Ferkans — Interactive Telecom Tutor

A Two-Variable Problem, Now With Gain

The optimization structure of active RIS is close to passive: AO between the BS precoder $\mathbf{W}$ and the RIS coefficients $\boldsymbol{\Psi}$ . The differences:

$\boldsymbol{\Psi}$ has complex (gain + phase) entries, not just unit-modulus.
The passive subproblem becomes a constrained quadratic in $\boldsymbol{\psi}$ with magnitude and sum-power bounds — more complex than the unit-modulus torus but often easier (the feasible set is convex!).
The objective includes the amplified noise term $\|\mathbf{h}_2^H \boldsymbol{\Psi}\|^2$ , which couples differently to $\boldsymbol{\psi}$ .

This section adapts the AO framework to active RIS.

Definition:
Joint Active-RIS Beamforming Problem

The active-RIS joint optimization problem is

$\boxed{\;\max_{\mathbf{W}, \boldsymbol{\Psi}} \sum_k \log_2(1 + \text{SINR}_k^{\text{active}})\;}$

subject to

BS power: $\text{tr}(\mathbf{W}^{H} \mathbf{W}) \leq P_t$ ,
Per-element gain: $|\psi_n| \leq g_{\max}$ ,
Total RIS power: $P_t\|\boldsymbol{\Psi}\mathbf{H}_1 \mathbf{W}\|_F^2 + \sigma^2_{\text{RIS}} \|\boldsymbol{\Psi}\|_F^2 \leq P_{\text{RIS}}$ .

The $\text{SINR}_k^{\text{active}}$ uses the active-RIS noise term from Theorem 9.1. Unlike passive RIS (where $|\psi_n| = 1$ is an equality), active RIS has inequality constraints on the magnitudes — giving a convex feasible set in $\boldsymbol{\psi}$ (but with a non-convex objective due to SINR ratios).

Theorem: The Active Passive Subproblem Has Convex Constraints

The active-RIS passive subproblem (fix $\mathbf{W}$ , optimize $\boldsymbol{\Psi}$ ) has a convex feasible set and a non-convex objective (rate = log of SINR ratio). The non-convexity is only in the objective; the constraints are well-behaved. This allows:

WMMSE reformulation of the active rate objective → a block-convex problem (Lagrangian duality applicable).
Global optimum of the WMMSE surrogate via SOCP / QCQP solvers — no need for SDR relaxation.
Faster convergence of the AO outer loop than in the passive case.

In short: active RIS is algorithmically easier than passive RIS because the feasible set is convex.

The constraint $|\psi_n| \leq g_{\max}$ defines a disk in $\mathbb{C}$ , a convex set. The total-power constraint is a quadratic $\leq$ bound, also convex. The feasible set of $\boldsymbol{\psi}$ is the intersection of $N$ disks and one ellipsoid — convex! This is in sharp contrast to the passive case where $|\psi_n| = 1$ defines a non-convex torus.

Proof

Disk constraint

$|\psi_n| \leq g_{\max}$ is a closed disk — convex.

Power constraint

Total RIS power is a convex quadratic $\leq$ constraint in $\boldsymbol{\psi}$ (since both $|\psi_n|^2$ terms are convex).

Intersection

Intersection of convex sets is convex. The feasible set is convex. $\blacksquare$

AO for Active RIS

Complexity:

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

,

T \sim 10

Input: channels, BS power

P_t

, RIS power

P_{\text{RIS}}

, max gain

g_{\max}

.

Output:

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

.

1. Initialize

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

(e.g., all

\psi_n = 1

, i.e., unit-gain reflection).

2. Repeat

t = 0, 1, \ldots

:

3.

\quad

Compute

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

using active-RIS

\boldsymbol{\Psi}^{(t)}

.

4.

\quad

Active update: WMMSE on effective channels

(with UE noise augmented by active-RIS noise term).

5.

\quad

Passive update: solve the convex-constraint QCQP-like problem

with WMMSE surrogate; gradient projection with line search,

or SOCP solver directly.

6.

\quad

Check

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

.

7. return

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

.

Passive update cost: $O(N^3)$ for SOCP; or $O(T_{\text{GP}} N)$ for gradient projection. Both scale better than passive RIS's SDR ( $N^{6.5}$ ), making active RIS surprisingly cheap at the algorithm level despite added modeling complexity.

Example: Active RIS for Single-User: Closed-Form $|\psi_n|$

Single-user active RIS with matched-filter BS beamformer, optimal phase alignment. Solve for the optimal $|\psi_n|$ per element under total power constraint $\sum_n |\psi_n|^2 \leq P_{\text{RIS}}/P_t/|(\mathbf{H}_1\mathbf{v})_n|^2 + \ldots$ (simplifying with $|\psi_n|^2 \leq P_{\text{RIS}}/(P_t(\text{signal power at element } n) + \sigma^2_{\text{RIS}})$ ).

Solution

Per-element gain

With total power $P_{\text{RIS}}$ and equal-strength channels, the optimum water-fills $|\psi_n|^2$ inversely to noise. For identical elements: $|\psi_n|^2 = P_{\text{RIS}}/(N \cdot (P_t\alpha^2 + \sigma^2_{\text{RIS}}))$ .

Optimum SINR

Substituting into the SINR formula and optimizing gives the active-RIS SINR. The max grows with $P_{\text{RIS}}$ up to a ceiling set by the signal-to-amplifier-noise ratio.

Compared to passive

Active SINR = passive SINR + $\log$ -term for amplifier contribution. Active wins when: (i) path loss is severe, (ii) passive SNR is below UE noise floor, (iii) $P_{\text{RIS}}$ exceeds the amplifier noise cost. Section 9.3 gives the precise crossover. $\blacksquare$

Passive Problem: NP-hard; Active Problem: Tractable

A quick sanity check: why is the active problem easier to optimize than the passive one?

Passive: unit-modulus constraint $|\phi_n| = 1$ is non-convex. NP-hard QCQP. Requires SDR, manifold methods, etc.
Active: magnitude $\leq g_{\max}$ is convex. The RIS power budget is convex (quadratic inequality). WMMSE surrogate gives a convex-constraint block-convex problem. Global optimum via SOCP.

So active RIS is more expensive to build and operate (amplifiers, power) but computationally easier to optimize. In trade-offs for realistic deployments, the algorithmic easiness is a secondary advantage — the main story is still the link-budget gain.

Active vs. Passive AO Convergence

Compare AO convergence traces for active and passive RIS on the same channel instance. Active AO typically converges faster (5-10 iterations) thanks to the convex inner problem; passive AO needs more iterations (10-30) due to non-convex unit-modulus constraint.

Parameters

RIS elements

N

64

BS antennas

N_t

4

Users

K

2

Max gain

g_{\max}

(dB)15

BS SNR (dB)0

Active-Passive Joint Optimization