Exercises

(a) Full CSIT: DoF = $5 + 4 = 9$. (b) No CSIT: DoF = $5 + 1 = 6$. Caching gain retained; spatial gain reduced by factor $L = 4$.

$L(1 - L/T_c) = 8(1 - 0.04) = 7.68$. The 4% pilot fraction costs about 0.32 DoF relative to the no-overhead $L = 8$.

$L^* = T_c/2 = 150$. Max spatial DoF: $L^* (1 - L^*/T_c) = 75$.

Adding antennas beyond $T_c/2$ hurts: pilot overhead exceeds spatial gain. A genuine pilot wall.

The MAN delivery transmits an XOR of subfiles; each user XOR- cancels using its cached side information. The cancellation operation requires only the user's local cache, not any knowledge of the channel. The transmitter similarly does not need CSI to form the XOR — it only needs to know which subfiles each user cached (combinatorial, not channel-dependent). Hence caching gain survives the no-CSIT regime.

$\mathrm{DoF}(\sigma_e^2) = \min(t + L(1 - \sigma_e^2/\sigma^2)_+, K)$. Linear degradation of spatial DoF with estimation error; caching gain preserved.

Per-user effective DoF: $\min(t + L(1 - L/T_c)_+, K)/K$. At high SNR, per-user rate $\approx$ DoF/K $\cdot \log \text{SNR}$.

$T_c \to \infty$: rate $\to (t+L)/K \cdot \log \text{SNR}$ (matches Lampiris-Caire '17). $T_c \to L$: rate $\to (t+1)/K \cdot \log \text{SNR}$ (pilot wall).

The cache-aided design provides a "floor" of $t/K \cdot \log\text{SNR}$ per-user rate that survives even the pilot wall. Without caching, the rate at the pilot wall is 0.

Without CSIT, the transmitter sends a single-stream signal. The effective channel is a scalar MAC with capacity $\log_2(1 + \text{SNR})$ per channel use. A cache-aided scheme on a single link achieves at most MAN DoF: $t + 1$ via XOR multicast.

Apply single-antenna MAN: DoF = $t + 1$. Matches the upper bound. $\blacksquare$

$\rho = 10$, $K = 10$. $-\ln 0.95/10 = 0.00513$. $C_{0.05}^{(10)} = \log_2(1 + 0.0513) \approx 0.072$ bits.

$C_{0.05}^{(1)} = \log_2(1 + (-\ln 0.95) \cdot 10) = \log_2(1.513) \approx 0.598$ bits.

Multicast penalty: factor $C_{0.05}^{(1)}/C_{0.05}^{(10)} \approx 8.3$ in rate. This is the motivation for multi-antenna schemes that keep group sizes small.

$R_\epsilon = (t+L) C_\epsilon^{(t+L)}/K$. As $L$ grows, DoF grows linearly but $C_\epsilon^{(t+L)}$ shrinks logarithmically (via the $1/K$ factor in the outage formula).

Taking derivative and setting to zero: optimal $L^* \approx T_c/2$ (matches pilot-wall analysis). In practice, $L \in [2, 8]$ is common.

$T_c = 20 \text{ ms} \cdot 10^6 = 2 \times 10^4$ symbols. $L = 4$ pilots per block: fraction $4/2 \times 10^4 = 2 \times 10^{-4}$, or 0.02% — negligible.

At moderate speeds and sub-6 GHz, pilot overhead is not the bottleneck. At mmWave and high speed ($T_c \to 100$), pilot fraction becomes 4% — still modest but growing. Pure-MIMO design at $L = 64$ would have pilot fraction 64% — the pilot wall.

Maddah-Ali-Tse: transmit $L$ block-long "symbols"; use delayed CSI to form retrospective interference-alignment corrections during later blocks. Asymptotic DoF $\sim (2L-1)/L$ per pair of blocks.

Add the MAN placement and XOR delivery layer on top. The retrospective scheme delivers its per-block DoF; the cached XOR delivers an additional $t$ DoF per coherence block (CSIT-free).

DoF $\approx t + (2L-1)/L$. Interpolates between no-CSIT ($t + 1$ at $L = 1$) and full-CSIT ($t + L$ as delayed DoF approaches $L$).

The exact characterization of delayed-CSIT cache-aided DoF is open for general $K$. Shariatpanahi et al. give partial results.

With partial CSIT of error $\sigma_e^2$, transmit with power $P(\sigma_e^2)$. Interference leakage: $P \sigma_e^2 \cdot L$.

Effective DoF = $t + L(1 - \sigma_e^2/\sigma^2)_+$ as before; power adaptation scales the SNR but not the DoF slope. No DoF recovery from power alone.

At finite SNR, power adaptation can recover some rate — but DoF (high-SNR slope) is invariant.

Pre-populate caches via backhaul before satellite pass (offline placement). During pass ($\approx 10$ min), deliver coded-XOR broadcasts at ~conservative MCS (targeting 5-10% FER). Group sizes limited to 20-30 users (beam coverage). HARQ not feasible due to latency; rely on forward error correction.

$T_c$ small at mm-wave/high-mobility. Effective spatial DoF small; caching DoF $t$ dominates. Per-user rate: $R_\epsilon \sim t \cdot C_\epsilon$ where $C_\epsilon$ is the outage-limited link rate.

LEO coded caching is a 6G research topic. Starlink-style constellations would benefit enormously. Cache sizing depends on content catalog turnover and satellite pass schedules.

Random vector quantization of an $L$-dim channel with $b$ bits: quantization error variance $\sim 2^{-b/(L-1)}$ (Jindal 2006).

$\mathrm{DoF}(b) = t + L(1 - 2^{-b/(L-1)})_+$. At $b = O(L \log \text{SNR})$, quantization error scales with $1/\text{SNR}$, preserving full DoF. At $b = O(1)$, spatial DoF $\to 0$.

Fewer feedback bits needed per user when caching provides $t$ extra DoF. For target DoF $t + D$, feedback required: $b \geq (L-1) \log_2(L/D)$ roughly. Caching reduces the required feedback by $t$ equivalent bits per user.

Maximize $\mathrm{DoF} = t + L(1 - L/T_c)$ subject to $L \leq L_{\max}$ (hardware), $M \leq M_{\max}$ (storage).

$L^* = T_c/2$ (spatial-optimal); $M$ at max (caching always helps).

In practice both $L$ and $M$ are constrained. The optimization is a joint resource-allocation problem; see Shariatpanahi-Caire '19 for algorithmic solutions.

At realistic deployments: $L \in [2, 8]$, $T_c \in [100, 10000]$, $M/N \in [0.05, 0.3]$. The sum $t + L$ rarely exceeds 20. Design goal: pick $M$ as large as storage allows to maximize the CSIT-independent floor.