---
title: Hybrid Beamforming Optimization for MIMO ISAC
url: https://www.emergentmind.com/topics/hybrid-beamforming-optimization-for-mimo-isac
type: topic
---

# Hybrid Beamforming Optimization for MIMO ISAC

Hybrid beamforming optimization for MIMO integrated sensing and communication (ISAC) encompasses the design and implementation of analog/digital precoders under hardware, spectral, and joint performance constraints, enabling simultaneous high-capacity communications and high-resolution sensing. State-of-the-art approaches focus on maximizing rate-based and sensing metrics by leveraging system, channel/statistical, and hardware structure, as well as exploiting algorithmic advances in manifold, convex, and data-driven optimization. This article provides a comprehensive account of hybrid beamforming methods for MIMO ISAC, emphasizing optimization formulations, key solution principles, algorithmic frameworks, and distinctive performance trade-offs.

## 1. System and Signal Modeling Paradigms

Hybrid MIMO ISAC systems deploy transmit and receive arrays partitioned into analog (typically phase-shifter or metasurface) and digital domains, often limited by cost and energy constraints on the number of radio-frequency (RF) chains. Analog precoders are generally constrained to constant-modulus (phase-only) structures, e.g., block-diagonal for partially-connected or fully-populated for fully-connected implementations. The digital precoder(s) offer additional flexibility, though overall DoFs are limited by the RF chain bottleneck.

Signal models incorporate:

- Dual-functional transmit signals: \(\mathbf{x} = \mathbf{F}_{\mathrm{RF}}\,\mathbf{F}_{\mathrm{BB}}\,\mathbf{s}\), with \(\mathbf{F}_{\mathrm{RF}}\) (analog, often unit-modulus/block-diagonal), \(\mathbf{F}_{\mathrm{BB}}\) (digital), and independent streams \(\mathbf{s}\).
- Separate propagation channels: user MIMO wireless channel (typically Saleh–Valenzuela/clustered for mmWave/THz, or MIMO LoS/NLoS for sub-6 GHz) and, for sensing, modeled physical echo paths (target and/or clutter, often parametrized by spatial angle, location, and random reflection coefficient).
- Sensing and communication receive signals: each composed of the aggregate transmit signal, respective channel, and additive white Gaussian noise.
- Practical impairment models: including nonlinear PA distortion (modeled with Bussgang decomposition and higher-order moments as in [2507.14018]) and hardware-specific responses (e.g., electromagnetic reconfigurability, metasurface admittance).

## 2. Design Objectives and Optimization Problem Formulations

MIMO ISAC beamforming optimization targets joint maximization of communication and sensing performance under hardware and regulatory constraints. Representative metrics and constraints include:

- **Sum-rate or weighted user rate:**  \(R = \sum_k \log_2(1 + \gamma_k)\), where \(\gamma_k\) is the receive SINR at user \(k\).
- **Sensing metrics:** mutual information (\(I_s = \log_2(1 + \gamma_s)\)), SCNR, mean-squared error lower bounds (e.g., PCRB [2506.07869], [2406.00689]), or position error bounds [2502.11446].
- **Joint objectives:** weighted trade-off, e.g.,
  \[
  J(\mathbf{F}_{\mathrm{RF}}, \mathbf{F}_{\mathrm{BB}}) = \alpha R_{\mathrm{c}} + (1-\alpha) I_{\mathrm{s}}
  \]
- **Energy efficiency:** bits-per-joule (rate over total hardware + transmit power) [2406.03737].
- **Beampattern control:** prescribed power radiated in selected spatial directions, often for radar tasks [2203.06324], [2504.07988], [2406.12323].

Constraints arise from:

- Power budgets: total or per-stream, post-distortion, or post-PA.
- Hardware structure: constant-modulus, block-diagonal, or metasurface/ERA-specific forms.
- QoS: minimum SINR or rate for comm users, minimum sensing gain or maximum estimation error.
- Analog/digital DoF: total number of available RF chains (and their allocation in transmit/receive).
- Non-linear physical constraints: e.g., enforcement of sparse connectivity, or distributed inference in cell-free settings [2509.25385].

## 3. Algorithmic Solution Frameworks

Hybrid beamforming optimization for MIMO ISAC is inherently non-convex and multi-objective, due to analog hardware constraints, coupled rate and sensing metrics, and, often, practical nonlinear impairments. Key solution techniques include:

### 3.1 Alternating Optimization (AO) and Block Coordinate Descent

- AO schemes decompose the original problem into subproblems per variable block—typically alternating between analog (RF) and digital (baseband) precoders—using either direct convex relaxations or surrogate approximations.
- Standard updates use closed-form, convex, or approximated (e.g., convexified via SCA, MM, or FPP-SCA) subproblems for the digital stage, and manifold-optimization for the analog stage [2507.14018], [2503.12129], [2406.00689], [2506.07869].
- For hardware-imposed decompositions, a stage-wise procedure is common: (1) globally or semi-globally optimize a full-digital beamforming matrix based on a relaxed or surrogate version of the joint objective; (2) decompose the full-digital solution into feasible analog/digital hybrids via alternating least-squares and element-wise projection onto the constraint set [2203.06324], [2406.03737].

### 3.2 Manifold and Riemannian Optimization

- Analog “phase-only” precoders correspond to points on complex tori (product of unit circles, or Stiefel manifolds for more general constraints), motivating Riemannian conjugate-gradient, steepest descent, or trust-region algorithms.
- In [2507.14018], the full-digital beamforming matrix is optimized over the complex Frobenius sphere (norm-constraint manifold), and analog precoders are refined via phase-projection.
- Similar approaches arise for dynamic metasurfaces (Neumann-approximated linearizations) or high-dimensional digital-only (rank-constrained) digital beamforming [2504.07988], [2406.12323].

### 3.3 Convex and Semidefinite Programming (SDP/SDR)

- SDR techniques relax the rank constraints of digital-only or BCD-based digital beamforming problems, often dropping the single-rank/dimension constraint, leading to tractable convex optimization that is then rounded or randomized to recover feasible solutions [2406.03737], [2406.12323].
- QCQP, SOCP, and sequential convex surrogate (SCA/FPP-SCA/MM) methods are extensively applied for per-iteration subproblem tractability in AO schemes [2503.12129], [2506.07869], [2406.00689].

### 3.4 Data-Driven and Graph-Based Approaches

- For large-scale or cell-free deployments, distributed optimization and graph neural network (GNN) architectures enable learning-based inference of beamformers from local channel state only [2509.25385].
- Post-training, inference is performed on FPGA accelerators for low-latency per-BS deployment.

### 3.5 Specialized/Hybrid Algorithms

- Position error bound (PEB)-driven designs leverage Riemannian trust-region (RTR) optimization for the analog precoder, with low-complexity OMP for practical, scalable designs [2502.11446].
- ERA/EM-enabled tri-hybrid schemes use fractional programming to split the coupled rate/SCNR objectives, combined with per-antenna manifold updates for the EM configuration [2510.14530].

## 4. Representative Formulations and Algorithm Details

The following table summarizes representative formulations and solution highlights from canonical references:

| Reference      | Objective/Constraints                                 | Key Algorithm/Decomposition                     |
|----------------|-------------------------------------------------------|-------------------------------------------------|
| [2507.14018]   | Rate+Mutual Info, PA distortion-aware, partial HBF    | MO + AO (digital on sphere, analog block-wise)  |
| [2506.07869], [2406.00689]  | PCRB minimization w/ rate, prior info         | AO: digital (SDP), analog (element-wise closed-form), FPP-SCA for non-convexity |
| [2406.12323]   | SE + SCNR, modular/XL-MIMO, near-field                | Closed-form analog via subarray response structure, digital on Stiefel manifold or SDR |
| [2503.12129]   | PD-max, GMR-max, sinr/rate fairness                   | AO: SCA for digital, Riemannian CG for analog   |
| [2401.17681]   | Weighted sum-rate + SCNR (mmWave/THz MU)              | Subspace-based BCD, explicit BD-style solution, hybrid via manifold AO |
| [2510.14530]   | Rate+SCNR, tri-hybrid (digital, analog, EM)           | FP + MO, AO over blocks with closed-form updates|
| [2504.07988]   | Fair max beampattern, SINR, power (DMA arrays)        | AO over digital (SDP) and analog (QCQP)         |
| [2502.11446]   | SE-PEB (bistatic, OFDM), gain/position error coupling | RTR (analog), SCA (digital), OMP (greedy)       |

## 5. Performance Trade-offs and Insights

Quantitative and structural analysis across studies reveals:

- **Optimality Conditions & Hybrid Sufficiency:** Hybrid architectures with enough RF chains can, in principle, match fully digital designs if the RF chain count exceeds the number of dominant scattering paths or the underlying subspace dimension [2401.17681]. In starved scenarios, prioritization between comm/sensing is imposed by power/rate constraints.
- **Allocation of Transmit/Receive Chains:** For ISAC sensing under a total RF chain budget, allocating more chains to the receiver sharply improves estimation bounds (PCRB), while increasing transmit RF chains becomes advantageous mainly as comm rate demands rise [2506.07869].
- **Impact of Hardware Impairments:** Explicitly accounting for hardware nonidealities such as PA-induced distortion is essential, especially at high SNR, where their effect dominates traditional beamforming losses. Distortion-aware designs produce notches in beam patterns in directions of both users and targets to mitigate interference from nonlinear artifacts [2507.14018].
- **Array Structure and Geometry:** Modular/XL array layouts and design exploiting the HSPM subarray response can yield closed-form analog beamformers and substantial joint comm-sensing gains [2406.12323]. Electromagnetic reconfiguration (beyond analog/digital) introduces an additional DoF, providing up to 10 dB gain vs comparable non-EM systems [2510.14530].
- **Algorithmic Efficiency & Scalability:** Manifold algorithms (CG, trust-region) are competitive for moderate dimensions, but complexity matters for large arrays or OFDM settings [2406.12323], [2502.11446]. Data-driven (GNN) methods are uniquely suited to distributed, cell-free ISAC, enabling msec-level inference [2509.25385].
- **Joint Optimization Regimes:** Pareto-optimal tradeoffs exist between comm and sensing metrics (rate/SCNR or SE/PEB), with intermediate scalarization parameters ensuring balanced performance. Resource allocation (e.g., waterfilling in analytical BD-style solutions) tracks classical comm/sensing trade-offs [2401.17681].

## 6. Open Challenges and Future Perspectives

Future research themes for hybrid beamforming in MIMO ISAC include:

- **Robustness to Imperfect CSI and Hardware Nonidealities:** Most current works assume perfect knowledge; emerging robust and adaptive algorithms will incorporate estimation uncertainty, dynamic reconfigurability, and quantization, particularly for metasurface and tri-hybrid arrangements [2504.07988].
- **Wideband and Near-Field Extensions:** Practical ISAC deployments will require scalable methods for hybrid beamforming under OFDM operation and non-far-field regimes, especially for XL-MIMO and holographic arrays [2406.12323].
- **Joint RF Chain/User/Target Scheduling:** Dynamic resource allocation across comm and sensing, possibly under real-time constraints, remains an area of active investigation, including distributed methods for large cell-free networks [2509.25385].
- **Algorithmic Generalization:** Fractional programming, Riemannian optimization, and data-driven methods are expected to further unify, simplify, and accelerate the solution of non-convex ISAC beamforming tasks, even in the presence of nonlinear coupling terms, non-standard hardware, or multi-objective performance surfaces.

## 7. Summary Table of Key Algorithms

| Work                | Key Algorithmic Techniques                   | ISAC Metric      | Hardware Constraint            |
|---------------------|----------------------------------------------|------------------|-------------------------------|
| [2507.14018]        | Manifold AO (nonlinear PA), block-diag HBF   | Sum-rate/MI      | Partial-hybrid, PA distortion |
| [2509.25385]        | GNN, FPGA inference, distributed graph       | Joint rate/SCNR  | Fully-connected, cell-free    |
| [2510.14530]        | Tri-hybrid, FP + MO, EM-weights              | Rate + SCNR      | Digital/Analog/EM-reconfig.   |
| [2506.07869]        | AO (digital-SDP, analog closed-form)         | PCRB             | Hybrid Tx/Rx, prior-aware     |
| [2406.12323]        | Subarray-level AO, manifold, SDR             | SE+SCNR          | Modular XL, near-field        |
| [2503.12129]        | SCA + Riemannian CG, bisection, MM           | PD, GMR          | Fully-connected HBF           |
| [2401.17681]        | BCD, subspace, BD, closed-form wireless-null | WSR+SCNR         | mmWave/THz, user-side hybrid  |

Hybrid beamforming optimization for MIMO ISAC thus encompasses a multidimensional design space, integrating physical, mathematical, and hardware constraints to realize joint operation across communications and radar in next-generation wireless systems. Foundational advances across optimization, statistical modeling, and robust learning continue to propel this field toward practical, scalable, and computationally efficient deployments.

Source: https://www.emergentmind.com/topics/hybrid-beamforming-optimization-for-mimo-isac