic DMT can significantly overestimate the level of diversity that is achievable for practical error rates and SNRs. We also present the finite-SNR DMT for decode-and-forward (DF) relaying protocols, and show the interesting result that fixed-gain AF outperforms DF at low SNR.
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