Ferkans — Interactive Telecom Tutor

Fix the Channel, Choose the Precoder

Once $\boldsymbol{\Phi}$ is fixed, the RIS-aided system looks identical to a standard MU-MIMO downlink with the effective channel $\mathbf{H}_{k,\text{eff}}^H = \mathbf{h}_{k,d}^H + \mathbf{h}_{k,2}^H \boldsymbol{\Phi} \mathbf{H}_1$ . The active beamforming subproblem is therefore any known MU-MIMO precoder design: MRT for single-user, ZF for interference-nulling, WMMSE for sum-rate maximization, etc. We recap the WMMSE scheme here because it is the default choice inside the AO loop and because it has an elegant closed-form update structure.

Definition:
WMMSE Reformulation of Sum-Rate Maximization

Consider the sum-rate maximization $\max_{\mathbf{W}: \text{tr}(\mathbf{W}^{H}\mathbf{W}) \leq P_t} \sum_k \log_2(1 + \text{SINR}_k)$ with $\text{SINR}_k$ as in DThe Joint Active-Passive Beamforming Problem. Introducing auxiliary variables $\mathbf{u}_k \in \mathbb{C}^{N_t}$ (combiner weights) and $w_k > 0$ (WMMSE weights), the sum-rate problem is equivalent to the WMMSE problem

$\min_{\mathbf{W}, \{\mathbf{u}_k\}, \{w_k\}} \sum_k w_k\, e_k(\mathbf{W}, \mathbf{u}_k) - \log_2 w_k \quad \text{s.t.}\ \text{tr}(\mathbf{W}^{H} \mathbf{W}) \leq P_t,$

where the MSE at user $k$ is $e_k = \mathbb{E}[|\mathbf{u}_k^H y_k - s_k|^2]$ . The two problems have the same KKT conditions; the WMMSE reformulation is block-convex in $(\mathbf{W}, \{\mathbf{u}_k\}, \{w_k\})$ and admits closed-form coordinate updates.

The WMMSE identity is a classical result from MU-MIMO optimization (Christensen et al. 2008, Shi et al. 2011). It is reused here unchanged: once $\boldsymbol{\Phi}$ fixes the effective channels, the RIS disappears from the active-beamformer problem entirely.

Theorem: Closed-Form WMMSE Coordinate Updates

At block-coordinate optima of the WMMSE problem:

Combiner update (MMSE receiver): $\mathbf{u}_k^\star = (\sum_j \mathbf{H}_{k,\text{eff}}^H \mathbf{v}_{j} \mathbf{v}_{j}^{H} \mathbf{H}_{k,\text{eff}} + \sigma^2 \mathbf{I})^{-1} \mathbf{H}_{k,\text{eff}}^H \mathbf{v}_{k}$ .
Weight update: $w_k^\star = 1/(1 - \mathbf{u}_k^{\star H} \mathbf{H}_{k,\text{eff}}^H \mathbf{v}_{k})$ . Equivalently, $w_k^\star = 1 / e_k^\star$ .
Precoder update (regularized ZF): $\mathbf{v}_{k}^\star = w_k \big(\sum_j w_j \mathbf{H}_{j,\text{eff}} \mathbf{u}_j \mathbf{u}_j^H \mathbf{H}_{j,\text{eff}}^H + \mu \mathbf{I}\big)^{-1} \mathbf{H}_{k,\text{eff}} \mathbf{u}_k$ , where $\mu \geq 0$ is the Lagrange multiplier for the power constraint, found by bisection.

The algorithm converges monotonically to a local optimum of the original sum-rate problem.

Each of the three variable blocks ( $\mathbf{u}_k$ , $w_k$ , $\mathbf{W}$ ) has a closed-form optimum given the others: MMSE combiner, rate weight, and a dual-variable-regulated linear precoder.

Proof

MMSE receiver from orthogonality

The optimal $\mathbf{u}_k$ that minimizes $\mathbb{E}|\mathbf{u}_k^H y_k - s_k|^2$ is the MMSE estimator: $\mathbf{u}_k = \mathbf{R}^{-1} \mathbf{H}_{k,\text{eff}}^H \mathbf{v}_{k}$ , where $\mathbf{R}$ is the total received covariance at user $k$ .

Weight update from Lagrangian

At the WMMSE optimum, the Lagrangian gradient gives $w_k = 1/e_k$ . This turns the objective into $\sum_k [1/e_k \cdot e_k - \log_2(1/e_k)] = \sum_k [1 + \log_2 e_k]$ , and $\log_2(1/e_k) = \log_2(1 + \text{SINR}_k)$ — the sum rate.

Precoder update from KKT

$\nabla_{\mathbf{v}_{k}} [\sum_j w_j e_j + \mu \text{tr}(\mathbf{W}^{H}\mathbf{W})] = 0$ gives a linear equation in $\mathbf{v}_{k}$ ; solving and applying the power constraint via the Lagrange multiplier $\mu$ gives the stated form. $\blacksquare$

WMMSE Sum-Rate Maximization (Inner Loop of AO)

Complexity:

O(T_{\text{WMMSE}} \cdot K N_t^{3})

per outer AO iteration;

T_{\text{WMMSE}} \sim 5

-

10

typical

Input: effective channels

\mathbf{H}_{k,\text{eff}}

(fixed, given

\boldsymbol{\Phi}

);

power budget

P_t

; tolerance

\epsilon_{\text{in}}

.

Output: precoder

\mathbf{W}^\star

.

1. Initialize

\mathbf{W}^{(0)}

, e.g., zero-forcing on direct channels, scaled to satisfy power.

2. For

t = 0, 1, 2, \ldots

:

3.

\quad

Combiner update:

\mathbf{u}_k^{(t+1)} =

MMSE receiver given

\mathbf{W}^{(t)}

.

4.

\quad

Weight update:

w_k^{(t+1)} = 1 / e_k^{(t+1)}

.

5.

\quad

Precoder update: Bisection on

\mu

to satisfy

\text{tr}(\mathbf{W}^{(t+1),H} \mathbf{W}^{(t+1)}) = P_t

; each inner problem is a linear system.

6.

\quad

If sum-rate change

< \epsilon_{\text{in}}

: break.

7. return

\mathbf{W}^{(t+1)}

.

Each WMMSE inner loop is $\mathcal{O}(KN_t^{3})$ per iteration, dominated by the $N_t \times N_t$ matrix inversion in the precoder update. For moderate $N_t$ (say $\leq 64$ ), this is sub-millisecond on modern hardware. The bisection on $\mu$ converges in $\sim 10$ iterations.

Example: Zero-Forcing as a Simplified Alternative

At high SNR, the optimal precoder approaches zero-forcing (ZF), which nulls inter-user interference. Derive the ZF precoder given the RIS-adjusted effective channel matrix $\mathbf{H}_{\text{eff}}^{\text{MU}} \in \mathbb{C}^{K \times N_t}$ (rows = per-user effective channels), and state the SINR under ZF.

Solution

Right inverse

ZF sets $\mathbf{W}^{\text{ZF}} = \alpha\,\mathbf{H}_{\text{eff}}^{\text{MU},H} (\mathbf{H}_{\text{eff}}^{\text{MU}} \mathbf{H}_{\text{eff}}^{\text{MU},H})^{-1}$ , where $\alpha$ normalizes total power. This zeros the inter-user terms: $\mathbf{H}_{\text{eff}}^{\text{MU}} \mathbf{W}^{\text{ZF}} = \alpha \mathbf{I}$ , so $\text{SINR}_k = P_t \alpha^2 / (K \sigma^2)$ (equal among users under equal power allocation).

RIS benefit

The RIS affects ZF indirectly by changing $\mathbf{H}_{\text{eff}}^{\text{MU}}$ . Increasing the Frobenius norm of the effective channel reduces the power inflation factor $\alpha^{-2} \propto \text{tr}((\mathbf{H}_{\text{eff}}^{\text{MU}}\mathbf{H}_{\text{eff}}^{\text{MU},H})^{-1})$ — the "ZF power penalty." An RIS that improves per-user channel norms thus directly improves the ZF rate.

Suboptimality

ZF is strictly suboptimal at finite SNR (WMMSE does better). But ZF is a closed-form one-shot computation — easily used as a warm-start for the WMMSE inner loop or as the active update inside AO when compute is constrained. $\blacksquare$

Rate vs. Transmit Power: With/Without RIS

Compare the sum-rate achievable by alternating optimization (WMMSE

RIS) against a no-RIS baseline (direct-channel WMMSE only). The RIS curve shows the joint optimization gain. Change $N$ to see how larger RIS widens the gap.

Parameters

RIS elements

N

64

BS antennas

N_t

8

Users

K

2

Min transmit SNR (dB)-10

Max transmit SNR (dB)30

Single-User: MRT + Element-Wise = Closed Form

For $K = 1$ , the WMMSE reduces to MRT (matched filter) — a closed-form expression with no iteration. The passive subproblem reduces to element-wise matching (Chapter 6). AO converges in 3-5 iterations for single-user, and the rate is within $< 0.1\text{ bits/s/Hz}$ of the global optimum. This is the clean case where RIS optimization is essentially solved. Multi-user complications require the full WMMSE machinery and Chapter 6's SDR/manifold methods on the passive side.

Common Mistake: WMMSE Initialization Matters

Mistake:

"Initialize $\mathbf{W}^{(0)} = \mathbf{0}$ to satisfy the power constraint trivially."

Correction:

WMMSE diverges from $\mathbf{W}^{(0)} = \mathbf{0}$ : the MSE is 1 per user, the weights are 1, and the combiner update has no signal to lock onto. Reasonable initializations: (i) ZF scaled to the power budget; (ii) MRT with equal power per user; (iii) random precoder with power constraint satisfied. All converge to the same local optimum within AO, but (i) is typically fastest.

The Active-Beamforming Subproblem