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Why Heterogeneous Channels Matter

The symmetric MIMO BC of Chapter 5 assumes every user has the same (up to rotation) channel. Real systems are asymmetric: users at the cell edge face weaker channels than users near the base station; high- mobility users face additional fading; indoor users experience penetration loss. In short, the channel qualities $\rho_k$ differ substantially across users in any deployed system.

This asymmetry has immediate operational consequences for coded caching. Multicasting — the natural delivery mode of MAN — is bottlenecked by the worst user. Transmitting a single coded message at rate $C_{\min} = \log_2(1 + \rho_{\min})$ means users with stronger channels waste their capacity. This chapter studies how caching interacts with channel heterogeneity and what the optimal delivery strategy looks like.

Definition:
Degraded Gaussian Broadcast Channel

The degraded Gaussian broadcast channel with $K$ users and a single-antenna transmitter has observations $y_k \;=\; x + \mathbf{w}_{k}, \quad \mathbf{w}_{k} \sim \mathcal{CN}(0, \sigma_k^2),$ with $\sigma_1^2 \leq \sigma_2^2 \leq \ldots \leq \sigma_K^2$ (equivalently, $\rho_1 \geq \rho_2 \geq \ldots \geq \rho_K$ , where $\rho_k = P/\sigma_k^2$ is user $k$ 's received SNR at transmit power $P$ ).

The channel is called "degraded" because user 2's observation can be constructed from user 1's observation by adding independent Gaussian noise — every downstream user sees a noisier version. User 1 is the strongest user; user $K$ is the weakest.

Many deployed wireless channels are approximately degraded: line-of-sight users with good SNR receive everything the non-line-of-sight users receive (modulo the residual noise difference). In a multi-antenna setting, "degradation" is more subtle — users with orthogonal channels are not degraded versions of one another. Chapter 6 focuses on the degraded scalar BC.

Definition:
Cache-Aided Degraded Broadcast Channel

The cache-aided degraded BC adds to the preceding model:

A library $\mathcal{W} = \{W_1, \ldots, W_{N}\}$ at the transmitter.
Per-user cache of size $M F$ bits, populated in the off-peak placement phase.
Demand vector $\mathbf{d} \in [N]^K$ revealed at delivery time.

The objective is the per-user symmetric rate $R_{\text{sym}} = \min_k R_k$ — the common rate at which all users can be served. In the worst-case-demand framework, this is the relevant figure of merit. Alternative metrics (sum-rate, weighted sum-rate) apply in different settings.

Theorem: Capacity Region of the Degraded BC (No Caching)

For the $K$ -user degraded Gaussian BC with user SNRs $\rho_1 \geq \ldots \geq \rho_K$ , the capacity region is $\mathcal{C}_{\text{BC}} \;=\; \bigl\{ (R_1, \ldots, R_K) : R_k \leq \log_2\!\left(1 + \frac{\alpha_k \rho_k}{1 + \rho_k \sum_{j < k} \alpha_j}\right), \sum_k \alpha_k \leq 1, \alpha_k \geq 0 \bigr\}.$ The region is achieved by superposition coding with rate-split parameters $(\alpha_1, \ldots, \alpha_K)$ .

Superposition coding layers messages for different users. The strongest user decodes everything; each subsequent user decodes only its layer, treating higher layers as noise. The rate-split $(\alpha_k)$ determines how power is allocated across layers.

Proof

Achievability via superposition

Let $x = \sum_k \sqrt{\alpha_k P} u_k$ , where $u_k \sim \mathcal{CN}(0, 1)$ are independent. User $j$ decodes layers $k = K, K-1, \ldots, j$ successively. Layer $k$ 's rate is $\log_2(1 + \alpha_k \rho_j/(1 + \rho_j \sum_{i < k} \alpha_i))$ .

Converse via cut-set

The degraded BC capacity region equals the superposition coding region by Cover's classical result — no other scheme (DPC, etc.) achieves more for degraded BC. Converse via EPI + chain rule.

Example: Two-User Degraded BC Capacity

For $\rho_1 = 20$ dB, $\rho_2 = 5$ dB, compute the capacity region and identify the sum-rate-maximizing and symmetric-rate-maximizing operating points.

Solution

Linearize SNRs

$\rho_1 = 100$ , $\rho_2 = 3.16$ . $C_1 = \log_2 101 \approx 6.66$ bits. $C_2 = \log_2 4.16 \approx 2.06$ bits.

Capacity region

With split $(\alpha, 1-\alpha)$ : $R_1 = \log_2(1 + \alpha \rho_1) \approx \alpha \cdot 6.66$ approximately (high-SNR). $R_2 = \log_2(1 + (1-\alpha)\rho_2/(1 + \alpha \rho_2)) \approx \log_2(1 + (1-\alpha) \cdot 3.16)$ (low-SNR limit).

Sum-rate max

$\alpha = 1$ : all power to user 1. $R_{\text{sum}} = C_1 \approx 6.66$ . (User 2 gets nothing; not symmetric.)

Symmetric max

Find $\alpha$ such that $R_1 = R_2$ : $\log_2(1 + \alpha \rho_1) = \log_2(1 + (1-\alpha)\rho_2/(1 + \alpha \rho_2))$ . Solving numerically: $\alpha \approx 0.05$ , $R_{\text{sym}} \approx 2.26$ bits — both users get this rate. Compare to multicast: $R_{\text{mc}} = C_2 = 2.06$ bits.

Observation

Symmetric operation via BC coding (2.26 bits) beats pure multicast (2.06 bits) by about 10%. This is the gain from channel-aware scheduling.

Degraded Gaussian Broadcast Channel with Caches — Single-antenna transmitter broadcasting to $K$ users with heterogeneous channel qualities $\rho_1 \geq \ldots \geq \rho_K$ . Each user has cache $\mathcal{Z}_k$ . The challenge: coded-caching multicast is bottlenecked by the worst-user channel.

The Caching Challenge on Heterogeneous BC

Coded caching's delivery phase in Chapter 2 sends XOR messages at a single rate. On a degraded BC, this single rate must be the worst-user rate — otherwise the weakest user cannot decode. Hence the naive cache-aided BC rate is $R_{\text{naive}} \;=\; C_{\min} \cdot \frac{1}{R_{\text{MAN}}} \;=\; \frac{\log_2(1 + \rho_{\min})}{K(1 - \mu)/(1 + K \mu)} \text{ per user}.$ Stronger users are unused.

Two possible fixes: (i) layered delivery — superposition coding with one coded-caching layer per user class; (ii) user grouping — cluster users by channel quality and run MAN within each group. Both exploit the non-degenerate structure of the BC.

The full theoretical characterization — including the case of mixed cacheable + uncacheable content — was established by Joudeh, Lampiris, Elia, and Caire (2019), a CommIT contribution treated in §6.3.

Key Takeaway

Channel heterogeneity is the new bottleneck. On a homogeneous channel (Ch 5), DoF = $t + L$ . On a degraded channel, the naive MAN-multicast rate is limited by the weakest user. The research question: can we design a cache-aided delivery that gracefully handles heterogeneous channels? Answer: yes, via superposition + JLEC separation (CommIT 2019).

⚠️Engineering Note

Channel Heterogeneity in Deployed Systems

In 5G NR deployments the per-user SNR typically spans 20–30 dB across a cell. Causes:

Distance / path loss. Cell edge vs cell center, up to 20 dB.
Shadowing. Building penetration, up to 10 dB.
Multipath fading. Short-term fluctuations, up to 10 dB.
Blockage (mmWave). Can be 30+ dB.
Mobility-induced Doppler. Indirect effect via CSIT aging.

Scheduling policies (PF, MaxWeight) already exploit this heterogeneity for uncoded delivery. Cache-aided delivery adds a new dimension: the cache content is shared across users but the delivery rate is channel-dependent.

The JLEC 2019 framework addresses this in a clean asymptotic regime (GDoF). Realistic-SNR schemes are an active research area.

Practical Constraints

•
Per-user SNR spread in 5G NR: 10–30 dB typical, 40+ dB at mmWave
•
Proportional-fair scheduling is the baseline for heterogeneous BC
•
Cache contents cannot adapt to instantaneous CSI in the placement phase
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Cache-aided delivery must balance multicast efficiency with user fairness

The Degraded Broadcast Channel with Caches