Joint Active and Passive Beamforming

• Joint active and passive beamforming is defined as the coordinated design of active transmit precoders and RIS phase shifts to enhance spectral efficiency and reduce power consumption.
• It employs iterative algorithms such as block coordinate descent, SDR, and manifold optimization to efficiently handle nonconvex constraints in complex wireless environments.
• Performance gains include up to 10–20 dB transmit power savings and SNR improvements scaling with the square of the number of RIS elements, even with low-resolution phase control.

Joint active and passive beamforming refers to the coordinated design of both traditional active (transmit) beamforming at a multiple-antenna transmitter (e.g., base station, access point, or radar) and passive (reflection-based) beamforming at a reconfigurable intelligent surface (RIS) or intelligent reflecting surface (IRS). In such systems, the direct electromagnetic propagation between transmitter and receivers is supplemented or shaped by an engineered reflective interface. This paradigm enables the wireless environment to be actively controlled to enhance spectral efficiency, reduce transmit power, manage interference, or enable integrated sensing and communication. The field has rapidly developed, yielding a range of algorithmic, architectural, and application results spanning wireless communications, radar, ISAC, and SWIPT models.

1. System Models and Problem Formulation

The canonical joint active/passive beamforming problem involves a multiple-antenna transmitter (MIMO/MISO), an RIS/IRS comprising $N$ passive reflecting elements each imposing a controllable phase shift (sometimes also amplitude), and one or more users or targets. The baseband channel between transmitter and RIS is typically denoted as $H\in\mathbb{C}^{N\times M}$, and between RIS and a $K$-antenna user as $G\in\mathbb{C}^{K\times N}$, with the overall end-to-end channel $\mathbf{G}\mathbf{\Theta}\mathbf{H}$, where $$\mathbf{\Theta}=\mathrm{diag}(e^{j\phi_1},...,e^{j\phi_N})$$ represents the RIS phase shifts.

The base station applies an active precoder $\mathbf{W}$; the RIS phase vector (passive beamformer) $\boldsymbol{\phi}$ is chosen from a continuous or quantized (low-resolution) set. The achievable rate (for Gaussian signaling) is typically given by

$$R(\mathbf{W},\mathbf{\Theta}) = \log_2\det\left[\mathbf{I}_K + \frac{1}{\sigma_n^2} \mathbf{G}\mathbf{\Theta}\mathbf{H}\mathbf{W}\mathbf{W}^H\mathbf{H}^H\mathbf{\Theta}^H\mathbf{G}^H \right]$$

subject to transmit power and unit-modulus constraints, and potentially discrete phase quantization on the RIS phase shifts. The problem is highly nonconvex due to the coupling and nonconvexity of the passive phase constraints (Souto, 2022).

In multiuser MISO/IRS settings, an AP equipped with $M$ antennas serves $K$ users, and the joint design aims to minimize total transmit power under individual SINR constraints and unit-modulus phase constraints at the IRS, with the SINR for user $k$ given by

$$\mathrm{SINR}_k = \frac{|\left(\mathbf{h}^H_{r,k} \mathbf{\Theta}\mathbf{G} + \mathbf{h}_{d,k}^H\right)\mathbf{w}_k|^2}{\sum_{j\ne k} |\left(\mathbf{h}^H_{r,k} \mathbf{\Theta}\mathbf{G} + \mathbf{h}_{d,k}^H\right)\mathbf{w}_j|^2 + \sigma^2}$$

(Wu et al., 2018). Variants include multi-IRS, multiuser MIMO, integrated sensing and communications (ISAC), and SWIPT.

2. Algorithmic Methodologies

The joint design is fundamentally nonconvex and high-dimensional. The prevailing methodology is block or alternating optimization, with each iteration optimizing the active beamforming (transmit weights) and the passive phase shifts sequentially, conditioned on the other. The major algorithmic archetypes:

• Block Coordinate Descent (BCD):

For fixed passive phases, the active beamforming subproblem reduces to a conventional waterfilling or SOCP/SDP power minimization under SINR or MSE constraints, which is convex (Souto, 2022, Zhao et al., 2021). For fixed active beams, the phase optimization is a quadratically-constrained quadratic program (QCQP) over unit or quantized moduli, typically relaxed to an SDP and solved via semidefinite relaxation (SDR) (Wu et al., 2018, Wu et al., 2018).

• Majorization-Minimization/Successive Convex Approximation (SCA):

For highly nonconvex relationships (e.g., mutual information, ISAC), first-order surrogates or convex minorizers are constructed and iterated to stationary points (Li et al., 2024, Xing et al., 2022).

• Manifold and Optimization on Unit-Modulus Constraints:

Manifold-based optimization leverages the structure of the unit-modulus set (the complex circle), providing gradient-based methods directly over the phase variables (Zhao et al., 2021). Majorization-minimization and matrix fractional programming enable lower-complexity, locally convergent updates.

• Proximal-Gradient for Quantized Phases:

To accommodate discrete phase shifts at the RIS, probability vectors are introduced for each element. The resulting smooth approximation enables the use of simple projection-based block coordinate gradient algorithms, enabling per-element optimal quantization (Souto, 2022).

• Low-Complexity and Structure-Exploiting Solutions:

For MIMO with Kronecker-structured (URA) channels, joint optimization decouples into lower-dimensional SVDs or tensor approximations, greatly reducing complexity at minimal SE loss (Ribeiro et al., 2023).

• Reinforcement Learning and Data-Driven Design:

Deep reinforcement learning (DRL), in particular soft actor-critic (SAC), optimizes joint analog and passive beamforming via reward functions (sum-rate), offering adaptation and robustness to dynamic environments (Zhu et al., 2022).

3. Performance, Hardware Constraints, and Scaling Laws

Theoretical and empirical findings across the literature reveal:

• Scaling Laws:

In single-user configurations with optimized phases, the SNR (and thus required transmit power for a target SINR) increases proportionally to $N^2$, the square of the number of RIS elements (passive array gain plus coherent combining), outperforming traditional amplify–forward or active-relay architectures (which scale linearly) (Wu et al., 2018, Li et al., 2019).

• Low-Resolution Phases:

Hardware constraints often limit phase resolution to a few bits. Simulations show 2-bit quantization (i.e., four phase levels) in each RIS element achieves $\ge$95% of the achievable rate of fully continuous phase control, losing less than 0.5 bps/Hz for moderate array sizes; even 1-bit quantization loses only 1–2 bps/Hz (Souto, 2022).

• Complexity vs. Performance:

Block-coordinate algorithms exploiting closed-form waterfilling, eigen/singular-value decompositions, and per-element projections reduce per-iteration complexity considerably (cubic in active antennas, linear in RIS size), making the approach scalable for large arrays (Souto, 2022, Ribeiro et al., 2023).

• Multi-User Interference and Association:

In massive-MIMO/IRS regimes, the “asymptotic automatic interference cancellation” (AIC) property emerges: when an IRS is optimally tuned to a user, its reflected gain to all others vanishes as $N\to\infty$. In multi-IRS deployments, this property reduces the general joint optimization to an IRS-user association combinatorial instance, solved efficiently by greedy or enumeration algorithms (Li et al., 2019).

• Trade-offs:

Joint active/passive optimization is consistently superior to isolated active or passive designs, achieving up to 10–20 dB transmit power savings versus active (MMSE/ZF) baselines, especially for users near the RIS and at higher densities (Wu et al., 2018, Li et al., 2019).

4. Extensions and Advanced Scenarios

ISAC and Multitask Objectives

RIS/IRSs have been incorporated in integrated sensing and communication (ISAC) systems. When maximizing a combination of sensing mutual information and user QoS, the optimal solution may not require a dedicated sensing beam in high-SNR rank-1 LoS regimes, but in the presence of clutter and Rician fading, alternating convex optimization over both beamformers is required (Li et al., 2024). Sensing performance metrics (probability of detection, beampattern, ultimate detection resolution) are rigorously handled via convex reformulations and SCA (Xing et al., 2022).

Energy and SWIPT

In simultaneous wireless information and power transfer (SWIPT) models, joint active/passive design copes with power splitting, interference, and subjected harvested energy constraints. Dual transforms and block decomposition (including MM, SCA) enable efficient iterative solutions with monotonic convergence. The inclusion of RIS/IRS substantially extends RF-based WPT range and reduces required transmit power, even when only coarse phase resolution or nonideal channel knowledge is available (He et al., 2021, Wu et al., 2019, Zhao et al., 2020).

Hybrid Analog/Digital and Deep Learning

Hybrid analog/digital architectures with RIS assistance, especially at mmWave, present formidable complexity; DRL, particularly with stochastic policies (SAC), matches or outperforms deterministic and FP-based baselines with improved adaptation to channel variation (Zhu et al., 2022).

5. Hardware Constraints and Beyond-Diagonal RIS

Beyond-diagonal RIS structures, where the scattering matrix is symmetric unitary (not just diagonal), further generalize the design space. Alternating minimization using closed-form updates and projection onto the symmetric-unitary manifold enables low-complexity, near-optimal performance for weighted sum-rate maximization, compared to computationally heavier quasi-Newton, manifold, or penalty-based approaches (Zhou et al., 17 Jan 2025).

6. Practical Implications and Outlook

• Practicality: The state-of-the-art demonstrates that, even subject to coarse phase quantization, moderate IRS sizes, unknown channels, and implementation constraints, near-optimal rate or power efficiency is practically attainable with joint active/passive designs (Souto, 2022, Ji et al., 2024).
• Adaptivity: Emerging frameworks leverage reinforcement learning and limited feedback to perform joint beamforming in dynamic or unmodeled scenarios, reducing stationary requirements and CSI acquisition burdens (Zhu et al., 2022, Ji et al., 2024).
• Architectural Trade-offs: The trade-off between RIS size, resolution, and hardware complexity directly influences system performance. Small increases in phase resolution or element count yield disproportionate gains in rate or power; additional RIS deployment, with association and partitioning, can further boost network capacity and coverage (Li et al., 2019, Wei et al., 2022).
• Extensions: Intersections with ISAC, radar, WPT/SWIPT, and new RIS topologies (e.g., STARS, BD-RIS) are prominent future directions (Shen et al., 22 Jul 2025, Li et al., 2024).

7. Summary Table of Key Methodologies

Methodology	Active Beamforming	Passive Beamforming	Notable References
Block Coordinate Descent (BCD)	Waterfilling/WMMSE/MMSE	SDR/SCA/projection (unit-modulus)	(Souto, 2022, Wu et al., 2018)
Manifold Optimization	Waterfilling/WMMSE	Riemannian gradient on circle	(Zhao et al., 2021)
Proximal Gradient (quantized RIS)	Waterfilling	Per-element quantization	(Souto, 2022)
SDR + Randomization	Power minimization	QCQP → SDP → randomization	(Wu et al., 2018, Wu et al., 2018)
Deep Reinforcement Learning	Neural network (SAC/FP)	Neural network (SAC)	(Zhu et al., 2022)
Tensor and Kronecker Approaches	Factorized SVD/HOSVD	Factorized SVD/HOSVD	(Ribeiro et al., 2023)
Matrix Fractional Programming	WMMSE-based	Fractional programming + MM	(Zhao et al., 2021)

The body of research confirms that the optimal exploitation of reconfigurable intelligent surfaces requires coherent joint active (transmitter) and passive (surface) beamforming design. Advances in algorithmic structure, problem-specific relaxations, hardware-aware modeling, and data-driven methods continue to shape the field's evolution.