Ferkans — Interactive Telecom Tutor

Two Beamformers, One Channel

The RIS introduces a new kind of beamforming to the wireless toolkit: the passive beamformer, expressed through the phase-shift matrix $\boldsymbol{\Phi}$ . It coexists with the classical active beamformer $\mathbf{W}$ at the BS. Both aim to shape the same effective channel, but they do so differently — active beamforming spends transmit power to focus energy; passive beamforming reshapes the propagation environment at no RF-power cost. The joint design question: how do we choose both simultaneously?

A naive guess might be that the two decouple — optimize $\mathbf{W}$ given the channel, then tune $\boldsymbol{\Phi}$ to make the channel look nicer. This is wrong: the effective channel depends on $\boldsymbol{\Phi}$ , so the optimal $\mathbf{W}$ depends on $\boldsymbol{\Phi}$ too, and vice versa. The problem is irreducibly coupled.

Definition:
The Joint Active-Passive Beamforming Problem

Consider a $K$ -user MISO downlink: BS with $N_t$ antennas, $K$ single-antenna UEs, single $N$ -element RIS. The BS transmits $\mathbf{x} = \sum_{k=1}^K \mathbf{v}_{k} s_k$ , where $\mathbf{v}_{k} \in \mathbb{C}^{N_t}$ is user $k$ 's beamformer and $s_k$ is the data symbol with $\mathbb{E}|s_k|^2 = 1$ . Stack the beamformers: $\mathbf{W} = [\mathbf{v}_{1}, \ldots, \mathbf{v}_{K}] \in \mathbb{C}^{N_t \times K}$ .

User $k$ 's received signal is $y_k = \mathbf{h}_{k,\text{eff}}^H \mathbf{x} + w_k$ , with $\mathbf{h}_{k,\text{eff}}^H = \mathbf{h}_{k,d}^H + \mathbf{h}_{k,2}^H \boldsymbol{\Phi} \mathbf{H}_1$ . The per-user SINR is

$\text{SINR}_k(\mathbf{W}, \boldsymbol{\Phi}) = \frac{|\mathbf{h}_{k,\text{eff}}^H \mathbf{v}_{k}|^2}{\sum_{j \neq k} |\mathbf{h}_{k,\text{eff}}^H \mathbf{v}_{j}|^2 + \sigma^2}.$

The joint sum-rate problem is

$\boxed{ \max_{\mathbf{W}, \boldsymbol{\Phi}} \sum_{k=1}^K \log_2(1 + \text{SINR}_k) \quad \text{s.t.} \quad \text{tr}(\mathbf{W}^{H} \mathbf{W}) \leq P_t,\ |\phi_n| = 1\ \forall n. }$

This is the central optimization of the book. Chapters 5–8 solve variants of it (single-user, multi-user, max-min fairness, discrete phases); Chapter 11 solves it under the array-fed architecture. Understanding its structure — bilinear objective, two constraint sets, one convex and one not — is the foundation for everything that follows.

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Theorem: The Joint Problem Is Non-Convex

The feasible set $\{(\mathbf{W}, \boldsymbol{\Phi}) : \text{tr}(\mathbf{W}^{H} \mathbf{W}) \leq P_t,\ |\phi_n| = 1\}$ is non-convex: any convex combination of two feasible $\boldsymbol{\Phi}^{(1)}, \boldsymbol{\Phi}^{(2)}$ violates the unit-modulus constraint (by Eex-ris-ch01-09). Moreover, for fixed $\mathbf{W}$ , the objective $\sum_k \log_2(1 + \text{SINR}_k)$ is non-concave in $\boldsymbol{\phi}$ due to the SINR denominator's dependence on inter-user interference.

As a consequence, no polynomial-time algorithm is known to produce the global optimum. All practical algorithms (alternating optimization, SDR, manifold methods) produce local optima and rely on multiple random initializations to find good solutions.

The unit-modulus constraint $|\phi_n| = 1$ defines a circle in $\mathbb{C}$ , which is non-convex. Its $N$ -dimensional product is a torus — also non-convex. Beyond the feasibility set, even the objective is non-concave in the joint variable. Non-convex problems can have multiple local optima and no efficient way to find the global one.

Proof

Non-convex feasibility set

Take $\boldsymbol{\phi}^{(1)} = \mathbf{1}$ and $\boldsymbol{\phi}^{(2)} = -\mathbf{1}$ . Both are feasible ( $|\phi_n| = 1$ ). Their midpoint is $\mathbf{0}$ , which has $|\phi_n| = 0 \neq 1$ . Non-convex.

Non-concave SINR ratio

For a 2-user example, the SINR denominator $|\mathbf{h}_{k,\text{eff}}^H \mathbf{v}_{j}|^2 + \sigma^2$ is a quadratic function of $\boldsymbol{\phi}$ (through $\mathbf{h}_{k,\text{eff}}$ ). The log of a ratio of quadratics is not concave in general — can be verified by checking Hessians at specific points. $\blacksquare$

Sub-problems Can Be Convex

The joint problem is non-convex, but its coordinate sub-problems have better structure:

Fix $\boldsymbol{\Phi}$ , optimize $\mathbf{W}$ : the effective channel is known, so this reduces to a standard MU-MIMO precoding problem — convex when reformulated as WMMSE (weighted MMSE), solvable in closed form or semidefinite programming.
Fix $\mathbf{W}$ , optimize $\boldsymbol{\Phi}$ : the effective channel depends linearly on $\boldsymbol{\phi}$ , so the objective is a quadratic in $\boldsymbol{\phi}$ subject to unit-modulus — a quadratic with unit-modulus constraint, still non-convex but amenable to SDR, manifold methods, etc. (Chapter 6).

This alternating-convex structure is what the alternating optimization algorithm of Section 5.2 exploits. It is also the organizing principle of the whole optimization portion of the book.

Example: Single-User MISO-RIS: The Clean Case

Consider a single-user MISO-RIS system with $K = 1$ . The SINR simplifies to $\text{SNR} = |\mathbf{h}_{\text{eff}}^H \mathbf{v}|^2 / \sigma^2$ (no interference term). Derive the optimal $(\mathbf{v}^\star, \boldsymbol{\Phi}^\star)$ .

Solution

Separate the two optimizations

For fixed $\boldsymbol{\Phi}$ , the optimal BS beamformer is the MRT: $\mathbf{v}^\star = \mathbf{h}_{\text{eff}}/\|\mathbf{h}_{\text{eff}}\|$ , achieving $\text{SNR}^\star(\boldsymbol{\Phi}) = P_t\|\mathbf{h}_{\text{eff}}(\boldsymbol{\Phi})\|^2/\sigma^2$ .

Reduce to passive-only

The joint problem reduces to $\max_{\boldsymbol{\Phi}} \|\mathbf{h}_d + \mathbf{H}_1^H \boldsymbol{\Phi} \mathbf{h}_2\|^2$ , subject to $|\phi_n| = 1$ .

Quadratic in $\boldsymbol{\phi}$

Using the diagonal-product identity (Theorem 3.1): $\mathbf{H}_1^H \boldsymbol{\Phi} \mathbf{h}_2 = \mathbf{G}^H \boldsymbol{\phi}$ , where $\mathbf{G} = \text{diag}(\mathbf{h}_2^*)\mathbf{H}_1$ . The objective becomes $\|\mathbf{h}_d + \mathbf{G}^H \boldsymbol{\phi}\|^2$ — a quadratic in $\boldsymbol{\phi}$ , optimized over the torus. For this single-user case, the element-wise optimization (Chapter 6) gives the closed-form solution $\phi_n^\star = e^{-j \arg([\mathbf{G} \mathbf{h}_{\text{eff}}^\star]_n)}$ — matched-filter on each element. $\blacksquare$

Key Takeaway

The joint problem has bilinear structure. For any fixed $\boldsymbol{\Phi}$ , the active subproblem reduces to standard MU-MIMO precoding (convex). For any fixed $\mathbf{W}$ , the passive subproblem reduces to unit-modulus quadratic optimization (non-convex but tractable). This separation is the algorithmic DNA of alternating optimization and of nearly every RIS optimization algorithm in the literature.

The Bilinear Structure of Joint RIS Beamforming — The two-variable optimization landscape. Black curves are level sets of the sum-rate objective. Red vertical lines show the passive sub-problem (fix $\mathbf{W}$ , optimize $\boldsymbol{\Phi}$ ); blue horizontal lines show the active sub-problem. Alternating optimization walks a staircase through this landscape, taking one conditional optimum at a time.

Common Mistake: Don't Decouple Active and Passive Beamforming

Mistake:

"Set $\boldsymbol{\Phi} = \mathbf{I}$ (all zeros), optimize $\mathbf{W}$ for the direct channel. Then re-optimize $\boldsymbol{\Phi}$ for the resulting beamformer."

Correction:

The active and passive beamformers are coupled through the cascaded channel. Decoupling produces suboptimal solutions that can miss the $N^2$ coherent gain entirely. Specifically, the optimal $\boldsymbol{\Phi}$ depends on $\mathbf{W}$ , and the optimal $\mathbf{W}$ depends on $\boldsymbol{\Phi}$ . The alternating approach iterates between the two sub-problems, converging to a joint local optimum — the minimum-effort correct approach. A one-shot decoupled design typically loses $3$ – $10$ dB of coherent gain.

Why This Matters: The MU-MIMO Precoder Analogy

If you already understand MU-MIMO precoding (MIMO Ch. 6, Telecom Ch. 17), the RIS joint problem is a generalization: in MU-MIMO, we choose the precoder $\mathbf{W}$ to shape the transmit signal for fixed channel. In RIS-MU-MIMO, we additionally choose $\boldsymbol{\Phi}$ to shape the channel itself. The WMMSE iteration (the standard MU-MIMO workhorse) becomes one step of an alternating scheme; the other step is the unit-modulus RIS subproblem. Think of RIS beamforming as "MU-MIMO precoding plus one extra variable" — the mental model scales.

See full treatment in Sum-Rate Maximization via WMMSE

The Joint Beamforming Problem