Ferkans — Interactive Telecom Tutor

Sum Rate: The Utilitarian Objective

Sum rate $R_{\text{sum}} = \sum_k \log_2(1 + \text{SINR}_k)$ is the natural metric when the goal is aggregate throughput — the operator's perspective on spectral efficiency. It rewards giving more resources to stronger users (water-filling), and the RIS can amplify this effect by coherently aligning the strongest user's channel. Sum-rate maximization is the right objective for hot-spot deployments where average throughput matters more than per-user fairness. We develop the WMMSE-AO algorithm in detail, since it is reused throughout this chapter and the advanced architectures (Ch. 11–13).

Definition:
The RIS-Aided Sum-Rate Problem

The RIS-aided sum-rate maximization problem is

$\boxed{\;\max_{\mathbf{W}, \boldsymbol{\Phi}} \sum_{k=1}^{K} \log_2\!\left(1 + \frac{|\mathbf{h}_{k,\text{eff}}^H \mathbf{v}_{k}|^2}{\sum_{j \neq k} |\mathbf{h}_{k,\text{eff}}^H \mathbf{v}_{j}|^2 + \sigma^2}\right)\;}$

subject to $\text{tr}(\mathbf{W}^{H} \mathbf{W}) \leq P_t$ and $|\phi_n| = 1$ for all $n$ . The outer optimization is over the joint variable $(\mathbf{W}, \boldsymbol{\Phi})$ ; the problem is non-convex in both directions.

Theorem: WMMSE Equivalence for MU-RIS

Fix $\boldsymbol{\Phi}$ . The RIS sum-rate problem is equivalent to the WMMSE problem

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

where $e_k = \mathbb{E}[|u_k^* y_k - s_k|^2]$ is user $k$ 's MSE under scalar combiner $u_k$ , and $w_k > 0$ is the WMMSE weight. The optima satisfy

$u_k^\star = \mathbf{h}_{k,\text{eff}}^H \mathbf{v}_{k} / (\sum_j |\mathbf{h}_{k,\text{eff}}^H \mathbf{v}_{j}|^2 + \sigma^2)$ ,
$w_k^\star = 1 / e_k^\star$ ,
$\mathbf{v}_{k}^\star = w_k u_k^\star (\sum_j w_j |u_j|^2 \mathbf{h}_{j,\text{eff}} \mathbf{h}_{j,\text{eff}}^H + \mu \mathbf{I})^{-1} \mathbf{h}_{k,\text{eff}}$ ,

where $\mu \geq 0$ is the Lagrange multiplier for the power constraint (found by bisection).

The WMMSE identity (Chapter 5) applies unchanged when the channels are any complex matrices, including those reshaped by the RIS. We reuse the WMMSE trick here: introduce auxiliary variables $(\mathbf{u}_k, w_k)$ and convert the hard log-sum problem into a block-convex problem with closed-form updates.

Proof

Per-user WMMSE

By the classical MMSE-rate identity, $\max R_k$ at fixed precoder is equivalent to $\min \log_2 e_k^{\text{min}}$ , where $e_k^{\text{min}}$ is the MMSE. The auxiliary weight $w_k$ converts this into a jointly block-convex form.

Sum over users

Summing gives the stated problem. Optimality conditions for each block (combiner, weight, precoder) follow by setting the partial gradient to zero, exactly as in Ch. 5.

RIS plugs in

The update for $\mathbf{v}_{k}$ uses the RIS-aware effective channels $\mathbf{h}_{k,\text{eff}}$ . The RIS itself enters nowhere else in the active subproblem — it is entirely absorbed into the effective channels, as required by the AO separation. $\blacksquare$

,

AO for MU-RIS Sum-Rate Maximization

Complexity:

O(T \cdot (T_{\text{WMMSE}} K N_t^{3} + C_{\text{passive}}))

,

T \sim 20

Input: channels

\{\mathbf{h}_{k,d}, \mathbf{h}_{k,2}, \mathbf{H}_1\}

, power

P_t

, tolerance

\epsilon

.

Output:

(\mathbf{W}^\star, \boldsymbol{\Phi}^\star)

achieving a local optimum.

1. Initialize

\boldsymbol{\Phi}^{(0)}

(all ones or random unit-modulus).

2. Repeat

t = 0, 1, 2, \ldots

:

3.

\quad

Compute effective channels

\mathbf{h}_{k,\text{eff}}^{(t)}

.

4.

\quad

Active (WMMSE) inner loop:

5.

\quad\quad

Update

\{u_k\}

(MMSE),

\{w_k\}

(from MSE),

\{\mathbf{v}_{k}\}

(regularized least-squares + power bisection).

6.

\quad\quad

Iterate until sum-rate plateau (

\sim 5

-

10

inner iterations).

7.

\quad

Passive (RIS) update: use Ch. 6 algorithm on the multi-user QCQP

\max_{|\phi_n|=1} \sum_k w_k \{2\Re(...) - ...\}

, a sum of weighted quadratics.

8.

\quad

Check convergence:

|R_{\text{sum}}^{(t+1)} - R_{\text{sum}}^{(t)}| < \epsilon

.

9. return

(\mathbf{W}, \boldsymbol{\Phi})

.

The passive subproblem in multi-user setting has rank- $K$ quadratic form (one term per user in the objective) — higher rank than single-user, so element-wise BCD has larger gaps. Manifold or SDR preferred for $K \geq 4$ .

Sum Rate vs. Number of Users

Sweep the number of users $K$ from 1 to 16 and plot the RIS-aided sum rate alongside the no-RIS baseline. The RIS gain grows with $K$ up to a saturation point (where the DoF floor $\min(N_t, K)$ caps the multiplexing gain). Increase $N$ to see the saturation shift upward.

Parameters

RIS elements

N

64

BS antennas

N_t

8

Max users

K

10

Transmit SNR (dB)15

Example: WMMSE Iteration for a 2-User System

A 2-user RIS-aided MISO system has $N_t = 4$ , $N = 16$ , $P_t / \sigma^2 = 10\text{ dB}$ . Given initial $\boldsymbol{\Phi}^{(0)} = \mathbf{I}$ , execute one WMMSE inner iteration and one RIS element-wise sweep; check that the sum rate increased.

Solution

Compute $\mathbf{h}_{k,\text{eff}}^{(0)}$

$\mathbf{h}_{k,\text{eff}}^{(0)} = \mathbf{h}_{k,d} + \mathbf{H}_1^H \cdot \mathbf{1} \cdot \mathbf{h}_{k,2}$ . (Since $\boldsymbol{\Phi}^{(0)} = \mathbf{I}$ means $\phi_n = 1$ — not OFF; the RIS sums the incident wave coherently with the direct path.)

Initialize $\ntn{prec}^{(0)}$

Use regularized ZF: $\mathbf{W}^{(0)} = \alpha \mathbf{H}_{\text{eff}}^H (\mathbf{H}_{\text{eff}} \mathbf{H}_{\text{eff}}^H + \rho \mathbf{I})^{-1}$ with $\alpha$ chosen to meet the power constraint, $\rho = \sigma^2 K / P_t$ .

WMMSE: combiner update

Compute $u_k = \mathbf{h}_{k,\text{eff}}^H \mathbf{v}_{k} / (\|\mathbf{H}_{\text{eff}}^H \mathbf{W}\|^2 + \sigma^2)$ using current $\mathbf{W}^{(0)}$ .

WMMSE: weight update

$w_k = 1 / e_k$ , where $e_k = 1 - |\mathbf{h}_{k,\text{eff}}^H \mathbf{v}_{k} u_k|^2$ (the MMSE).

WMMSE: precoder update

Bisect over $\mu$ to find the value such that $\text{tr}(\mathbf{W}^{(1), H} \mathbf{W}^{(1)}) = P_t$ . The resulting $\mathbf{W}^{(1)}$ gives higher sum rate than $\mathbf{W}^{(0)}$ (monotone guarantee).

RIS element-wise sweep

For each $n$ , compute $\alpha_n$ (aggregated signal contribution at coordinate $n$ ) and update $\phi_n = e^{j\arg \alpha_n}$ . After one full sweep, $R_{\text{sum}}^{(1)} \geq R_{\text{sum}}^{(0)}$ by the monotonicity of Chapter 6's element-wise BCD. $\blacksquare$

Sum Rate Favors the Strong

At sum-rate optimum, stronger users receive more power and their channels are further amplified by the RIS. The optimization is Matthew-effect: the rich get richer. Users with poor direct channels get virtually none of the power budget unless the RIS can make them competitive. This is not a flaw of the algorithm — it is the efficient-market outcome of sum-rate maximization. If you dislike the outcome, switch objectives (max-min fairness, Section 7.3, or proportional-fair).

🔧Engineering Note

WMMSE Runtime in MU-RIS Deployment

Each WMMSE inner iteration is $O(N_t^{3} + K N_t^{2})$ . For $N_t = 16, K = 8$ : $\sim 10^4$ flops, under $10\,\mu\text{s}$ on modern hardware. The RIS element-wise sweep cost dominates: $O(N K)$ for a sum-of-quadratics multi-user objective.

Typical total optimization time per coherence block for $N = 256, K = 4, N_t = 16$ : $\sim 2$ ms (element-wise), $\sim 50$ ms (manifold), $\sim 5$ s (SDR). Only the first two are real-time-feasible.

Practical Constraints

•
WMMSE inner iterations: 5-10 for $\epsilon = 10^{-3}$ convergence.
•
Outer AO iterations: 10-20; warm-starting reduces to 3-5.
•
Memory: $O(N N_t K)$ for effective-channel storage.

Common Mistake: Don't Drop Users at Sum-Rate Optimum

Mistake:

"At low SNR, sum rate drops weak users entirely (zero power). So I just remove them from the active list."

Correction:

The weak users are part of the problem: serving them eventually demands fairness, and their CSI is needed for active-inactive user tracking. More pragmatically: at even slightly higher SNR, dropped users might become feasible — and re-entry costs time. Modern RAN schedulers allocate a minimum per-user rate; WMMSE with a per-user rate floor enforces this without explicit user dropping. See Ch. 17 for deployment considerations.

Sum-Rate Maximization via WMMSE