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The Relay as a Wireless Repeater

The relay channel — where an intermediate node assists communication between a source and a destination — is one of the oldest open problems in information theory, first studied by van der Meulen (1971) and fundamentally characterised by Cover and El Gamal (1979). Unlike a simple repeater, an information-theoretic relay can process, compress, or re-encode the received signal, giving rise to a rich family of protocols. The practical motivation is clear: a relay placed between a distant source and destination can exploit the shorter propagation paths $S \to R$ and $R \to D$ to achieve rates far exceeding the direct link $S \to D$ , especially in high path-loss environments. LTE-Advanced introduced relay nodes (Type 1 and Type 2), and 5G NR supports integrated access and backhaul (IAB) relaying for coverage extension.

Definition:
The Three-Node Relay Channel

The relay channel consists of three nodes: source $S$ , relay $R$ , and destination $D$ . In each time slot, $S$ transmits signal $x_S$ and $R$ transmits signal $x_R$ (based on its past observations). The received signals are:

$y_R = h_{SR}\,x_S + n_R \quad\text{(at the relay)}$ $y_D = h_{SD}\,x_S + h_{RD}\,x_R + n_D \quad\text{(at the destination)}$

where $h_{SR}, h_{SD}, h_{RD}$ are the channel gains with path-loss $|h_{ij}|^2 \propto d_{ij}^{-\alpha}$ , and $n_R, n_D \sim \mathcal{CN}(0, \sigma^2)$ are AWGN.

The half-duplex constraint requires the relay to either transmit or receive in each time slot (not both simultaneously), dividing the protocol into two phases.

The full-duplex relay channel (where the relay transmits and receives simultaneously) achieves higher rates but requires self-interference cancellation, which remains a practical challenge. Most deployed systems use half-duplex relaying.

Definition:
Relay Protocols — DF, AF, CF

Three canonical half-duplex relay protocols are:

Decode-and-Forward (DF): The relay fully decodes the source message, re-encodes it, and forwards it to the destination. The achievable rate is:

$C_{\text{DF}} = \frac{1}{2}\min\!\left\{ \log_2(1 + \text{SNR}_{SR}),\; \log_2(1 + \text{SNR}_{SD} + \text{SNR}_{RD})\right\}$

Amplify-and-Forward (AF): The relay amplifies its received signal (including noise) and retransmits it. The achievable rate is:

$C_{\text{AF}} = \frac{1}{2}\log_2\!\left(1 + \text{SNR}_{SD} + \frac{\text{SNR}_{SR}\cdot\text{SNR}_{RD}} {1 + \text{SNR}_{SR} + \text{SNR}_{RD}}\right)$

Compress-and-Forward (CF): The relay quantises (compresses) its observation and forwards the compressed version. It does not decode the source message.

The factor $1/2$ reflects the half-duplex loss: each protocol requires two time slots for one message.

DF is optimal when the $S \to R$ link is strong (the relay can decode reliably). AF is simpler and works well when the relay cannot decode. CF is optimal when the relay observation is correlated with but not decodable as the source message, and it approaches the cut-set bound when the relay is close to the destination.

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Theorem: Cut-Set Upper Bound on the Relay Channel

The capacity of the relay channel is upper bounded by the cut-set bound:

$C \leq C_{\text{cut}} = \max_{p(x_S, x_R)} \min\!\left\{ I(X_S, X_R; Y_D),\; I(X_S; Y_R, Y_D \mid X_R)\right\}$

For the Gaussian half-duplex relay channel with optimal time-sharing between the two phases, this evaluates to:

$C_{\text{cut}} = \max_{0 \leq t \leq 1}\min\!\left\{ t\log_2(1 + \text{SNR}_{SR}) + (1-t)\log_2(1 + \text{SNR}_{SD}),\; t\log_2(1 + \text{SNR}_{SD}) + (1-t)\log_2(1 + \text{SNR}_{SD} + \text{SNR}_{RD})\right\}$

where $t$ is the fraction of time allocated to Phase 1 ( $S$ transmits, $R$ listens).

Decode-and-forward achieves the cut-set bound when $\text{SNR}_{SR} \geq \text{SNR}_{SD} + \text{SNR}_{RD}$ (strong relay condition).

The cut-set bound considers two cuts of the network: the cut separating $\{S, R\}$ from $\{D\}$ (limiting the total information flow to $D$ ) and the cut separating $\{S\}$ from $\{R, D\}$ (limiting the information that $S$ can convey). The capacity cannot exceed the minimum of these two cuts — a generalisation of the max-flow min-cut theorem to information theory.

Proof

Cut 1 — information to the destination

The first cut separates $(S, R)$ from $D$ . The maximum information flow across this cut is:

$I(X_S, X_R; Y_D) = \log_2(1 + \text{SNR}_{SD} + \text{SNR}_{RD} + 2\sqrt{\text{SNR}_{SD}\,\text{SNR}_{RD}}\,\rho)$

where $\rho$ is the correlation between $X_S$ and $X_R$ (coherent combining at $D$ ). For independent $X_S, X_R$ :

$I(X_S, X_R; Y_D) = \log_2(1 + \text{SNR}_{SD} + \text{SNR}_{RD})$

Cut 2 — broadcast from source

The second cut separates $S$ from $(R, D)$ . The maximum information that $S$ can convey is:

$I(X_S; Y_R, Y_D \mid X_R) = \log_2(1 + \text{SNR}_{SR} + \text{SNR}_{SD})$

where $R$ and $D$ can jointly process their observations (given knowledge of $X_R$ ).

Half-duplex specialisation

Under the half-duplex constraint with time-sharing parameter $t$ , Phase 1 ( $R$ receives) contributes the broadcast rate $\sim \text{SNR}_{SR}$ , and Phase 2 ( $R$ transmits) contributes the cooperative rate $\sim \text{SNR}_{RD}$ . The cut-set bound becomes the maximum over $t$ of the minimum of the two phase-weighted cuts.

DF achieves this when the relay can always decode (strong $S \to R$ link), since it effectively creates a virtual antenna at $D$ that coherently combines $S$ and $R$ transmissions. $\blacksquare$

Decode-and-Forward vs Amplify-and-Forward

A step-by-step animation of the DF and AF relay protocols. Watch how DF removes noise at the relay before forwarding, while AF amplifies both signal and noise.

In DF, the relay fully decodes the source message (removing noise) and re-encodes before forwarding. In AF, the relay linearly amplifies its received signal, propagating noise to the destination.

Relay Protocol Comparison

Compare the achievable rates of DF, AF, and the cut-set bound as a function of SNR and relay position. Adjust the source- destination distance $d_{SD}$ and the path-loss exponent to see how the relay protocols perform relative to the direct link and the upper bound. Observe that DF approaches the cut-set bound when the relay is close to the source, while AF provides diminishing gains when the $S \to R$ link is weak.

Parameters

Transmit SNR (dB)15

d_{SD}

(source-destination distance)2

Path-loss exponent

\alpha

3

Example: Comparing DF, AF, and Direct Transmission

A source communicates with a destination at distance $d_{SD} = 2$ km with path-loss exponent $\alpha = 4$ and transmit SNR $= 20$ dB (at reference distance 1 km). A relay is placed at the midpoint ( $d_{SR} = d_{RD} = 1$ km).

(a) Compute $\text{SNR}_{SD}$ , $\text{SNR}_{SR}$ , and $\text{SNR}_{RD}$ . (b) Compute the direct transmission rate. (c) Compute the DF and AF achievable rates. (d) Which protocol should be used, and what is the gain over direct transmission?

Solution

SNR computation

(a) With reference SNR $= 100$ (20 dB) at $d_0 = 1$ km:

$\text{SNR}_{SD} = 100 \times 2^{-4} = 100/16 = 6.25$ (8.0 dB) $\text{SNR}_{SR} = 100 \times 1^{-4} = 100$ (20.0 dB) $\text{SNR}_{RD} = 100 \times 1^{-4} = 100$ (20.0 dB)

Direct rate

(b) $C_{\text{direct}} = \log_2(1 + 6.25) = \log_2(7.25) = 2.86$ bits/s/Hz.

DF and AF rates

(c) DF: $C_{\text{DF}} = \frac{1}{2}\min\{ \log_2(101), \log_2(1 + 6.25 + 100)\} = \frac{1}{2}\min\{6.66, 6.74\} = 3.33$ bits/s/Hz.

AF: $C_{\text{AF}} = \frac{1}{2}\log_2(1 + 6.25 + \frac{100 \times 100}{1 + 100 + 100}) = \frac{1}{2}\log_2(1 + 6.25 + 49.75) = \frac{1}{2}\log_2(57.0) = 2.92$ bits/s/Hz.

Protocol selection

(d) DF achieves 3.33 bits/s/Hz — a 16% gain over direct transmission (2.86 bits/s/Hz). AF provides only a 2% gain due to the half-duplex penalty partially offsetting the relay benefit.

DF is preferred here because the $S \to R$ link is strong (20 dB), ensuring reliable decoding at the relay. $\blacksquare$

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Quick Check

In decode-and-forward (DF) relaying, what determines the achievable rate?

The rate of the weaker of the $S \to R$ and $S \to D + R \to D$ links (bottleneck principle)

The sum of all three link capacities

Only the $R \to D$ link capacity

The direct link capacity $C_{SD}$ only

Correction:

The rate of the weaker of the

S \to R

and

S \to D + R \to D

links (bottleneck principle)

DF requires the relay to fully decode, so the rate is limited by $\min\{C_{SR}, C_{SD} + C_{RD}\}$ . If the $S \to R$ link is weak, the relay cannot decode and DF is bottlenecked by this link.

Historical Note: Cover and El Gamal's Relay Channel

1971--1979

The relay channel was introduced by van der Meulen in 1971 as one of the simplest multi-terminal information theory problems. Despite its apparent simplicity, the capacity of the general relay channel remains unknown to this day. In their landmark 1979 paper, Cover and El Gamal established the decode-and-forward and compress-and-forward achievability schemes and the cut-set upper bound. They showed that DF achieves capacity for the degraded relay channel (where the relay's observation is a degraded version of the destination's) and that the capacity is within 0.5 bits of the cut-set bound for the Gaussian relay channel. This 0.5-bit gap has never been closed for the general case, making the relay channel one of the most celebrated open problems in information theory.

Decode-and-Forward (DF)

A relay protocol where the relay fully decodes the source message, re-encodes it (possibly with a different codebook), and transmits it to the destination. The achievable rate is limited by the weaker of the $S \to R$ link (decoding constraint) and the combined $S + R \to D$ link. DF is optimal when the relay is close to the source.

Amplify-and-Forward (AF)

A relay protocol where the relay linearly amplifies its received signal (including noise) and retransmits it without decoding. AF is simple to implement but amplifies noise, making it less efficient than DF when the $S \to R$ link is strong. AF is preferred when the relay lacks the processing capability to decode or when the $S \to R$ link quality is unknown.

Common Mistake: Assuming DF Always Outperforms AF

Mistake:

"Decode-and-forward always achieves higher rates than amplify-and-forward because it fully decodes and removes noise at the relay."

Correction:

DF outperforms AF only when the $S \to R$ link is strong enough for reliable decoding. When the $S \to R$ link is weak (relay far from the source or in deep fade), DF fails entirely — the relay cannot decode and the achievable rate drops to zero. AF, by contrast, always provides some benefit because it forwards an amplified version of the signal (plus noise).

Quantitatively: DF achieves rate $R_{\mathrm{DF}} = \min\{C_{SR}, C_{SD} + C_{RD}\}$ , while AF achieves $R_{\mathrm{AF}} = \log_2\!\left(1 + \text{SNR}_{SD} + \frac{\text{SNR}_{SR}\,\text{SNR}_{RD}}{\text{SNR}_{SR} + \text{SNR}_{RD} + 1}\right)$ .

When $\text{SNR}_{SR}$ is low, $C_{SR}$ bottlenecks DF, but the AF harmonic mean term still contributes. In practice, adaptive relay selection (choosing DF or AF based on instantaneous CSI) is standard in LTE-Advanced and 5G IAB.

Why This Matters: Relay Channels as Distributed MIMO

The relay channel is intimately connected to MIMO systems studied in Chapters 15--18. A single-antenna relay node that helps a single-antenna source communicate with a single-antenna destination creates a virtual $2 \times 1$ MISO channel — the destination sees two independent signal copies (direct and relayed), just as an MRC receiver combines signals from two transmit antennas.

The key difference is the half-duplex constraint: unlike a co-located $2 \times 1$ MISO, the relay must listen before transmitting, consuming a fraction of the transmission time. This is why the cooperative DMT (Section 22.2) resembles the MISO DMT $d(r) = M(1 - r)$ but with a rate penalty.

Book MIMO (Chapters 5--8) develops the co-located MIMO theory that underlies this analogy, including the capacity expressions, beamforming strategies, and the diversity-multiplexing tradeoff that Section 22.2 extends to the distributed setting.

Key Takeaway

Three relay protocols, three regimes. (1) Decode-and-forward is optimal when the relay is close to the source ( $S \to R$ link strong); the achievable rate equals the cut-set bound for the degraded relay channel. (2) Compress-and-forward excels when the relay is close to the destination and cannot decode the source message. (3) Amplify-and-forward is the simplest protocol and provides gain in all regimes, but is suboptimal compared to DF and CF in their respective strong-link regimes. The capacity of the general relay channel remains unknown — the gap to the cut-set bound is at most 0.5 bits for the Gaussian case.

Cut-Set Bound

An information-theoretic upper bound on the capacity of multi-terminal networks, obtained by considering all possible partitions (cuts) of the network nodes into two sets and bounding the information flow across each cut. For the relay channel, the cut-set bound involves two cuts and is achieved by DF under the strong relay condition.

Related: Decode-and-Forward (DF)

Relay Channel Fundamentals