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Joint Design of $\mathbf{W}$ and $\boldsymbol{\phi}$

With the cascaded channel model of Section 21.3 in hand, the multiuser problem is now clear: we must design a digital precoder $\mathbf{W} \in \mathbb{C}^{N_a \times K}$ on the $N_a$ -element active feed together with a passive phase profile $\boldsymbol{\phi} \in [0,2\pi)^{N_{\text{RIS}}}$ on the RIS. The digital precoder picks the right combination of the $N_a$ spatial modes; the RIS phase profile picks which $N_a$ modes the feed has access to.

A clean analytical derivation is not available — the problem is bilinear in $\mathbf{W}$ and $\boldsymbol{\phi}$ and has unit-modulus phase constraints, which are non-convex. The standard recipe is alternating optimization: (i) fix $\boldsymbol{\phi}$ , compute the optimal linear precoder via ZF or MMSE on the resulting effective channel; (ii) fix $\mathbf{W}$ , update $\boldsymbol{\phi}$ by solving the per-element sub-problem or by an SDR/majorization relaxation. The procedure converges quickly in practice even on non-convex landscapes, and we report numerical evidence that 5–15 iterations reach within $0.5$ dB of a much more expensive branch-and-bound baseline.

,

Definition:
Multiuser Downlink with Array-Fed RIS

The active-array feed of an array-fed RIS base station serves $K$ single-antenna users. The transmitted baseband signal is

$\mathbf{x}_a = \mathbf{W}\, \mathbf{s} = \sum_{k=1}^{K} \mathbf{v}_{k}\, s_k,$

where $\mathbf{s} = (s_1, \ldots, s_{K})^T$ is the vector of independent unit-power user symbols and $\mathbf{v}_{k} \in \mathbb{C}^{N_a}$ is the active-array precoder for user $k$ , with the sum-power constraint $\sum_k \|\mathbf{v}_{k}\|^2 \leq P_t$ . User $k$ 's received signal is

$y_k = h_{\text{eff},k}^H(\boldsymbol{\phi})\, \mathbf{v}_{k}\, s_k + \sum_{j\neq k} h_{\text{eff},k}^H(\boldsymbol{\phi})\, \mathbf{v}_{j}\, s_j + w_k,$

where $h_{\text{eff},k}^H(\boldsymbol{\phi}) = \mathbf{H}_{\text{RIS-Rx},k}^H\, \text{diag}(\boldsymbol{\phi})\, \mathbf{G}_f$ is user $k$ 's effective channel. The SINR of user $k$ is

$\gamma_k = \frac{|h_{\text{eff},k}^H \mathbf{v}_{k}|^2} {\sum_{j\neq k} |h_{\text{eff},k}^H \mathbf{v}_{j}|^2 + \sigma^2},$

and the downlink sum rate is $\sum_k \log_2(1 + \gamma_k)$ .

Theorem: Optimal Linear Precoder for Fixed RIS Phase Profile

Fix $\boldsymbol{\phi}$ and define the effective multiuser channel matrix

$\mathbf{H}(\boldsymbol{\phi}) = [h_{\text{eff},1}(\boldsymbol{\phi}), \ldots, h_{\text{eff},K}(\boldsymbol{\phi})]^H \in \mathbb{C}^{K \times N_a}.$

If $N_a \geq K$ and $\mathbf{H}(\boldsymbol{\phi})$ has full row rank, the zero-forcing precoder

$\mathbf{W}_{\text{ZF}}(\boldsymbol{\phi}) = \mathbf{H}(\boldsymbol{\phi})^H \left(\mathbf{H}(\boldsymbol{\phi})\mathbf{H}(\boldsymbol{\phi})^H\right)^{-1} \mathbf{\Lambda}^{1/2}$

nulls all interference, where $\mathbf{\Lambda} = \text{diag}(p_1, \ldots, p_{K})$ sets the per-user powers. The regularized (MMSE) variant replaces the inverse by $(\mathbf{H}(\boldsymbol{\phi})\mathbf{H}(\boldsymbol{\phi})^H + \alpha \mathbf{I})^{-1}$ with $\alpha = K\sigma^2/P_t$ , yielding a higher sum rate at finite SNR.

ZF reduces the problem to $K$ parallel scalar channels at the price of a power penalty set by the condition of $\mathbf{H} \mathbf{H}^H$ . A well-chosen $\boldsymbol{\phi}$ shapes $\mathbf{H}(\boldsymbol{\phi})$ to be well-conditioned — essentially picking a set of $K$ near-orthogonal directions on the $N_a$ -dimensional active-feed manifold — which is the design objective for the outer $\boldsymbol{\phi}$ loop.

Proof

Interference cancellation condition

Require $h_{\text{eff},k}^H \mathbf{v}_{j} = 0$ for $j \neq k$ . This is $K(K-1)$ linear constraints on the $N_a K$ entries of $\mathbf{W}$ . For $N_a \geq K$ the feasibility is generic.

Right pseudo-inverse

The Moore–Penrose pseudo-inverse $\mathbf{H}^H (\mathbf{H} \mathbf{H}^H)^{-1}$ is the right inverse of a full-row-rank $\mathbf{H}$ , so $\mathbf{H} \mathbf{W}_{\text{ZF}} = \mathbf{\Lambda}^{1/2}$ , which is diagonal. Interference is zero by construction, and the per-user effective channel becomes $\sqrt{p_k}$ .

SINR of ZF

$\gamma_k^{\text{ZF}} = p_k / \sigma^2$ , and the sum-power constraint $\text{tr}(\mathbf{W}_{\text{ZF}}^H \mathbf{W}_{\text{ZF}}) = \text{tr}(\mathbf{\Lambda} (\mathbf{H}\mathbf{H}^H)^{-1}) \leq P_t$ determines the feasible $\{p_k\}$ by waterfilling. $\blacksquare$

,

Alternating Optimization for Array-Fed RIS Sum-Rate

Complexity:

\mathcal{O}(T (K N_a^2 + N_{\text{RIS}}\, K N_a))

with

T \sim 5

–

20

Input: Effective channel factors

\mathbf{H}_{\text{RIS-Rx},k}

for

k = 1,\ldots,K

, feed coupling

\mathbf{G}_f

, sum-power

budget

P_t

, tolerance

\epsilon

.

Output:

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

maximizing the sum rate.

1. Initialize

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

uniformly at random.

2. for

t = 0, 1, 2, \ldots

do

3.

\quad

Form

\mathbf{H}^{(t)} \leftarrow

rows

[\mathbf{H}_{\text{RIS-Rx},k}^H \text{diag}(\boldsymbol{\phi}^{(t)}) \mathbf{G}_f]

.

4.

\quad \mathbf{W}^{(t)} \leftarrow

MMSE precoder for

\mathbf{H}^{(t)}

with budget

P_t

.

5.

\quad

for

n = 1, \ldots, N_{\text{RIS}}

do

6.

\quad\quad

Hold all

\phi_{m \neq n}^{(t)}

fixed. The sum-rate

as a function of

\phi_n

is of the form

f(\phi_n) = \sum_k a_k |b_k + c_k e^{j\phi_n}|^2 / \text{noise}

,

which is a sinusoid in

\phi_n

.

7.

\quad\quad

Solve

\max_{\phi_n \in [0,2\pi)} f(\phi_n)

in closed

form (derivative is a single sinusoid).

8.

\quad\quad

\phi_n^{(t+1)} \leftarrow

optimizer.

9.

\quad

end for

10.

\quad

if

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

then break

11. end for

12. return

(\mathbf{W}^{(t+1)}, \boldsymbol{\phi}^{(t+1)})

Each per-element update in lines 6–7 is exact because the dependence of the sum rate on a single $\phi_n$ (with all others fixed) is a single sinusoid — Caire and collaborators' key structural observation. This makes the outer loop monotonically non-decreasing and allows certification of local optimality at convergence.

Example: Two-User Array-Fed RIS: Numerical Walk-Through

Consider an array-fed RIS with $N_a = 4$ , $N_{\text{RIS}} = 128$ , $K = 2$ single-antenna users at angles $(\phi_1^u, \phi_2^u) = (-10^\circ, +25^\circ)$ in the far field of the RIS. Assume LOS reflected channels $\mathbf{H}_{\text{RIS-Rx},k} = \sqrt{G_{\text{RIS}}} \cdot \mathbf{a}_{\text{RIS}}(\phi_k^u) / d_2$ with $d_2 = 15$ m. The sum-power budget is $P_t = 0.1$ W. Compute (a) the ZF precoder given the ideal RIS profile aligning both user directions, (b) the SINR of each user, and (c) the sum rate at $f_0 = 28$ GHz.

Solution

Ideal RIS phase profile

The RIS aligns its aperture between the two user steering vectors. Setting $\phi_n = \arg[\alpha \mathbf{a}_{\text{RIS}}(\phi_1^u)_n + (1-\alpha)\mathbf{a}_{\text{RIS}}(\phi_2^u)_n]$ for some $\alpha \in [0,1]$ maximizes the joint illumination. For the symmetric case $\alpha = 1/2$ , the effective per-user array gains are $\approx N_{\text{RIS}}/2$ in each direction.

Effective channel matrix

With the near-field coupler $\mathbf{G}_f$ approximated as orthonormal, $\mathbf{H}(\boldsymbol{\phi}) \in \mathbb{C}^{2\times 4}$ has row norms $\sim \sqrt{G_{\text{RIS}}/2}/d_2$ . The two rows are nearly orthogonal because the angular separation $35^\circ > 2/\sqrt{N_a}$ exceeds the active-feed beamwidth.

ZF SINR and sum rate

With $N_a = 4 > K = 2$ , ZF is feasible. Equal-power allocation gives $p_k = P_t/2$ . With $\|h_{\text{eff},k}\|^2 \sim 128/2 \cdot 1/15^2 \approx 0.284$ in the chosen normalization, the post-ZF SNR per user is $\gamma_k \approx 0.1/2 \cdot 0.284 / \sigma^2$ . At $\sigma^2 = -90$ dBm over $B = 100$ MHz, $\gamma_k \approx 14$ dB, and the sum rate is approximately $2 \log_2(1 + 10^{1.4}) \approx 9.5$ bits/s/Hz — vastly better than a passive RIS of the same size. $\blacksquare$

Multiuser Sum Rate vs $N_{\text{RIS}}$

Compute the multiuser sum rate of an array-fed RIS with $N_a$ active elements serving $K$ users as $N_{\text{RIS}}$ grows. The curves use the analytical alt-opt upper bound; compare with a passive RIS baseline at the same $N_{\text{RIS}}$ .

Parameters

N_a

8

K

4

per-user SNR (dB)10

Precoder

Estimating the Cascaded Channel

All the algorithms in this section assume perfect knowledge of $\mathbf{H}(\boldsymbol{\phi})$ , which is itself a function of the RIS phase profile. In practice, the BS must probe the cascaded channel using a sequence of pilot training patterns $\{\boldsymbol{\phi}^{(t)}\}$ , each producing a measurement of a different linear combination of the $N_{\text{RIS}}$ rank-one terms in the sum decomposition of $\mathbf{H}_{\text{eff}}$ . The number of training phases required scales as $\mathcal{O}(N_{\text{RIS}}/N_a)$ to resolve all cascaded entries, and the pilot overhead can be reduced further by exploiting angular sparsity of the reflected channels — a direct generalization of the compressed channel estimation techniques of Chapter 8 and FSI Chapter 12. The bottom line is that the channel estimation problem is non-trivial but tractable; we leave its full treatment to Chapter 22 and the RIS book.

⚠️Engineering Note

Convergence and Real-Time Operation

The alternating optimization of Algorithm AAlternating Optimization for Array-Fed RIS Sum-Rate converges in $T \in [5, 20]$ outer iterations for typical geometries. Each iteration requires one MMSE precoder update ( $\mathcal{O}(N_a^3 + K N_a^2)$ ) and $N_{\text{RIS}}$ per-element phase updates ( $\mathcal{O}(N_{\text{RIS}} K N_a)$ ). At $N_a = 8$ , $N_{\text{RIS}} = 1024$ , $K = 8$ , $T = 10$ , this is roughly $10^6$ flops per coherence block — milliseconds on a commodity DSP. The bottleneck in real deployments is channel estimation, not precoding.

Three practical observations from the CommIT prototype:

Warm-starting $\boldsymbol{\phi}$ from the previous coherence block reduces $T$ to 2–3.
Per-element updates can be parallelized, because (after a Taylor expansion) the cross-effects are weak when $N_a$ is small.
Phase quantization is applied only at the final step; quantizing inside the loop causes instability.

Practical Constraints

•
Coherence block of ~ 1 ms at mmWave (large Doppler)
•
Phase resolution in hardware: 1–3 bits (Section 21.1 engineering note)
•
Channel estimation pilot overhead scales as $\mathcal{O}(N_{\text{RIS}}/N_a)$

📋 Ref: 3GPP TR 38.843 (smart radio environments), IEEE 802.11bf (WLAN sensing)

Common Mistake: Alt-Opt Is Not Globally Optimal

Mistake:

Because each step of the alternating procedure monotonically increases the sum rate, it is tempting to claim that Algorithm AAlternating Optimization for Array-Fed RIS Sum-Rate converges to the global optimum.

Correction:

Monotone convergence only guarantees a local stationary point. The joint problem is non-convex in $\boldsymbol{\phi}$ (unit-modulus constraints) and multi-modal. Empirically, alt-opt reaches a good local optimum within 0.5 dB of the branch-and-bound global solution, but rare initializations converge to inferior critical points. In production, it is common to run a small number of random restarts (5–10) and pick the best. We will see similar caveats when we generalize to SDR and majorization-based algorithms in Chapter 22.

Quick Check

An array-fed RIS with $N_a = 8$ , $N_{\text{RIS}} = 512$ serves $K = 12$ users. Using the ZF precoder alone is infeasible. Which of the following is the most reasonable remedy?

Increase $N_{\text{RIS}}$ until $N_{\text{RIS}} \geq K$

Apply MMSE + user scheduling so that at most $N_a$ users are served per resource block

Use random RIS phases and ZF

Drop 4 users permanently

Correction:

Apply MMSE + user scheduling so that at most

N_a

users are served per resource block

The rank of $\mathbf{H}_{\text{eff}}$ is capped at $N_a = 8$ , so at most 8 users can be served on orthogonal spatial modes at a single time-frequency resource. The 12 users must share via time-frequency scheduling, each resource block serving at most $N_a$ of them. MMSE precoding handles the residual interference between scheduled users better than ZF when SINR is moderate. Increasing $N_{\text{RIS}}$ helps array gain but does not unlock more DoF.

Multiuser Multibeam Beamforming

Joint Design of W\mathbf{W}W and ϕ\boldsymbol{\phi}ϕ

Definition: Multiuser Downlink with Array-Fed RIS

Theorem: Optimal Linear Precoder for Fixed RIS Phase Profile

Interference cancellation condition

Right pseudo-inverse

SINR of ZF

Alternating Optimization for Array-Fed RIS Sum-Rate

Example: Two-User Array-Fed RIS: Numerical Walk-Through

Ideal RIS phase profile

Effective channel matrix

ZF SINR and sum rate

Multiuser Sum Rate vs NRISN_{\text{RIS}}NRIS​

Parameters

Estimating the Cascaded Channel

Convergence and Real-Time Operation

Common Mistake: Alt-Opt Is Not Globally Optimal

Quick Check

Joint Design of $\mathbf{W}$ and $\boldsymbol{\phi}$

Definition:
Multiuser Downlink with Array-Fed RIS

Multiuser Sum Rate vs $N_{\text{RIS}}$