Joint Active and Passive Beamforming

• Joint active and passive beamforming is a design approach that combines active multi-antenna transmit beamforming with passive IRS phase control to enhance signal quality and energy efficiency.
• It employs block coordinate descent to alternate between convex AP beamforming and nonconvex IRS phase optimization using techniques like SDR and Gaussian randomization.
• The design achieves significant improvements including quadratic scaling of passive array gain, effective interference suppression, and extended network coverage in diverse wireless scenarios.

Joint active and passive beamforming design refers to the co-optimization of the transmit (active) beamforming at a multi-antenna access point or base station (AP/BS) and the reflect (passive) beamforming via the adjustable phase shifters at an intelligent reflecting/reconfigurable intelligent surface (IRS/RIS) or similar large-array surface. This unified design paradigm is central to IRS/RIS-assisted wireless networks, as it enables the system to leverage the high-dimensional phase control of the IRS/RIS alongside transmitter-side spatial processing to achieve sharp beamforming, interference suppression, and substantial power and coverage gains across a variety of settings (Wu et al., 2018).

1. Fundamental System Model and Problem Formulation

In the canonical IRS-aided multiuser MISO downlink, an AP with $M$ antennas communicates through (and possibly directly with) $K$ single-antenna users, assisted by an IRS with $N$ passive reflecting elements. The key baseband expressions are:

• Channels: Direct (AP→user): $h_{d,k}\in\mathbb{C}^{M}$; AP→IRS: $G\in\mathbb{C}^{N\times M}$; IRS→user: $h_{r,k}\in\mathbb{C}^{N}$.
• IRS phase matrix: $$\Theta = \mathrm{diag}(e^{j\theta_1}, ..., e^{j\theta_N})$$, $|e^{j\theta_n}|=1$.
• Transmit signal: $x = \sum_{k=1}^K w_k s_k$, where $w_k$ is the AP beamforming vector for user $k$.
• Received signal at user $k$: $$y_k = (h_{r,k}^H\Theta G + h_{d,k}^H)\sum_{j=1}^K w_j s_j + n_k$$, with $n_k\sim\mathcal{CN}(0,\sigma_k^2)$.
• SINR at user $k$:

$$\mathrm{SINR}_k = \frac{|\left(h_{r,k}^H\Theta G + h_{d,k}^H\right)w_k|^2}{\sum_{j\neq k}|\left(h_{r,k}^H\Theta G + h_{d,k}^H\right)w_j|^2+\sigma_k^2}.$$

The central joint design problem is often cast as AP transmit power minimization (or sum-rate maximization) under per-user SINR requirements, with the IRS restricted to unit-modulus constraints:

$$\min_{\{w_k\},\{\theta_n\}} \sum_{k=1}^K \|w_k\|^2 \quad \text{s.t.}~ \mathrm{SINR}_k\ge\gamma_k,~\forall k;~|e^{j\theta_n}|=1,~\forall n.$$

This problem is intrinsically nonconvex due to coupled bilinear terms and the unimodular constraints (Wu et al., 2018), and serves as the unifying template for various operational extensions (sum-rate, SWIPT, ISAC, etc.).

2. Core Solution Methodologies

The prevailing methodology is block coordinate descent (alternating optimization, AO): A. Active Beamforming Subproblem (fixed IRS phases):

• Reduces to classical MISO downlink power minimization under SINR constraints, which is convex and efficiently solved via SOCP or MMSE fixed-point recursion; optimality in single iteration per IRS phase setting.

B. Passive Beamforming Subproblem (fixed AP beams):

• Becomes a nonconvex QCQP (quadratically constrained quadratic program) over unit-modulus IRS phases.
• Standard approach is semidefinite relaxation (SDR): lift phases to $v\in\mathbb{C}^N$ and relax rank-1 constraint on $V=vv^H$ to $V\succeq0$ with $V_{ii}=1$.
• The resulting SDP may not return rank-1, so Gaussian randomization reconstructs a feasible (near-optimal) phase vector.
• Various low-complexity approximations: penalty-CCP (convex-concave procedure), SOCP, or minorization–maximization (MM).

Convergence and Complexity: Each AO iteration monotonically reduces the cost and guarantees convergence to a stationary point. Active subproblem: $O(M^3K^3)$; passive (SDP/SDR): $O(N^6)$ per iteration (Wu et al., 2018).

3. Theoretical Performance and Scaling Laws

Passive Array Gain

• When the IRS is optimally tuned and user is near the IRS, the transmit power (or received SNR) benefit scales as $O(N^2)$ with the number of reflecting elements (quadratic gain)—in contrast with random/zero IRS which scale $O(N)$, and with the $O(N)$ scaling provided by amplify-and-forward relays (Wu et al., 2018).

Multiuser Interference Suppression

• IRS reflecting phases can be jointly tuned with AP beams to induce constructive/destructive combining at users, mitigating multiuser interference beyond the spatial degrees of freedom offered by the AP alone. This effect is pronounced for "near-IRS" users and in moderate to large $N$.

Coverage Extension

• IRSs can extend the coverage envelope at fixed transmit power—e.g., 20-element IRS extends coverage from $\sim$33 m to $>$50 m for a 10 dBm AP budget (Wu et al., 2018). Similar findings hold for SNR and rate scaling (Wu et al., 2018).

Comparative Analysis

• IRS outperforms both HD/FD AF relays at sufficiently large $N$; the latter only achieve $O(N)$ receive SNR scaling and require more hardware complexity.

4. Extensions and Generalizations

A. Alternative Objectives and System Architectures:

• Sum-rate, min-rate, SWIPT (simultaneous wireless information and power transfer) (Zhao et al., 2020), ISAC (integrated sensing and communication) (AlaaEldin et al., 2023Li et al., 2024Xing et al., 2022), distributed/multi-IRS networks (Wei et al., 2022Li et al., 2019).
• Multi-antenna receivers and full MIMO (not just MISO) setups (Rehman et al., 2021Zhao et al., 2021Ribeiro et al., 2023).
• BD-RIS (beyond-diagonal RIS): relaxation of the traditional diagonal reflection model, enabling more general symmetric-unitary constraints and richer interactions (Zhou et al., 17 Jan 2025).
• Hybrid active/passive RIS with both amplifier and phase control (Sankar et al., 2022).
• Discrete/quantized phase shift implementations (Souto, 2022), robust designs under channel uncertainty (AlaaEldin et al., 2023), and hardware impairment-aware formulations.

B. Algorithmic Innovations:

• Learning-based approaches, e.g., deep reinforcement learning (soft actor-critic) for stochastic policy exploration in high-dimensional joint active/passive design (Zhu et al., 2022), or graph neural networks that predict both AP beams and IRS configurations in a single forward pass, achieving near-iterative performance at much lower computational cost (Le et al., 2024).
• Specialized methods for low-complexity/large-scale settings: Kronecker-factored SVD/tensor methods (Ribeiro et al., 2023), VAMP-based iterative solvers (Rehman et al., 2021), penalty methods and MM for large networks (Zhao et al., 2021).

5. Applications: Communication, Sensing, and SWIPT

Application	Problem Objective	Special Constraints / Approach
Communication	Power, rate, SINR	Classical AO/SDR/SOCP/MM; handles large $N$, $K$
ISAC	MIMO MI, radar-SINR	New constraints (Frobenius, cross-correlation, detection probability); alternates AO/MM/SDR (AlaaEldin et al., 2023Li et al., 2024Xing et al., 2022Hua et al., 2022)
SWIPT	R-E region (rate-energy)	Nonlinear harvester models; BCD/GP, waveform/precoding joint design (Zhao et al., 2020)
WET	Harvested energy	One-bit feedback (ACCPM/distributed beam), minimal ER complexity (Ji et al., 2024)

In ISAC, joint design must balance comms-SINR and radar metric (e.g., echo power, detection resolution)—often requiring additional feasibility analysis, SCA for nonconvex constraints, and new theoretical detection-complexity tradeoffs (Li et al., 2024Xing et al., 2022). In SWIPT, nonlinear RF/electronic effects mandate waveform-aware co-design and new optimization decompositions (Zhao et al., 2020). Large-scale systems benefit from low-complexity heuristics (greedy IRS-user association (Li et al., 2019), Kronecker/tensorized solvers (Ribeiro et al., 2023)).

6. Practical Considerations and Implementation Insights

• Scalability and Complexity: Although AO+SDR is near-optimal, its per-iteration complexity is high for large $N$ (e.g., $O(N^6)$ for SDR). Low-complexity approximations, tensor factorizations, learning-based surrogates, and distributed implementations are under active investigation (Ribeiro et al., 2023Le et al., 2024).
• CSI Acquisition: Global CSI is assumed in most centralized designs; distributed algorithms (alternating AP/IRS updates) require only local (composite) CSI and converge within a few transmissions, substantially reducing channel estimation and backhaul requirements (Wu et al., 2018).
• Phase Quantization and Hardware Nonidealities: Continuous phase assumption yields performance upper-bounds; algorithmic adaptations for discrete ($b$-bit) phase are available, with even 1–2 bit phase quantizers yielding most of the IRS gain for large $N$ (Souto, 2022).
• Deployment Guidelines: IRSs are best deployed at cell-edges or in coverage holes to maximize $N^2$ scaling; for multiuser setups, pure LoS AP–IRS links can reduce spatial rank/multiplexing (Wu et al., 2018). Hybrid architectures (with some active elements) further extend the power/radar region for ISAC (Sankar et al., 2022).
• Convergence: Monotonic power/rate descent and bounded feasible sets ensure convergence for AO/SDR, MM, FP, and VAMP variants; learning-based methods demonstrate fast empirical convergence with generalization to unseen large-scale topologies (Zhu et al., 2022Le et al., 2024).

7. Impact, Open Problems, and Future Directions

Joint active and passive beamforming is foundational to IRS/RIS-enabled wireless systems, offering orders-of-magnitude improvements in power efficiency, spectral coverage, and environmental control. Key open research areas include:

• Real-time, CSI-robust scalable algorithms, especially under fast-fading or mobility;
• Fully distributed and feedback-efficient implementations (e.g., one-bit feedback, self-configuration, groupwise beam management);
• Robust joint designs for integrated communication, sensing, and SWIPT under hardware constraints;
• Quantized/hardware-constrained designs for low-cost, large-scale IRSs and BD-RIS architectures;
• Theoretical capacity and scaling laws in multi-cell, multi-IRS, and multi-user regimes, especially in heterogeneous environments with blockage, interference, and realistic propagation effects.

These directions are actively under exploration, aiming to make joint active/passive beamforming deployable and practical for 6G and beyond (Wu et al., 2018, Li et al., 2019, Li et al., 2024, Souto, 2022, Zhu et al., 2022, Le et al., 2024, Ribeiro et al., 2023).