Exercises

Sensing-assisted communication uses the outputs of a sensing subsystem (target position, velocity, geometry) to compute channel estimates and drive comms operations (precoder, beam, scheduler).

Classical uses dedicated pilot symbols embedded in the frame to estimate channel coefficients directly.

Classical estimates the channel; SAC estimates the geometry and derives the channel deterministically. SAC gives short-horizon prediction; classical gives only current-frame estimate.

$\mathbf{H}_{DD}[\ell, k] = \sum_i a_i \mathbf{a}_r(\theta_i) \mathbf{a}_t(\phi_i)^H \delta[\ell - \ell_i, k - k_i] e^{-j2\pi \nu_i \ell_i/(MN)}$

Each path: rank-one spatial filter + DD shift + Doppler phase. Sum across paths.

Deterministic given $\Theta$. No additional noise.

$\sigma_a = 0.1$ m/s². $T_{\text{pred}} = 10 \cdot 0.01 / 0.2 = 500$ ms.

$\sigma_a = 0.5$ m/s². $T_{\text{pred}} = 10 \cdot 0.01 / 1.0 = 100$ ms.

$\sigma_a = 1$ m/s² (orbit perturbations). $T_{\text{pred}} = 10 \cdot 0.01 / 2 = 50$ ms. At these horizons, LEO and vehicle are comparable; pedestrian is much easier.

Classical: one pilot per coherence time. Pilot fraction = $T_{\text{fr}}/T_c$. Rate = $(1 - T_{\text{fr}}/T_c) R_{\text{peak}}$. SAC: one pilot per prediction horizon. Pilot fraction = $T_{\text{fr}}/T_{\text{pred}}$.

$G = \text{rate}_{\text{SAC}}/\text{rate}_{\text{classical}} = (1 - T_{\text{fr}}/T_{\text{pred}})/(1 - T_{\text{fr}}/T_c)$

$T_{\text{pred}} \to T_c$: $G = 1$. No benefit. $T_{\text{pred}} \gg T_c$: $G \to 1/(1 - T_{\text{fr}}/T_c)$. For mobility ($T_{\text{fr}}/T_c = 0.2$): $G \leq 1.25$.

$v_{\text{break-even}} = c/(2 f_0 T_{\text{fr}} G_{\text{min}}) = 3 \times 10^8 / (2 \cdot 28 \times 10^9 \cdot 10^{-3} \cdot 0.01) = 5.4 \times 10^{-1}$ m/s = 1.9 km/h.

Very low! Most users (vehicular, UAV) easily exceed this threshold. SAC is essentially always beneficial at 28 GHz.

Break-even scales as $1/f_0$. At 3 GHz: ~5 m/s. At 77 GHz: <1 m/s.

$k$ candidate motion models (CV, CA, CT). Each runs its own Kalman filter. Weights $\mu_i^{(t)}$ track the likelihood of each model given observations.

At each step: mixed initial conditions (weighted average of states). Filters don't diverge during quiet periods.

When a maneuver occurs, CA weight rises, CV weight falls. The weighted average estimate adapts quickly without needing offline model selection.

Single-CV filter lags acceleration events by several frames. IMM catches up within 1-2 frames via weight adjustment. Compute cost: $k\times$ single-filter cost.

$B = 2/N_t = 2/64 = 0.03$ rad = $1.8°$.

Tangential velocity at distance $d$: $\dot\phi = v/d$. For $v = 1$ m/s, $d = 20$ m: $\dot\phi = 0.05$ rad/s $= 2.9°$/s.

Over 50 ms: $\dot\phi T_{\text{pred}} = 2.9° \cdot 0.05 = 0.15°$. Tiny compared to beamwidth.

$P_{\text{correct}} = \Phi((B/2 - \dot\phi T_{\text{pred}})/\sigma_\phi) = \Phi((0.9 - 0.15)/2) = \Phi(0.38) = 64\%$. Poor! Sensing CRB is the dominant issue, not drift. Need $\sigma_\phi \leq 0.5°$ for 95% reliability.

For pedestrian at short range, the BS needs tight sensing (large array aperture or SNR) to track reliably.

$\sigma_h^2 \leq \epsilon \|\mathbf{h}\|^2$. Budget for sensing noise and process noise: each $\leq \epsilon/2 \|\mathbf{h}\|^2$.

$\mathrm{CRB}(\theta) \propto 1/(\mathrm{pilot power})$. Pilot power fraction $\eta_{\text{pilot}}$: $\mathrm{CRB} \sim 1/\eta_{\text{pilot}}$.

Process noise per frame: $\sigma_a^2 T_{\text{fr}}^2$. Accumulated over $T_{\text{pred}}$ frames: $\sigma_a^2 T_{\text{fr}}^2 T_{\text{pred}}$.

$\epsilon/2 \geq \sigma_a^2 T_{\text{fr}}^2 T_{\text{pred}}$. With pilot refresh every $T_{\text{pred}}$ frames: $T_{\text{pred}} \leq \epsilon/(\sigma_a^2 T_{\text{fr}}^2)$. Pilot fraction: $\eta \sim T_{\text{fr}}/T_{\text{pred}} = \sigma_a^2 T_{\text{fr}}^3/\epsilon$.

For $\sigma_a = 1$, $T_{\text{fr}} = 0.01$, $\epsilon = 0.1$: $\eta \approx 10^{-5}$. Tiny — far below classical 10%.

Reserve worst-case: per-URLLC-frame, at least enough for $10^{-5}$ reliability. For rate 1 Mbps, channel variation means reserving $\sim 10$ Mbps of BS capacity continuously. Fraction: 10%.

Sensing horizon 50 ms. URLLC budget 1 ms. Reservation = $1/50 \cdot 10\% = 0.2\%$.

Classical: 10%. PRA: 0.2%. $50\times$ reduction. Released capacity: 9.8% = 9.8 Mbps — available for eMBB.

Short-horizon sensing compounds into massive URLLC efficiency gains. Critical for 5G/6G mixed-load scenarios.

$\max \sum U_k(\rho_k)$ s.t. $\sum_k \rho_k \leq R_{\text{total}}$, $\rho_k \leq R_{\text{max},k}$.

$L = \sum U_k(\rho_k) + \mu(R_{\text{total}} - \sum_k \rho_k) + \sum_k \lambda_k (R_{\text{max},k} - \rho_k)$.

$g(\mu, \lambda) = \max_{\rho} L$, which gives $U_k(\rho_k^*) - \mu - \lambda_k = 0$ at the stationary point.

$\mu$ = shadow price of total capacity. $\lambda_k$ = shadow price of user $k$'s max rate constraint. Users with high $U_k'$ get more resources (standard water-filling).

On a curve of radius $r$: lateral acceleration $v^2/r$, angular velocity $\dot\phi = v/r$, angular acceleration $\ddot\phi = a_{\text{tang}}/r$.

Over prediction window $T_{\text{pred}}$: angular drift error = $\ddot\phi T^2/2$. Adds to velocity drift.

$T_{\text{pred, circular}} \leq \sqrt{2 B / (2\pi \ddot\phi)}$. For $\ddot\phi = 1$ rad/s², $B = 0.03$ rad: $T_{\text{pred}} \leq 0.07$ s = 7 frames at 100 Hz.

On a curve, prediction horizon shrinks by factor 2-5 compared to straight-line motion. IMM with CT model essential.

LOS: direct path. Angle depends on UE position only. Sensing CRB: small. Prediction horizon: long. NLOS: reflected path. Angle depends on UE position AND scatterer properties. Sensing CRB: larger.

Each scatterer has position error $\sigma_d$ (parking lot, building). Angle estimation from scatterer is imperfect.

$T_{\text{pred, NLOS}} \sim \sigma_\theta^{-1} \cdot B$. NLOS: larger $\sigma_\theta$ → shorter horizon.

Track multiple paths (not just dominant). When LOS disappears, NLOS path 2 takes over with shorter horizon. Fallback to pilot-based in heavy NLOS.

$\mathrm{CRB}(\theta) \propto 1/\mathrm{SNR} \propto 1/d^4$ (free-space pathloss twice: BS → target → BS).

$\mathrm{CRB}(\theta) \leq \gamma$ iff $d^4 \leq P_t/\sigma_w^2/\gamma$, i.e. $d \leq (P_t/\sigma_w^2/\gamma)^{1/4}$.

SAC-useful range: up to the above distance. Beyond that, sensing is too noisy; fallback to pilots needed.

For $P_t/\sigma_w^2 = 10^{10}$ (100 dB), $\gamma = 0.0001$ rad²: $d \leq (10^{10}/10^{-4})^{1/4} = 10^{3.5}$ m $\approx 3$ km. Plenty for urban cells. For sub-mm-wave, range drops.

$\hat\Theta^{(t+1)} = f(\mathbf{y}^{(t+1)}; \mathbf{F}^{(t+1)})$ where $\mathbf{F}^{(t+1)} = g(\hat\Theta^{(t+1|t)})$ is the predicted precoder. Substituting: $\hat\Theta^{(t+1)} = f(\mathbf{y}; g(\hat\Theta^{(t)} + \text{kinematic}))$.

Near the fixed point, define Jacobian $\mathbf{J}_{\text{loop}} = \partial f/\partial \hat\Theta \cdot \partial g/\partial \hat\Theta$. Loop stable iff $\|\mathbf{J}_{\text{loop}}\| < 1$.

$\partial f/\partial \hat\Theta$: small when sensing is accurate (CRB tight). Good sensing → stable loop.

$\partial g/\partial \hat\Theta$: small when precoder is robust to channel errors. Good robust design → stable loop.

Loop stable iff $\mathrm{CRB}(\text{sensing}) \cdot \|\text{precoder Jacobian}\| < 1$. For typical mmWave: met at SNR > 10 dB.

Sensing bootstrapping: 2 pilots per connection setup. Steady-state: 1 pilot per 10 frames (1%). Maneuver trigger: adaptive.

URLLC users: $T_{\text{pred}} = 50$ ms (vehicular). eMBB: $T_{\text{pred}} = 500$ ms (pedestrian).

Classical: 20% of BS capacity. PRA: 20% × 1 ms/50 ms = 0.4%. Gain: 98% reduction.

Released: 19.6% of capacity for eMBB. eMBB baseline: 20 users × 20 Mbps = 400 Mbps. With released capacity: 480 Mbps. Throughput gain: 20%.

20% eMBB gain + 98% URLLC efficiency + pilot overhead 9-fold reduction. Feasible on 2028-era 5G/6G BS.

$\partial G/\partial T_{\text{pred}} = T_{\text{fr}}/ (T_{\text{pred}}^2 (1 - T_{\text{fr}}/T_c)) > 0$ for $T_{\text{pred}} > 0$.

Larger horizon always increases gain, confirming that better sensing → more SAC benefit.

As $T_{\text{pred}} \to \infty$: $G \to 1/(1 - T_{\text{fr}}/T_c)$. Finite ceiling set by $T_{\text{fr}}/T_c$.

SAC benefit saturates. Beyond some horizon (10-100× $T_c$), further sensing improvement adds marginal rate. Optimal design: sensing SNR matched to just beyond this threshold.