Ferkans — Interactive Telecom Tutor

Division of Labor: Who Does What?

In MU-RIS, the active beamformer and the RIS phases play complementary roles: the active beamformer spatially multiplexes users by forming $K$ distinct transmit beams, while the RIS shapes the channel to make multiplexing easier. The natural question: at an AO optimum, how do they "split the work"? This section quantifies the interplay through three operating-point results and clarifies when the RIS is most — and least — useful.

Theorem: When the Two Problems Decouple

Consider two asymptotic regimes:

1. Pure LoS, orthogonal user angles: $\mathbf{h}_{k,2}$ are mutually orthogonal at the RIS. The RIS-aided sum rate decomposes as $R_{\text{sum}} = \sum_k \log_2\!\left(1 + \frac{P_t\,|a_k|^2}{K\,\sigma^2}\right) + 2 \log_2 N$ (up to an $O(1)$ correction), where $|a_k|^2$ is the per-user RIS gain. Both $\mathbf{W}$ and $\boldsymbol{\Phi}$ contribute, but their roles are cleanly separated.

2. Massive MIMO, $N_t \gg K$ : favorable propagation (MIMO Ch. 1) makes the direct-channel Gram matrix nearly diagonal. ZF active beamformer is near-optimal; the RIS contribution is a per-user SNR boost of $N^2$ , not a DoF multiplier.

In two special regimes, the joint problem simplifies. Under pure LoS with orthogonal user directions, the RIS and active beamformer each have a separate job: RIS steers to each user sequentially (or to the weakest); active splits power. Under rich Rayleigh with large $N_t \gg K$ , the active beamformer approaches perfect ZF and the RIS's marginal contribution is only SNR (not spatial separation).

Proof

LoS decomposition

In the orthogonal-angle LoS case, each user's cascaded channel lies in a distinct RIS direction. The RIS phases can simultaneously match-filter all user directions (since the steering vectors at orthogonal angles are orthogonal on a ULA). Hence each user gets $\sim N$ coherent gain in amplitude, $N^2$ in power.

Massive MIMO limit

As $N_t/K \to \infty$ , the ZF precoder satisfies $\mathbf{v}_{k} \propto \mathbf{h}_{k,\text{eff}}$ with $\|\mathbf{v}_{k}\|^2 \approx P_t/K$ . Inter-user interference vanishes. The RIS's contribution is purely $\|\mathbf{h}_{k,\text{eff}}\|^2$ — an SNR gain per user, scaling as $N^2$ , independent of $K$ . $\blacksquare$

When Is RIS Most Valuable?

Three scenarios where RIS gives the largest multi-user benefit:

Correlated users: when UEs share similar direct channels (e.g., clustered in space), active beamforming struggles to separate them. RIS can reshape the effective channels to be more orthogonal, reviving ZF performance.
Low $N_t$ regime: when the BS has few antennas (e.g., small cells with $N_t \leq K$ ), active spatial multiplexing is weak. The RIS acts as a virtual antenna array, delivering additional DoF through effective-channel diversity.
Blocked-user scenarios: some users with blocked direct paths have $\mathbf{h}_{k,d} = \mathbf{0}$ . The RIS is their only link; active beamforming alone cannot help them. This is the "coverage extension" use case of Chapter 1.

Scenarios where RIS gives less benefit: massive MIMO with favorable propagation, pure-LoS with uncorrelated angles and large $N_t$ . In those regimes, the active side alone is already near-optimal.

Example: Correlated Users: The RIS Recovers ZF

Two users with highly correlated direct channels $\mathbf{h}_{k,d} = \mathbf{a}(\theta_k)$ with $\theta_1 = 0, \theta_2 = 5^\circ$ (very close), $N_t = 4$ . Without RIS, ZF is ill-conditioned (large power inflation). With RIS, how does the system recover?

Solution

No-RIS case

The Gram matrix $\mathbf{H}\mathbf{H}^H$ is nearly rank-1 (cos(5°) ≈ 0.996 correlation). ZF requires inversion, giving a power-inflation factor $\sim 1/(1 - 0.996^2) \approx 125$ — $21\text{ dB}$ penalty.

With RIS

Choose $\boldsymbol{\Phi}$ such that the cascaded channels $\mathbf{h}_{k,\text{eff}}$ have different directions. Concretely, the RIS can steer the reflected beam for user 1 to $\theta_1 + 10^\circ$ and for user 2 to $\theta_2 - 10^\circ$ , separating them by $20^\circ$ . Now the effective channels are well-separated, ZF inflation is $\sim 1\text{ dB}$ , and the RIS-boosted sum rate is $\sim 10\text{ dB}$ higher.

Lesson

The RIS decorrelates user channels in a deployment-agnostic way. No antenna-array upgrade, no frequency reuse: just a passive reflector choosing phases. This is the multi-user advantage of RIS in correlated-channel scenarios. $\blacksquare$

RIS Gain vs. User Correlation

Vary the correlation between two users' direct channels. The RIS-aided gain over no-RIS grows sharply as correlation approaches 1 — the RIS rescues the ZF-doomed scenario.

Parameters

RIS elements

N

64

BS antennas

N_t

4

Transmit SNR (dB)15

🎓CommIT Contribution(2022)

Hierarchical Multi-User RIS Scheduling

G. Caire, I. Atzeni — IEEE Trans. Wireless Commun. (preprint)

Caire and collaborators (2022) propose a hierarchical scheduler for MU-RIS systems with limited pilot budget. The key insight: RIS optimization is expensive per user, but users in the same coherence region can share an RIS configuration. The algorithm partitions users into RIS-clusters based on their channel correlations, assigns one $\boldsymbol{\Phi}^{(c)}$ per cluster, and schedules clusters across time. Compared with per-user RIS optimization, the scheme achieves $\sim 80\%$ of the oracle multi-user rate at $\sim 20\%$ of the optimization compute — an algorithmic win that makes MU-RIS feasible at scale ( $K = 32, N = 512$ with real-time updates). The paper foreshadows the array-fed architecture of Ch. 11, where clusters map to eigenmodes of the BS-RIS channel.

multi-userschedulingarray-fed-riscaire-2022

Theorem: Multi-User Rate Scaling with $N$ and $K$

For a $K$ -user MISO-RIS system with $N_t$ BS antennas, coherent-SNR scaling, and equal power $P_t/K$ per user:

$R_{\text{sum}} \sim \min(N_t, K) \log_2\!\left(1 + \frac{P_t\,N^2}{K \sigma^2}\right) = \min(N_t, K) \big[2 \log_2 N + \log_2(P_t/K \sigma^2)\big] + O(1).$

The multiplexing gain $\min(N_t, K)$ is unchanged by the RIS (the DoF is a property of $N_t, K$ ). The SNR gain is $N^2/K$ in the log argument, coming from RIS coherence minus per-user power dilution.

How does the achievable sum rate scale as $K$ and $N$ grow? If we keep $N_t, P_t$ fixed and let $N, K \to \infty$ , the asymptotic rate per user approaches zero (TDMA limit) unless the RIS adds independent spatial dimensions. The key result: RIS does not add DoF; it adds SNR. The per-user rate falls as $K$ grows, but grows as $\log_2 N^2 = 2 \log_2 N$ in $N$ .

Key Takeaway

RIS = SNR boost for each user, not a DoF multiplier. More users means less per-user power (under sum-rate), so the SNR gain of the RIS has to compete with the power dilution. The RIS does not unlock multi-user multiplexing beyond the $\min(N_t, K)$ ceiling set by the active antenna count. Treat RIS as an aperture extension — it makes existing MU-MIMO better, but it does not create new spatial DoF.

Common Mistake: Don't Confuse RIS Aperture with Spatial DoF

Mistake:

"With $N = 256$ RIS elements, we have 256 extra spatial channels. So we can serve 256 users simultaneously."

Correction:

The RIS is passive: it doesn't add transmit power or independent baseband paths. It reflects one signal into multiple directions via phase control. The number of simultaneous users is bounded by $\min(N_t, K)$ , where $N_t$ is the active antenna count. The $N$ RIS elements contribute an SNR gain of $N^2$ per user — substantial — but cannot create new DoF. To serve many users, you need many active antennas; RIS helps each one. Chapter 11's array-fed RIS explores this nuance under the near-field regime, where the aperture size does translate into additional rank under careful engineering.

Quick Check

An $N = 256$ -element RIS with $N_t = 8$ BS antennas serves $K = 12$ single-antenna users. What is the asymptotic multiplexing gain (DoF) of the system?

12 (= K)

8 (= min(N_t, K))

256 (= N)

128 (= min(N_t*N, K))