Ferkans — Interactive Telecom Tutor

When the Relay Cannot Decode

Decode-and-forward requires the relay to fully decode the source message, which limits the rate to the source-relay link capacity. But what if the source-relay link is weak? The relay still receives a noisy observation of the source signal, and this observation is correlated with the destination's received signal. Even if the relay cannot decode the message, it can compress its observation and forward the compressed version to the destination, providing useful side information.

This is compress-and-forward (CF), and the key insight is that the relay uses Wyner-Ziv coding — lossy compression with decoder side information — because the destination already has its own observation $Y^n$ , which is correlated with the relay's observation $Y_r^n$ . The compression can exploit this correlation, requiring fewer bits to describe $Y_r$ than if the destination had no side information.

Definition:
Compress-and-Forward Strategy

In compress-and-forward, the relay performs the following operations:

Quantize: compress its observation $Y_r^n$ to a quantized version $\hat{Y}_r^n$ using a Wyner-Ziv codebook, exploiting the correlation between $Y_r$ and $Y$ (the destination's observation serves as decoder side information).
Bin: map the quantized sequence to a bin index using Wyner-Ziv binning.
Forward: encode the bin index and transmit it to the destination over the relay-destination link.

The destination uses its own observation $Y^n$ together with the bin index to reconstruct $\hat{Y}_r^n$ , then jointly decodes the source message from $(Y^n, \hat{Y}_r^n)$ .

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Compress-and-Forward

A relay strategy where the relay compresses its channel observation using Wyner-Ziv coding (exploiting decoder side information) and forwards the compressed description to the destination, rather than attempting to decode the source message.

Related: Wyner-Ziv Coding

Wyner-Ziv Coding

Lossy source coding with decoder side information. The encoder compresses a source without access to the side information available at the decoder, achieving a rate equal to the conditional rate-distortion function.

Related: Compress-and-Forward

Theorem: Compress-and-Forward Achievable Rate

For the discrete memoryless relay channel $p(y, y_r | x, x_r)$ , the following rate is achievable using compress-and-forward: $R_{\text{CF}} = \max_{p(x) p(x_r) p(\hat{y}_r | y_r, x_r)} I(X; \hat{Y}_r, Y \mid X_r)$ subject to the constraint $I(Y_r; \hat{Y}_r \mid Y, X_r) \leq I(X_r; Y).$

Here $\hat{Y}_r$ is the compressed version of the relay's observation $Y_r$ . The constraint ensures the destination can decode the compression index sent by the relay.

The achievable rate $I(X; \hat{Y}_r, Y | X_r)$ is the mutual information between the source and the combined observation at the destination (its own $Y$ plus the relay's compressed $\hat{Y}_r$ ). The constraint $I(Y_r; \hat{Y}_r | Y, X_r) \leq I(X_r; Y)$ balances the compression rate against the relay-to-destination link capacity: the relay can only forward as many bits as the relay-destination link supports, and these bits must suffice to describe $\hat{Y}_r$ to within the Wyner-Ziv rate (given $Y$ as side information).

Intuitively, what happens is a trade-off: finer compression (smaller distortion) gives the destination more information about $Y_r$ and hence a higher rate $I(X; \hat{Y}_r, Y | X_r)$ , but requires more bits to describe, potentially violating the relay-destination link constraint.

Proof

Codebook generation

Fix $p(x)$ , $p(x_r)$ , and $p(\hat{y}_r | y_r, x_r)$ .

Generate $2^{nR}$ i.i.d. source codewords $x^n(w)$ .
Generate $2^{nR_0}$ i.i.d. relay codewords $x_r^n(s)$ , $s \in [1:2^{nR_0}]$ .
For each relay codeword $x_r^n(s)$ , generate $2^{n\hat{R}}$ compression codewords $\hat{y}_r^n(l | s)$ , each i.i.d. $\sim \prod_i p(\hat{y}_{r,i} | x_{r,i}(s))$ .
Partition the compression codewords into $2^{nR_0}$ bins of size $2^{n(\hat{R} - R_0)}$ .

Relay operation

In block $b$ , the relay receives $y_r^n$ and finds a compression codeword $\hat{y}_r^n(l | s)$ jointly typical with $(y_r^n, x_r^n(s))$ . This succeeds if $\hat{R} > I(Y_r; \hat{Y}_r | X_r)$ . The relay then transmits the bin index of $l$ in the next block using codeword $x_r^n(\text{bin}(l))$ .

Destination decoding

The destination knows $x_r^n(s)$ (from the previous block's relay transmission). It searches the bin for the unique $\hat{y}_r^n(l | s)$ jointly typical with $(y^n, x_r^n(s))$ . This succeeds if $\hat{R} - R_0 < I(\hat{Y}_r; Y | X_r)$ , i.e., $I(Y_r; \hat{Y}_r | X_r) - R_0 < I(\hat{Y}_r; Y | X_r)$ . Rearranging: $R_0 > I(Y_r; \hat{Y}_r | Y, X_r)$ .

Having recovered $\hat{y}_r^n$ , the destination jointly decodes $w$ from $(y^n, \hat{y}_r^n, x_r^n)$ , succeeding if $R < I(X; \hat{Y}_r, Y | X_r)$ .

Rate constraint

The relay must be able to transmit the bin index: $R_0 \leq I(X_r; Y)$ (the relay-destination link capacity). Combining with $R_0 > I(Y_r; \hat{Y}_r | Y, X_r)$ : $I(Y_r; \hat{Y}_r | Y, X_r) \leq I(X_r; Y).$ This is the Wyner-Ziv constraint. The achievable rate is $R_{\text{CF}} = I(X; \hat{Y}_r, Y | X_r)$ .

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Example: When CF Outperforms DF

Consider a relay channel where the source-relay link has capacity $I(X; Y_r | X_r) = 0.3$ bits (weak relay link) but the relay-destination link has capacity $I(X_r; Y) = 2$ bits (strong relay-destination link). The direct link gives $I(X; Y) = 0.5$ bits. Suppose that with CF and optimal compression, $I(X; \hat{Y}_r, Y | X_r) = 0.7$ bits and $I(Y_r; \hat{Y}_r | Y, X_r) = 1.5$ bits. Compare DF and CF.

Solution

DF rate

$R_{\text{DF}} = \min\{0.3, I(X, X_r; Y)\}$ . Since the relay must decode, the rate is at most $0.3$ bits — the weak source-relay link is the bottleneck.

CF rate

Check the Wyner-Ziv constraint: $I(Y_r; \hat{Y}_r | Y, X_r) = 1.5 \leq I(X_r; Y) = 2$ . Satisfied. The CF rate is $R_{\text{CF}} = 0.7$ bits.

Comparison

CF achieves $0.7$ bits vs. DF's $0.3$ bits — more than double the rate. CF wins because it does not require the relay to decode: even a noisy observation is useful as compressed side information. The strong relay-destination link allows the relay to forward a fine-grained compressed description.

Common Mistake: Source and Relay Inputs Are Independent in Standard CF

Mistake:

Trying to optimize $R_{\text{CF}}$ over joint distributions $p(x, x_r)$ as in decode-and-forward.

Correction:

In the standard CF scheme, the source and relay inputs are independent: $p(x, x_r) = p(x) p(x_r)$ . This is because the relay does not decode the source message, so it cannot coordinate its transmission with the source's current codeword. The relay's codeword carries only the compression index from the previous block. (More advanced schemes like "partial decode-and-compress" allow dependent inputs.)

CF Compression-Rate Tradeoff

Explore how the compression quality at the relay affects the CF achievable rate. Finer compression provides better side information but requires more bits on the relay-destination link.

Parameters

Source-Relay SNR (dB)0

Source-Destination SNR (dB)5

Relay-Destination SNR (dB)15

Compress-and-Forward Signal Flow

Animated walkthrough of compress-and-forward: the relay receives a noisy observation, compresses it via Wyner-Ziv coding, and forwards the compressed version to the destination, which jointly decodes using both its own observation and the relay's compressed signal.

CF as 'Hash-and-Forward'

An alternative viewpoint, due to Ahlswede and Han, interprets compress-and-forward as "hash-and-forward": the relay sends a hash (random bin index) of its observation sequence. The destination, having its own correlated observation, can use the hash to disambiguate among the possible relay observations that fall in the same bin. This viewpoint emphasizes that the relay need not produce an explicit reconstruction of $Y_r$ ; it only needs to send enough information for the destination to jointly decode with its own observation. This perspective generalizes more naturally to networks with multiple relays.

Key Takeaway

Compress-and-forward does not require the relay to decode the source message — it only compresses its observation using Wyner-Ziv coding and forwards the compressed version. CF is superior to DF when the source-relay link is weak but the relay-destination link is strong, because even noisy side information helps the destination.

Compress-and-Forward

When the Relay Cannot Decode

Definition: Compress-and-Forward Strategy

Compress-and-Forward

Wyner-Ziv Coding

Theorem: Compress-and-Forward Achievable Rate

Codebook generation

Relay operation

Destination decoding

Rate constraint

Example: When CF Outperforms DF

DF rate

CF rate

Comparison

Common Mistake: Source and Relay Inputs Are Independent in Standard CF

CF Compression-Rate Tradeoff

Parameters

Compress-and-Forward Signal Flow

CF as 'Hash-and-Forward'

Key Takeaway

Definition:
Compress-and-Forward Strategy