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Fix the Precoder, Choose the Phases

The second half of each AO iteration holds $\mathbf{W}$ fixed and optimizes $\boldsymbol{\Phi}$ . Unlike the active subproblem, the passive one is genuinely non-convex — the unit-modulus constraint $|\phi_n| = 1$ defines a non-convex torus. This chapter frames the passive subproblem abstractly; Chapter 6 develops the three workhorse algorithms (SDR, manifold optimization, element-wise optimization) that solve it.

Definition:
The Passive Beamforming Subproblem

Given fixed $\mathbf{W} = [\mathbf{v}_{1}, \ldots, \mathbf{v}_{K}]$ , the passive subproblem is

$\boxed{\;\max_{\boldsymbol{\phi} \in \mathbb{C}^N: |\phi_n| = 1}\; f_{\boldsymbol{\phi}}(\boldsymbol{\phi})\;}$

where $f_{\boldsymbol{\phi}}$ is the sum-rate (or another utility) with $\boldsymbol{\Phi} = \text{diag}(\boldsymbol{\phi})$ plugged into the per-user SINRs. Using the diagonal-product identity from Chapter 3, $\mathbf{h}_{k,\text{eff}}^H \mathbf{v}_{j} = \mathbf{h}_{k,d}^H \mathbf{v}_{j} + \boldsymbol{\phi}^T (\text{diag}(\mathbf{h}_{k,2}^*) \mathbf{H}_1 \mathbf{v}_{j})$ is an affine function of $\boldsymbol{\phi}$ , so the SINRs are ratios of quadratics in $\boldsymbol{\phi}$ .

The passive subproblem has the structure "maximize a ratio of quadratic forms subject to unit-modulus constraints." This problem class is called the QCQP on the complex torus and has a substantial optimization literature — most of which Chapter 6 surveys.

Theorem: Passive Objective as a Ratio of Quadratic Forms

For fixed $\mathbf{W}$ , the $k$ -th SINR can be written as

$\text{SINR}_k(\boldsymbol{\phi}) = \frac{\boldsymbol{\tilde{\phi}}^H \mathbf{A}_k \boldsymbol{\tilde{\phi}}}{\boldsymbol{\tilde{\phi}}^H \mathbf{B}_k \boldsymbol{\tilde{\phi}}}$

where $\boldsymbol{\tilde{\phi}} = [\boldsymbol{\phi}^T, 1]^T \in \mathbb{C}^{N+1}$ (augmented to handle the direct path's constant term) and $\mathbf{A}_k, \mathbf{B}_k \in \mathbb{C}^{(N+1) \times (N+1)}$ are Hermitian PSD matrices constructed from the channels and $\mathbf{W}$ . The passive sum-rate subproblem is

$\max_{\boldsymbol{\phi}: |\phi_n| = 1} \sum_k \log_2\!\left(1 + \frac{\boldsymbol{\tilde{\phi}}^H \mathbf{A}_k \boldsymbol{\tilde{\phi}}}{\boldsymbol{\tilde{\phi}}^H \mathbf{B}_k \boldsymbol{\tilde{\phi}}}\right)$

with the additional constraint $|\tilde{\phi}_{N+1}| = 1$ (set to 1 wlog by phase reference).

Each SINR is a ratio whose numerator and denominator are both quadratic in $\boldsymbol{\phi}$ , because the effective channel is affine in $\boldsymbol{\phi}$ and products of affines give quadratics. The sum of $\log$ s of ratios of quadratics is the full objective.

Proof

Expand the effective channel

$\mathbf{h}_{k,\text{eff}}^H \mathbf{v}_{j} = \mathbf{h}_{k,d}^H \mathbf{v}_{j} + \boldsymbol{\phi}^T \mathbf{g}_{k,j}$ , where $\mathbf{g}_{k,j} = \text{diag}(\mathbf{h}_{k,2}^*) \mathbf{H}_1 \mathbf{v}_{j}$ . Combine into affine: $\mathbf{h}_{k,\text{eff}}^H \mathbf{v}_{j} = \boldsymbol{\tilde{\phi}}^T \mathbf{b}_{k,j}$ , where $\mathbf{b}_{k,j} = [\mathbf{g}_{k,j}; \mathbf{h}_{k,d}^H \mathbf{v}_{j}]$ .

Quadratic forms

$|\mathbf{h}_{k,\text{eff}}^H \mathbf{v}_{j}|^2 = \boldsymbol{\tilde{\phi}}^H (\mathbf{b}_{k,j} \mathbf{b}_{k,j}^H)^T \boldsymbol{\tilde{\phi}}$ , so numerator $\boldsymbol{\tilde{\phi}}^H \mathbf{A}_k \boldsymbol{\tilde{\phi}}$ with $\mathbf{A}_k = \mathbf{b}_{k,k}^* \mathbf{b}_{k,k}^T$ and denominator similarly from the interference + noise.

Augmented feasibility

$|\phi_n| = 1$ for $n = 1, \ldots, N$ becomes $|\tilde{\phi}_n| = 1$ for $n = 1, \ldots, N+1$ (adding the convention $\tilde{\phi}_{N+1} = 1$ ). This is the standard SDR setup of Chapter 6. $\blacksquare$

Three Ways to Attack the Subproblem

The unit-modulus quadratic problem has three canonical algorithmic approaches, each explored in Chapter 6:

Semidefinite Relaxation (SDR): Lift $\boldsymbol{\tilde{\phi}}$ to a PSD matrix $\mathbf{X} = \boldsymbol{\tilde{\phi}} \boldsymbol{\tilde{\phi}}^H$ , relax the rank-1 constraint, solve an SDP, then use Gaussian randomization to recover $\boldsymbol{\phi}$ . Strong theoretical guarantees; $O(N^{6.5})$ solve time.
Manifold optimization: View the torus as a Riemannian manifold, perform gradient descent in its tangent space, retract to the manifold. Fast $O(N \log N)$ per iteration; globally convergent under smoothness.
Element-wise optimization: Update one $\phi_n$ at a time while fixing the others. Each update is a 1D problem with closed-form solution (maximize a sinusoid). Simplest, linear in $N$ per full sweep, often competitive in practice.

For the AO framework of this chapter, any of the three works as the passive update step.

Example: Element-wise Update for a Single-User Problem

Derive the closed-form element-wise update for the single-user MISO-RIS problem: $\max_{\boldsymbol{\phi}: |\phi_n| = 1} |\mathbf{h}_{\text{eff}}^H \mathbf{v}^\star|^2$ where $\mathbf{h}_{\text{eff}}^H \mathbf{v}^\star = a + \boldsymbol{\phi}^T \mathbf{b}$ for some $a, \mathbf{b}$ .

Solution

Single-coordinate objective

Fix all $\phi_m$ for $m \neq n$ . The objective as a function of $\phi_n$ is $|a + \phi_n b_n + \sum_{m \neq n} \phi_m b_m|^2 = |(a + \sum_{m \neq n} \phi_m b_m) + \phi_n b_n|^2$ . Let $c = a + \sum_{m \neq n} \phi_m b_m$ (known).

Maximize $|c + \phi_n b_n|^2$

Under $|\phi_n| = 1$ , the maximum of $|c + \phi_n b_n|$ is achieved when $c$ and $\phi_n b_n$ are aligned: $\phi_n^\star = e^{j(\arg(c) - \arg(b_n))}$ .

Convergence

Cycling over $n = 1, \ldots, N$ gives a full sweep. Each update monotonically increases the objective, so the sweep converges. In practice, 3–5 sweeps suffice for $\epsilon = 10^{-3}$ convergence. $\blacksquare$

Element-wise Passive Update: One Element at a Time

Visualize the element-wise update on a complex-plane diagram. Each

\phi_n

rotates to align with the combined contribution of the other elements and the direct path. After one sweep, the phases converge to near-matched-filter alignment; after 3-5 sweeps, the objective plateau.

Common Mistake: The Rank-1 Assumption Is Single-User-Only

Mistake:

"The element-wise matched-filter rule (Example 5.4) is the optimal passive update for any scenario."

Correction:

The closed-form element-wise rule is optimal only for single-user scenarios (or more generally, when the passive subproblem reduces to maximizing a single $|\boldsymbol{\phi}^T \mathbf{b}|^2$ ). For multi-user sum-rate with non-trivial interference, the objective is a ratio of quadratics, not a single squared inner product, and element-wise updates become a heuristic. Use SDR or manifold methods (Chapter 6) for multi-user.

🔧Engineering Note

Which Passive Algorithm to Use?

Rule of thumb for selecting the passive-update algorithm:

Single-user: element-wise closed-form. Fastest.
$K \leq 4$ , modest $N$ ( $\leq 128$ ): SDR. Strong theoretical solution quality.
Large $N$ ( $> 256$ ): manifold optimization. SDR becomes too expensive for larger RIS due to $O(N^{6.5})$ SDP cost.
Real-time deployment: element-wise if acceptable; otherwise manifold with warm-starting from the previous iteration.
Discrete phases (Chapter 8): project element-wise onto the finite phase grid at each step, or use projected gradient in the manifold framework.

Practical Constraints

•
SDR time for $N = 256$ via CVX/MOSEK: $\sim 1$ s per solve. Prohibitive in real time.
•
Element-wise sweep cost per step: $\sim 0.1$ ms at $N = 256$ in NumPy.
•
Manifold gradient descent convergence: typically 50-200 inner iterations.

The Passive-Beamforming Subproblem