Exercises

$\rho_k$: 1000, 100, 10, 1. $C_k$: 9.97, 6.66, 3.46, 1.00 bits. $C_{\min} = 1.00$ bits — bottleneck for multicast.

Worst-user rate is 10x less than best-user rate in linear units. Multicasting at 1 bit per channel use leaves 9 bits of capacity unused at the strongest user.

$\{(R_c, R_u) : R_c \leq (t+L), R_u \leq L, R_c/(t+L) + R_u/L \leq 1\}$ in sum-GDoF coordinates. Time-sharing achieves the entire boundary between the two corner points.

$C_{\min} = \log_2(1 + 3.16) = 2.06$ bits. $R_{\text{MAN-file}} = 8 \cdot 0.75/(1 + 2) = 2$. Per-user: $2.06/2 = 1.03$ bits per channel use. Also matches $C_{\min} (1+t)/(K - t) = 2.06 \cdot 3/6 = 1.03$. ✓

Uncacheable content (live streams, chat, real-time data) is generated at delivery time, after caches are populated. No cached bits correlate with uncacheable messages; no XOR decoding at users is possible. Hence pure caching cannot help; only spatial multiplexing can.

DoF assumes all users share the same high-SNR scaling $\log \rho$. GDoF tracks individual SNR exponents: user $k$ has SNR $\rho^{\alpha_k}$ and contributes $\mathrm{GDoF}_k = R_k/\log \rho$, which can range over $[0, \alpha_k]$. For symmetric channels (all $\alpha_k = 1$), GDoF reduces to DoF. GDoF is the natural framework for degraded channels.

$R_{\text{sym}} = \log_2(1 + 3.16) = 2.06$ bits.

With split $(\alpha, 1-\alpha)$: $R_w = \log_2(1 + (1-\alpha) \cdot 3.16)$. For $R_w = 2.06$: $(1-\alpha) \cdot 3.16 = 2.16$, so $1-\alpha = 0.68$, $\alpha = 0.32$. $R_s = \log_2(1 + 0.32 \cdot 100) = \log_2(33) \approx 5.04$ bits. Set $R_s = R_w \Rightarrow$ further iterate: at $\alpha = 0.03$, $R_s = \log_2(4) = 2$, $R_w = \log_2(1 + 0.97 \cdot 3.16/(1 + 0.03 \cdot 3.16)) \approx 2.03$ bits. Symmetric rate $\approx 2.03$. Slight improvement over multicast.

Marginal gain in this case. For more skewed SNRs (or larger $K$), superposition's gain over multicast is larger.

$t = 5$. $t + L = 9$; $L = 4$.

$\theta \cdot 9/(1-\theta) \cdot 4 = 2/1 \Rightarrow \theta/ (1-\theta) = 8/9$, $\theta = 8/17 \approx 0.47$.

$R_c = 0.47 \cdot 9/20 = 0.21$ GDoF. $R_u = 0.53 \cdot 4/20 = 0.106$ GDoF. Ratio $R_c/R_u = 2.0$. ✓ Total per-user GDoF: 0.316 (= $0.47 \cdot 9/20 + 0.53 \cdot 4/20 = (4.23 + 2.12)/20 = 0.318$).

Group 1 (5 highest SNR, ~15–20 dB): $C_{\min,1} = \log_2(1 + 10^{1.5}) \approx 5.07$. Group 2 (~10–15 dB): $C_{\min,2} \approx 3.46$. Group 3 (~5–10 dB): $C_{\min,3} \approx 2.06$. Group 4 (0–5 dB): $C_{\min,4} \approx 1.00$.

Within-group $K_g = 5$, $\mu$-cache: $t_g = 5 \cdot 0.25 = 1.25$ (non-integer; memory-share to $t = 1$ and $t = 2$). Approx $R_{\text{MAN-file},g} = 5 \cdot 0.75/(1 + 1.25) \approx 1.67$ files.

Group 1: $5.07/1.67 \approx 3.04$. Group 2: $2.07$. Group 3: $1.23$. Group 4: $0.60$.

Global MAN: $C_{\min,\text{global}} = 1.00$ bit, $R_{\text{MAN-file}} = 20 \cdot 0.75/6 = 2.5$. Per-user: $1.00/2.5 = 0.40$ bits.

Groups-average: $(3.04+2.07+1.23+0.60)/4 = 1.73$ bits per user — 4x better than global MAN via grouping. Channel heterogeneity is devastating for global MAN; grouping mitigates it.

For cacheable content: by Lampiris-Caire (Ch 5) converse, $\mathrm{DoF}_c \leq t + L$. For uncacheable content: placement cannot help, so effective channel is $L$-antenna BC with no cache contribution; $\mathrm{DoF}_u \leq L$. Joint time-sharing: if fraction $\theta$ is cacheable mode, total cacheable GDoF = $\theta (t+L)$; uncacheable GDoF = $(1-\theta) L$. Eliminating $\theta$: $R_c/(t+L) + R_u/L = 1$. Any operating point inside this tradeoff is achievable; the boundary cannot be exceeded.

JLEC 2019 is a GDoF result: it holds asymptotically as $\rho \to \infty$. At finite SNR, schemes can benefit from: (i) Correlated cache contents and messages. Joint designs exploit overlap between the XOR'd cacheable bits and the uncacheable bits. (ii) Pre-log constants. Some joint schemes achieve the same DoF but with better constants. (iii) Interference alignment at finite SNR. Aligned interference is reduced even when not eliminated.

Thus finite-SNR joint schemes can strictly outperform the JLEC separation at operational SNRs, though they still cannot exceed the GDoF ceiling.

Let $\theta \in [0, 1]$ be the fraction of channel uses in cacheable mode. During this mode, use Lampiris-Caire achieving sum-DoF $t + L$.

During cacheable mode ($\theta T$ channel uses): $R_c^{(k)} \cdot \log \rho = (t+L)/K \cdot \theta T \cdot \log \rho$. Per-user GDoF: $\mathrm{GDoF}_c = \theta (t+L)/K$.

During uncacheable mode ($(1-\theta) T$): MU-MIMO achieves $\mathrm{DoF}_u = L$ per user (if $L \geq K$ via DPC; MMSE-SIC for the scalar case). Per-user GDoF: $\mathrm{GDoF}_u = (1 - \theta) L/K$.

$K \mathrm{GDoF}_c/(t+L) + K \mathrm{GDoF}_u/L = \theta + (1 - \theta) = 1$, giving the pentagon boundary. By choosing $\theta$, any point on this line is achievable. Together with the corners, the full pentagon is achieved. $\blacksquare$

With $M_k$ varying, the effective caching gain is per-user: $t_k = K M_k/N$. The Lampiris-Caire scheme needs a refined placement (e.g., random uniform over size-$M_k$).

A weighted-average caching gain $\bar t = \sum_k t_k = KM/N$ can be achieved via a suitable scheme. Separation from uncacheable traffic remains GDoF-optimal.

For specific cache distributions (e.g., half users with $M = 0$, half with $M = 2N/K$), tighter converse bounds may be possible. Sengupta-Tandon-Clancy (2017) give partial characterizations; the fully general case is open.

Superimpose: let $x = \sqrt{\beta P} X_{\text{cache-XOR}} + \sqrt{(1-\beta) P} X_{\text{uncache}}$. Strong users decode both layers; weak users decode only the cacheable XOR layer (higher- power, lower-rate).

At finite SNR, superposition adds a small constant improvement over strict time-sharing. The JLEC GDoF region is not exceeded, but the pre-log constant is larger.

Characterizing the exact finite-SNR capacity region for mixed cache-aided BC is open. Joudeh-Caire 2021+ have explored this; see the references.

Replace "uncacheable traffic" with "sensing task" (e.g., localizing targets). Sensing consumes DoF too: approximately $\mathrm{DoF}_{\text{sens}} = L$ (orthogonal beams scan).

GDoF region: $\mathrm{GDoF}_c/(t+L) + \mathrm{GDoF}_{\text{sens}}/L \leq K$, as before but with sensing replacing uncacheable rate. The JLEC separation theorem extends naturally.

Caire-Liu-Elia-Caire on ISAC and coded caching (2024+) formalize this tradeoff. Chapter 19 develops it in detail.

$K, M/N, L,$ CSIT quality (from full to none), channel spread (degraded vs symmetric), and coherence time.

$\mathrm{DoF}_{\text{effective}} = f(t, L, \text{CSIT}, \text{het})$. In the symmetric, perfect-CSIT, static regime: $t + L$. Degradation reduces the spatial component; imperfect CSIT further reduces it; fading adds multiplicative pilot overhead. Caching gain is robust across all of these.

The cache-aided wireless theory centers on a single insight: caching is a CSIT-free and heterogeneity-robust enabler. Where spatial multiplexing requires good channels and CSIT, caching delivers gain unconditionally. This design principle drives most of the CommIT research program.

A unified analytical framework has not been fully developed. Existing results are piecemeal: Ch 5 for clean DoF, Ch 6 for heterogeneity, Ch 7 for fading. The integrated theory is a research goal.