---
title: Joint Active and Passive Beamforming
url: https://www.emergentmind.com/topics/joint-active-and-passive-beamforming-design
type: topic
---

# Joint Active and Passive Beamforming

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 [1810.03961].

## 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 [1810.03961], 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 [1810.03961].

## 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 [1810.03961].

### 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 [1810.03961]. Similar findings hold for SNR and rate scaling [1809.01423].

### 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) [2012.05646], ISAC (integrated sensing and communication) [2309.00978,2407.16543,2206.00525], distributed/multi-IRS networks [2210.01309,1912.00728].
- Multi-antenna receivers and full MIMO (not just MISO) setups [2102.01232,2101.10071,2305.14650].
- BD-RIS (beyond-diagonal RIS): relaxation of the traditional diagonal reflection model, enabling more general symmetric-unitary constraints and richer interactions [2501.10227].
- Hybrid active/passive RIS with both amplifier and phase control [2203.05902].
- Discrete/quantized phase shift implementations [2208.00717], robust designs under channel uncertainty [2309.00978], 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 [2202.11702], 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 [2405.01979].
- Specialized methods for low-complexity/large-scale settings: Kronecker-factored SVD/tensor methods [2305.14650], VAMP-based iterative solvers [2102.01232], penalty methods and MM for large networks [2101.10071].

## 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 [2309.00978,2407.16543,2206.00525,2203.14532]      |
| SWIPT             | R-E region (rate-energy) | Nonlinear harvester models; BCD/GP, waveform/precoding joint design [2012.05646]                  |
| WET               | Harvested energy         | One-bit feedback (ACCPM/distributed beam), minimal ER complexity [2404.05418]                     |

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 [2407.16543,2206.00525]. In SWIPT, nonlinear RF/electronic effects mandate waveform-aware co-design and new optimization decompositions [2012.05646]. Large-scale systems benefit from low-complexity heuristics (greedy IRS-user association [1912.00728], Kronecker/tensorized solvers [2305.14650]).

## 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 [2305.14650,2405.01979].
- **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 [1809.01423].
- **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 $N$ [2208.00717].
- **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 [1810.03961]. Hybrid architectures (with some active elements) further extend the power/radar region for ISAC [2203.05902].
- **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 [2202.11702,2405.01979].

## 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 [1810.03961, 1912.00728, 2407.16543, 2208.00717, 2202.11702, 2405.01979, 2305.14650].

Source: https://www.emergentmind.com/topics/joint-active-and-passive-beamforming-design