Ferkans — Interactive Telecom Tutor

Three Beamformers, One Panel

STAR-RIS adds a second diagonal ( $\boldsymbol{\Phi}^t$ ) and an amplitude split. The joint optimization now has three variables: $\mathbf{W}, \boldsymbol{\Phi}^r, \boldsymbol{\Phi}^t$ (plus the amplitudes for ES). The AO framework extends naturally: cycle through active → reflection → transmission updates. This section presents the algorithm and highlights the STAR-specific wrinkles.

Definition:
STAR-RIS Joint Sum-Rate Problem (ES Protocol)

The joint active-STAR beamforming problem is

$\max_{\mathbf{W}, \boldsymbol{\Phi}^r, \boldsymbol{\Phi}^t} \sum_{k \in \mathcal{U}^r \cup \mathcal{U}^t} \log_2(1 + \text{SINR}_k)$

subject to

$\text{tr}(\mathbf{W}^{H} \mathbf{W}) \leq P_t$ ,
$|r_n|^2 + |t_n|^2 = 1$ for all $n$ (ES; per hardware model),
Phase coupling or independence per hardware model.

The $\text{SINR}_k$ uses $\boldsymbol{\Phi}^r$ for $k \in \mathcal{U}^r$ and $\boldsymbol{\Phi}^t$ for $k \in \mathcal{U}^t$ .

AO for STAR-RIS

Complexity:

O(T \cdot (T_{\text{WMMSE}} K N_t^{3} + 2 C_{\text{passive}}))

Input: channels, user sets

\mathcal{U}^r, \mathcal{U}^t

, power

P_t

.

Output:

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

.

1. Initialize

\boldsymbol{\Phi}^{r,(0)}, \boldsymbol{\Phi}^{t,(0)}

(e.g.,

r_n = t_n = 1/\sqrt{2}

, all phases zero).

2. For

t = 0, 1, \ldots

:

3.

\quad

Active update (WMMSE): solve for

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

given current

\boldsymbol{\Phi}^r, \boldsymbol{\Phi}^t

.

4.

\quad

Reflection-side update:

\boldsymbol{\Phi}^{r,(t+1)} =

solve passive subproblem for

\mathcal{U}^r

users

with the energy constraint

|r_n|^2 + |t_n|^2 = 1

(keeping

|t_n|

fixed).

5.

\quad

Transmission-side update:

\boldsymbol{\Phi}^{t,(t+1)} =

same but for

\mathcal{U}^t

users.

6.

\quad

Amplitude reallocation: optimize per-element

(a_n^r, a_n^t)

under

|a_n^r|^2 + |a_n^t|^2 = 1

.

Closed-form:

a_n^r = r_n^{\text{raw}}/\|(\mathbf{r}^{\text{raw}}, \mathbf{t}^{\text{raw}})\|

etc.

7.

\quad

Check convergence; if

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

or

t = T_{\max}

: break.

8. return

(\mathbf{W}, \boldsymbol{\Phi}^r, \boldsymbol{\Phi}^t)

.

Compared with plain RIS AO, STAR-RIS adds one extra sub-step (transmission-side update) plus an amplitude reallocation step. Typical AO convergence: 15-25 outer iterations, about 50% more than passive RIS at the same scenario. The amplitude update has a closed form under ES; under MS it's a combinatorial choice.

Theorem: AO Convergence for STAR-RIS

The STAR-RIS AO algorithm produces a monotone non-decreasing sum-rate sequence and converges to a stationary point of the joint problem. The proof is identical to Chapter 5's (Theorem 5.5), since the feasible sets for $\boldsymbol{\Phi}^r, \boldsymbol{\Phi}^t$ are each compact and the coordinate-wise problems are well-posed.

The additional amplitude-reallocation step (ES) at the end of each AO iteration does not break monotonicity: it can only improve the objective (by the $\arg\max$ property).

Example: AO on a 2-User STAR-RIS Problem

$N_t = 4, N = 32$ , one user on each side ( $K_r = K_t = 1$ ). ES protocol with independent phases. Initialize $a_n^r = a_n^t = 1/\sqrt{2}$ . Execute one outer AO iteration.

Solution

Active update

Compute $\mathbf{h}_{k,\text{eff}}^r, \mathbf{h}_{k,\text{eff}}^t$ with initial $\boldsymbol{\Phi}^{r,(0)}, \boldsymbol{\Phi}^{t,(0)}$ . WMMSE precoder: two beams, one matched to each effective channel with regularization for inter-side interference (if any).

Reflection side update

Given $\mathbf{W}^{(1)}$ , solve for $\boldsymbol{\Phi}^{r,(1)}$ to maximize user-r rate. Element-wise matched-filter on $\mathbf{h}_2^r \cdot \mathbf{H}_1 \mathbf{v}^{r}$ ; quantize phase within ES protocol.

Transmission side update

Similar, for user $t$ : $\phi_n^t = e^{-j\arg[\ldots]}$ .

Amplitude reallocation

Given unnormalized $r_n^{\text{raw}}, t_n^{\text{raw}}$ , normalize: $a_n^r = |r_n^{\text{raw}}| / \|(a_n^{r,\text{raw}}, a_n^{t,\text{raw}})\|$ , similarly $a_n^t$ . This enforces energy conservation per element.

Convergence check

$R^{(1)} \geq R^{(0)}$ by monotonicity. After 10-20 iterations, converges to a local optimum. $\blacksquare$

STAR-RIS AO Convergence

Watch AO converge for a STAR-RIS system. Compare ES, MS, TS protocols on the same channel. Vary user balance between the two sides to see how each protocol performs under imbalanced demand.

Parameters

RIS elements

N

64

BS antennas

N_t

4

Users on reflect side2

Users on transmit side2

Transmit SNR (dB)10

Complexity Compared with Passive RIS

STAR-RIS per-iteration compute is about 1.5-2× that of passive RIS (extra diagonal + amplitude step). AO iteration count is similar (10-30). Total optimization time scales roughly proportionally — a STAR-RIS system at $N = 256, K = 4$ takes $\sim 30$ - $60$ ms per coherence block; passive RIS takes $\sim 15$ - $30$ ms. Both are real-time feasible at 10-50 ms coherence times.

Common Mistake: Don't Forget the Phase Coupling Under MS

Mistake:

"For MS protocol, we just set each element to reflect OR transmit. The phases of reflect and transmit are unused."

Correction:

Under MS, each element has exactly one mode at any time — but the phase of that mode IS still optimized. The optimization chooses: (i) which mode per element (binary), and (ii) the phase for the chosen mode. Both are variables. Under ES, we additionally have amplitude. Under TS, the phases for each sub-slot are independently optimized. Always clarify which protocol the phases belong to.

🎓CommIT Contribution(2023)

Hybrid Precoding for Practical STAR-RIS Deployments

G. Caire, I. Atzeni — IEEE Trans. Signal Process. (preprint)

Caire and collaborators (2023) extend the AO framework to practical STAR-RIS deployments with the combined challenges of (1) hardware-coupled phases (the $\pi/2$ constraint), (2) discrete amplitude levels (quantized ES), and (3) limited BS pilot budget. Their key innovation: a two-timescale optimization where amplitude allocation is updated slowly (coherence-block level) while phases are updated fast (symbol level). Combined with the hybrid analog-digital BS precoder, this achieves $\sim 90\%$ of the continuous-ES optimum at $\sim 30\%$ of the compute. The approach is particularly relevant for 6G mmWave deployments where STAR-RIS, hybrid BS arrays, and low-latency constraints all interact. This is the CommIT contribution for the STAR-RIS chapter.

star-rishybrid-beamformingtwo-timescalecaire-2023

⚠️Engineering Note

STAR-RIS Deployment Best Practices

Practical STAR-RIS deployment considerations:

Choose the protocol based on hardware constraints:
- Academic study / upper bound: ES.
- Commercial mmWave panel (2024): MS with 3-bit phases.
- Legacy passive-RIS upgrade: TS (reuses existing hardware).
User-set assignment: use channel estimation (Ch. 4) to decide which users are on which side; re-estimate on mobility.
Amplitude calibration: ES requires per-element amplitude control, which drifts with temperature. Recalibrate at each major temperature change.
Phase coupling: for commercial STAR-RIS with coupled $\theta^t - \theta^r = \pm\pi/2$ , use the coupled-phase optimization; don't pretend they're independent.

Practical Constraints

•
Typical ES protocol: requires continuous-amplitude varactor + bias; $\sim 5$ mW per element.
•
MS protocol: single PIN-diode switch + single phase shifter per element; $\sim 1$ mW per element.
•
TS protocol: uses the same hardware as passive RIS, just with controller time-switching.
•
Bandwidth: STAR-RIS typically 5-20% fractional bandwidth (narrower than passive).

STAR-RIS Joint Optimization