Summary

Chapter 22 Summary: Relay, Cooperative, and Multi-Hop Communications

Key Points

• 1.

  Relay channel fundamentals: The three canonical relay protocols — decode-and-forward (DF), amplify-and-forward (AF), and compress-and-forward (CF) — trade off decoding complexity against achievable rate. DF achieves the cut-set bound under the strong relay condition ($\text{SNR}_{SR} \geq \text{SNR}_{SD} + \text{SNR}_{RD}$), while CF approaches it when the relay is close to the destination. The capacity of the general relay channel remains open, but all protocols are within 0.5 bits of the cut-set bound for Gaussian channels.

• 2.

  Cooperative diversity: Single-antenna nodes create virtual MIMO arrays through cooperative protocols. With $L$ relays, the system achieves diversity order $L+1$, matching the performance of an $(L+1)$-antenna MISO system. Dynamic decode-and-forward (DDF) achieves the optimal DMT $d(r) = (L+1)(1-r)$, eliminating the multiplexing loss inherent in simpler protocols. The cooperation gain at $p_{\text{out}} = 10^{-3}$ is approximately $10(L+1)/(L+2)$ dB per additional relay.

• 3.

  Relay position optimisation: The optimal DF relay position satisfies $f^{-\alpha} = 1 + (1-f)^{-\alpha}$, placing the relay slightly closer to the source than the midpoint. This asymmetry arises because the destination benefits from both the direct and relay links. For $\alpha = 4$, the optimal position is at $f \approx 0.435$. The relay gain is maximised at moderate SNR and high path-loss exponents.

• 4.

  Network coding: Physical-layer network coding (PNC) achieves 2$\times$ the spectral efficiency of conventional relaying in the two-way relay channel by reducing the exchange from 4 phases to 2. The relay decodes the XOR of the two messages from the superimposed signal, exploiting each end node's side information. Digital NC (3 phases) provides an intermediate $4/3\times$ gain.

• 5.

  Scaling laws: The Gupta--Kumar law establishes that per-node throughput with multi-hop scales as $\Theta(1/\sqrt{n \log n})$ — vanishing as the network grows. The bottleneck is relay congestion: each node must forward traffic for $\Theta(\sqrt{n})$ other pairs. Hierarchical MIMO cooperation (Ozgur--Leveque--Tse) breaks this barrier, achieving $\Theta(1)$ per-node throughput through distributed spatial multiplexing.

• 6.

  Relay rate analysis: Half-duplex relaying is most beneficial in the low-to-moderate SNR regime (coverage extension). At high SNR, the half-duplex penalty of $1/2$ dominates, and direct transmission is superior unless the relay provides a quadratic SNR improvement. The crossover SNR depends on the relay position and path-loss exponent. Full-duplex relaying eliminates this crossover but requires self-interference cancellation.