Ferkans — Interactive Telecom Tutor

Beyond Linear Precoding — Iterative Optimization

ZF and RZF precoding are closed-form but suboptimal: they do not directly maximise the weighted sum rate. The weighted sum-rate maximisation problem is NP-hard in general, but a remarkable reformulation — the weighted minimum mean-square error (WMMSE) approach — transforms it into a sequence of convex subproblems that converges to a stationary point. The WMMSE algorithm is the workhorse of modern multi-user MIMO optimisation, bridging the gap between simple linear precoding and the unattainable DPC bound.

Definition:
Weighted Sum-Rate Maximisation and WMMSE Reformulation

Consider the $K$ -user MISO BC where user $k$ 's rate is:

$R_k = \log_2\!\left(1 + \frac{|\mathbf{h}_k^H\mathbf{w}_k|^2} {\sigma^2 + \sum_{j \neq k}|\mathbf{h}_k^H\mathbf{w}_j|^2}\right)$

The weighted sum-rate (WSR) maximisation problem is:

$\max_{\{\mathbf{w}_k\}} \sum_{k=1}^{K} \mu_k R_k \quad \text{s.t.} \quad \sum_{k=1}^{K}\|\mathbf{w}_k\|^2 \leq P$

where $\mu_k > 0$ are user weights. This is non-convex due to the interference terms in $R_k$ .

The WMMSE reformulation introduces auxiliary variables:

Receive filter $u_k \in \mathbb{C}$ (MMSE equaliser)
MSE weight $w_k > 0$

The MSE for user $k$ with receive filter $u_k$ is:

$e_k(u_k, \{\mathbf{w}_j\}) = |1 - u_k^*\mathbf{h}_k^H\mathbf{w}_k|^2 + |u_k|^2\!\left(\sum_{j \neq k}|\mathbf{h}_k^H\mathbf{w}_j|^2 + \sigma^2\right)$

The key identity (Shi et al., 2011) is:

$\mu_k R_k = \max_{w_k > 0, u_k} \left[\mu_k\log_2(w_k) - \mu_k w_k e_k + \mu_k\right]$

which transforms the WSR problem into:

$\max_{\{u_k, w_k, \mathbf{w}_k\}} \sum_{k=1}^{K} \left[\mu_k\log_2(w_k) - \mu_k w_k e_k + \mu_k\right] \quad \text{s.t.} \quad \sum_k\|\mathbf{w}_k\|^2 \leq P$

This reformulated problem is convex in each block of variables $(u_k, w_k, \mathbf{w}_k)$ when the others are fixed.

WMMSE Algorithm

Complexity: Each iteration requires

O(KN_t^2 + N_t^3)

operations for the matrix inversion in Step 3. Convergence typically occurs within 10-30 iterations. Total complexity:

O(T_{\text{iter}}(KN_t^2 + N_t^3))

.

Input: Channel vectors

\{\mathbf{h}_k\}_{k=1}^K

, weights

\{\mu_k\}

, power

P

, noise

\sigma^2

, tolerance

\epsilon

Initialise: Precoding vectors

\{\mathbf{w}_k^{(0)}\}

(e.g., from RZF), iteration

t = 0

Repeat:

\quad

t \leftarrow t + 1

\quad

Step 1 — Update receive filters (MMSE receivers):

\quad\quad

u_k^{(t)} = \frac{\mathbf{h}_k^H\mathbf{w}_k^{(t-1)}}{\sum_{j=1}^{K}|\mathbf{h}_k^H\mathbf{w}_j^{(t-1)}|^2 + \sigma^2}, \quad \forall k

\quad

Step 2 — Update MSE weights:

\quad\quad

e_k^{(t)} = 1 - u_k^{(t)*}\mathbf{h}_k^H\mathbf{w}_k^{(t-1)}

\quad\quad

w_k^{(t)} = 1/e_k^{(t)}, \quad \forall k

\quad

Step 3 — Update precoding vectors (weighted MMSE):

\quad\quad

\mathbf{w}_k^{(t)} = \left(\sum_{j=1}^{K}\mu_j w_j^{(t)}|u_j^{(t)}|^2 \mathbf{h}_j\mathbf{h}_j^H + \lambda\mathbf{I}\right)^{-1} \mu_k w_k^{(t)} u_k^{(t)} \mathbf{h}_k, \quad \forall k

\quad\quad

where

\lambda \geq 0

is chosen to satisfy

\sum_k\|\mathbf{w}_k^{(t)}\|^2 = P

\quad

Compute:

f^{(t)} = \sum_k \mu_k R_k(\{\mathbf{w}_k^{(t)}\})

Until

|f^{(t)} - f^{(t-1)}| < \epsilon

Output: Precoding vectors

\{\mathbf{w}_k^{(t)}\}

The WMMSE algorithm can be initialised with ZF or RZF precoders, which provide a good starting point and reduce the number of iterations needed. The algorithm extends naturally to multi-cell and multi-antenna user scenarios.

WMMSE Convergence Animation

Watch the WMMSE algorithm converge iteration by iteration: the weighted sum rate increases monotonically from the ZF initialisation toward the DPC bound. The monotonic increase is guaranteed by the block coordinate descent structure.

WMMSE convergence for a 4-user, 8-antenna system at 15 dB SNR. The algorithm converges within ~15 iterations, closing most of the gap between ZF and DPC.

WMMSE Algorithm Convergence

Observe the convergence behaviour of the WMMSE algorithm. The plot shows the weighted sum rate versus iteration number. Vary the number of BS antennas, users, and SNR to see how they affect convergence speed and the final sum rate relative to ZF and RZF baselines.

Parameters

N_t

(BS antennas)8

K

(users)4

SNR (dB)15

Theorem: WMMSE Convergence Guarantee

The WMMSE algorithm generates a sequence of precoding matrices $\{\mathbf{w}_k^{(t)}\}_{t=0}^{\infty}$ satisfying:

Monotonic non-decrease: The weighted sum rate is non-decreasing across iterations:

$\sum_{k=1}^{K}\mu_k R_k^{(t+1)} \geq \sum_{k=1}^{K}\mu_k R_k^{(t)}$
Convergence to stationary point: Every limit point of the sequence $\{\mathbf{w}_k^{(t)}\}$ satisfies the KKT conditions of the original WSR maximisation problem.

Each WMMSE step optimises one block of variables while fixing the others, and the WMMSE objective is an exact surrogate for the WSR. Since the WSR is bounded above (by the DPC rate), a monotonically non-decreasing bounded sequence must converge. The block coordinate descent structure, combined with the smoothness of the objective and the exactness of the rate-MSE relationship, guarantees KKT stationarity at the limit.

Proof

Monotonicity from block optimality

In Step 1, $u_k^{(t)}$ is the MMSE receiver for the current precoders, minimising $e_k$ . In Step 2, $w_k^{(t)} = 1/e_k^{(t)}$ maximises $\mu_k[\log_2(w_k) - w_k e_k + 1]$ over $w_k > 0$ . In Step 3, $\mathbf{w}_k^{(t)}$ solves a convex quadratic programme.

Each step either increases or maintains the WMMSE objective. Since the WMMSE objective equals the WSR at the jointly optimal $(u_k, w_k)$ for given $\{\mathbf{w}_k\}$ :

$\text{WSR}^{(t+1)} \geq \text{WMMSE}^{(t+1)} \geq \text{WMMSE}^{(t)} = \text{WSR}^{(t)}$

where the first inequality holds because the WMMSE objective is a lower bound on the WSR (tight at the optimal $u_k, w_k$ ).

Convergence to KKT point

Since the WSR is upper-bounded (by the DPC capacity) and non-decreasing, it converges. At any limit point, the optimality conditions of each block subproblem are satisfied simultaneously. These block-optimal conditions are equivalent to the KKT conditions of the original WSR problem because the WMMSE reformulation is an exact surrogate (no relaxation gap). $\blacksquare$

Example: One WMMSE Iteration

Consider $K = 2$ users, $N_t = 2$ , $\sigma^2 = 1$ , equal weights $\mu_1 = \mu_2 = 1$ , channels $\mathbf{h}_1 = [1, 0.5]^T$ , $\mathbf{h}_2 = [0.5, 1]^T$ , and initial precoders $\mathbf{w}_1^{(0)} = [1, 0]^T$ , $\mathbf{w}_2^{(0)} = [0, 1]^T$ (with $P = 2$ ). Perform one WMMSE iteration.

Solution

Step 1: Compute MMSE receivers

User 1: $\mathbf{h}_1^H\mathbf{w}_1^{(0)} = 1$ , $|\mathbf{h}_1^H\mathbf{w}_2^{(0)}|^2 = 0.25$

Denominator: $|1|^2 + 0.25 + 1 = 2.25$

$u_1^{(1)} = 1/2.25 = 0.444$

User 2: $\mathbf{h}_2^H\mathbf{w}_2^{(0)} = 1$ , $|\mathbf{h}_2^H\mathbf{w}_1^{(0)}|^2 = 0.25$

$u_2^{(1)} = 1/2.25 = 0.444$

Step 2: Compute MSE weights

$e_1 = 1 - 0.444 \times 1 = 0.556$

$w_1^{(1)} = 1/0.556 = 1.80$

$e_2 = 0.556$ , $w_2^{(1)} = 1.80$

Step 3: Update precoders

Matrix to invert:

$\mathbf{A} = 1.80 \times 0.444^2(\mathbf{h}_1\mathbf{h}_1^H + \mathbf{h}_2\mathbf{h}_2^H) + \lambda\mathbf{I}$

$= 0.355\begin{bmatrix} 1.25 & 1.0 \\ 1.0 & 1.25 \end{bmatrix} + \lambda\mathbf{I}$

The updated precoders $\mathbf{w}_k^{(1)} = \mathbf{A}^{-1}(1.80 \times 0.444)\mathbf{h}_k$ with $\lambda$ chosen to satisfy the power constraint.

After normalisation to $\sum\|\mathbf{w}_k\|^2 = 2$ , the updated precoders rotate slightly toward ZF directions, increasing the sum rate from the initial MRT-like configuration. $\blacksquare$

Quick Check

What is the key property that guarantees WMMSE convergence?

The WSR objective is convex

Each WMMSE step solves a convex subproblem, and the overall objective is monotonically non-decreasing and bounded above

The WMMSE objective is a convex relaxation of the WSR

WMMSE always converges to the global optimum

Correction:

Each WMMSE step solves a convex subproblem, and the overall objective is monotonically non-decreasing and bounded above

Block coordinate descent with convex subproblems ensures monotonic improvement. Boundedness (by DPC capacity) guarantees convergence. The WMMSE reformulation provides an exact surrogate that makes each block update tractable.

Historical Note: The WMMSE Algorithm

2011

The WMMSE algorithm was developed by Qingjiang Shi, Meisam Razaviyayn, Zhi-Quan Luo, and Chen He, published in IEEE Transactions on Signal Processing in 2011. The key insight was the rate-WMMSE relationship: $R_k = \max_{w_k} [\log(w_k) - w_k e_k + 1]$ , which transforms the non-convex WSR problem into a form amenable to block coordinate descent. This paper became one of the most cited works in signal processing for wireless communications, as the algorithm is remarkably versatile — it extends to MIMO users, multi-cell coordination, relay networks, and heterogeneous networks with minimal modifications.

Weighted Minimum Mean-Square Error (WMMSE)

An iterative algorithm for weighted sum-rate maximisation in multi-user MIMO systems. It alternates between updating receive filters, MSE weights, and precoding vectors, with each step solving a convex subproblem. Converges to a KKT stationary point.

Weighted Sum Rate (WSR)

The performance metric $\sum_k \mu_k R_k$ where $\mu_k > 0$ are user priority weights and $R_k$ are individual user rates. Different weight choices trace the Pareto boundary of the rate region.

WMMSE Algorithm and Iterative Optimization