Ferkans — Interactive Telecom Tutor

Definition:
Joint Active-Passive Beamforming Problem

The central optimisation problem for RIS-assisted communication is the joint active-passive beamforming problem: simultaneously optimise the BS precoder $\mathbf{w}$ ("active" beamforming) and the RIS phase shifts $\boldsymbol{\phi}$ ("passive" beamforming) to maximise the received signal quality. For a single-user MISO system:

$\max_{\mathbf{w}, \boldsymbol{\phi}} \quad \frac{|\mathbf{h}_{\mathrm{eff}}^H(\boldsymbol{\phi}) \mathbf{w}|^2}{\sigma^2}$

$\text{s.t.} \quad \|\mathbf{w}\|^2 \leq P, \quad |\phi_n| = 1, \; n = 1, \ldots, N$

where $\mathbf{h}_{\mathrm{eff}}(\boldsymbol{\phi}) = \mathbf{h}_d + \mathbf{G}^H \boldsymbol{\Theta} \mathbf{h}_r$ is the effective channel (with $\boldsymbol{\Theta} = \mathrm{diag}(\boldsymbol{\phi})$ ).

This problem is non-convex due to:

The unit-modulus constraints $|\phi_n| = 1$ , which define the complex circle manifold $\mathcal{S}^1$ — a non-convex set.
The coupling between $\mathbf{w}$ and $\boldsymbol{\phi}$ in the objective function.

For multi-user systems with $K$ single-antenna users, the problem generalises to:

$\max_{\{\mathbf{w}_k\}, \boldsymbol{\phi}} \quad \sum_{k=1}^K \log_2\!\left(1 + \frac{|\mathbf{h}_{\mathrm{eff},k}^H(\boldsymbol{\phi}) \mathbf{w}_k|^2}{\sum_{j \neq k} |\mathbf{h}_{\mathrm{eff},k}^H(\boldsymbol{\phi}) \mathbf{w}_j|^2 + \sigma^2}\right)$

$\text{s.t.} \quad \sum_{k=1}^K \|\mathbf{w}_k\|^2 \leq P, \quad |\phi_n| = 1, \; \forall n$

The single-user problem admits a simple closed-form optimal active beamformer: given $\boldsymbol{\phi}$ , the optimal $\mathbf{w}$ is maximum ratio transmission (MRT) along the effective channel, $\mathbf{w}^* = \sqrt{P}\, \mathbf{h}_{\mathrm{eff}}(\boldsymbol{\phi}) / \|\mathbf{h}_{\mathrm{eff}}(\boldsymbol{\phi})\|$ . The difficulty lies entirely in optimising $\boldsymbol{\phi}$ .

Alternating Optimisation for Joint Active-Passive Beamforming

Input: Channels

\mathbf{h}_d

,

\mathbf{G}

,

\mathbf{h}_r

;

power budget

P

; tolerance

\epsilon > 0

; max iterations

I_{\max}

Output: Optimised

\mathbf{w}^*

,

\boldsymbol{\phi}^*

1. Initialise

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

(e.g., random unit-modulus or all-ones)

2. For

i = 0, 1, 2, \ldots, I_{\max} - 1

:

a. Active beamforming (fix $\boldsymbol{\phi}^{(i)}$ , optimise $\mathbf{w}$ ):

Compute effective channel:

\mathbf{h}_{\mathrm{eff}}^{(i)} = \mathbf{h}_d + \mathbf{G}^H \mathrm{diag}(\boldsymbol{\phi}^{(i)}) \mathbf{h}_r

Set MRT beamformer:

\mathbf{w}^{(i+1)} = \sqrt{P} \, \mathbf{h}_{\mathrm{eff}}^{(i)} / \|\mathbf{h}_{\mathrm{eff}}^{(i)}\|

b. Passive beamforming (fix $\mathbf{w}^{(i+1)}$ , optimise $\boldsymbol{\phi}$ ):

Define

a_n = h_{r,n}^* \mathbf{g}_n^T \mathbf{w}^{(i+1)}

for

n = 1, \ldots, N

and

c = \mathbf{h}_d^H \mathbf{w}^{(i+1)}

.

The objective becomes

|c + \sum_{n=1}^N \phi_n a_n|^2

.

Element-wise closed-form:

\phi_n^{(i+1)} = e^{j(\angle c^* - \angle a_n)} \cdot e^{j \angle(\sum_{m \neq n} \phi_m^{(i)} a_m + c)}

Simplified: align each

\phi_n a_n

with the sum of all other terms.

Or full closed-form (when $c$ is treated as fixed):

\phi_n^{(i+1)} = e^{-j\angle a_n} \cdot e^{j\angle c}

(align all terms to phase of

c

).

c. Check convergence:

\text{SNR}^{(i+1)} = P\|\mathbf{h}_{\mathrm{eff}}^{(i+1)}\|^2 / \sigma^2

.

If

|\text{SNR}^{(i+1)} - \text{SNR}^{(i)}| / \text{SNR}^{(i)} < \epsilon

: break

3. Return

\mathbf{w}^* = \mathbf{w}^{(i+1)}

,

\boldsymbol{\phi}^* = \boldsymbol{\phi}^{(i+1)}

Complexity per iteration:

O(NM)

for active BF,

O(N)

for passive BF.

Typical convergence: 5--15 iterations.

Theorem: Convergence of Alternating Optimisation

Let $\{(\mathbf{w}^{(i)}, \boldsymbol{\phi}^{(i)})\}_{i=0}^{\infty}$ be the sequence generated by the alternating optimisation algorithm for the single-user joint active-passive beamforming problem. Then:

Monotonicity: The objective value (received SNR) is non-decreasing: $\text{SNR}^{(i+1)} \geq \text{SNR}^{(i)}$ for all $i \geq 0$ .
Convergence: The sequence $\{\text{SNR}^{(i)}\}$ converges to a finite limit $\text{SNR}^*$ .
Stationary point: Every limit point $(\mathbf{w}^*, \boldsymbol{\phi}^*)$ of the sequence satisfies the first-order necessary optimality conditions (KKT conditions) for the joint problem, i.e., it is a stationary point on the product manifold $\mathbb{C}^M \times (\mathcal{S}^1)^N$ .

Caveat: The algorithm is not guaranteed to converge to the global optimum due to the non-convexity of the unit-modulus constraints. Multiple random initialisations are recommended to improve the solution quality.

Each subproblem optimisation can only improve (or maintain) the objective, since it finds the optimal solution within its variable while holding the other fixed. Since the SNR is bounded above (by the power constraint and finite channel gains), a non-decreasing bounded sequence must converge.

Proof

Monotonicity

At iteration $i$ , the active beamforming step computes $\mathbf{w}^{(i+1)} = \arg\max_{\|\mathbf{w}\|^2 \leq P} |\mathbf{h}_{\mathrm{eff}}^H(\boldsymbol{\phi}^{(i)}) \mathbf{w}|^2 / \sigma^2$ . Since $\mathbf{w}^{(i)}$ is feasible but $\mathbf{w}^{(i+1)}$ is optimal:

$f(\mathbf{w}^{(i+1)}, \boldsymbol{\phi}^{(i)}) \geq f(\mathbf{w}^{(i)}, \boldsymbol{\phi}^{(i)})$

The passive beamforming step computes $\boldsymbol{\phi}^{(i+1)} = \arg\max_{|\phi_n|=1} f(\mathbf{w}^{(i+1)}, \boldsymbol{\phi})$ . Since $\boldsymbol{\phi}^{(i)}$ is feasible:

$f(\mathbf{w}^{(i+1)}, \boldsymbol{\phi}^{(i+1)}) \geq f(\mathbf{w}^{(i+1)}, \boldsymbol{\phi}^{(i)})$

Combining: $\text{SNR}^{(i+1)} \geq \text{SNR}^{(i)}$ .

Boundedness and convergence

The SNR is bounded above:

$\text{SNR} \leq \frac{P(\|\mathbf{h}_d\| + \|\mathbf{G}^H\| \|\mathbf{h}_r\| N)^2}{\sigma^2} < \infty$

By the monotone convergence theorem, the non-decreasing bounded sequence $\{\text{SNR}^{(i)}\}$ converges. $\blacksquare$

Stationarity (sketch)

At convergence, $\mathbf{w}^*$ is the global optimum of the active subproblem (convex) and $\boldsymbol{\phi}^*$ satisfies the KKT conditions of the passive subproblem (with Lagrange multipliers for the unit-modulus constraints). Together, these conditions imply that the gradient of the Lagrangian of the joint problem vanishes at $(\mathbf{w}^*, \boldsymbol{\phi}^*)$ on the product manifold. A rigorous proof uses the framework of block coordinate descent on Riemannian manifolds (see Absil et al., "Optimization Algorithms on Matrix Manifolds").

SDR and Manifold Optimisation for Phase Optimisation

Semidefinite relaxation (SDR). The passive beamforming subproblem can be reformulated as:

$\max_{\boldsymbol{\phi}} \; |\mathbf{c}^H \bar{\boldsymbol{\phi}}|^2 \quad \text{s.t.} \; |\bar{\phi}_n| = 1, \; n = 0, 1, \ldots, N$

where $\bar{\boldsymbol{\phi}} = [1; \boldsymbol{\phi}]$ and $\mathbf{c}$ absorbs the channel and active beamformer. Lifting $\bar{\boldsymbol{\Phi}} = \bar{\boldsymbol{\phi}} \bar{\boldsymbol{\phi}}^H$ (a rank-one PSD matrix with unit diagonal), this becomes:

$\max_{\bar{\boldsymbol{\Phi}} \succeq \mathbf{0}} \; \mathrm{tr}(\mathbf{C} \bar{\boldsymbol{\Phi}}) \quad \text{s.t.} \; \bar{\Phi}_{nn} = 1, \; \mathrm{rank}(\bar{\boldsymbol{\Phi}}) = 1$

Dropping the rank-one constraint yields a convex SDP, solvable in polynomial time. If the SDP solution $\bar{\boldsymbol{\Phi}}^*$ has rank one, it is globally optimal. Otherwise, Gaussian randomisation extracts a feasible rank-one solution: generate $\bar{\boldsymbol{\phi}} \sim \mathcal{CN}(\mathbf{0}, \bar{\boldsymbol{\Phi}}^*)$ , project onto the unit circle ( $\phi_n \leftarrow e^{j\angle \bar{\phi}_n}$ ), and keep the best among $L$ random draws. Wu and Zhang (2019) showed that the SDR achieves an approximation ratio of $\pi/4 \approx 0.785$ for the single-user case.

Manifold optimisation. The constraint $|\phi_n| = 1$ defines the complex circle $\mathcal{S}^1 = \{z \in \mathbb{C} : |z| = 1\}$ . The product manifold $(\mathcal{S}^1)^N$ is a smooth Riemannian manifold. Riemannian gradient descent and Riemannian conjugate gradient methods optimise directly on this manifold, respecting the geometry of the constraint set.

The Riemannian gradient at $\boldsymbol{\phi}$ is the projection of the Euclidean gradient onto the tangent space:

$\mathrm{grad}_{\mathcal{M}} f(\boldsymbol{\phi}) = \nabla f(\boldsymbol{\phi}) - \mathrm{Re}(\nabla f(\boldsymbol{\phi}) \odot \boldsymbol{\phi}^*) \odot \boldsymbol{\phi}$

followed by a retraction step (projection back onto $\mathcal{S}^1$ ): $\phi_n \leftarrow e^{j\angle(\phi_n + \alpha \, g_n)}$ where $\alpha$ is the step size and $g_n$ is the $n$ -th Riemannian gradient component.

Complexity comparison:

Alternating opt with element-wise update: $O(NI)$ per outer iteration
SDR: $O(N^{3.5})$ (interior-point SDP solver) — prohibitive for large $N$
Manifold optimisation: $O(N)$ per iteration, typically 20--50 iterations

For practical systems with $N > 100$ , manifold optimisation provides the best trade-off between solution quality and computational cost.

Rate Gain with Joint Active-Passive Beamforming

Compare the achievable rate (bits/s/Hz) as a function of the number of RIS elements $N$ for three schemes: (1) no RIS (direct link only), (2) RIS with random phases, and (3) RIS with jointly optimised active-passive beamforming. The channels are generated with Rayleigh fading for all links. Observe that optimised RIS phases provide a significant rate gain over random phases, and that the gain increases with $N$ due to the coherent combining effect. The BS uses $M$ antennas with MRT beamforming in all cases.

Parameters

M

BS antennas4

N_\text{max}

RIS elements128

SNR (dB)10

Alternating Optimisation Convergence

Watch the alternating optimisation algorithm converge for joint active-passive beamforming. Blue steps optimise the BS beamformer (active); red steps optimise the RIS phases (passive). The SNR increases monotonically and typically converges within 5--10 iterations, confirming the theoretical monotone convergence guarantee.

Alternating optimisation converges monotonically in 5--10 iterations.

Alternating Optimisation Convergence

Animate the convergence of the alternating optimisation algorithm for joint active-passive beamforming. At each iteration, the algorithm alternates between optimising the BS beamformer $\mathbf{w}$ (active step, shown in blue) and the RIS phases $\boldsymbol{\phi}$ (passive step, shown in red). The plot shows the achieved SNR versus iteration number, with both the active and passive step improvements visible. Observe the rapid convergence: typically 5--10 iterations suffice for near-optimal performance. Random channel realisations are generated for each parameter setting.

Parameters

M

BS antennas4

N

RIS elements32

Iterations15

Unit-Modulus Constraint

The constraint $|\phi_n| = 1$ on each RIS phase-shift coefficient, reflecting that passive elements can only change the phase (not amplitude) of the reflected signal. This constraint defines the complex circle manifold $\mathcal{S}^1$ and makes the beamforming optimisation non-convex.

Semidefinite Relaxation (SDR)

A convex relaxation technique that lifts the rank-one, unit-modulus constrained problem into a semidefinite program (SDP) by dropping the rank constraint. For RIS phase optimisation, SDR achieves an approximation ratio of $\pi/4 \approx 0.785$ in the single-user case.

Related: Unit-Modulus Constraint

Quick Check

In the alternating optimisation algorithm for joint active-passive beamforming, why is the active beamforming subproblem (optimising $\mathbf{w}$ for fixed $\boldsymbol{\phi}$ ) easy to solve?

Because the unit-modulus constraint on $\boldsymbol{\phi}$ is relaxed during this step

Because with $\boldsymbol{\phi}$ fixed, the effective channel $\mathbf{h}_{\mathrm{eff}}$ is a known vector, and the optimal $\mathbf{w}$ is simply MRT along $\mathbf{h}_{\mathrm{eff}}$

Because the problem is always feasible when $P > 0$

Because the BS has more antennas than users in the single-user case

Correction:

Because with

\boldsymbol{\phi}

fixed, the effective channel

\mathbf{h}_{\mathrm{eff}}

is a known vector, and the optimal

\mathbf{w}

is simply MRT along

\mathbf{h}_{\mathrm{eff}}

When $\boldsymbol{\phi}$ is fixed, the effective channel $\mathbf{h}_{\mathrm{eff}} = \mathbf{h}_d + \mathbf{G}^H \boldsymbol{\Theta} \mathbf{h}_r$ is a known deterministic vector. The problem reduces to $\max_{\|\mathbf{w}\|^2 \leq P} |\mathbf{h}_{\mathrm{eff}}^H \mathbf{w}|^2$ , which is a standard MISO beamforming problem with the well-known closed-form solution $\mathbf{w}^* = \sqrt{P}\,\mathbf{h}_{\mathrm{eff}} / \|\mathbf{h}_{\mathrm{eff}}\|$ (maximum ratio transmission). This is a rank-one optimisation that requires no iterative solver.

Beamforming Optimisation

Definition: Joint Active-Passive Beamforming Problem

Alternating Optimisation for Joint Active-Passive Beamforming

Theorem: Convergence of Alternating Optimisation

Monotonicity

Boundedness and convergence

Stationarity (sketch)

SDR and Manifold Optimisation for Phase Optimisation

Rate Gain with Joint Active-Passive Beamforming

Parameters

Alternating Optimisation Convergence

Alternating Optimisation Convergence

Parameters

Unit-Modulus Constraint

Semidefinite Relaxation (SDR)

Quick Check

Definition:
Joint Active-Passive Beamforming Problem