Exercises

$\lambda = 3 \times 10^8 / (28 \times 10^9) \approx 0.0107$ m $\approx 1.07$ cm.

$d_F = 2 \cdot 1.2^2 / 0.0107 \approx 269$ m.

$r = 30$ m $< d_F = 269$ m, so the user is deep in the near field. The spherical wavefront curvature across the 1.2 m aperture is easily observable and the polar dictionary of Section 18.4 is essential.

$|\hat{\mathcal{V}}_k|/|\mathcal{V}_k| = 400/200 = 2$. Term (a) $= 2/10 = 0.2$.

Miss count $= 200 - 180 = 20$. Miss fraction $= 20/200 = 0.1$. Term (b) $= 0.1$.

$\text{NMSE} = 0.2 + 0.1 = 0.3$. Compare with the genie NMSE of $1/10 = 0.1$: the mismatched detector suffers a 3x penalty. $\blacksquare$

$s = 1 + 1 + 1 - 1 = 2$.

$\Pr[m = 1 \mid \text{neighbours}] = 1/(1 + \exp(-2 J s - 2 h)) = 1/(1 + \exp(-2 \cdot 0.9 \cdot 2)) = 1/(1 + e^{-3.6}).$

$e^{-3.6} \approx 0.0273$, so $\Pr \approx 1/1.0273 \approx 0.973$. The site is almost certain to be active, driven by its three active neighbours. $\blacksquare$

$K \cdot N_t^{3} = 32 \cdot 4096^3 \approx 2.2 \times 10^{12}$ flops.

$M = N_t/S = 256$; per-subarray $M^3 = 1.68 \times 10^7$; per-user $S M^3 = 16 \cdot 1.68 \times 10^7 \approx 2.68 \times 10^8$; per-block $32 \cdot 2.68 \times 10^8 \approx 8.6 \times 10^9$ flops.

$2.2 \times 10^{12} / 8.6 \times 10^9 \approx 256 = S^2$, matching Theorem TComplexity Reduction from Subarray Decomposition. $\blacksquare$

Width $= 2\pi/3 \approx 2.094$ rad. $\Delta\theta = 2/\sqrt{128} \approx 0.177$ rad. $G_\theta = \lceil 2.094/0.177 \rceil = 12$.

$\log_{1.3}(50/5) = \log(10)/\log(1.3) \approx 2.303/0.262 \approx 8.8$, so $G_r = \lceil 8.8 \rceil + 1 = 10$ range bins.

$G = G_\theta \cdot G_r = 12 \cdot 10 = 120$.

$\tau_p \geq 2 \cdot 3 \cdot \log_2 120 \approx 6 \cdot 6.91 \approx 42$ symbols. The far-field counterpart would need $\tau_p \geq N_t = 128$, so the polar estimator saves about 3x pilot overhead. $\blacksquare$

$\mathbb{E}[\ell_n \mid m = 0] = \frac{3}{4} \cdot 1 - \log 4 = 0.75 - 1.386 \approx -0.636.$

$\mathbb{E}[\ell_n \mid m = 1] = \frac{3}{4} \cdot 4 - \log 4 = 3 - 1.386 \approx 1.614.$

The gap is $\approx 2.25$ nats: about $3.3$ dB of log-likelihood separation between true and false antennas. The MRF coupling $J$ amplifies this gap by letting neighbouring antennas pool their evidence. $\blacksquare$

Let $\pi(\sigma) = Z^{-1} \exp(-H(\sigma))$ be the Ising distribution with Hamiltonian $H(\sigma) = -J \sum_{\langle \cdot \rangle} \sigma\sigma' - h \sum \sigma$. Let $\sigma'$ differ from $\sigma$ at exactly one site $(i,j)$. The Gibbs transition probability is $\Pr[\sigma \to \sigma'] = \pi(\sigma_{ij}' \mid \sigma_{-ij})$.

$\pi(\sigma) \Pr[\sigma \to \sigma'] = \pi(\sigma) \pi(\sigma_{ij}' \mid \sigma_{-ij})$. By the definition of conditional probability, $\pi(\sigma) \pi(\sigma_{ij}' \mid \sigma_{-ij}) = \pi(\sigma_{-ij}) \pi(\sigma_{ij}, \sigma_{-ij}) / \pi(\sigma_{-ij}) \cdot \pi(\sigma_{ij}' \mid \sigma_{-ij}) = \pi(\sigma_{ij}, \sigma_{-ij}) \pi(\sigma_{ij}' \mid \sigma_{-ij})$. A symmetric calculation for $\sigma' \to \sigma$ gives the same joint factor, so detailed balance holds.

The chain is reversible with respect to $\pi$, aperiodic (self-loops possible when $p \in (0,1)$), and irreducible (any configuration is reachable via single-spin flips). Therefore $\pi$ is the unique stationary distribution and the sampler converges to it. $\blacksquare$

Non-central $\chi^2_2$ with non-centrality $\rho$ and scale $1$ has density $e^{-(T + \rho)} I_0(2\sqrt{T \rho})$. Averaging over exponential $\rho$ gives $p(T \mid m=1) = \int_0^\infty e^{-T - \rho} I_0(2\sqrt{T\rho}) \frac{1}{\bar\rho} e^{-\rho/\bar\rho} d\rho = \frac{1}{1 + \bar\rho} e^{-T/(1 + \bar\rho)}$ via the standard Laplace transform of the Bessel function.

$p(T \mid m=0) = e^{-T}$ (central $\chi^2_2$, mean 1).

$\ell_n = \log \frac{p(T \mid m=1)}{p(T \mid m=0)} = -\log(1 + \bar\rho) + T \left(1 - \frac{1}{1 + \bar\rho}\right) = -\log(1 + \bar\rho) + \frac{\bar\rho}{1 + \bar\rho} T.$

This matches the fixed-$\rho$ formula of Definition DPosterior MRF from Pilot Observations with $\rho_n$ replaced by its mean $\bar\rho$. The Rayleigh mixture does not change the LLR form — a particular gift of the exponential prior. $\blacksquare$

The relevant ELBO term is $\mathbb{E}_q[\log p(\mathbf{Y}_p \mid \mathbf{m}_k, \mathbf{z})] = -\frac{1}{\sigma^2}\sum_n q_n \|(\mathbf{Y}_p {\mathbf{S}_{i,k}}_{k}^* /\|{\mathbf{S}_{i,k}}_{k}\|^2)_n - (\mathbf{A}_{\text{polar}}\mathbf{z})_n\|^2 + \text{const}$.

With a Laplace prior $\Pr[\mathbf{z}] \propto e^{-\lambda \|\mathbf{z}\|_1}$, the M-step minimizes $\sum_n w_n |\tilde{y}_n - (\mathbf{A}_{\text{polar}}\mathbf{z})_n|^2 + \lambda \|\mathbf{z}\|_1$ where $\tilde{\mathbf{y}} = \mathbf{Y}_p {\mathbf{S}_{i,k}}_{k}^* /\|{\mathbf{S}_{i,k}}_{k}\|^2$ and $w_n = q_n/\sigma^2$.

The objective is a positive-weighted quadratic plus an $\ell_1$ term — convex. ISTA iterations are $\mathbf{z} \leftarrow \mathcal{S}_{\lambda/L}(\mathbf{z} + \frac{1}{L} \mathbf{A}^H \mathbf{W}(\tilde{\mathbf{y}} - \mathbf{A}\mathbf{z}))$ with $L$ = Lipschitz constant of the gradient and $\mathcal{S}_\tau(\cdot)$ the soft-thresholding operator. Convergence follows from standard proximal-gradient theory. $\blacksquare$

With identical pilot ${\mathbf{S}_{i,k}}_{1} = {\mathbf{S}_{i,k}}_{2} = \mathbf{s}$, the matched-filter output at antenna $n$ is $T_n = \mathbf{H}_{1,n} + \mathbf{H}_{2,n} + \text{noise}$. Since at most one of $\mathbf{H}_{1,n}, \mathbf{H}_{2,n}$ is non-zero (disjoint VRs), the per- antenna observation carries user $k$'s channel uncontaminated by user $k' \neq k$.

The joint EM of Section 18.5 detects each user's VR mask separately; once detected, the masked observation $\mathbf{m}_k \odot T_n$ is clean.

With orthogonal pilots, two users need $\tau_p = 2$. With shared pilots, $\tau_p = 1$. The factor-of-2 saving compounds for $K$ users whose VRs are pairwise disjoint: the achievable pre-log factor $(\tau_c - \tau_p)/\tau_c$ improves by the orthogonal-pilot savings $\tau_p^{\text{orth}} - \tau_p^{\text{shared}} = K - 1$ symbols.

Perfectly disjoint VRs are rare; in practice the MRF posterior has to resolve partially overlapping regions, and the pilot saving is smaller. But the principle — VR structure doing the work that pilot orthogonality does in the stationary regime — is exactly the spatial pilot decontamination of Section 18.5. $\blacksquare$

Focus on a single boundary cell between active and inactive regions, containing $n_b$ antennas. The net LLR favoring the wrong decision is $\Delta = \sum_{n \in \partial} \ell_n + 4 J \sum_{n \in \partial} (2\mathbb{1}\{\text{wrong}\} - 1)$. For attractive coupling ($J > 0$) the second term is always negative on the boundary of a coherent cluster.

Under the central-limit approximation, $\Delta$ is approximately Gaussian with mean $\bar\Delta = n_b \bar\ell$ and variance $\sigma^2_\Delta = n_b \sigma^2_\ell$ where $\bar\ell, \sigma_\ell$ are the LLR mean and standard deviation on a single antenna.

The probability that loopy BP flips the boundary cell is $\Pr[\Delta < 0] \leq Q\!\left(\frac{n_b \bar\ell}{\sqrt{n_b}\sigma_\ell}\right) = Q(\sqrt{n_b}\,\bar\ell/\sigma_\ell)$. This decays exponentially in $n_b$ — the longer the cluster boundary, the harder it is for BP to mislabel it.

With damping factor $\alpha \in [0,1]$, each BP sweep moves messages only $\alpha$-fraction toward the fixed point, so the effective boundary signal strength is $\alpha \bar\ell$. The error bound becomes $Q(\alpha \sqrt{n_b}\,\bar\ell/\sigma_\ell)$: aggressive damping ($\alpha < 1$) stabilizes the iteration but slows the error decay. The Xu–Caire paper recommends $\alpha \in [0.5, 0.8]$. $\blacksquare$

As $r \to \infty$, $x_n^2 \cos^2\theta / (2r) \to 0$ for fixed $x_n, \theta$, so $r_n - r \to -x_n \sin\theta$. The near-field steering vector becomes $(\mathbf{a})_n = (1/\sqrt{N_t}) e^{j k_0 x_n \sin\theta}$, which is exactly the far-field steering vector $\mathbf{a}(\theta)$.

For any sufficiently large range $r^*$, the columns of $\mathbf{A}_{\text{polar}}(\theta_q, r_q)$ with $r_q \geq r^*$ are numerically indistinguishable. The dictionary degenerates: the entire range dimension maps to a single column per angle. The polar dictionary therefore contains the far-field DFT as a subset, differing only in having extra columns at short ranges.

A far-field user's channel has zero correlation with the short-range columns (they are near-orthogonal to the far-field steering vectors), and its OMP run picks the angular DFT columns exclusively. Thus polar-OMP reduces gracefully to far-field OMP in the far-field limit. $\blacksquare$

$14 = 8 + 6$: the VR occupies 2 subarrays along each axis, covering $2 \times 2 = 4$ subarrays. $|\mathcal{A}_k|_{\text{best}} = 4$.

$14$ antennas straddle 3 subarrays along each axis (e.g., 2 in subarray 1, 8 in subarray 2, 4 in subarray 3). $|\mathcal{A}_k|_{\text{worst}} = 3 \times 3 = 9$.

Plain subarray uses all $S = 64$ tiles. VR-pruned uses only $|\mathcal{A}_k|$. Best-case speedup: $64/4 = 16$x. Worst-case speedup: $64/9 \approx 7.1$x. Combined with the $S^2 = 4096$ factor over full-aperture MMSE, VR pruning delivers $> 30,000$x total speedup in the best case. $\blacksquare$

$\text{KL}(q \| p) = q_n \log(q_n/p_1) + (1 - q_n)\log((1-q_n)/p_0)$ where $p_1/p_0 = e^{\ell_n}$ from the local posterior.

$\partial_{q_n} \text{KL} = \log(q_n/(1-q_n)) - \ell_n = 0$ gives $q_n^* = \sigma(\ell_n)$. This is exactly the per-antenna BP marginal without coupling.

Substituting $q_n^*$ into the ELBO yields $-\log(1 + e^{-\ell_n}) + \text{const}$, confirming that BP maximizes the ELBO's local contribution. Adding coupling $J$ couples neighbouring updates and is precisely what loopy BP does on the full grid. $\blacksquare$

Sample a VR mask from the 2D Ising prior with $(J, h) = (0.9, -0.2)$. Draw the channel on the mask from $\mathcal{CN}(0, \mathbf{I})$ — or from the polar dictionary with 4 random scatterers for a near-field test. Corrupt with Gaussian noise at the chosen SNR.

Implement the E-step as 10 sweeps of loopy sum-product on the Ising graph. Implement the M-step as 100 iterations of ISTA on the weighted LASSO. Start from a polar-OMP estimate.

Sequential baseline: run the VR detector once (threshold the raw LLRs), then do LS on the detected support. Full-aperture LS: ignore the mask.

Plot NMSE vs SNR (e.g., $-5$ to $20$ dB in $1$-dB steps). The joint EM should lie within $1$ dB of the genie, the sequential detector should lag by $3$–$6$ dB at moderate SNR, and the LS baseline should be much worse. Plot the ELBO vs EM iteration and verify monotonicity. $\blacksquare$

$\mathbf{R}_k = \mathbf{U}\boldsymbol{\Lambda}\mathbf{U}^H$ with $\mathbf{U} \in \mathbb{C}^{N_t \times |\mathcal{V}_k|}$, $\boldsymbol{\Lambda} \in \mathbb{R}^{|\mathcal{V}_k| \times |\mathcal{V}_k|}_{++}$.

$\hat{\mathbf{H}}_k = \mathbf{R}_k (\mathbf{R}_k + \sigma^2 \mathbf{I})^{-1} \mathbf{y}$. Using Woodbury, $(\mathbf{R}_k + \sigma^2\mathbf{I})^{-1} = \frac{1}{\sigma^2} \mathbf{I} - \frac{1}{{\sigma^2}^{2}} \mathbf{U}(\mathbf{I} + \frac{1}{\sigma^2}\boldsymbol{\Lambda})^{-1}\mathbf{U}^H$.

Plugging back gives $\hat{\mathbf{H}}_k = \mathbf{U}(\boldsymbol{\Lambda}^{-1} + \frac{1}{\sigma^2}\mathbf{I})^{-1}\frac{1}{\sigma^2} \mathbf{U}^H \mathbf{y}$, which depends on $\mathbf{y}$ only through $\mathbf{U}^H \mathbf{y}$ — the projection onto the VR support. The remaining $N_t - |\mathcal{V}_k|$ dimensions of $\mathbf{y}$ carry only noise and are discarded exactly by the MMSE. $\blacksquare$