Exercises

Minimum pilot length: $\tau_p^{\min} = K = 15$.

$$f_p^{\min} = \tau_p^{\min}/\tau_c = 15/300 = 5\%$$

Data samples: $\tau_c - \tau_p = 300 - 15 = 285$.

Data fraction: $285/300 = 95\%$.

$\tau_p^{\min} = 30$, so $f_p^{\min} = 30/300 = 10\%$.

Doubling users doubles the pilot overhead and reduces data to 90%.

$\rho_p = 10^{0.5} \approx 3.16$.

$$\text{MSE}^{\text{LS}} = \frac{N_t}{\rho_p} = \frac{64}{3.16} \approx 20.3$$

All eigenvalues $\lambda_i = \beta_k = 1$:

$$\text{MSE}^{\text{MMSE}} = N_t \cdot \frac{\beta_k}{\rho_p\beta_k + 1} = 64 \cdot \frac{1}{3.16 + 1} \approx 15.4$$

MMSE does better than LS by factor $(1 + 1/\rho_p)$ — a modest gain. When all eigenvalues are equal, MMSE reduces to a scalar shrinkage $\hat{h}_i = \rho_p/(1+\rho_p) \cdot h_i^{\text{LS}}$, providing no subspace gain.

With $r_k = 8$ equal eigenvalues $\lambda_i = N_t\beta_k/r_k = 64/8 = 8$:

$$\text{MSE}^{\text{MMSE}} = 8 \cdot \frac{8}{3.16 \times 8 + 1} \approx \frac{64}{26.3} \approx 2.43$$

MMSE gain: $20.3/2.43 \approx 8.3 \approx 9.2$ dB. The gain scales with $N_t/r_k = 8$.

$\mathbf{y}_k = \sqrt{p_u\tau_p}\mathbf{h}_k + \sqrt{p_u\tau_p}\sum_{j\neq k}c_{kj}^*\mathbf{h}_j + \mathbf{n}_k$$where$\mathbf{n}_k = \mathbf{N}_p\boldsymbol{\phi}k^*/\sqrt{\tau_p} \sim \mathcal{CN}(\mathbf{0},\sigma^2\mathbf{I})$. Non-orthogonality introduces an effective interference term that scales with$|c{kj}|$.

Cross-covariance: $\boldsymbol{\Sigma}_{\mathbf{h}_k\mathbf{y}_k} = \sqrt{p_u\tau_p}\mathbf{R}_k$

Observation covariance: $\boldsymbol{\Sigma}_{\mathbf{y}_k\mathbf{y}_k} = p_u\tau_p\mathbf{R}_k + p_u\tau_p\sum_{j\neq k}|c_{kj}|^2\mathbf{R}_j + \sigma^2\mathbf{I}$

$\hat{\mathbf{h}}_k = \sqrt{p_u\tau_p}\mathbf{R}_k\boldsymbol{\Sigma}_{\mathbf{y}_k\mathbf{y}_k}^{-1}\mathbf{y}_k$$The error covariance is$\mathbf{C}_k = \mathbf{R}_k - p_u\tau_p\mathbf{R}k\boldsymbol{\Sigma}{\mathbf{y}_k\mathbf{y}_k}^{-1}\mathbf{R}k$. Non-orthogonal pilots increase$\boldsymbol{\Sigma}{\mathbf{y}_k\mathbf{y}k}$, enlarging the error covariance. Orthogonal pilots ($c{kj} = 0$,$j\neq k$) recover the standard MMSE formula.$\blacksquare$

$\hat{\mathbf{h}}_{1,1} = \sqrt{p_u\tau_p}\mathbf{R}_{1,1}(p_u\tau_p(\mathbf{R}_{1,1}+\mathbf{R}_{2,1}) + \sigma^2\mathbf{I})^{-1}\mathbf{y}_{1,1}$$

With $\mathbf{R}_{2,1} = \alpha\mathbf{R}_{1,1}$:

$$\mathbf{C}_{1,1} = \mathbf{R}_{1,1} - p_u\tau_p\mathbf{R}_{1,1}(p_u\tau_p(1+\alpha)\mathbf{R}_{1,1} + \sigma^2\mathbf{I})^{-1}\mathbf{R}_{1,1}$$

As $\sigma^2\to 0$ (high SNR), this approaches:

$$\mathbf{C}_{1,1} \to \mathbf{R}_{1,1} - \frac{1}{1+\alpha}\mathbf{R}_{1,1} = \frac{\alpha}{1+\alpha}\mathbf{R}_{1,1} \neq \mathbf{0}$$

The error remains of order $\mathbf{R}_{1,1}$ — the contamination floor is proportional to $\alpha/(1+\alpha)$.

When $\mathbf{R}_{1,1}\mathbf{R}_{2,1} = \mathbf{0}$, the projected observation is:

$$\mathbf{P}_{1,1}\mathbf{y}_{1,1} = \sqrt{p_u\tau_p}\mathbf{h}_{1,1} + \mathbf{P}_{1,1}\mathbf{n}$$

The MMSE estimator from this clean observation has error covariance:

$$\mathbf{C}_{1,1}^{\text{proj}} = \mathbf{R}_{1,1} - p_u\tau_p\mathbf{R}_{1,1}(p_u\tau_p\mathbf{R}_{1,1} + \sigma^2\mathbf{P}_{1,1})^{-1}\mathbf{R}_{1,1}$$

As $p_u\tau_p/\sigma^2 \to \infty$, $\mathbf{C}_{1,1}^{\text{proj}} \to \mathbf{0}$. $\blacksquare$

The MMSE estimator takes the form $\hat{\mathbf{h}}_k = \mathbf{A}\mathbf{y}_k$ where $\mathbf{A} = \sqrt{p_u\tau_p}\mathbf{R}_k(p_u\tau_p\mathbf{R}_k + \sigma^2\mathbf{I})^{-1}$.

$\mathbb{E}[\tilde{\mathbf{h}}_k\mathbf{y}_k^H] = \mathbb{E}[(\mathbf{h}_k - \mathbf{A}\mathbf{y}_k)\mathbf{y}_k^H]KATEXPLACEHOLDER0END= \mathbb{E}[\mathbf{h}_k\mathbf{y}_k^H] - \mathbf{A}\mathbb{E}[\mathbf{y}_k\mathbf{y}_k^H]KATEXPLACEHOLDER1END= \boldsymbol{\Sigma}_{\mathbf{h}_k\mathbf{y}_k} - \mathbf{A}\boldsymbol{\Sigma}_{\mathbf{y}_k\mathbf{y}_k}$$By definition of the LMMSE estimator,$\mathbf{A} = \boldsymbol{\Sigma}_{\mathbf{h}_k\mathbf{y}k}\boldsymbol{\Sigma}{\mathbf{y}_k\mathbf{y}_k}^{-1}$, so the expression equals$\mathbf{0}$.

Since $\hat{\mathbf{h}}_k = \mathbf{A}\mathbf{y}_k$ is a linear function of $\mathbf{y}_k$:

$$\mathbb{E}[\tilde{\mathbf{h}}_k\hat{\mathbf{h}}_k^H] = \mathbb{E}[\tilde{\mathbf{h}}_k\mathbf{y}_k^H]\mathbf{A}^H = \mathbf{0} \cdot \mathbf{A}^H = \mathbf{0}$$

$\blacksquare$

This property is critical: it means estimation error and estimate are uncorrelated, which is what makes the "use-and-then-forget" bound (Ch. 4) tight.

$[\boldsymbol{\Phi}\boldsymbol{\Phi}^H]_{kj} = \frac{1}{K}\sum_{n=1}^{K}e^{j2\pi(k-1)(n-1)/K}e^{-j2\pi(j-1)(n-1)/K}KATEXPLACEHOLDER0END= \frac{1}{K}\sum_{n=0}^{K-1}e^{j2\pi(k-j)n/K} = \delta_{kj}$$by the orthogonality of complex exponentials. So$\boldsymbol{\Phi}\boldsymbol{\Phi}^H = \mathbf{I}_K$.

Each entry has $|[\boldsymbol{\Phi}]_{k,n}| = 1/\sqrt{K}$ — constant across $n$. Maximum instantaneous power = average power, so $\text{PAPR} = 1$ (0 dB).

DFT rows are constant-amplitude sequences with PAPR = 1.

Zadoff-Chu sequences have:

1. Constant amplitude (PAPR = 1, same as DFT)
2. Ideal cyclic autocorrelation: $|r_{zc}(\tau)| = 0$ for $\tau \neq 0$ — no inter-symbol interference in OFDM
3. Multiple sequences: different root indices give mutually low-correlation sequences
4. Robustness to frequency offset: flat spectrum enables timing/frequency estimation

In 5G NR, ZC sequences are used for PRACH preambles and SRS pilots because their cyclic properties enable efficient FFT-based correlation at the receiver.

The covariance matrix $\mathbf{R}$ is a Hermitian Toeplitz matrix with generating sequence $r[m] = e^{j\pi m\sin\theta_0}\text{sinc}(m\Delta\theta_\text{rad})$. Its spectrum is the angular-domain spectrum concentrated in the interval $[\sin\theta_0 - \Delta\theta_\text{rad}/\pi, \sin\theta_0 + \Delta\theta_\text{rad}/\pi]$.

The number of significant eigenvalues (Szego's theorem) is approximately: $$r_k \approx N_t \cdot \frac{2\Delta\theta_\text{rad}/\pi}{2} = \frac{N_t\Delta\theta_\text{rad}}{\pi}$$

User $k$: angular interval $[5°, 15°]$ in degrees, $[0.087, 0.262]$ rad. User $k'$: angular interval $[35°, 45°]$ in degrees, $[0.611, 0.785]$ rad.

These intervals are disjoint: min separation $= 35° - 15° = 20° \gg 0$. The covariance spectra are supported on non-overlapping sets, implying approximately orthogonal eigenbases for large $N_t$. Formally: $\text{tr}(\mathbf{R}_{1,k}\mathbf{R}_{2,k'})/(\|\mathbf{R}_{1,k}\|_F\|\mathbf{R}_{2,k'}\|_F) \to 0$.

Exact orthogonality requires the angular windows to be disjoint:

$$|\theta_1 - \theta_2| > \Delta\theta_1 + \Delta\theta_2$$

With $\Delta\theta_1 = \Delta\theta_2 = 5°$: separation $> 10°$.

For the specific numbers above: $|10° - 40°| = 30° \gg 10°$ — well orthogonal. Two users at $10°$ and $18°$ with $5°$ spread each would barely satisfy the $> 10°$ condition and would have partial subspace overlap.

Let $\rho_p = p_u/\sigma^2$ (fixed). Define $f(\tau_p) = \tau_p\rho_p/(1+\tau_p\rho_p)$. This is increasing and concave in $\tau_p$ (positive second derivative is negative).

The product $(\tau_c-\tau_p)\log_2(1+N_t f(\tau_p))$ is the product of a decreasing linear function and a concave increasing function — jointly concave, so a unique maximum exists.

Setting $d(\text{ESE})/d\tau_p = 0$:

$$-\log_2(1+N_t f(\tau_p)) + (\tau_c-\tau_p) \cdot \frac{N_t f'(\tau_p)}{(1+N_t f(\tau_p))\ln 2} = 0$$

where $f'(\tau_p) = \rho_p/(1+\tau_p\rho_p)^2$. This transcendental equation must be solved numerically for general parameters.

At high SNR, $f(\tau_p) \to 1$ quickly and rate saturates at $\log_2(1+N_t)$. Further pilot investment provides diminishing rate returns while the pre-log factor $(1-\tau_p/\tau_c)$ still decreases linearly.

The optimal $\tau_p^*$ converges to the minimum feasible: $\tau_p^* = K$ (just enough for orthogonal pilots). Pre-log factor: $(1-K/\tau_c) = 190/200 = 95\%$. The high-SNR ESE is approximately $0.95\log_2(65) \approx 5.76$ bits/s/Hz.

Let $C^* = \sum_{k,j:\phi^*(j)=\phi^*(k),j<k}\rho_{k,j}$ be the optimal total cost and $C^G$ be the greedy cost. User $k$ assigned pilot $p$ by greedy incurs cost $C_k^G = \sum_{j:\phi^G(j)=p,j<k}\rho_{k,j}$.

When assigning user $k$, greedy picks the pilot minimizing $C_k^G$. The average cost over all $\tau_p$ pilots is at most $\bar{C}_k = \frac{1}{\tau_p}\sum_p\sum_{j:\phi^G(j)=p}\rho_{k,j}$. Greedy achieves $C_k^G \leq \bar{C}_k$.

Summing over all $k$:

$$C^G = \sum_k C_k^G \leq \sum_k \frac{1}{\tau_p}\sum_{j<k}\rho_{k,j} = \frac{1}{\tau_p}\sum_{k<j}\rho_{k,j}$$

Since the optimal assignment can only do better: $C^* \leq C^G \leq \frac{1}{\tau_p}C_{\text{all}}$,

where $C_{\text{all}} = \sum_{k<j}\rho_{k,j}$ is the cost when all users share one pilot.

The approximation ratio is at most $\tau_p$. $\blacksquare$

From Exercise 7(a): the one-ring covariance has rank $r_k \approx N_t\Delta\theta_\text{rad}/\pi$. In the DFT domain, $\mathbf{R} = \mathbf{F}\tilde{\mathbf{R}}\mathbf{F}^H$ where $\tilde{\mathbf{R}}$ is approximately block-diagonal with $r_k$ nonzero entries (the angular window).

Therefore $\tilde{\mathbf{h}} = \mathbf{F}^H\mathbf{h}$ is supported on $r_k$ angular bins — approximately $r_k$-sparse.

Pilot observation: $\mathbf{y}_k = \sqrt{p_u\tau_p}\mathbf{h}_k + \mathbf{n} = \sqrt{p_u\tau_p}\boldsymbol{\Phi}\mathbf{F}\tilde{\mathbf{h}}_k + \mathbf{n}$

Wait — pilot matrix applied to $h$: actually observation model for CS is: $\mathbf{y}_k = \sqrt{p_u}\boldsymbol{\Phi}\mathbf{h}_k + \mathbf{n} = \sqrt{p_u}\boldsymbol{\Psi}\tilde{\mathbf{h}}_k + \mathbf{n}$

where $\boldsymbol{\Psi} = \boldsymbol{\Phi}\mathbf{F}$ is the effective sensing matrix. CS recovers $\tilde{\mathbf{h}}_k$ if $\boldsymbol{\Psi}$ satisfies RIP with sparsity $r_k$.

Random Gaussian $\boldsymbol{\Phi}$ gives a random $\boldsymbol{\Psi}$ that satisfies RIP with high probability when $\tau_p = \mathcal{O}(r_k\log(N_t/r_k))$.

• CS requirement: $\tau_p = \mathcal{O}(r_k\log(N_t/r_k))$
• Standard MMSE: $\tau_p = K$ (orthogonal pilots, one per user)

For $N_t = 128$, $r_k = 8$, $K = 10$: CS needs $\tau_p \approx 8\log(16) \approx 32$. Standard MMSE needs $\tau_p = 10$.

CS is actually worse than standard MMSE for the intra-cell estimation problem! The CS advantage appears in compressed feedback scenarios (FDD massive MIMO, Ch. 8) where the channel dimension $N_t$ is large and the sparsity enables compression below the Nyquist dimension.

With reuse factor $f$, the pilot pool is split into $f$ groups. Each cell uses one group. In a 7-cell cluster, each group covers $7/f$ cells. The fraction of cells sharing the same pilot pool is $1/f$.

• $f=1$: all 7 cells share one pool (universal reuse, maximum contamination)
• $f=7$: each cell has a unique pool (no reuse, maximum pilot overhead)

Pilot overhead per cell increases with $f$ (need more pilots): $\tau_p = fK$, so pre-log factor: $(1 - fK/\tau_c)$.

Co-pilot interferers: $(L/f - 1)$ cells, so SINR floor $\propto 1/(L/f - 1)$.

Rate model (simplified): $R(f) \approx (1 - fK/\tau_c)\log_2(1 + c/(L/f - 1))$ for some constant $c$ depending on $N_t$, SNR, and covariance structure.

For 7-cell: $L = 7$, $K = 10$, $\tau_c = 200$:

• $f=1$: $\tau_p = 10$, pre-log $= 0.95$, 6 co-pilot cells, low SINR floor
• $f=3$: $\tau_p = 30$, pre-log $= 0.85$, 1.33 co-pilot cells, medium floor
• $f=7$: $\tau_p = 70$, pre-log $= 0.65$, 0 co-pilot cells (no contamination!)

At high SNR with spatial correlation, $f=7$ often wins due to zero contamination. At low SNR, $f=1$ wins since pilot efficiency (high pre-log) dominates.

The contaminated estimate $\hat{\mathbf{h}}_{1,k}^{\text{cont}} \approx \mathbf{h}_{1,k} + \mathbf{h}_{2,k}$ (simplified 2-cell case). BS 1 "knows" something about $\mathbf{h}_{2,k}$ through contamination — it can exploit this information to coordinate with BS 2.

$y_k = \mathbf{h}_{1,k}^H\hat{\mathbf{h}}_{1,k}^{\text{cont}}s_1 + \mathbf{h}_{2,k}^H\hat{\mathbf{h}}_{2,k}^{\text{cont}}s_2 + n$$As$N_t \to \infty$:$\frac{1}{N_t}\mathbf{h}{\ell,k}^H\hat{\mathbf{h}}{\ell,k}^{\text{cont}} \to $a deterministic constant$c_\ell$.

Setting $s_2 = -c_1/c_2 \cdot s_1$ would cancel interference, but this reduces user 2's signal. The PCP idea instead chooses $s_1, s_2$ to jointly maximize sum rate across both cells — a multi-cell DPC-like approach. The key insight is that the "side information" about the other cell's channel, obtained via contamination, can be exploited for coordinated precoding.

High contamination pairs: $(1,1)=0.8$, $(2,2)=0.9$, $(3,3)=0.7$, $(3,4)=0.8$, $(5,5)=0.85$. Focus on: $(2,5)$ with 0.9 and 0.85, and $(3,4)$ both high at 0.7/0.8.

1. User 1 → Pilot A
2. User 2 → Pilot B (would conflict with user 1 if assigned A: $\rho_{1,2}=0.1$ — small, but assign B anyway as most problematic)
3. User 5 (high cross-contamination with user 1, $\rho_{1,5}=0.6$) → Pilot C
4. User 3 → Pilot B ($\rho_{3,2}=0.2$ — smallest remaining cost)
5. User 4 → Pilot A ($\rho_{4,1}=0.3$, $\rho_{4,3}=0.8$ avoided)

Assignment: $\{1,4\}\to A$, $\{2,3\}\to B$, $\{5\}\to C$. Total contamination: $\rho_{1,4}=0.3 + \rho_{2,3}=0.2 = 0.5$ (much better than random).

With $\mathbf{R}_{1,1}=\mathbf{R}_{2,1}=\beta\mathbf{I}$:

$$\hat{\mathbf{h}}_{1,1} = \frac{\sqrt{p_u\tau_p}\beta\mathbf{I}}{2p_u\tau_p\beta + \sigma^2}\mathbf{y}_{1,1} \triangleq \alpha\mathbf{y}_{1,1}$$

$\mathbb{E}[\hat{\mathbf{h}}_{1,1}^H\mathbf{h}_{1,1}] = \alpha\mathbb{E}[\mathbf{y}_{1,1}^H\mathbf{h}_{1,1}] = \alpha\sqrt{p_u\tau_p}\beta N_t$

(squaring gives the numerator)

$\mathbb{E}[\hat{\mathbf{h}}_{1,1}^H\mathbf{h}_{2,1}] = \alpha\mathbb{E}[\mathbf{y}_{1,1}^H\mathbf{h}_{2,1}] = \alpha\sqrt{p_u\tau_p}\beta N_t$

Same magnitude! Both signal and contamination grow as $N_t$ with the same coefficient.

Both numerator and contamination scale as $N_t^2$, so the SINR converges to:

$$\text{SINR}^{\infty} = \frac{|\alpha\sqrt{p_u\tau_p}\beta N_t|^2}{|\alpha\sqrt{p_u\tau_p}\beta N_t|^2} = 1$$

(plus noise term which becomes negligible as $N_t\to\infty$).

With only one contaminator and identical covariances, the floor is SINR = 1 (0 dB). For $L$ co-pilot cells: SINR$^\infty = 1/(L-1)$. $\blacksquare$

• $L = 7$ hexagonal cells, $K = 10$ users per cell
• Channel model: one-ring with $\beta_{\ell k} = 1$, angular spread $\Delta\theta = 5°$
• $N_t \in \{8, 16, 32, 64, 128, 256, 512\}$
• Universal pilot reuse ($\tau_p = K = 10$)
• MRC combining using contaminated MMSE estimate

for each N_t in range:
  sinr_trials = []
  for trial = 1:N_mc:
    Generate h[l,k] ~ CN(0, R[l,k]) for all l,k
    Compute contaminated observations y[k] = sqrt(p*tau_p)*sum_l(h[l,k]) + n
    Compute MMSE estimates h_hat[k]
    Compute MRC output: r_k = h_hat[k]^H * (sqrt(p)*sum(h[1,:]*s) + noise)
    Compute SINR_k = |E[h_hat^H h]|^2 / (interference + noise)
    sinr_trials.append(SINR_k)
  sinr_vs_nt.append(mean(sinr_trials))
Plot sinr_vs_nt vs N_t (should plateau at SINR_floor)

SINR converges to the floor as $N_t \to \infty$. For large $N_t$, self-averaging reduces variance — the law of large numbers over $N_t$ antenna elements.

For $\pm 0.1$ dB ($\approx 1.15\%$ relative SINR accuracy), by central limit theorem, needed trials $N_{\text{mc}} \approx (Z_{0.975}\sigma_{\text{SINR}}/0.0115)^2$. With $\sigma_{\text{SINR}} \approx 0.5$ (typical): $N_{\text{mc}} \approx 7500$ trials.