Exercises

$R = 1 \cdot K(1-\mu)/(1+K\mu) = R_\text{MAN}$. Standard MAN.

$R = 0 \cdot R_\text{MAN} = 0$. No data delivery. CRB $= c_0/P$. Best sensing.

$R_\text{MAN} = 10 \cdot 0.8 / 3 = 2.67$. $R = 0.5 \cdot 2.67 = 1.33$ files/use.

$\text{CRB} = 2 \cdot c_0/P$. Twice the floor.

Use $\rho_1$ for fraction $\beta$ of time; $\rho_2$ for $1-\beta$. Average rate: $\beta R(\rho_1) + (1-\beta) R(\rho_2)$. Average sensing precision: $\beta/\text{CRB}(\rho_1) + (1-\beta)/\text{CRB}(\rho_2)$.

Any point on the line connecting $(R_1, 1/\text{CRB}_1)$ and $(R_2, 1/\text{CRB}_2)$ achievable via time-sharing.

Convex hull of $\{(\rho, R(\rho), 1/\text{CRB}(\rho))\}_{\rho \in [0,1]}$ is achievable. Pareto frontier is concave in $(R, 1/\text{CRB})$.

$R_\text{MAN} = K(1-\mu)/(1+K\mu)$. For $K\mu = 5$: depends on $\mu$. With $\mu = 0.2, K = 25$: $R = 25 \cdot 0.8 / 6 = 3.33$.

$(1 - 0.8) \cdot 3.33 = 0.67$. Savings: 80%.

Sensing-aware predictive caching + coded caching: combined savings of $\sim 5\times$ vs uncoded, and another $5\times$ vs static MAN caching. Total $\sim 25\times$ vs uncoded.

Maximize $(1-\rho) R_\text{MAN}$ subject to $c_0/(\rho P) \leq \text{CRB}_\text{target}$.

$\rho \geq c_0/(P \cdot \text{CRB}_\text{target})$. Maximize rate by choosing $\rho$ exactly at this minimum.

$\rho^* = c_0/(P \cdot \text{CRB}_\text{target})$ — depends on the CRB target, not generally $0.5$. Equal split is optimal only for one particular CRB target.

Broadcast $X$ from BS to user $k$ (cut): $R \leq C_\text{comm}((1-\rho) P)$ where $C_\text{comm}$ is channel capacity.

CRB $\geq 1/\text{FI}(\rho P)$ where $\text{FI}$ is sensing Fisher information.

$R \leq (1 - \rho) R_\text{MAN}$ and CRB $\geq c_0/(\rho P)$. Matches achievable (§19.2) — so for orthogonal allocation the characterization is tight.

Superposition can exceed the orthogonal bound. Converse with non-orthogonal allocation is open.

Per-vehicle: GPS position, velocity, heading. BS combines to form spatial density map + mobility predictions.

For each vehicle $k$ and tile $n$: $\phi_n(\mathbf{s}_k) =$ probability of needing tile $n$ in next 10 seconds based on projected trajectory + tile's geographic boundary.

Every $\Delta t = 5$ seconds: rank tiles by $\phi_n$; retain top-$M$ in cache. Evict lowest-ranked.

For uncached tiles, BS runs MAN delivery over vehicles with common uncached-tile overlap. Coded gain from MAN preserved.

No data service. Not a fair comparison for a network. Combined always strictly better than nothing.

No sensing capability — missing safety/automation benefits. Combined dominates if sensing value > comm. tradeoff cost.

If network requires both, combined is the only feasible option. Coded caching expands both — it strictly dominates uncached ISAC.

$\mathbb{E}[R] = (1 - \bar\alpha) R_\text{MAN}$ — linear in $\alpha$, so only mean matters at first order.

Variance of rate: $\text{Var}(R) = \sigma_\alpha^2 R_\text{MAN}^2$. Larger variance = more unpredictable outcomes (but same mean).

For risk-averse operators (prioritize low-variance rate), prefer stable predictors $\sigma_\alpha^2 \approx 0$ over high-variance ones with the same mean.

Foundation: the $K(1-\mu)/(1+K\mu)$ rate formula. Directly plugged into ISAC rate expression.

ISAC uses MIMO waveforms. The $t + L$ DoF framework applies to joint sensing-communication beamforming.

Distributed ISAC in multi-cell settings. NDT framework could be extended with a sensing metric.

Random caching suitable for dynamic vehicular settings where coordination is hard.

Practical implementation (finite $F$) interacts with real-time sensing constraints.

Known target direction: $L$-antenna beamforming gives gain $L$ in SNR, no loss in rate.

Effective DoF = $t + L$ (Lampiris-Elia-Caire). Rate scaled proportionally.

$R = (1 - \rho) (t + L) / (t + 1) \cdot R_\text{MAN}$ — extra $L$-factor from beamforming. Sensing CRB unchanged at first order.

Neighboring BSs share: (i) sensing observations → multi-view target tracking; (ii) caches → virtual library expansion; (iii) coded transmissions → distributed MAN.

Cell-free MIMO coded caching + multi-view radar fusion. Rate: approximately $N_\text{BS} \times$ single-cell rate (if cells cooperate fully).

Scaling of NDT-like metric with $N_\text{BS}$ for ISAC setting. Active research.

Standard 5G. Moderate sensing performance. Standard MAN applied at the file level works with any waveform.

Better delay-Doppler sensing. Enables higher-accuracy predictive caching.

Coded caching is waveform-agnostic at file level; better waveform → better sensing → better predictive placement. Layered complementarity.

Targets widely separated in (range, Doppler, angle). CRB per target approximately independent. Total CRB $\sim M / (\rho P)$ where $M$ = number of targets.

Sensing ambiguity degrades CRB for each. Not additive; super-linear growth.

Independent of target count — caching affects only the communication side. More targets → need more sensing budget → tighter caching requirement to compensate.

What is the optimal $(R, \text{CRB}, M)$ region under non-orthogonal allocation? Important because current results are loose; closing the gap may reveal significant gains.

If sensing predictions are adversarial (attack on ML predictor), how does caching degrade? Important for robust 6G systems.

Battery-constrained ISAC + caching: three-way tradeoff becomes four-way with energy. Important for IoT / V2X.