Hybrid Beamforming Optimization for MIMO ISAC

• Hybrid beamforming optimization for MIMO ISAC is a method that designs analog and digital precoders to jointly enhance high-capacity communications and high-resolution sensing while respecting hardware constraints.
• Algorithmic frameworks such as alternating optimization, manifold methods, and convex programming are used to manage non-convex designs and balance competing rate and sensing metrics.
• Emerging solutions leverage data-driven and distributed techniques to improve real-time performance and robustness against practical impairments in integrated sensing and communication systems.

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 (Zhang et al., 18 Jul 2025)) 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 (Wang et al., 9 Jun 2025, Wang et al., 2024)), or position error bounds (Mao et al., 17 Feb 2025).
• 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) (Singh et al., 2024).
• Beampattern control: prescribed power radiated in selected spatial directions, often for radar tasks (Qi et al., 2022, Khosroshahi et al., 7 Apr 2025, Meng et al., 2024).

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 (Du et al., 29 Sep 2025).

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 (Zhang et al., 18 Jul 2025, Singh et al., 15 Mar 2025, Wang et al., 2024, Wang et al., 9 Jun 2025).
• 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 (Qi et al., 2022, Singh et al., 2024).

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 (Zhang et al., 18 Jul 2025), 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 (Khosroshahi et al., 7 Apr 2025, Meng et al., 2024).

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 (Singh et al., 2024, Meng et al., 2024).
• QCQP, SOCP, and sequential convex surrogate (SCA/FPP-SCA/MM) methods are extensively applied for per-iteration subproblem tractability in AO schemes (Singh et al., 15 Mar 2025, Wang et al., 9 Jun 2025, Wang et al., 2024).

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 (Du et al., 29 Sep 2025).
• 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 (Mao et al., 17 Feb 2025).
• 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 (Chen et al., 16 Oct 2025).

4. Representative Formulations and Algorithm Details

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

Reference	Objective/Constraints	Key Algorithm/Decomposition
(Zhang et al., 18 Jul 2025)	Rate+Mutual Info, PA distortion-aware, partial HBF	MO + AO (digital on sphere, analog block-wise)
(Wang et al., 9 Jun 2025, Wang et al., 2024)	PCRB minimization w/ rate, prior info	AO: digital (SDP), analog (element-wise closed-form), FPP-SCA for non-convexity
(Meng et al., 2024)	SE + SCNR, modular/XL-MIMO, near-field	Closed-form analog via subarray response structure, digital on Stiefel manifold or SDR
(Singh et al., 15 Mar 2025)	PD-max, GMR-max, sinr/rate fairness	AO: SCA for digital, Riemannian CG for analog
(Wang et al., 2024)	Weighted sum-rate + SCNR (mmWave/THz MU)	Subspace-based BCD, explicit BD-style solution, hybrid via manifold AO
(Chen et al., 16 Oct 2025)	Rate+SCNR, tri-hybrid (digital, analog, EM)	FP + MO, AO over blocks with closed-form updates
(Khosroshahi et al., 7 Apr 2025)	Fair max beampattern, SINR, power (DMA arrays)	AO over digital (SDP) and analog (QCQP)
(Mao et al., 17 Feb 2025)	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 (Wang et al., 2024). 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 (Wang et al., 9 Jun 2025).
• 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 (Zhang et al., 18 Jul 2025).
• 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 (Meng et al., 2024). Electromagnetic reconfiguration (beyond analog/digital) introduces an additional DoF, providing up to 10 dB gain vs comparable non-EM systems (Chen et al., 16 Oct 2025).
• Algorithmic Efficiency & Scalability: Manifold algorithms (CG, trust-region) are competitive for moderate dimensions, but complexity matters for large arrays or OFDM settings (Meng et al., 2024, Mao et al., 17 Feb 2025). Data-driven (GNN) methods are uniquely suited to distributed, cell-free ISAC, enabling msec-level inference (Du et al., 29 Sep 2025).
• 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 (Wang et al., 2024).

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 (Khosroshahi et al., 7 Apr 2025).
• 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 (Meng et al., 2024).
• 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 (Du et al., 29 Sep 2025).
• 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
(Zhang et al., 18 Jul 2025)	Manifold AO (nonlinear PA), block-diag HBF	Sum-rate/MI	Partial-hybrid, PA distortion
(Du et al., 29 Sep 2025)	GNN, FPGA inference, distributed graph	Joint rate/SCNR	Fully-connected, cell-free
(Chen et al., 16 Oct 2025)	Tri-hybrid, FP + MO, EM-weights	Rate + SCNR	Digital/Analog/EM-reconfig.
(Wang et al., 9 Jun 2025)	AO (digital-SDP, analog closed-form)	PCRB	Hybrid Tx/Rx, prior-aware
(Meng et al., 2024)	Subarray-level AO, manifold, SDR	SE+SCNR	Modular XL, near-field
(Singh et al., 15 Mar 2025)	SCA + Riemannian CG, bisection, MM	PD, GMR	Fully-connected HBF
(Wang et al., 2024)	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.