The Relay Channel Model

A discrete memoryless relay channel consists of:

• A source alphabet $\mathcal{X}$, a relay input alphabet $\mathcal{X}_r$,
• A destination alphabet $\mathcal{Y}$, a relay observation alphabet $\mathcal{Y}_r$,
• A channel transition probability $p(y, y_r | x, x_r)$.

At time $i$, the source sends $X_i \in \mathcal{X}$, the relay sends $X_{r,i} \in \mathcal{X}_r$ (which can depend only on its past observations $Y_r^{i-1}$), the destination receives $Y_i$, and the relay receives $Y_{r,i}$.

The strictly causal constraint on the relay is essential: at time $i$, the relay input $X_{r,i}$ is a function of $(Y_{r,1}, \ldots, Y_{r,i-1})$ only — not of $Y_{r,i}$.

A relay operates in full-duplex mode if it can transmit and receive simultaneously on the same channel. The channel model $p(y, y_r | x, x_r)$ allows arbitrary dependence between the relay's received and transmitted signals.

The full-duplex assumption is the standard model in the information-theoretic literature. It simplifies the analysis but is often difficult to realize in practice due to self-interference: the relay's own transmitted signal overwhelms its receiver.

A relay operates in half-duplex mode if it can either transmit or receive at any given time, but not both. The channel use is divided into two phases:

• Listen phase: relay receives $Y_{r,i}$ (sets $X_{r,i} = 0$ or idle),
• Transmit phase: relay sends $X_{r,i}$ (does not receive).

Let $\alpha \in [0, 1]$ denote the fraction of time the relay listens. The achievable rate depends on the optimization over $\alpha$, introducing a scheduling dimension absent in the full-duplex model.

A $(2^{nR}, n)$ code for the relay channel consists of:

1. A message $W$ uniformly distributed on $\{1, 2, \ldots, 2^{nR}\}$.
2. An encoder that maps $W$ to a codeword $X^n(W) \in \mathcal{X}^n$.
3. A set of relay functions $\{f_i\}_{i=1}^n$ where $X_{r,i} = f_i(Y_{r,1}, \ldots, Y_{r,i-1})$.
4. A decoder that maps $Y^n$ to an estimate $\hat{W}$.

The probability of error is $P_e^{(n)} = \Pr(\hat{W} \neq W)$. A rate $R$ is achievable if there exists a sequence of $(2^{nR}, n)$ codes with $P_e^{(n)} \to 0$. The capacity $C$ is the supremum of all achievable rates.

The relay channel is physically degraded if $X \to (Y_r, X_r) \to Y$ forms a Markov chain, i.e., the destination's observation is a degraded version of what the relay sees. In this case, the relay has a "better view" of the source than the destination.

The relay channel is reversely degraded if $X \to (Y, X_r) \to Y_r$, i.e., the destination has a better channel than the relay.

For the degraded relay channel, decode-and-forward achieves capacity. For the reversely degraded channel, the relay is essentially useless: the capacity equals the direct-link capacity $\max_{p(x)} I(X; Y)$.