Ferkans — Interactive Telecom Tutor

Why Optimize the CPU Weights?

In Levels 1 and 2, the CPU simply sums the local estimates: $\hat{s}_k = \sum_m \hat{s}_{mk}$ . This equal-weight combining ignores a key fact: the quality of $\hat{s}_{mk}$ varies enormously across APs. An AP with strong path loss to user $k$ provides a reliable estimate; a distant AP provides mostly noise. Equal combining is wasteful — it allows noisy APs to degrade the overall estimate. LSFD solves this by computing optimal weights that account for each AP's signal quality, interference level, and correlation with other APs' estimates. The remarkable fact is that these optimal weights depend only on large-scale statistics, making them practical to compute and update.

Theorem: SINR Under LSFD (Level 3)

Let $\hat{s}_{mk} = \mathbf{a}_{mk}^H \mathbf{y}_m$ be the local estimate of user $k$ 's symbol at AP $m$ using arbitrary local combining. Define the $M \times 1$ vector of local estimates $\hat{\mathbf{s}}_k = [\hat{s}_{1k}, \ldots, \hat{s}_{Mk}]^T$ and the CPU combining vector $\boldsymbol{\alpha}_k \in \mathbb{C}^M$ . The final estimate is $\hat{s}_k = \boldsymbol{\alpha}_k^H \hat{\mathbf{s}}_k$ .

Under the UatF bound, the SINR of user $k$ is

$\text{SINR}_k^{(3)} = \frac{p_k |\boldsymbol{\alpha}_k^H \mathbf{b}_k|^2}{\boldsymbol{\alpha}_k^H \left( \sum_{j=1}^{K} p_j \mathbf{D}_{kj} - p_k \mathbf{b}_k \mathbf{b}_k^H \right) \boldsymbol{\alpha}_k}$

where the $M \times 1$ vector $\mathbf{b}_k$ and the $M \times M$ matrices $\mathbf{D}_{kj}$ are defined as

$[\mathbf{b}_k]_m = \mathbb{E}[\mathbf{a}_{mk}^H \mathbf{g}_{mk}], \qquad [\mathbf{D}_{kj}]_{m,m'} = \mathbb{E}[(\mathbf{a}_{mk}^H \mathbf{g}_{mj})(\mathbf{g}_{m'j}^H \mathbf{a}_{m'k})]$

These quantities depend only on the large-scale fading coefficients and the local combining statistics.

The vector $\mathbf{b}_k$ captures the expected beamforming gain from each AP to user $k$ . The matrices $\mathbf{D}_{kj}$ capture the inter-AP correlation of the interference from user $j$ as seen through the local combiners. The SINR has the form of a generalized Rayleigh quotient in $\boldsymbol{\alpha}_k$ .

Proof

Signal decomposition

Write $\hat{s}_{mk} = \mathbf{a}_{mk}^H \mathbf{y}_m$ . Under the UatF approach, the desired signal component is $\sqrt{p_k} \, \mathbb{E}[\mathbf{a}_{mk}^H \mathbf{g}_{mk}] \, s_k$ , and everything else is treated as uncorrelated effective noise.

Desired signal power

The desired signal power after CPU combining is $|\boldsymbol{\alpha}_k^H \mathbf{b}_k|^2 p_k |\mathbb{E}[s_k]|^2 = p_k |\boldsymbol{\alpha}_k^H \mathbf{b}_k|^2$ where $[\mathbf{b}_k]_m = \mathbb{E}[\mathbf{a}_{mk}^H \mathbf{g}_{mk}]$ .

Interference-plus-noise power

The variance of the effective noise is $\boldsymbol{\alpha}_k^H \mathbf{E}_k \boldsymbol{\alpha}_k$ where $\mathbf{E}_k = \sum_{j=1}^{K} p_j \mathbf{D}_{kj} - p_k \mathbf{b}_k \mathbf{b}_k^H$ captures inter-user interference, beamforming gain uncertainty, and thermal noise. The off-diagonal entries of $\mathbf{D}_{kj}$ capture the correlation of interference across different APs.

SINR expression

Combining yields $\text{SINR}_k^{(3)} = p_k |\boldsymbol{\alpha}_k^H \mathbf{b}_k|^2 / (\boldsymbol{\alpha}_k^H \mathbf{E}_k \boldsymbol{\alpha}_k)$ . This is a generalized Rayleigh quotient in $\boldsymbol{\alpha}_k$ . $\blacksquare$

Theorem: Optimal LSFD Weights

The LSFD weight vector that maximizes $\text{SINR}_k^{(3)}$ is the solution to the generalized eigenvalue problem

$\boldsymbol{\alpha}_k^{\star} = \mathbf{E}_k^{-1} \mathbf{b}_k$

where $\mathbf{E}_k = \sum_{j=1}^{K} p_j \mathbf{D}_{kj} - p_k \mathbf{b}_k \mathbf{b}_k^H$ . The resulting optimal SINR is

$\text{SINR}_k^{(3,\star)} = p_k \, \mathbf{b}_k^H \mathbf{E}_k^{-1} \mathbf{b}_k$

The optimal LSFD weights are the MMSE solution to the problem of estimating $s_k$ from the local estimates $\hat{\mathbf{s}}_k$ . An AP with strong signal quality (large $[\mathbf{b}_k]_m$ ) and low interference (small diagonal of $\mathbf{E}_k$ ) gets a large weight. An AP contributing mostly interference gets a small or even negative weight (partial interference cancellation across APs).

Proof

Generalized Rayleigh quotient optimization

The SINR is a generalized Rayleigh quotient: $\text{SINR}_k^{(3)} = p_k \frac{|\boldsymbol{\alpha}_k^H \mathbf{b}_k|^2}{\boldsymbol{\alpha}_k^H \mathbf{E}_k \boldsymbol{\alpha}_k}$

This is maximized by $\boldsymbol{\alpha}_k^{\star} = c \, \mathbf{E}_k^{-1} \mathbf{b}_k$ for any scalar $c > 0$ (the SINR is scale-invariant). Setting $c = 1$ gives $\boldsymbol{\alpha}_k^{\star} = \mathbf{E}_k^{-1} \mathbf{b}_k$ .

Optimal SINR value

Substituting back: $\text{SINR}_k^{(3,\star)} = p_k \frac{|\mathbf{b}_k^H \mathbf{E}_k^{-1} \mathbf{b}_k|^2}{\mathbf{b}_k^H \mathbf{E}_k^{-1} \mathbf{E}_k \mathbf{E}_k^{-1} \mathbf{b}_k} = p_k \, \mathbf{b}_k^H \mathbf{E}_k^{-1} \mathbf{b}_k$

since $\mathbf{E}_k^{-1} \mathbf{E}_k \mathbf{E}_k^{-1} = \mathbf{E}_k^{-1}$ . $\blacksquare$

Example: LSFD Weights for a Two-AP Scenario

Consider $M = 2$ single-antenna APs ( $N = 1$ ) serving $K = 2$ users with Level 1 (local MRC) at the APs. The large-scale fading coefficients are $\beta_{11} = 1$ , $\beta_{12} = 0.1$ , $\beta_{21} = 0.2$ , $\beta_{22} = 0.8$ . Transmit powers $p_1 = p_2 = 1$ , noise variance $\sigma^2 = 0.01$ . Pilot length $\tau_p = 2$ (orthogonal pilots). Compute the optimal LSFD weights for user 1 and compare the SINR with and without LSFD.

Solution

Compute channel estimate variances

With orthogonal pilots and MMSE estimation: $\gamma_{mk} = \frac{\tau_p p_k \beta_{mk}^{2}}{\tau_p p_k \beta_{mk} + \sigma^2}$

$\gamma_{11} = \frac{2 \times 1 \times 1}{2 \times 1 + 0.01} = \frac{2}{2.01} \approx 0.995$

$\gamma_{21} = \frac{2 \times 0.04}{2 \times 0.2 + 0.01} = \frac{0.08}{0.41} \approx 0.195$

$\gamma_{12} = \frac{2 \times 0.01}{2 \times 0.1 + 0.01} = \frac{0.02}{0.21} \approx 0.095$

$\gamma_{22} = \frac{2 \times 0.64}{2 \times 0.8 + 0.01} = \frac{1.28}{1.61} \approx 0.795$

Compute $\mathbf{b}_1$ and $\mathbf{D}_{1j}$

For MRC with $N = 1$ : $\mathbf{a}_{mk} = \hat{g}_{mk}$ , so $[\mathbf{b}_1]_m = \mathbb{E}[|\hat{g}_{m1}|^2] = \gamma_{m1}$ .

$\mathbf{b}_1 = \begin{bmatrix} 0.995 \\ 0.195 \end{bmatrix}$

The matrices $\mathbf{D}_{1j}$ are diagonal (APs have independent noise): $[\mathbf{D}_{1j}]_{mm} = \mathbb{E}[|\hat{g}_{m1}|^2 |g_{mj}|^2] = \gamma_{m1} (\gamma_{mj} + \beta_{mj})$ for $j = 1$ , and $= \gamma_{m1} \beta_{mj}$ for $j \neq 1$ .

Actually, for the diagonal case with $N=1$ : $[\mathbf{D}_{1j}]_{mm} = \gamma_{m1} \beta_{mj} + \gamma_{m1}^2 \delta_{j1}$

Let us compute $\mathbf{E}_1$ : $[\mathbf{E}_1]_{mm} = \sum_j p_j \gamma_{m1} \beta_{mj} + \sigma^2 \gamma_{m1}$

Compute $\mathbf{E}_1$ and optimal weights

$[\mathbf{E}_1]_{11} = 0.995 \times (1 + 0.1) + 0.01 \times 0.995 = 0.995 \times 1.11 = 1.104$

$[\mathbf{E}_1]_{22} = 0.195 \times (0.2 + 0.8) + 0.01 \times 0.195 = 0.195 \times 1.01 = 0.197$

Since the APs are independent, $\mathbf{E}_1$ is diagonal.

$\boldsymbol{\alpha}_1^{\star} = \mathbf{E}_1^{-1} \mathbf{b}_1 = \begin{bmatrix} 0.995/1.104 \\ 0.195/0.197 \end{bmatrix} = \begin{bmatrix} 0.901 \\ 0.990 \end{bmatrix}$

The weights are similar because, despite AP 1 having much better path loss to user 1, AP 2's signal-to-interference ratio is also favorable (user 2 is closer to AP 2, creating less interference at AP 2 for user 1).

SINR comparison

With LSFD: $\text{SINR}_1^{(3,\star)} = \mathbf{b}_1^H \mathbf{E}_1^{-1} \mathbf{b}_1 = 0.995^2/1.104 + 0.195^2/0.197 = 0.897 + 0.193 = 1.090$ (linear), i.e., $0.37$ dB.

Without LSFD (equal weights): $\boldsymbol{\alpha}_1 = [1, 1]^T$ : $\text{SINR}_1^{(2)} = \frac{(0.995 + 0.195)^2}{1.104 + 0.197} = \frac{1.416}{1.301} = 1.088$ (linear), i.e., $0.37$ dB.

In this simple example, the gain from LSFD is negligible because the interference structure is already favorable. The LSFD gain becomes significant when interference correlations across APs are strong, which requires more users and APs.

LSFD Rate Improvement over Equal-Weight Combining

Compare the per-user achievable rate with optimal LSFD weights (Level 3) vs. equal-weight combining (Level 2) as a function of the number of users. LSFD provides the largest gain when the network is interference-limited (many users, high SNR).

Parameters

M

(APs)100

Number of access points

N

(antennas/AP)1

Antennas per access point

K_{\\max}

40

Maximum number of users

\text{SNR}

(dB)10

Average transmit SNR in dB

Common Mistake: Ignoring Off-Diagonal Terms in LSFD

Mistake:

Approximating $\mathbf{E}_k$ as diagonal (ignoring inter-AP correlation of interference) and computing LSFD weights as if the APs were independent.

Correction:

The off-diagonal entries $[\mathbf{D}_{kj}]_{m,m'}$ for $m \neq m'$ capture the correlation of interference from user $j$ across APs $m$ and $m'$ . These terms are nonzero when two APs share the same pilot contamination source. Ignoring them leads to suboptimal LSFD weights and can degrade performance by 10–20% in pilot-contaminated scenarios. Always compute the full $\mathbf{E}_k$ matrix.

Definition:
LSFD Combined with Local MMSE (Level 3 Proper)

When Level 3 uses local MMSE combining (Level 2) at the APs followed by optimal LSFD weights at the CPU, the quantities become:

$[\mathbf{b}_k]_m = \mathbb{E}\left[ \hat{\mathbf{g}}_{mk}^H \left( \sum_{j \in \mathcal{D}_m} p_j \hat{\mathbf{g}}_{mj} \hat{\mathbf{g}}_{mj}^H + \mathbf{Z}_m \right)^{-1} \mathbf{g}_{mk} \right]$

where $\mathbf{Z}_m = \sum_{j \in \mathcal{D}_m} p_j \mathbf{C}_{mj} + \sigma^2 \mathbf{I}_N$ . These expectations can be computed in closed form for Rayleigh fading using the matrix inversion lemma and properties of Wishart distributions.

The Level 3 SINR with local MMSE satisfies

$\text{SINR}_k^{(2)} \leq \text{SINR}_k^{(3)} \leq \text{SINR}_k^{(4)}$

where the first inequality is strict whenever the interference structure is non-uniform across APs.

Level 3 is the recommended operating point for most practical deployments. It combines the local interference suppression of MMSE with the macro-diversity exploitation of LSFD, requiring only large-scale statistics at the CPU.

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Key Takeaway

LSFD is the "sweet spot" of cell-free processing. By optimizing the CPU weights using only large-scale fading statistics, Level 3 captures the macro-diversity gain of the distributed architecture without requiring instantaneous CSI at the CPU. The optimal weights are the solution to an $M \times M$ linear system that changes only when users move significantly. In practice, LSFD closes 70–90% of the gap between equal-weight combining (Level 2) and centralized MMSE (Level 4), at negligible additional computational cost.

Quick Check

How often do the LSFD weights $\boldsymbol{\alpha}_k^{\star}$ need to be recomputed?

Every coherence interval (every few milliseconds)

Every time the large-scale fading changes (every 100–1000 ms)

Only once during network deployment

Every time a new user joins the network

Correction:

Every time the large-scale fading changes (every 100–1000 ms)

Correct. The LSFD weights depend on $\{\beta_{mk}\}$ and the local combining statistics, which change on the large-scale fading timescale. For pedestrian users, this is on the order of 100 ms to 1 s.

Generalized Rayleigh Quotient

A ratio of the form $R(\mathbf{x}) = \frac{\mathbf{x}^H \mathbf{A} \mathbf{x}}{\mathbf{x}^H \mathbf{B} \mathbf{x}}$ where $\mathbf{A}$ and $\mathbf{B}$ are Hermitian matrices and $\mathbf{B}$ is positive definite. The maximum of $R(\mathbf{x})$ over $\mathbf{x} \neq \mathbf{0}$ equals the largest generalized eigenvalue of $(\mathbf{A}, \mathbf{B})$ , attained by the corresponding eigenvector. The LSFD SINR optimization is a special case with rank-one $\mathbf{A}$ .

Large-Scale Fading Decoding

Why Optimize the CPU Weights?

Theorem: SINR Under LSFD (Level 3)

Signal decomposition

Desired signal power

Interference-plus-noise power

SINR expression

Theorem: Optimal LSFD Weights

Generalized Rayleigh quotient optimization

Optimal SINR value

Example: LSFD Weights for a Two-AP Scenario

Compute channel estimate variances

Compute $\mathbf{b}_1$ and $\mathbf{D}_{1j}$

Compute $\mathbf{E}_1$ and optimal weights

SINR comparison

LSFD Rate Improvement over Equal-Weight Combining

Parameters

Common Mistake: Ignoring Off-Diagonal Terms in LSFD

Definition: LSFD Combined with Local MMSE (Level 3 Proper)

Key Takeaway

Quick Check

Generalized Rayleigh Quotient

Definition:
LSFD Combined with Local MMSE (Level 3 Proper)