Ferkans — Interactive Telecom Tutor

Broadcasting a Source to Multiple Receivers

We now turn the problem around: instead of multiple encoders and one decoder, we consider one encoder broadcasting a common source to multiple receivers, each of which has different side information. This is the dual of the MAC source coding problem, and it arises naturally in broadcasting, caching, and content distribution systems.

The central question remains the same: can we separate source compression from channel coding, or must we design them jointly?

Definition:
Broadcasting a Source over a BC

A transmitter observes a source $S^n$ drawn i.i.d. $\sim P_S$ and wishes to communicate it to two receivers through a degraded broadcast channel $P_{Y_1 Y_2 | X}$ . Receiver $k$ has side information $T_k^n$ correlated with $S^n$ through a joint distribution $P_{S T_1 T_2}$ .

A $(2^{nR}, n)$ joint source–channel code consists of:

Encoder: a mapping $f: \mathcal{S}^n \to \mathcal{X}^n$
Decoder $k$ : a mapping $g_k: \mathcal{Y}_k^n \times \mathcal{T}_k^n \to \hat{\mathcal{S}}^n$

For lossless reconstruction, the error criterion is $P_{e,k}^{(n)} = \Pr\bigl[S^n \neq g_k(Y_k^n, T_k^n)\bigr] \to 0$ for $k = 1, 2$ .

For lossy reconstruction, we require $\mathbb{E}\bigl[d(S^n, \hat{S}_k^n)\bigr] \leq D_k$ for each receiver.

Theorem: Separation for the Degraded BC

For a degraded broadcast channel $X \to Y_1 \to Y_2$ and a source $S$ with side information $T_1, T_2$ at the receivers (where $T_1$ is a degraded version of $T_2$ , i.e., $S \to T_2 \to T_1$ forms a $X \multimap Y \multimap Z$ chain), separation of source and channel coding is optimal.

Specifically, the source is transmissible at distortions $(D_1, D_2)$ if and only if there exist rates $R_1 \leq R_2$ such that:

$R_k \geq R_{S|T_k}(D_k)$ for $k = 1, 2$ (the conditional rate-distortion functions)
$(R_1, R_2)$ is achievable on the degraded BC via superposition coding

The degradedness conditions on both the channel and the side information create a natural nesting: receiver 2 (with better side information $T_2$ ) needs fewer bits, and receiver 1 (with worse side information $T_1$ ) needs more. This nesting matches the layered structure of superposition coding on the degraded BC, where the "base layer" (decoded by both receivers) carries the coarse description and the "enhancement layer" (decoded only by the stronger receiver) carries the refinement.

Proof

Source coding layer

Use successive refinement source coding (Chapter 6): encode $S^n$ into a base description at rate $R_1$ (decodable with side information $T_1$ ) and a refinement at additional rate $R_2 - R_1$ (decodable with side information $T_2$ ). The degradedness of the side information $S \to T_2 \to T_1$ ensures that successive refinement is optimal (Cover and Thomas, Theorem 13.5.1).

Channel coding layer

Transmit the base layer using the "cloud center" codebook of superposition coding on the degraded BC, and the refinement layer using the "satellite" codebook. Receiver 1 decodes only the base layer; receiver 2 decodes both layers.

Optimality

The converse follows from the converse for the degraded BC capacity region (Chapter 15) combined with the converse for the conditional rate-distortion function. The degradedness of both the channel and the side information ensures that no joint scheme can outperform the separated design. $\blacksquare$

When Degradedness Fails

The degradedness conditions are essential. When either the channel or the side information structure is not degraded, separation can fail. Consider the following scenario: a transmitter broadcasts a binary source to two receivers with complementary side information (receiver 1 knows the odd-indexed bits, receiver 2 knows the even-indexed bits). The optimal strategy may involve a joint design that interleaves source and channel coding in a way that exploits the complementary side information structure — something that separate coding cannot do.

The general non-degraded case remains an active area of research, with only partial results known.

Example: Broadcasting a Binary Source with Erasure Side Information

A transmitter observes $S^n$ with $S_i \sim \text{Bern}(1/2)$ i.i.d. and broadcasts over a degraded BC. Receiver 1 has no side information ( $T_1 = \emptyset$ ). Receiver 2 observes $S^n$ through a binary erasure channel with erasure probability $\epsilon$ , i.e., $T_{2,i} = S_i$ with probability $1 - \epsilon$ and $T_{2,i} = \text{?}$ with probability $\epsilon$ .

Both receivers want lossless reconstruction. Determine the minimum channel rates needed under separation.

Solution

Rate for receiver 1

Receiver 1 has no side information, so it needs the full source entropy: $R_1 \geq H(S) = 1 \text{ bit/symbol}.$

Rate for receiver 2

Receiver 2 knows $S_i$ at positions that are not erased. The conditional entropy is $H(S | T_2) = \epsilon \cdot 1 + (1-\epsilon) \cdot 0 = \epsilon \text{ bits/symbol}.$ So $R_2 \geq \epsilon$ .

Channel requirement

Under separation, we need the degraded BC to support rates $(R_1, R_2)$ with $R_1 \geq 1$ and $R_2 \geq \epsilon$ . The base layer (at rate $\epsilon$ ) is decoded by both receivers, and the enhancement layer (at rate $1 - \epsilon$ ) is decoded only by receiver 1. Since $T_2$ is a degraded version of $S$ and receiver 1 has no side information (trivially degraded), separation is optimal by the theorem above.

Common Mistake: Separation Always Holds for the BC

Mistake:

Assuming that separation is optimal for any broadcast scenario because it works for the degraded case.

Correction:

Separation is optimal for the degraded BC with degraded side information. For general (non-degraded) broadcast channels or non-degraded side information structures, joint source–channel coding can strictly outperform separation. The general characterization remains open.

Definition:
Hybrid Digital–Analog Coding

A hybrid coding scheme combines digital (separate) and analog (uncoded) transmission. The encoder splits the source into two parts: one is digitally compressed and channel-coded, the other is transmitted using an analog mapping (e.g., linear scaling for Gaussian sources over Gaussian channels).

Formally, $X^n = \alpha \cdot \phi(S^n) + \beta \cdot c(M)$ where $\phi$ is an analog mapping, $c(M)$ is a channel codeword carrying the compressed message $M$ , and $\alpha, \beta$ control the power allocation.

Hybrid coding can outperform pure digital (separate) coding in multi-terminal settings because the analog component preserves the source structure that digital compression discards.

Hybrid digital–analog coding

A joint source–channel coding strategy that combines digital compression/channel coding with an analog (uncoded) mapping of the source to the channel input. Particularly effective for broadcasting to heterogeneous receivers.

Quick Check

For a degraded BC with degraded side information at the receivers, which coding strategy is optimal?

Separate source and channel coding (successive refinement + superposition coding)

Joint source–channel coding is always needed

Uncoded (analog) transmission

Correction:

Separate source and channel coding (successive refinement + superposition coding)

The degradedness of both the channel and the side information creates a natural nesting that matches the layered structure of superposition coding. Separation is optimal in this case.

🔧Engineering Note

Layered Coding in DVB and ATSC 3.0

The separation result for degraded BCs provides the theoretical foundation for layered coding in broadcast standards. DVB-T2 and ATSC 3.0 use layered division multiplexing (LDM), where a base layer (robust, low-rate) is superimposed with an enhancement layer (high-rate, for receivers with better channel conditions). This is precisely the superposition coding strategy predicted by information theory.

However, these standards use separation: the source (video) is compressed using H.265/HEVC independently of the channel code. The information-theoretic results here confirm that for the degraded broadcast scenario typical of terrestrial broadcasting, this separation architecture is indeed optimal — there is no performance penalty from the modular design.

Key Takeaway

Separation is optimal for the degraded BC with degraded side information, matching the layered structure of superposition coding with successive refinement source coding. For non-degraded settings, joint source–channel coding can provide strict gains, and hybrid digital–analog schemes offer a practical middle ground.

Source Coding for the Broadcast Channel