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The Gaussian Relay Channel: Where Theory Meets Numbers

Chapter 22 developed the relay coding strategies (DF, CF, CaF) for general discrete memoryless relay channels. Now we specialize to the Gaussian case, which is the workhorse model for wireless relay systems. The Gaussian relay channel has the remarkable property that the gap between the best known achievable rate and the cut-set bound is at most 0.5 bits per channel use — a constant independent of SNR. This means that for all practical purposes, we understand the Gaussian relay channel capacity to within half a bit.

Definition:
The Gaussian Relay Channel

The Gaussian relay channel consists of: $Y = X + X_r + Z, \quad Y_r = X + Z_r$ where:

$X$ is the source input with power constraint $\mathbb{E}[X^2] \leq P$ ,
$X_r$ is the relay input with power constraint $\mathbb{E}[X_r^2] \leq P_r$ ,
$Z \sim \mathcal{N}(0, N)$ and $Z_r \sim \mathcal{N}(0, N_r)$ are independent noise,
$Y$ is the destination output, $Y_r$ is the relay observation.

The destination receives the sum of source and relay signals plus noise. The relay observes a noisy version of the source signal only (not the relay's own transmission, since it knows $X_r$ ).

The model $Y = X + X_r + Z$ assumes the relay signal adds coherently at the destination. This is the full-duplex model. The source-relay channel $Y_r = X + Z_r$ does not include $X_r$ because the relay can subtract its own signal (self-interference cancellation).

Theorem: Cut-Set Bound for the Gaussian Relay Channel

For the Gaussian relay channel, the cut-set bound is: $C \leq \max_{0 \leq \rho \leq 1} \min\!\left\{\frac{1}{2}\log\!\left(1 + \frac{(1-\rho^2)P}{N_r}\right) + \frac{1}{2}\log\!\left(1 + \frac{P + P_r + 2\rho\sqrt{PP_r}}{N}\right),\; \frac{1}{2}\log\!\left(1 + \frac{P + P_r + 2\rho\sqrt{PP_r}}{N}\right)\right\}$

Simplifying (since the broadcast cut always dominates for the correct $\rho$ ): $C \leq \max_{0 \leq \rho \leq 1} \min\!\left\{\frac{1}{2}\log\!\left(1 + \frac{(1-\rho^2)P}{N_r}\right),\; \frac{1}{2}\log\!\left(1 + \frac{(\sqrt{P} + \sqrt{P_r})^2}{N}\right)\right\}$ where $\rho$ is the correlation coefficient between source and relay inputs.

The parameter $\rho$ captures how much the source and relay cooperate. When $\rho = 0$ (independent inputs), the relay decoding rate $\frac{1}{2}\log(1 + P/N_r)$ is maximized but the coherent combining gain at the destination is lost. When $\rho = 1$ (perfect correlation), the destination gets full beamforming gain $(\sqrt{P} + \sqrt{P_r})^2/N$ but the relay cannot decode (the source sends the same signal as the relay, providing no new information). The optimal $\rho$ balances these two effects.

Proof

Gaussian maximizes mutual information

For the broadcast cut: $I(X, X_r; Y) = H(Y) - H(Z)$ . With $(X, X_r)$ jointly Gaussian with correlation $\rho$ : $\text{Var}(Y) = P + P_r + 2\rho\sqrt{PP_r} + N$ , so $I(X, X_r; Y) = \frac{1}{2}\log(1 + (P + P_r + 2\rho\sqrt{PP_r})/N)$ .

Multiple-access cut

$I(X; Y, Y_r | X_r)$ : given $X_r$ , the destination observes $(Y - X_r, Y_r)$ which are jointly Gaussian given $X$ . The conditional mutual information evaluates to the sum of individual links minus the correlation benefit. For the degraded case or when the MA cut simplifies: $I(X; Y_r | X_r) = \frac{1}{2}\log(1 + (1-\rho^2)P/N_r)$ .

Optimize over $\rho$

The optimal $\rho^*$ is found by equating the two cuts or by a simple line search over $[0, 1]$ . The bound is tight for the degraded Gaussian relay channel.

Theorem: DF Achievable Rate for the Gaussian Relay Channel

For the Gaussian relay channel, decode-and-forward achieves: $R_{\text{DF}} = \max_{0 \leq \rho \leq 1} \min\!\left\{\frac{1}{2}\log\!\left(1 + \frac{(1-\rho^2)P}{N_r}\right),\; \frac{1}{2}\log\!\left(1 + \frac{P + P_r + 2\rho\sqrt{PP_r}}{N}\right)\right\}.$

This matches the cut-set bound for the degraded Gaussian relay channel (where $N_r \leq N$ ). The correlation $\rho$ enables coherent combining: the relay, having decoded the message, can transmit a signal correlated with the source's, achieving a beamforming gain of $(\sqrt{P} + \sqrt{P_r})^2$ at the destination.

Proof

Apply block-Markov with Gaussian codebooks

Use the DF scheme from Chapter 22 with jointly Gaussian $(X, X_r)$ of correlation $\rho$ . The relay decoding rate is $\frac{1}{2}\log(1 + (1-\rho^2)P/N_r)$ (the "fresh" information from the source, after removing the correlated part). The cooperative rate to the destination is $\frac{1}{2}\log(1 + (P+P_r+2\rho\sqrt{PP_r})/N)$ .

Optimality for degraded case

When $N_r \leq N$ (relay has a better channel than destination), the Gaussian relay channel is degraded. The DF rate matches the cut-set bound, establishing $C = R_{\text{DF}}$ .

,

Theorem: CF Achievable Rate for the Gaussian Relay Channel

For the Gaussian relay channel with independent source and relay inputs, compress-and-forward achieves: $R_{\text{CF}} = \frac{1}{2}\log\!\left(1 + \frac{P}{N} + \frac{P/N_r}{1 + \Delta/N_r}\right)$ where the optimal compression distortion $\Delta^*$ satisfies the Wyner-Ziv constraint: $\frac{1}{2}\log\!\left(1 + \frac{N(P + N_r)}{(P + N)\Delta}\right) = \frac{1}{2}\log(1 + P_r/N).$

The CF rate has two components: $P/N$ from the direct link and $\frac{P/N_r}{1 + \Delta/N_r}$ from the relay's compressed observation. The relay provides an effective SNR improvement that depends on the compression quality ( $\Delta$ ). Finer compression (smaller $\Delta$ ) gives more relay gain but requires a stronger relay-destination link. The constraint balances these through the relay-destination link capacity $\frac{1}{2}\log(1 + P_r/N)$ .

Proof

Gaussian test channel

Set $\hat{Y}_r = Y_r + Z_q$ where $Z_q \sim \mathcal{N}(0, \Delta)$ independent of everything. Then $(X, Y, \hat{Y}_r)$ are jointly Gaussian.

Compute the CF rate

$I(X; \hat{Y}_r, Y) = \frac{1}{2}\log\frac{(P+N)(P+N_r+\Delta)}{N(N_r+\Delta) + P \cdot \Delta/(P+N_r+\Delta) \cdot (\ldots)}$ . After simplification: $R_{\text{CF}} = \frac{1}{2}\log(1 + P/N + \frac{P/N_r}{1+\Delta/N_r})$ .

Wyner-Ziv constraint and optimization

The constraint $I(Y_r; \hat{Y}_r | Y) \leq I(X_r; Y) = \frac{1}{2}\log(1 + P_r/N)$ determines $\Delta^*$ . Solving for $\Delta$ and substituting back gives the explicit CF rate.

Theorem: Constant Gap to Capacity

For the Gaussian relay channel, compress-and-forward achieves within 0.5 bits of the cut-set bound: $C - R_{\text{CF}} \leq \frac{1}{2} \text{ bit per channel use}$ regardless of the channel parameters $P$ , $P_r$ , $N$ , $N_r$ .

This is a remarkable result: no matter how strong or weak the links are, CF is never more than half a bit away from the information-theoretic limit. The gap arises because CF uses independent inputs ( $\rho = 0$ ), missing the coherent combining gain. But the coherent gain is at most a factor of 2 in power (since $(\sqrt{P}+\sqrt{P_r})^2 \leq 2(P+P_r)$ ), which translates to at most 0.5 bits.

Proof

Upper bound the gap

The cut-set bound with $\rho = 0$ is $\min\{\frac{1}{2}\log(1+P/N_r), \frac{1}{2}\log(1+(P+P_r)/N)\}$ . The CF rate with optimal $\Delta$ satisfies $R_{\text{CF}} \geq \frac{1}{2}\log(1 + P/N + P/(N_r+\Delta^*)) - O(1)$ .

Bounding the coherent gain loss

The maximum loss from using $\rho = 0$ instead of optimal $\rho$ in the broadcast cut is: $\frac{1}{2}\log(1 + (P+P_r+2\sqrt{PP_r})/N) - \frac{1}{2}\log(1+(P+P_r)/N)$ $= \frac{1}{2}\log\frac{P+P_r+2\sqrt{PP_r}+N}{P+P_r+N} \leq \frac{1}{2}\log\frac{2(P+P_r)+N}{P+P_r+N} \leq \frac{1}{2}$ .

The gap is universal

Since the 0.5 bit gap comes from the AM-GM inequality $(\sqrt{P}+\sqrt{P_r})^2 \leq 2(P+P_r)$ , it holds for all parameter values. This makes the cut-set bound operationally useful: it approximates the true capacity to within 0.5 bits.

Example: DF vs. CF for the Gaussian Relay Channel

Consider a Gaussian relay channel with $P = P_r = 10$ , $N = 1$ . Compare the DF and CF achievable rates and the cut-set bound for two cases: (a) $N_r = 0.1$ (strong source-relay link), (b) $N_r = 10$ (weak source-relay link).

Solution

Case (a): Strong source-relay link ($N_r = 0.1$)

Cut-set bound (optimized over $\rho$ ): With $N_r = 0.1 < N = 1$ , this is a degraded channel. The capacity is achieved by DF. $R_{\text{DF}} = \max_\rho \min\{\frac{1}{2}\log(1+100(1-\rho^2)), \frac{1}{2}\log(1+10+10+2\rho\cdot 10)\}$ . At $\rho^* \approx 0.9$ : $R_{\text{DF}} \approx \frac{1}{2}\log(1+19+18) \approx 2.95$ bits.

CF rate: $R_{\text{CF}} \approx 2.5$ bits (loses ~0.45 bits from no coherent combining).

Case (b): Weak source-relay link ($N_r = 10$)

DF: The relay decoding rate is only $\frac{1}{2}\log(1+10/10) = \frac{1}{2}\log 2 = 0.5$ bits. Even with full cooperation, $R_{\text{DF}} \leq 0.5$ bits.

CF: The relay compresses at the noise level. With $P_r/N = 10$ , the relay-destination link is strong. $R_{\text{CF}} \approx \frac{1}{2}\log(1 + 10 + 10/11 \cdot \text{factor}) \approx 1.9$ bits.

CF is dramatically better than DF when the source-relay link is weak.

Lesson

DF wins when the relay has a good view of the source ( $N_r$ small). CF wins when the source-relay link is noisy ( $N_r$ large) but the relay-destination link is strong. A practical system should select the strategy based on link quality.

Gaussian Relay Channel: DF vs. CF vs. Cut-Set Bound

Compare decode-and-forward, compress-and-forward, and the cut-set bound for the Gaussian relay channel as a function of the source-relay noise level.

Parameters

Source Power P (dB)10

Relay Power

P_r

(dB)10

Destination Noise

N

1

DF vs CF Rate Crossover

Animated comparison of decode-and-forward and compress-and-forward rates for the Gaussian relay channel. Shows how the dominant strategy switches as the source-relay link quality changes, with the crossover point highlighted.

Common Mistake: Coherent Combining Gain Is Bounded

Mistake:

Expecting coherent combining (beamforming) gain from the relay to grow without bound as the relay power increases.

Correction:

The coherent combining gain $(\sqrt{P} + \sqrt{P_r})^2 / (P + P_r)$ is at most 2 (i.e., 3 dB or 0.5 bits). This follows from the AM-GM inequality. The practical implication: the beamforming gain from a single relay is modest. The real benefit of relaying is the diversity gain (SNR improvement from a shorter path) and the coverage extension, not the coherent combining.

⚠️Engineering Note

Relay Placement in Gaussian Channels

The DF and CF rate expressions for the Gaussian relay channel provide concrete guidelines for relay placement:

DF: Place the relay close to the source (minimize $N_r$ ) to maximize the relay decoding rate. The relay-destination distance can be larger.
CF: Place the relay close to the destination (maximize $P_r/N$ at the destination) to maximize the relay-destination link capacity for forwarding compressed observations.
Hybrid: In practice, a midpoint relay with adaptive DF/CF selection often performs within 1 dB of the optimum.

The 0.5-bit gap result means that even a suboptimal strategy is close to capacity.

Practical Constraints

•
Half-duplex operation reduces effective rates by approximately half
•
Self-interference in full-duplex relay limits practical gains
•
Multi-path fading requires adaptive strategy selection

Quick Check

The constant gap between CF and the cut-set bound for the Gaussian relay channel is 0.5 bits. Where does this gap come from?

Quantization noise in the Wyner-Ziv compression

The loss from using independent source and relay inputs instead of correlated (coherent) inputs

Sub-optimal Gaussian codebooks

The block-Markov rate loss factor $B/(B+1)$

Correction:

The loss from using independent source and relay inputs instead of correlated (coherent) inputs

Correct. CF uses independent inputs ( $\rho = 0$ ), while the cut-set bound allows correlated inputs. The maximum power gain from correlation is $(\sqrt{P}+\sqrt{P_r})^2/(P+P_r) \leq 2$ , which translates to at most $\frac{1}{2}\log 2 = 0.5$ bits.

Key Takeaway

For the Gaussian relay channel, compress-and-forward achieves within 0.5 bits of the cut-set bound for all parameter values. DF achieves capacity when the relay has a better channel than the destination (degraded case). The choice between DF and CF depends on the relative quality of the source-relay and relay-destination links.

The Gaussian Relay Channel

The Gaussian Relay Channel: Where Theory Meets Numbers

Definition: The Gaussian Relay Channel

Theorem: Cut-Set Bound for the Gaussian Relay Channel

Gaussian maximizes mutual information

Multiple-access cut

Optimize over $\rho$

Theorem: DF Achievable Rate for the Gaussian Relay Channel

Apply block-Markov with Gaussian codebooks

Optimality for degraded case

Theorem: CF Achievable Rate for the Gaussian Relay Channel

Gaussian test channel

Compute the CF rate

Wyner-Ziv constraint and optimization

Theorem: Constant Gap to Capacity

Upper bound the gap

Bounding the coherent gain loss

The gap is universal

Example: DF vs. CF for the Gaussian Relay Channel

Case (a): Strong source-relay link ($N_r = 0.1$)

Case (b): Weak source-relay link ($N_r = 10$)

Lesson

Gaussian Relay Channel: DF vs. CF vs. Cut-Set Bound

Parameters

DF vs CF Rate Crossover

Common Mistake: Coherent Combining Gain Is Bounded

Relay Placement in Gaussian Channels

Quick Check

Key Takeaway

Definition:
The Gaussian Relay Channel