Ferkans — Interactive Telecom Tutor

Why Fading Changes Everything

Chapters 5–6 worked with a static MIMO BC: the channel was fixed and perfectly known to the transmitter. Real wireless channels are neither. They vary on the coherence timescale $T_c$ and are known only approximately (via pilots) or not at all. These two realities interact with coded caching in interesting ways.

The key observation: caching is pilot-free. Coded caching's gain comes from the pre-placed cache contents and XOR multicasting, neither of which requires channel knowledge at delivery time. By contrast, MIMO's spatial multiplexing gain $L$ collapses without CSIT. On a CSIT-expensive channel — high-mobility, FDD, mmWave with aging beamforming — caching becomes disproportionately valuable. This chapter formalizes this intuition.

Definition:
Cache-Aided Block-Fading Broadcast Channel

The cache-aided block-fading BC extends the MIMO BC of Chapter 5 to a time-varying channel. Per coherence block of $T_c$ channel uses:

The channel vector $\mathbf{h}_k[b]$ for user $k$ and block $b$ is drawn i.i.d. from $\mathcal{CN}(\mathbf{0}, \mathbf{I}_L)$ .
The channel is constant within a block, independent across blocks.
Each coherence block allocates $\tau$ channel uses to pilots (for CSIT estimation) and $T_c - \tau$ channel uses to data.

The received signal at user $k$ , channel use $m$ of block $b$ : $y_k[m, b] \;=\; \mathbf{h}_k[b]^H \mathbf{x}[m, b] + \mathbf{w}_{k}[m, b],$ $\mathbf{w}_{k} \sim \mathcal{CN}(0, \sigma^2)$ , transmit constraint $\mathbb{E}[\|\mathbf{x}\|^2] \leq P$ , $\text{SNR} = P/\sigma^2$ .

The library, caches, and delivery structure are as in Chapter 5.

The coherence block length $T_c$ depends on user mobility and carrier frequency. At 2 GHz, $v = 30$ m/s gives $T_c \approx 10^3$ symbols at typical symbol rates. At 28 GHz (mmWave), the same velocity yields $T_c \approx 70$ — much shorter. This makes CSIT acquisition progressively more expensive as we move up the spectrum.

Definition:
CSIT Regimes

We distinguish three regimes based on CSIT quality:

Full CSIT. The transmitter knows $\{\mathbf{h}_k[b]\}$ exactly before each block's data phase begins. This is the Chapter 5 assumption; DoF = $t + L$ .
Partial CSIT (estimation variance $\sigma_e^2$ ). The transmitter has $\hat{\mathbf{h}}_k = \mathbf{h}_k + \mathbf{e}_k$ with $\mathbf{e}_k \sim \mathcal{CN}(\mathbf{0}, \sigma_e^2 \mathbf{I})$ . DoF scales as $t + L(1 - \sigma_e^2/\sigma^2)_+$ .
No CSIT. The transmitter knows only statistics. Spatial multiplexing collapses; DoF = $t + 1$ (caching gain only).

The tradeoff parameters are the coherence time $T_c$ , the pilot allocation $\tau$ , and the feedback quality (in FDD) or reciprocity noise (in TDD).

Theorem: DoF under Imperfect CSIT

For the cache-aided $L$ -antenna BC with estimation error variance $\sigma_e^2/\sigma^2 \in [0, 1]$ : $\mathrm{DoF}(\sigma_e^2) \;\leq\; t + L \left(1 - \frac{\sigma_e^2}{\sigma^2}\right)_+,$ with saturation at $K$ . The bound is achieved by the Lampiris-Caire scheme with suitably degraded beamforming.

Imperfect CSIT degrades zero-forcing: residual interference at each null scales with $\sigma_e^2$ . Effective per-beam SINR reduces, shrinking spatial DoF by a factor $(1 - \sigma_e^2/\sigma^2)$ . Caching gain $t$ is unaffected because XOR cancellation does not use channel information.

Proof

Interference leakage

With estimate $\hat{\mathbf{h}}_k = \mathbf{h}_k + \mathbf{e}_k$ , zero-forcing beam $\mathbf{v}_k$ nulls $\hat{\mathbf{h}}_k$ but not $\mathbf{h}_k$ . Residual interference power at user $k$ from another beam: $|\mathbf{e}_k^H \mathbf{v}_{j}|^2 \propto \sigma_e^2$ .

Per-beam effective SINR

For transmit power $\text{SNR}$ , each beam's effective SINR is approximately $\mathrm{SINR}_{\text{eff}} \;=\; \frac{\text{SNR}}{1 + (L-1)\text{SNR}\sigma_e^2/\sigma^2}.$

Spatial DoF

High-SNR scaling: $\lim_{\text{SNR}\to\infty} \log_2(1 + \mathrm{SINR}_{\text{eff}})/\log_2 \text{SNR}$ $= (1 - \sigma_e^2/\sigma^2)_+$ per beam. Summing over $L$ beams: effective spatial DoF = $L(1 - \sigma_e^2/\sigma^2)_+$ .

Caching gain survives

The caching gain $t$ is realized via XOR cancellation at the receivers — no CSIT required. Hence $\mathrm{DoF} = t + L(1 - \sigma_e^2/\sigma^2)_+$ up to saturation $K$ . $\blacksquare$

Effective DoF vs CSIT Error Variance

Plot the cache-aided MIMO DoF as a function of CSIT estimation error variance $\sigma_e^2/\sigma^2$ . Blue: combined DoF = $t + L(1 - \sigma_e^2/\sigma^2)$ . Green dashed: CSIT-free caching baseline DoF = $t+1$ . Red dotted: pure MIMO (no caching). Notice that the combined scheme is always above both baselines; in particular, at $\sigma_e^2 = \sigma^2$ (no-CSIT regime), the combined curve drops to the $t+1$ level, matching the pure-caching baseline.

Parameters

Users K20

Antennas L4

Memory ratio M/N0.2

Cache-aided fading BC topology — Transmitter with $L$ antennas broadcasting to $K$ users over a block-fading channel. Per coherence block, $\tau$ pilots are sent to acquire CSIT before $T_c - \tau$ data symbols are transmitted. Each user's cache $\mathcal{Z}_k$ was populated in an earlier off-peak placement phase.

CSIT Degradation: $\mathrm{DoF} = t + L(1 - \sigma_e^2/\sigma^2)$

As the CSIT estimation error grows from 0 (perfect) to 1 (no CSIT), the spatial DoF component collapses linearly while the caching floor

t + 1

remains intact. Blue: combined cache + MIMO; green: pure caching floor; red: pure MIMO without cache.

Key Takeaway

Caching gain $t$ is CSIT-independent; spatial gain $L$ is not. On a CSIT-poor channel, the Lampiris-Caire DoF degrades from $t+L$ to $t+1$ — a large loss for $L \gg 1$ but still non-trivial. Caching provides a floor guarantee: no matter how bad the channel knowledge, the coded-multicast gain $t$ is retained.

🚨Critical Engineering Note

The Cost of CSIT in Real Systems

CSIT is expensive in several concrete ways:

Pilot overhead. In a $T_c = 500$ symbol block with $L = 4$ antennas, $L = 4$ pilots consume 0.8% of capacity — modest. At $T_c = 50$ (mmWave, high mobility) and $L = 64$ (massive MIMO), the overhead becomes infeasible.
Feedback overhead (FDD). Users must send CSIT back to the transmitter, consuming uplink capacity proportional to $L$ and the required precision.
Age of CSI. Between pilot and data phases, the channel has aged. At mmWave and high velocity, the aging can be 30%+ of the channel within a single coherence block.
Quantization. Practical CSI feedback uses finite-rate codebooks; 3GPP Type II codebooks consume ~50 bits per RB per user at $L = 8$ .

All of these conspire to make the no-CSIT regime increasingly relevant. Coded caching's CSIT-free caching gain is a practical lever against this. The CommIT group has argued in several papers that coded caching should be viewed as "free" spatial DoF in CSIT-poor settings.

Practical Constraints

•
5G NR Type II codebook: up to 64 bits CSI feedback per slot
•
mmWave coherence: 20-100 symbols at v=30 m/s, fc=28 GHz
•
FDD LTE/NR CSI feedback period: 5-40 ms
•
Massive MIMO pilot contamination bottleneck when L = K

Historical Note: The CSIT Research Thread

2012–2021

The CSIT-DoF question has a long history in MIMO-BC research. Davoodi and Jafar (2016) characterized the DoF of the MIMO BC under various CSIT assumptions (full, delayed, alternating); Yang, Kobayashi, Gesbert, Yi (2013) gave the delayed-CSIT schemes; Maddah-Ali-Tse (2012) proved the optimality of their "retrospective" scheme. All of this pre-dates coded caching.

The coded-caching extension came from the CommIT group: Lampiris, Caire and collaborators (2017+) showed that the caching gain $t$ survives the CSIT degradation, giving coded caching a unique robustness property. The 2021 paper of Lampiris-Bhattacharjee-Caire (in IEEE TWC) is the clearest statement of this idea in the fading context.

A related thread is imperfect CSIT pre-coding (Clerckx-Joudeh et al.); rate-splitting MA inherits the CSIT-dependence but provides finer-grained tradeoffs than time-sharing.

The Cache-Aided Fading Broadcast Channel