Exercises

$\text{SNR}_{\text{sens}} \propto N^4$ under coherent RIS alignment.

Signal traverses the RIS twice: BS→RIS→target (outgoing) and target→RIS→BS (return). Each pass: $N$ amplitude gain (coherent combining). Squared power: $N^2$ per pass. Total round-trip: $N^4$ in power.

Communication: one RIS pass ($N^2$). RIS is quadratically more effective for sensing than for communication. $\blacksquare$

$N^4 = 256^4 = 4.29 \times 10^9$. In dB: $10 \log_{10}(4.29 \times 10^9) = 96.3$ dB.

RIS adds $\sim 96$ dB to radar SNR — compensates for the severe two-way path loss and makes radar detection feasible at far greater range. $\blacksquare$

$\max (1-\lambda) R_{\text{comm}}(\mathbf{W}, \boldsymbol{\Phi}) + \lambda R_{\text{sens}}(\mathbf{W}, \boldsymbol{\Phi})$.

Comm-only optimization. All weight on users' rate. Chapter 5's problem.

Sensing-only. All weight on sensing SNR. Radar-oriented waveform.

Joint dual-function. Pareto-boundary tradeoff. Typical deployment: $\lambda \sim 0.3$-$0.5$ for comm-primary, sensing-secondary systems. $\blacksquare$

No-RIS case: $\boldsymbol{\Phi}$ does not contribute to either comm or sensing. Set $\boldsymbol{\Phi} = \mathbf{0}_{N \times N}$ (or block the RIS path entirely). This is a feasible choice under RIS-aided formulation.

Every $(\mathbf{W}, \boldsymbol{\Phi} = \mathbf{0})$ is a feasible point in the RIS-aided problem that achieves the same objective as the no-RIS problem. Hence the no-RIS Pareto region is contained in the RIS-aided one.

RIS-aided offers additional operating points (non-zero RIS contributions), so the Pareto region is strictly larger. $\blacksquare$

$\alpha$ (BS-RIS) = $\lambda/(4\pi d_1)$. Assume $d_1 = 20$ m. $\lambda = 1.07$ cm. $\alpha = 0.0107 / (4\pi \cdot 20) = 4.3 \times 10^{-5}$. $\beta$ (RIS-target) = $\lambda/(4\pi d_t) = 0.0107/(4\pi \cdot 100) = 8.5 \times 10^{-6}$.

$P_r/P_t = N^4 \cdot \alpha^4 \beta^4 \cdot \sigma_t$. $= 256^4 \cdot (4.3 \times 10^{-5})^4 \cdot (8.5 \times 10^{-6})^4 \cdot 1$. $= 4.3 \times 10^9 \cdot 3.4 \times 10^{-18} \cdot 5.2 \times 10^{-21}$ $\approx 7.6 \times 10^{-29}$.

$-280$ dB. Huge loss. Needs very high transmit power or integration to close the radar link. RIS makes it feasible via the $N^4 = 10^{9.6}$ factor, otherwise impossible. $\blacksquare$

RIS shapes both comm and sensing paths. Example: roadside panel serving V2X users while detecting incoming vehicles.

Radar uses direct-path beamforming; RIS handles user coverage. Example: BS-based radar with adjacent RIS extending comm range to indoor UEs.

RIS enables otherwise-impossible radar geometry. Example: over-the-hill target detection where RIS provides the only illumination path. $\blacksquare$

Lift $\mathbf{W}\mathbf{W}^{H} \to \mathbf{W} \succeq 0$, rank$\leq K$. Lift $\boldsymbol{\phi}\boldsymbol{\phi}^H \to \boldsymbol{\Phi}_{\text{lift}}$ with rank $\leq 1$ target.

For single-target radar and $K \leq N_t$, the SDP has a natural rank-bound on the optimum: the optimal $\mathbf{W}$ has rank at most $K$; optimal $\boldsymbol{\Phi}_{\text{lift}}$ has rank 1 (by beampattern optimality).

When the rank bound matches the problem's natural structure (no rank gap), the SDP optimum is achievable by the feasible set. Extract via eigendecomposition; no randomization needed. Tightness proven by Shapiro-Barvinok low-rank SDP theorem.

Multi-target or $K > N_t$: optimal rank may exceed feasibility bounds. Randomization recovers $\sim 0.5$-$1$ dB gap. $\blacksquare$

The BS transmits radar illumination AND receives the return. Its own transmit signal dominates the weak backscatter at the receive antennas — swamps the target return.

Pulse radar: transmit in pulse, listen during silent intervals. Loses continuous sensing.

Use $\boldsymbol{\Phi}$ to null transmit signal toward the BS's own radar receive antennas. Constraint added to joint optimization.

Separate radar receive antenna + analog cancellation. Hardware upgrade. $\blacksquare$

Received sensing signal: $y_{\text{sens}}(t) = \alpha \sigma_t \cdot s(t - \tau_t) + w(t)$. Parameter: $\tau_t = 2 d_t/c$, which determines $d_t$.

For $N_s$ samples with noise variance $\sigma^2$: $J(\tau_t) = (2 P_t |\alpha \sigma_t|^2 / \sigma^2) \cdot \|s'(t)\|^2$ (derivative of waveform vs. time).

$\text{CRB}(d_t) = (c^2 / 4) \cdot J^{-1}(\tau_t)$. With RIS: $|\alpha \sigma_t|^2$ → $N^4 |\alpha \beta|^2 \sigma_t$. CRB decreases as $N^4$. $\blacksquare$

The Pareto frontier is concave, not linear. At $\lambda = 0.5$, comm and sensing are each above 50% of their respective maxima — typically 70-85% of each.

Small weight in either direction gives disproportionately large gain in the weighted objective. The "middle" is efficient: both services simultaneously well-served.

For comm-primary systems: $\lambda = 0.2$-$0.4$. For sensing-primary: $\lambda = 0.6$-$0.8$. For equal weight: $\lambda = 0.5$ gives near-optimal balance, though actual operating point is typically slightly north/south. $\blacksquare$

Objective has $|\mathbf{h}^H \mathbf{v}|^2$ (quadratic in $\mathbf{v}$) and $|\mathbf{a}^H \boldsymbol{\Phi}|^2$ (quadratic in $\boldsymbol{\phi}$). Unit-modulus constraint $|\phi_n| = 1$ is non-convex.

Replace $\mathbf{v}\mathbf{v}^{H} \to \mathbf{W} \succeq 0$ and $\boldsymbol{\phi}\boldsymbol{\phi}^H \to \boldsymbol{\Phi}_{\text{lift}} \succeq 0$. Objective becomes linear in $\mathbf{W}, \boldsymbol{\Phi}_{\text{lift}}$.

Power: $\text{tr}(\mathbf{W}) \leq P_t$ — linear. PSD: $\mathbf{W}, \boldsymbol{\Phi}_{\text{lift}} \succeq 0$ — convex. Diagonal: $[\boldsymbol{\Phi}_{\text{lift}}]_{nn} = 1$ — linear. All convex. SDR is a convex SDP. $\blacksquare$

SDP at $N_t = 8, N = 256$: $\sim 1$-$3$ s (CVX/MOSEK). For offline calibration: acceptable.

After SDR warm-start: 3-5 AO iterations for convergence. Each $\sim 10$-$20$ ms. Total $\sim 50$-$100$ ms per coherence block.

For 20-ms coherence: $50$-$100$ ms is borderline. Use two-timescale: SDR once per second, AO per coherence block. This is the CommIT deployment recipe. $\blacksquare$

Pedestrian: $\sim 10$ ms coherence at 28 GHz. Vehicular: $\sim 2$ ms. Aircraft: $\sim 0.5$ ms.

CommIT AO: $\sim 50$ ms (full convergence) or $\sim 10$ ms (warm-started). SDR: seconds.

Pedestrian/moderate-mobility: AO fits in coherence block. Vehicular: tight but feasible with warm-starting. Aircraft: infeasible; coherence too short for RIS optimization.

Radar sensing itself needs moderate coherence: target must stay within a beam for multiple pulses. Fast mobility requires re-detection per new coherence block. Both comm and sensing favor pedestrian-to-urban mobility. $\blacksquare$

$N_t = 8$ active antennas, $N = 256$ passive elements at 28 GHz. Panel on a 5-m-tall pole.

$\lambda = 0.4$ (comm-primary, sensing secondary). Users are continuous service; vehicle detection is best-effort.

Offline: SDR once per deployment. Online: AO warm-started per coherence block. Two-timescale operation.

Comm: 8 users at $\sim 15$-$18$ dB SINR. Sensing: 98% detection probability at 50 m on a 1-m² RCS target. Combined: seamless dual service. $\blacksquare$

Goal: serve users on both sides of a panel. Communication-only; no sensing.

Goal: serve users AND sense targets from a single panel on one side. Multi-service; single-sided.

Coverage-critical (indoor-outdoor, dense urban) where users on both sides matter. No sensing need.

Automotive V2X, industrial safety, smart-city where targets and users are both on the same side. Dual-function hardware consolidation.

A combined STAR-RIS + ISAC architecture (triple-function) is a natural research direction but adds substantial hardware cost. Typically one of the two wins in specific deployments. $\blacksquare$