Exercises

$\partial(R_c + \mathrm{MI}_s)/\partial\rho = -P_t/(\sigma^2 + (1-\rho)P_t) + P_t/(\sigma^2 + \rho P_t) = 0$. This gives $\rho^* = 0.5$. Equal power splitting maximises the sum when the noise levels are equal. $\square$

With MRT: $\mathrm{SINR} = N_t P_c/\sigma^2 \geq 10$. Hence $P_c \geq 10\sigma^2/N_t = 1.25\sigma^2$.

At $\text{SNR} = 20$ dB ($P_t = 100\sigma^2$): $P_s = P_t - 1.25\sigma^2 = 98.75\sigma^2$. Only 1.25% of power is needed for communication; 98.75% is available for sensing --- the "free sensing" benefit of ISAC with many antennas. $\square$

Total path: $R_1 + R_2 = c\tau = 3 \times 10^8 \times 8 \times 10^{-6} = 2400$ m. Baseline: $d = 1000$ m. Bistatic range: $R_b = 2400 - 1000 = 1400$ m.

The target lies on an ellipse with foci at Tx and Rx, semi-major axis $a = 1200$ m, focal distance $c_e = 500$ m, semi-minor axis $b = \sqrt{1200^2 - 500^2} = 1091$ m. Without angle information, any point on this ellipse is a candidate. $\square$

$G_{\mathrm{SE}} = (100 + 100)/100 = 2$ (maximum gain).

The $2\times$ gain assumes both functions can fully utilise the shared band. In practice: (1) sensing pilots reduce communication throughput; (2) communication data creates "noise" for sensing (data-compensation penalty); (3) PAPR constraints limit effective waveform design. Typical practical gain: $1.5$--$1.8\times$. $\square$

$R_c = \log_2(1 + p_1/\sigma^2) + \log_2(1 + p_2/\sigma^2)$ with $p_2 = P_t - p_1$. Maximised at $p_1 = p_2 = P_t/2$.

$\mathrm{MI}_s = \log_2(1 + p_1/\sigma^2_{s})$. Maximised at $p_1 = P_t$ (all power to antenna 1).

As $p_1$ increases from $P_t/2$ to $P_t$: $\mathrm{MI}_s$ increases from $\log_2(1 + P_t/(2\sigma^2_{s}))$ to $\log_2(1 + P_t/\sigma^2_{s})$. $R_c$ decreases from $2\log_2(1 + P_t/(2\sigma^2))$ to $\log_2(1 + P_t/\sigma^2)$ (one antenna gets all power). The frontier is a curve parameterised by $p_1 \in [P_t/2, P_t]$. There IS a tradeoff because $\mathbf{a} = [1,0]^T$ is aligned with only one eigenmode. $\square$

$\mathbf{H} = [\mathbf{a}(-20^\circ), \mathbf{a}(30^\circ)]^H \in \mathbb{C}^{2 \times 8}$. ZF: $\mathbf{W}_c = \mathbf{H}^H(\mathbf{H}\mathbf{H}^H)^{-1}$, consuming 2 DOF.

$\mathbf{P}_\perp = \mathbf{I}_8 - \mathbf{H}^H(\mathbf{H}\mathbf{H}^H)^{-1}\mathbf{H}$, rank 6.

$\mathbf{v}_{s} = \mathbf{P}_\perp\mathbf{a}(50^\circ) / \|\mathbf{P}_\perp\mathbf{a}(50^\circ)\|$. Gain: $\|\mathbf{P}_\perp\mathbf{a}(50^\circ)\|^2 \approx 6/8 = 75\%$ ($-1.25$ dB loss) since $50^\circ$ is well-separated from users. $\square$

Total pilots: $4 \times 8 = 32$. Place uniformly every 32nd subcarrier. Assign each user a comb offset: user $k$ uses $\{k, k+32, k+64, \ldots\}$ for $k = 0, 1, 2, 3$.

All 32 pilots contribute to the range profile. Unambiguous range: $R_{\max} = c/(2 \times 32\Delta f)$. With $\Delta f = 15$ kHz: $R_{\max} = 312.5$ m. Sidelobes: $-13.2$ dB (Dirichlet kernel, 32 samples). Adding data subcarriers after compensation improves to $-30$ dB. $\square$

For 10 dB echo SNR after cancellation, residual direct path must be $-90$ dB relative to original. Required: 90 dB total.

Stage 1 (analog): Reference antenna + delay matching: $\sim 40$ dB. Stage 2 (digital): Adaptive filter (LMS, 256 taps): $\sim 50$ dB. Total: 90 dB. In practice, multipath limits digital stage to $\sim 45$ dB; CLEAN algorithm recovers the last 5 dB. $\square$

Bandwidth: $W = M\Delta f = 128 \times 15 \times 10^3 = 1.92$ MHz. Range resolution: $\Delta R = c/(2W) = 78$ m. Max range: $R_{\max} = c/(2\Delta f) = 10$ km.

Frame duration: $T_f = N/(M\Delta f) = 64/(1.92 \times 10^6) = 33.3\;\mu$s. Wait --- $T_s = 1/\Delta f = 66.7\;\mu$s per OFDM symbol; frame = $NT_s = 4.27$ ms. Velocity resolution: $\Delta v = \lambda/(2NT_s) = \lambda/(2 \times 4.27 \times 10^{-3})$. At $f_0 = 3.5$ GHz ($\lambda = 0.086$ m): $\Delta v = 10$ m/s. Max velocity: $v_{\max} = \lambda\Delta f / 2 = 0.086 \times 15000/2 = 643$ m/s. $\square$

$\mathbf{y} = \alpha\mathbf{A}\mathbf{x} + \mathbf{w}$, $\mathbf{w} \sim \mathcal{CN}(\mathbf{0}, \sigma^2\mathbf{I})$. With $\mathbf{x}$ random and independent of $\alpha$:

Conditional on $\mathbf{x}$: $J(\alpha|\mathbf{x}) = (2/\sigma^2)\|\mathbf{A}\mathbf{x}\|^2$. Taking expectation: $J(\alpha) = (2/\sigma^2)\mathbb{E}[\|\mathbf{A}\mathbf{x}\|^2] = (2/\sigma^2)\operatorname{tr}(\mathbf{A}\mathbf{R}_x\mathbf{A}^{H})$.

The FIM depends on $\mathbf{R}_x = \mathbb{E}[\mathbf{x}\mathbf{x}^H]$ only, not the specific distribution of $\mathbf{x}$. Gaussian, QAM, or any other distribution with the same covariance yields the same Fisher information. $\square$

$\max_{\mathbf{W}_c, \mathbf{W}_s \succeq 0} \operatorname{tr}(\mathbf{B}\mathbf{W}_s) + \operatorname{tr}(\mathbf{B}\mathbf{W}_c)$ s.t. $\operatorname{tr}(\mathbf{H}\mathbf{W}_c) \geq \gamma(\operatorname{tr}(\mathbf{H}\mathbf{W}_s) + \sigma^2)$, $\operatorname{tr}(\mathbf{W}_c + \mathbf{W}_s) \leq P_t$. Here $\mathbf{B} = \mathbf{a}(\theta_t)\mathbf{a}^{H}(\theta_t)$ and $\mathbf{H} = \mathbf{h}\mathbf{h}^H$.

The SDP has 2 inequality constraints (SINR + power). By the rank reduction theorem (Huang and Palomar, 2010): $\operatorname{rank}(\mathbf{W}_c^*)^2 \leq 2$, so $\operatorname{rank}(\mathbf{W}_c^*) = 1$. Similarly for $\mathbf{W}_s^*$.

For $N_t = 4 \geq K_u + K_t + 1 = 3$: SDR is tight. The beamforming vectors are $\mathbf{v}_{c} = \sqrt{\lambda_1}\mathbf{v}_1$ from the dominant eigenpair of $\mathbf{W}_c^*$. $\square$

$\mathbf{y}(f_n) = \alpha e^{-j4\pi f_n R/c}\mathbf{a}(\theta) + \mathbf{w}(f_n)$. Stacking: $\mathbf{y} = \alpha\mathbf{d}(R) \otimes \mathbf{a}(\theta) + \mathbf{w}$ where $[\mathbf{d}(R)]_n = e^{-j4\pi f_n R/c}$.

$[\mathbf{J}]_{11} = (2|\alpha|^2/\sigma^2) N_f \|\dot{\mathbf{a}}(\theta)\|^2$, $[\mathbf{J}]_{22} = (2|\alpha|^2/\sigma^2) N_t \|\dot{\mathbf{d}}(R)\|^2$, $[\mathbf{J}]_{12} = (2|\alpha|^2/\sigma^2) \dot{\mathbf{a}}^H\mathbf{a} \cdot \dot{\mathbf{d}}^H\mathbf{d}$.

$[\mathbf{J}]_{12} = 0$ when the array is symmetric ($\dot{\mathbf{a}}^H\mathbf{a} = 0$ at broadside) or frequencies are symmetric about $f_0$. Then: $\mathrm{CRB}(\theta) = \sigma^2/(2|\alpha|^2 N_f \|\dot{\mathbf{a}}\|^2)$, $\mathrm{CRB}(R) = \sigma^2/(2|\alpha|^2 N_t \|\dot{\mathbf{d}}\|^2)$. $\square$

$|\mathbf{g}_t^T\boldsymbol{\Phi}\mathbf{h}_t|^2 = |\sum_n g_{t,n} e^{j\phi_n} h_{t,n}|^2$. Maximised when all terms add coherently.

$\phi_n^* = -\angle(g_{t,n} h_{t,n})$ for each element $n$. This aligns all $N_{\mathrm{RIS}}$ terms in phase, giving: $|\mathbf{g}_t^T\boldsymbol{\Phi}^*\mathbf{h}_t|^2 = (\sum_n |g_{t,n}||h_{t,n}|)^2$.

If $|g_{t,n}| \approx |h_{t,n}| \approx \bar{g}$ for all $n$: gain $\approx N_{\mathrm{RIS}}^2 \bar{g}^4$. The $N_{\mathrm{RIS}}^2$ (not $N_{\mathrm{RIS}}$) scaling is the hallmark of coherent RIS beamforming --- each element contributes coherently. $\square$

$H_0$: $\mathbf{R}_0 = \sigma^2\mathbf{I}$. $H_1$: $\mathbf{R}_1 = |\alpha|^2\hat{\mathbf{a}}\mathbf{a}^{H}\mathbf{R}_x\mathbf{a}\hat{\mathbf{a}}^{H} + \sigma^2\mathbf{I}$.

$T = \frac{|\hat{\mathbf{a}}^{H}\mathbf{y}|^2}{\sigma^2} \overset{H_1}{\underset{H_0}{\gtrless}} \eta$. Under $H_0$: $T \sim \chi^2_2$. Under $H_1$: $T \sim \chi^2_2(2|\alpha|^2\mathbf{a}^{H}\mathbf{R}_x\mathbf{a}/\sigma^2)$.

The deflection coefficient is $d^2 = 2|\alpha|^2\mathbf{a}^{H}\mathbf{R}_x\mathbf{a}/\sigma^2$. This shows that sensing performance depends on the beampattern energy $\mathbf{a}^{H}\mathbf{R}_x\mathbf{a}$ --- the same quantity that appears in the CRB. $\square$

A single bistatic pair has FIM with eigenvalues proportional to $(W^{2}, D_{\mathrm{eff}}^{-2})$ where $D_{\mathrm{eff}}$ is the effective cross-range aperture. Condition number: $\kappa \sim W D_{\mathrm{eff}}$ (typically 10--100).

$\mathbf{J}_{\mathrm{total}} = \sum_{i<j} \mathbf{J}_{ij}$. For BSs on a square: pairs span $0^\circ, 45^\circ, 90^\circ$ baseline orientations. The sum averages the anisotropic FIMs, reducing the condition number.

With 6 pairs averaging diverse geometries: $\operatorname{tr}(\mathbf{J}_{\mathrm{total}}) \approx 6 \times \operatorname{tr}(\mathbf{J}_{\mathrm{single}})$. CRB improves by $\sqrt{6} \approx 2.4\times$ in each dimension. The condition number drops to $\sim 2$ (near-isotropic). $\square$

The echo from voxel $q$ at range $R_q$ and angle $\theta_q$: $\mathbf{y}(f_n) = \sum_q c_q e^{-j4\pi f_n R_q/c} \mathbf{a}(\theta_q) + \mathbf{w}$. Stacking over frequencies and antennas: $\mathbf{y} = \sum_q c_q [\mathbf{d}(R_q) \otimes \mathbf{a}(\theta_q)]$. Discretising on a grid: $\mathbf{A} = \mathbf{D} \otimes \mathbf{A}_\theta$.

$\mathbf{A}^{H}\mathbf{A} = (\mathbf{D}^H\mathbf{D}) \otimes (\mathbf{A}_\theta^H\mathbf{A}_\theta)$. Condition number: $\kappa(\mathbf{A}) = \kappa(\mathbf{D}) \cdot \kappa(\mathbf{A}_\theta)$. For critically sampled grids: $\kappa(\mathbf{D}) \approx 1$ and $\kappa(\mathbf{A}_\theta) \approx 1$, so $\kappa(\mathbf{A}) \approx 1$. For oversampled grids: both factors grow, and $\kappa$ can be $> 100$.

Range: $\Delta R = c/(2N_f\Delta f) = c/(2W)$. Angle: $\Delta\theta = 2/N_t$ (in units of $\sin\theta$). Cross-range at distance $R_0$: $\Delta x = R_0 \lambda / (N_t d)$. This is exactly the resolution predicted by the PSF analysis of Chapter 13, now derived from the ISAC waveform parameters. $\square$

$f_0 = 140$ GHz, $\lambda = 2.14$ mm. $W = 15$ GHz (130--145 GHz). $\Delta R = 1$ cm. Array: $N_t = 256$ (UPA $16 \times 16$), aperture $\approx 17$ mm.

Communication: beam-space MIMO, 4 streams. Rate: $4 \times \log_2(1 + \text{SNR})$ with $\text{SNR} \approx 25$ dB $\to 4 \times 8.3 = 33$ bits/s/Hz $\times 7.5$ GHz effective $\approx 250$ Gbps. Sensing: null-space beamforming, 252 DOF for sensing.

Path loss at 5 m: $87$ dB. Tx power: 20 dBm. Array gain: 24 dB each side. Communication SNR: $\sim 53$ dB. Sensing (two-way, $\sigma = 0.01$ m$^2$): $\sim -44$ dB per pulse. Coherent integration ($10^4$ pulses): $-4$ dB. Marginal.

(1) Path loss limits sensing to $< 10$ m. (2) Phase noise degrades integration. (3) ADC power at 15 GHz bandwidth. (4) Cross-range needs multi-view for 1 cm. $\square$

$\mathbf{y} = [\mathbf{y}_{12}^T, \ldots, \mathbf{y}_{34}^T]^T = \mathbf{A}_{\mathrm{joint}}\mathbf{c} + \mathbf{w}$ where $\mathbf{A}_{\mathrm{joint}} = [\mathbf{A}_{12}^{T}, \ldots, \mathbf{A}_{34}^{T}]^T$.

$\hat{\mathbf{c}} = \arg\min \|\mathbf{y} - \mathbf{A}_{\mathrm{joint}}\mathbf{c}\|^2 + \lambda\|\mathbf{c}\|_1$. The Gram matrix $\mathbf{A}_{\mathrm{joint}}^H\mathbf{A}_{\mathrm{joint}} = \sum_{ij}\mathbf{A}_{ij}^{H}\mathbf{A}_{ij}$ sums PSFs from different viewing angles.

Single pair: range $c/(2W)$, cross-range limited by bistatic geometry. With 6 pairs spanning diverse angles: effective angular aperture increases, cross-range improves $\sim 3\times$. Down-range unchanged. $\square$