Exercises

Both BSs are at distance 250 m. Path loss: $\beta = 250^{-4} = 2.56 \times 10^{-10}$.

$\text{SIR} = \beta / \beta = 1 = 0$ dB.

$\text{SIR} = 64 \cdot \beta / \beta = 64 = 18.1$ dB. Massive MIMO provides $10\log_{10}(64) = 18.1$ dB improvement at the cell edge.

Each AP estimates $K = 10$ scalar channels (one per user, since $N = 1$). Total across all APs: $L \cdot K = 500$ channel estimates.

The BS estimates $K = 10$ channel vectors of dimension 50. Total scalar channel coefficients: $50 \cdot 10 = 500$.

The total number of channel coefficients is identical: $M \cdot K = 500$. The difference is that in cell-free, estimation is distributed — each AP estimates only $K$ scalars, which is much simpler than estimating $K$ vectors of dimension $N_t$ at a centralized BS.

Each BS transmits with amplitude $\sqrt{P}$ and the channel gain is $\sqrt{\beta}$ (same for all). The coherent received signal is $B \sqrt{P \beta} \cdot s$, so the power is $(B\sqrt{P\beta})^2 = B^2 P \beta$.

If the phases are random, the powers add: total power $= B \cdot P \beta$.

Coherent: $B^2 P\beta$, non-coherent: $B P\beta$, single BS: $P\beta$. The coherent gain is $B^2$ and the non-coherent gain is $B$. $\blacksquare$

With $L = (D/d)^2$ APs on a grid, $\sum_l d_{lk}^{-\alpha} \approx \frac{L}{D^2} \int_0^D \int_0^D r^{-\alpha} \, dA$ where $r$ is the distance from the center. Converting to polar coordinates: $\sum_l d_{lk}^{-\alpha} \approx \frac{L}{D^2} \cdot 2\pi \int_0^{D/2} r^{1-\alpha} \, dr = \frac{2\pi L}{D^2} \cdot \frac{(D/2)^{2-\alpha}}{2-\alpha}$ (for $\alpha > 2$).

Similarly, $\sum_l d_{lk}^{-2\alpha} \approx \frac{2\pi L}{D^2} \cdot \frac{(D/2)^{2-2\alpha}}{2-2\alpha}$. The ratio: $\frac{\sum_l \beta_{lk}^{2}}{(\sum_l \beta_{lk})^2} \sim \frac{C_1 L}{C_2^2 L^2} = \frac{C_1}{C_2^2 L} \to 0$.

As $L \to \infty$, the normalized variance vanishes as $O(1/L)$, confirming channel hardening. $\blacksquare$

With $\eta_{lk} = \eta$: $\eta \sum_k \gamma_{lk} \leq 1$ for all $l$.

$\eta \leq 1 / \max_l \sum_k \gamma_{lk}$.

APs in dense user areas (large $\sum_k \gamma_{lk}$) limit the power. Max-min power control can allocate $\eta_{lk}$ non-uniformly to avoid this bottleneck.

With orthogonal pilots, the received pilot at AP $l$ on user $k$'s dimension is $\mathbf{y}_l^{(k)} = \sqrt{\tau_p p_p} \, \mathbf{H}_{lk} + \mathbf{w}_{l}$.

$\gamma_{lk} = \frac{\tau_p p_p \, \beta_{lk}^{2}}{\tau_p p_p \, \beta_{lk} + \sigma^2} = \frac{\rho_p \, \beta_{lk}}{1 + \rho_p}$ where $\rho_p = \tau_p p_p \beta_{lk} / \sigma^2$.

$\gamma_{lk} = \beta_{lk} \cdot \frac{\rho_p}{1 + \rho_p}$ is monotonically increasing in $\rho_p$ since $\frac{d}{d\rho_p} \frac{\rho_p}{1+\rho_p} = \frac{1}{(1+\rho_p)^2} > 0$. At high pilot SNR, $\gamma_{lk} \to \beta_{lk}$ (perfect estimation). $\blacksquare$

$\text{Numerator} = (\sum_l \gamma_{lk})^2$. If $\gamma_{lk} \sim \bar{\gamma}/L$ on average, $\sum_l \gamma_{lk} \sim \bar{\gamma}$, so the numerator is $O(1)$. But with fixed per-AP channel quality and increasing $L$, $\sum_l \gamma_{lk} = O(L)$ and the numerator is $O(L^2)$.

Each interference term: $\sum_l \gamma_{lj} \beta_{lk} = O(L)$. There are $K - 1$ such terms. The denominator is $O(L)$.

$\text{SINR}_k = O(L^2) / O(L) = O(L) \to \infty$ as $L \to \infty$. In contrast, in cellular systems the interference grows at the same rate as the signal (more BSs = more interferers), so SINR saturates. $\blacksquare$

$\mathbb{E}[\|\mathbf{x}_l\|^2] = P_t \sum_{k,j} \sqrt{\eta_{lk} \eta_{lj}} \, \mathbb{E}[\hat{\mathbf{H}}_{lk}^H \hat{\mathbf{H}}_{lj}] \, \mathbb{E}[s_k^* s_j]$.

Since $\mathbb{E}[s_k^* s_j] = \delta_{kj}$: $\mathbb{E}[\|\mathbf{x}_l\|^2] = P_t \sum_k \eta_{lk} \, \mathbb{E}[\|\hat{\mathbf{H}}_{lk}\|^2] = P_t \sum_k \eta_{lk} N \gamma_{lk}$.

Imposing $\mathbb{E}[\|\mathbf{x}_l\|^2] \leq P_t$ and dividing by $P_t$ gives $N \sum_k \eta_{lk} \gamma_{lk} \leq 1$. For $N = 1$ (single-antenna AP): $\sum_k \eta_{lk} \gamma_{lk} \leq 1$. $\blacksquare$

With AP spacing $\Delta = D/L$: $G \approx \frac{L}{D} \int_0^D (|x_0 - x| + d_0)^{-\alpha} dx$. By symmetry about $x_0$, this is approximately $\frac{2L}{D} \int_0^{D/2} (r + d_0)^{-\alpha} dr$.

$\int_0^{D/2} (r + d_0)^{-\alpha} dr = \frac{(d_0)^{1-\alpha} - (D/2 + d_0)^{1-\alpha}}{\alpha - 1}$. For large $D$, this is $\approx \frac{d_0^{1-\alpha}}{\alpha - 1}$ (constant, independent of $D$).

$G \approx \frac{2L}{D} \cdot \frac{d_0^{1-\alpha}}{\alpha-1}$. With $D$ fixed and $L \to \infty$: $G = O(L)$. With $D = L^{1/\alpha} d_0$ (scaling the area with $L$): $G = O(L^{1-1/\alpha})$.

1. No cell boundaries: Every user has nearby APs, eliminating the cell-edge problem.
2. Macro-diversity: Signals from multiple APs provide resilience against shadowing.
3. Fairness: Max-min power control achieves 5--10x higher 95%-likely rate.

1. Fronthaul: Connecting all APs to a CPU requires extensive fronthaul infrastructure.
2. Synchronization: Coherent transmission requires tight time/phase synchronization.
3. Scalability: The original formulation where every AP serves every user has $O(L \cdot K)$ complexity.

$\tau_p / \tau_c = 20 / 200 = 10\%$. Moderate overhead.

$\tau_p / \tau_c = 10 / 200 = 5\%$. Lower overhead, but introduces pilot contamination between users sharing the same pilot.

Reducing pilot overhead increases the fraction available for data ($\tau_d / \tau_c$) but degrades channel estimation quality due to pilot contamination. The optimal balance depends on the channel statistics and user geometry.

Let $u_{lk} = \sqrt{\eta_{lk}} \geq 0$. The desired signal strength for user $k$ is $S_k = \sum_l u_{lk} \gamma_{lk}$, which is linear in $\{u_{lk}\}$.

$\text{SINR}_k \geq t$ becomes: $S_k^2 \geq t \cdot (\text{interference}_k + \sigma^2/P_t)$. Taking square roots: $S_k \geq \sqrt{t} \cdot \|\mathbf{d}_k\|_2$ where $\mathbf{d}_k$ collects the interference and noise terms. This is a second-order cone constraint.

$\sum_k u_{lk}^2 \gamma_{lk} \leq 1$, which is $\|\mathbf{f}_l\|_2 \leq 1$ where $f_{lk} = u_{lk} \sqrt{\gamma_{lk}}$. This is also an SOC constraint. $\blacksquare$

In co-located massive MIMO, all antennas are at the same location and share the same path loss and shadowing to a given user. Hardening averages out small-scale fading only: $\frac{1}{N_t} \|\mathbf{H}_{k}\|^2 \to \beta_{k}$.

In cell-free, APs at different locations have different path losses $\beta_{lk}$ and independent shadowing. The aggregate channel $G_k = \sum_l \beta_{lk} |g_{lk}|^2$ averages out both small-scale fading ($g_{lk}$) and shadow fading variations across AP locations. This provides a stronger form of hardening.

$\frac{1}{L} \sum_l \gamma_{lk} \to \mathbb{E}[\gamma] \triangleq \bar{\gamma}$ by WLLN. The numerator is $(L \bar{\gamma})^2$.

The interference term: $\sum_{j \neq k} \sum_l \gamma_{lj} \beta_{lk} \to (K-1) L \cdot \mathbb{E}[\gamma \beta]$. Total denominator: $(K-1) L \, \mathbb{E}[\gamma \beta] + \sigma^2/(\eta P_t)$.

$\text{SINR}_k \to \frac{L^2 \bar{\gamma}^2}{(K-1) L \, \mathbb{E}[\gamma \beta] + \text{noise}} = \frac{L \bar{\gamma}^2}{(\rho^{-1} - 1/L) \, \mathbb{E}[\gamma \beta] + \text{noise}/L}$. For large $L$: $\text{SINR}_k \approx \frac{\rho \, \bar{\gamma}^2}{\mathbb{E}[\gamma \beta]}$.

$R_k \to \log_2\left(1 + \frac{\rho \, \bar{\gamma}^2}{\mathbb{E}[\gamma \beta]}\right)$, which is deterministic and depends only on the distribution of path losses through $\bar{\gamma}$ and $\mathbb{E}[\gamma \beta]$.

$\hat{\mathbf{H}}_{lk} = \frac{\sqrt{\tau_p p_p} \, \beta_{lk}}{\tau_p p_p (\beta_{lk} + \beta_{lk'}) + \sigma^2} \mathbf{y}_l^{(k)}$ where $k'$ is the user sharing $k$'s pilot.

The term $\beta_{lk'}$ in the denominator represents pilot contamination. If user $k'$ is close to AP $l$, the contamination is severe. If $k'$ is far, $\beta_{lk'} \approx 0$.

In cellular systems, pilot contamination comes from users in other cells at similar angular positions — they can be at the same distance as the desired user. In cell-free systems, pilot assignment can exploit geography: assign the same pilot to users that are far apart, ensuring $\beta_{lk'} \ll \beta_{lk}$ at APs near user $k$.

Per AP: $K$ complex scalars = $16 \times 2 \times 16 = 512$ bits. Total: $64 \times 512 = 32{,}768$ bits per coherence interval. Fronthaul rate: $32{,}768 \times (W / \tau_c) = 32{,}768 \times 100{,}000 = 3.28$ Gbit/s.

Per AP: $N \cdot \tau_c = 1 \times 200 = 200$ complex samples = $200 \times 2 \times 16 = 6{,}400$ bits. Total: $64 \times 6{,}400 = 409{,}600$ bits per coherence interval. Fronthaul rate: $409{,}600 \times 100{,}000 = 40.96$ Gbit/s.

Level 4 requires $\sim 12\times$ more fronthaul than Level 1. This motivates distributed processing strategies that keep computation local to the APs.

$\sum_l \mathbf{H}_{lk}^* \mathbf{H}_{lj} = \sum_l \sqrt{\beta_{lk} \beta_{lj}} \, g_{lk}^* g_{lj}$. Since $g_{lk}^* g_{lj}$ has zero mean (independence of $g_{lk}, g_{lj}$ for $k \neq j$), $\text{Var}(\sum_l \cdots) = \sum_l \beta_{lk} \beta_{lj}$.

$\sum_l |\mathbf{H}_{lk}|^2 = \sum_l \beta_{lk} |g_{lk}|^2 \to \sum_l \beta_{lk}$ (WLLN).

The normalized cross-correlation has variance $\frac{\sum_l \beta_{lk} \beta_{lj}}{(\sum_l \beta_{lk})(\sum_l \beta_{lj})} \leq \frac{\max_l \min(\beta_{lk}, \beta_{lj})}{\min(\sum_l \beta_{lk}, \sum_l \beta_{lj})} \to 0$. By Chebyshev, the ratio converges to 0. $\blacksquare$

With 3 BSs at equal distance, each contributing amplitude $A$: Coherent: $(3A)^2 = 9A^2$. Single-cell: $A^2$. Gain: $9\times$ or 9.5 dB.

1. Initialize $\tau_p$ empty pilot groups.
2. For each user $k$ (in order of decreasing max $\beta_{lk}$): a. For each pilot group $p$: compute the additional contamination if $k$ joins group $p$. b. Assign $k$ to the group with minimum additional contamination.

Additional contamination from assigning user $k$ to group $p$: $\Delta C = \sum_{k' \in p} \sum_l (\gamma_{lk} \beta_{lk'} + \gamma_{lk'} \beta_{lk})$.

$O(K \cdot \tau_p \cdot (K/\tau_p) \cdot L) = O(K^{2} L / \tau_p)$. For $L = 100$, $K = 40$, $\tau_p = 10$: $\sim 16{,}000$ operations. Fast enough for real-time.

$\beta_{lk} = \beta_{k}$, $\gamma_{lk} = \gamma_k$ for all $l$. Then $\sum_l \sqrt{\eta_{lk}} \gamma_{lk} = L \sqrt{\eta_k} \gamma_k$ (with $\eta_{lk} = \eta_k$).

$\text{SINR}_k = \frac{L^2 \eta_k \gamma_k^2}{\sum_{j \neq k} L \eta_j \gamma_j \beta_{k} + \sigma^2/P_t}$. This is the standard co-located massive MIMO SINR with $N_t = L$ antennas. $\blacksquare$