Exercises

$t = 20 \cdot 0.25 = 5$. $\mathrm{DoF} = \min(5 + 5, 20) = 10$.

$\mathrm{DoF} = \min(t + L, K)$. If $L \geq K - t$, then $t + L \geq K$, so $\mathrm{DoF} = K$. All users served at full DoF.

Once antenna count plus caching gain reaches the user count, additional antennas or caches yield no further DoF improvement — the system is DoF-saturated. This sets the optimal resource allocation: balance $L$ and $M$ to reach $t+L = K$ economically.

Both mechanisms produce "user-specific degrees of freedom" per channel use: spatial gain lets $L$ independent streams coexist via beamforming; caching gain lets each stream reach $t+1$ users via XOR decoding of cached side information. The net users served is $L(t+1) - Lt = L + t$ after accounting for overlap — each extra cached bit does not independently enable an extra stream. A multiplicative $(t+1)L$ gain would require distinct discrimination modes beyond what either mechanism alone can provide.

$\mathrm{DoF} = \min(1 + 2, 2) = 2$. Both users served at full DoF (one DoF each).

Pure MIMO with ZF already achieves $L = 2$ DoF. Caching is not needed in this regime — the antenna count dominates. At larger $K$ the caching contribution becomes non-trivial.

Without CSIT, zero-forcing cannot be computed. The transmitter resorts to single-beam multicast (like a single-antenna system). Coded caching still provides $\mathrm{DoF} = t + 1$ via XOR multicast. Spatial gain is lost.

Fix $s$ users. Let $T$ be the number of channel uses. Total DoF from the transmitter: $T L$. Total side info: $s M/N$ fraction of the library, i.e., $sMF/N$ bits.

To deliver $s$ files: $TL \cdot \log \rho \geq s F - sMF/N \log \rho$ (asymptotically). Rearranging: $T/F \geq s(1-M/N)/L/\log \rho$ per file unit.

$\mathrm{DoF} = \lim s/(T/F \cdot \log \rho) \leq sL/(s(1-M/N)) = L/(1-M/N)$... the precise combination with the cut-set yields $\mathrm{DoF} \leq t + L$ after optimizing $s$. Full details in Lampiris-Caire '17 §IV.

MAN, $t = 1$: subfiles $W_{n,\{1\}}, W_{n,\{2\}}, W_{n,\{3\}}$. User $k$ caches $W_{n,\{k\}}$.

Only one group $\mathcal{S}' = \{1,2,3\}$. Two beams: Beam 1 (target $\{1,2\}$, null at user 3): send $\tilde x_1 = W_{d_1, \{2\}} \oplus W_{d_2, \{1\}}$ via ZF nulling $\mathbf{h}_3$. Beam 2 (target $\{1,3\}$, null at user 2): send $\tilde x_2 = W_{d_1, \{3\}} \oplus W_{d_3, \{1\}}$ via ZF nulling $\mathbf{h}_2$.

User 1: receives both beams. Cancel $W_{d_2,\{1\}}$ (cached) from beam 1 → gets $W_{d_1,\{2\}}$. Cancel $W_{d_3,\{1\}}$ from beam 2 → gets $W_{d_1,\{3\}}$. Already has $W_{d_1,\{1\}}$. ✓ Users 2, 3 similarly decode their missing subfiles.

2 beams × 1 subfile/beam × 1 channel use = 2 DoF delivered. DoF = $t + L = 3$... wait, we have $\mathrm{DoF} = 3$ but only 2 beams. The reconciliation: the per-group DoF is $L = 2$; the caching gain $t = 1$ is accounted for by each beam serving 2 users. Net users served per channel use = $L(t+1)/|\mathcal{S}'| \cdot |\mathcal{S}'| = L + t$ after averaging over groups — not visible from a single group example. Full bookkeeping is cleaner for $K \geq 4$.

MAN: $\binom{20}{5} = 15504$ subfiles. Lampiris-Caire: $\binom{20}{8} = 125970$ subfiles. Ratio: $125970/15504 = 8.12$ — the multi-antenna scheme needs 8x more subfiles per file than MAN.

For a 1 GB file, MAN subfile is 65 KB (feasible); Lampiris-Caire subfile is 8 KB (still feasible but tighter). For 100 MB files the situation is near the limit. At $K = 100$, $L = 4$, $t = 25$ the subpacketization would exceed $10^{22}$ — infeasible. This is why polynomial-subpacketization multi-antenna schemes are active research.

All pairs give $L + t = 8 \leq K = 20$, so DoF = 8 for each allocation.

At equal resource budget $L + t$, the DoF is invariant to the split. The choice of $L$ vs $t$ comes from practical constraints:

• Pure $L$ (no caching, $(8, 0)$): no placement cost, relies on CSIT, subpacketization is 1 (no split).
• Pure $t$ ($(1, 7)$): placement cost = library copies, subpacketization $\binom{20}{7}$, CSIT-free.
• Mixed: intermediate tradeoffs on cost and complexity.

DoF-equivalence of antennas and caching at asymptotic regime does not imply practical equivalence. In CSIT-poor regimes (mmWave), cache is preferred; in storage-poor (edge nodes with little cache), antennas are preferred.

Without CSIT, ZF is impossible — beams cannot be steered. The transmitter must use a single isotropic beam (or equivalent open-loop strategy). With one beam, only one stream can be sent per channel use.

The single stream can be a coded-XOR message; cached side information still lets it benefit $t+1$ users via the MAN argument. Net DoF: $t + 1$. Spatial multiplexing is lost; coded caching is entirely retained.

Coded caching is a "CSIT-free" enabler — unusual in the MIMO-BC literature where DoF typically requires CSIT. This is one reason coded caching is practically attractive for CSIT-poor regimes.

In each channel use, $L$ beams transmit, each serving $t+1$ users in one $(t+L)$-subset. But different beams target different sub-subsets of the same group, and users in the targeted groups are shared across beams.

Each user appears in $\binom{K-1}{t+L-1}$ different $(t+L)$-subsets. Across delivery rounds, each user receives $L \cdot \binom{K-1}{t+L-1}/(t+L)$... this simplifies (using identities) to $\binom{K-1}{t} =$ number of subfiles missing. Each user collects all missing subfiles in $\binom{K}{t+L}/\binom{K-1}{t+L-1} \cdot T_{\text{beam}}$ channel uses.

Sum-DoF = $K F / T / \log \rho = t + L$ after algebraic simplification (Lampiris-Caire '17 §V). The key identity: $L \binom{K-1}{t+L-1} / \binom{K-1}{t} = (t+L)/(t+1) \cdot \binom{K}{t+L}/\binom{K}{t}$.

With estimation error $\hat{\mathbf{h}} = \mathbf{h} + \Delta$ where $\Delta \sim \mathcal{CN}(0, \sigma_e^2 \mathbf{I})$, the ZF nulling is imperfect. Residual interference at each null direction: $\rho \sigma_e^2$.

Signal: $\rho$. Noise + leakage: $1 + \rho \sigma_e^2 (L-1)$. Effective SINR: $\rho/(1 + \rho \sigma_e^2 (L-1))$.

As $\rho \to \infty$ with $\sigma_e^2$ fixed: effective SINR saturates at $1/\sigma_e^2 (L-1)$. Spatial DoF reduces: if $\sigma_e^2$ does not scale with SNR, spatial DoF → 0. Only if $\sigma_e^2 \propto \rho^{-1}$ (CSI quality scales with SNR) does spatial DoF stay at $L$.

$\mathrm{DoF}(\sigma_e^2) = t + L (1 - (L-1) \sigma_e^2/\sigma^2)_+$ approximately, capturing that imperfect CSIT scales the spatial DoF by a factor dependent on the error-to-noise ratio.

Under uncoded placement, $\mathrm{DoF}^*_{\text{unc}}(M, L) = \min(t + L, K)$ exactly (matching Lampiris-Caire achievability).

Generalize YMA '18: average over user permutations yields a symmetric scheme. The symmetric scheme's DoF is exactly $t + L$ via the achievability construction. Since the max over schemes is at least the max over permutations, and the max over permutations is $t + L$, the optimal DoF is $t + L$.

Coded placement for MIMO BC is largely open. Recent works (Salehi-Aghdam-Khalaj 2021) give small improvements for specific $(K, L)$, but no systematic coded-placement gain is known.

With $L$ BSes each with $L'$ antennas cooperating, effective antenna count is $L' L$. Caches at each BS aggregate to effective per-user cache $M$ (assuming uniform distribution).

$\mathrm{DoF} = \min(t + L' L, K)$ where $t = KM/N$. Cooperative network MIMO + caching yields a DoF that stacks all three resources.

Cache-aided cell-free massive MIMO is the CommIT frontier — Lampiris-Caire 2021+ treat this in the context of fog-based edge caching. Chapter 8 discusses the NDT framework that handles fronthaul capacity as well.

With $L$ antennas and $\tau = L$ uplink pilots per coherence block, effective spatial DoF $= L(1 - L/T_c)$. For $L \geq T_c/2$, spatial DoF falls off quadratically.

Caching gain $t$ does not require pilots (CSIT-free for the multicast part). Caching DoF is unaffected by $T_c$.

Pure caching beats pure MIMO when $t + 1 > L(1 - L/T_c)$. For small $T_c$ (high mobility, mmWave), the cache wins. For large $T_c$ (quasi-static), MIMO wins.

"In CSIT-poor regimes, caching is the right primitive." This is a design principle that emerges from this scaling analysis and is repeatedly emphasized in the CommIT group's papers.