Frequency-Selective Hybrid Beamforming

• Frequency-selective hybrid beamforming decomposes the beamforming process into a frequency-flat analog stage and frequency-dependent digital processing to reduce hardware and energy costs.
• Key algorithmic strategies include eigen-decomposition, Gram–Schmidt procedures, and data-driven methods that efficiently balance system performance and hardware constraints.
• This design enables near-digital performance in massive MIMO, mmWave, and ISAC applications while minimizing the number of required RF chains and phase shifters.

Frequency-selective hybrid beamforming design refers to methodologies and architectures that decompose the transmit or receive beamforming operation in wideband or multi-carrier massive MIMO systems into a two-stage structure: a high-dimensional analog (RF) beamformer—implemented with hardware-constrained phase shifters—and a lower-dimensional digital (baseband) beamformer. In frequency-selective (wideband) channels, the system must allow for the analog beamformer, typically frequency-flat, to be combined with frequency-dependent digital beamformers tailored to each subcarrier or time-domain tap. The primary motivation is to reduce hardware complexity (notably the number of costly RF chains and phase shifters) and energy consumption while still achieving the performance (in terms of sum rate, bit error rate, or other statistical metrics) close to that of fully digital solutions.

1. Hardware Architecture and Fundamental Decomposition

The elementary structure for frequency-selective hybrid beamforming leverages a cascade of two stages: an analog beamformer $\mathbf{F}_{\mathrm{RF}}$ that applies coarse spatial shaping (with constant-modulus constraints on each element) and a set of digital baseband beamformers $\{\mathbf{F}_{\mathrm{BB}}[k]\}$ adapted to the frequency subcarriers. The transmitted signal at subcarrier or time index $k$ is typically expressed as

$$\mathbf{x}[k] = \mathbf{F}_{\mathrm{RF}} \, \mathbf{F}_{\mathrm{BB}}[k] \, \mathbf{s}[k]$$

with $N_{\mathrm{RF}} \ll N$ (number of RF chains much less than the number of antennas). The analog stage is realized via a network of phase shifters; for each RF chain, only the phase (not amplitude) of each signal path can be controlled, imposing a non-convex constant-modulus constraint, typically $|\left[\mathbf{F}_{\mathrm{RF}}\right]_{m,n}| = 1$. In the frequency-selective (e.g., OFDM) setting, $\mathbf{F}_{\mathrm{RF}}$ is common across all subcarriers, while $\mathbf{F}_{\mathrm{BB}}[k]$ varies per subcarrier.

A central result (Bogale et al., 2014, Sohrabi et al., 2016) shows that, for a system with $r_t$ as the aggregate rank of the frequency-selective digital precoders, near-optimal performance can be attained with only $r_t$ RF chains and $2r_t(N - r_t + 1)$ phase shifters, or—if $N_{\mathrm{RF}} \geq 2N_s$—any fully digital beamformer of $N_s$ streams can be exactly implemented.

2. Algorithmic Methodologies and Design Strategies

Design and optimization of frequency-selective hybrid beamformers is inherently non-convex due to hardware constraints and the coupling between analog and digital stages. Several algorithmic frameworks have emerged:

• Two-Stage/Decoupled Design: The analog beamformer is derived to capture the dominant subspace (via eigen-decomposition of the spatial covariance, Gram–Schmidt procedures, or codebook search), and then a per-subcarrier digital precoder is computed (e.g., by SVD, water-filling, or WMMSE algorithms) for residual frequency selectivity (Alkhateeb et al., 2016, Zhu et al., 2016, Sohrabi et al., 2017). For instance, the Gram–Schmidt greedy algorithm selects RF beams incrementally to orthogonalize coverage of the dominant subspaces, ensuring high mutual information across all subcarriers.
• Unified Analog Beamforming: In multiuser or multi-group scenarios, a frequency-flat analog beamformer is optimized based on the aggregated spatial covariance matrices of all supported groups/users. This approach ensures that the analog beamformer “covers” all signal subspaces relevant to the served traffic (Zhu et al., 2016).
• Environmental and Location-Aware Optimization: Channel knowledge maps (CKM), such as channel angle maps (CAM) or beam index maps (BIM), allow the system to predict spatial statistics of the channel and restrict beam search and channel estimation to a small set of dominant directions, dramatically reducing real-time training overhead (Wu et al., 2022).
• Low-Complexity, Heuristic, and Data-Driven Approaches: Methods are proposed that use implicit CSI (pilot correlations) rather than explicit full channel estimation, low-complexity codebook design for compressed sensing-based channel recovery, and deep learning frameworks (CNNs) to regress from channel estimates to beamformer weights, achieving robust design for imperfect or limited feedback (Chiang et al., 2017, Chiang et al., 2018, Sung et al., 2019, Elbir et al., 2019).
• Integrated Sensing and Communication (ISAC): Joint optimization using performance metrics such as the posterior Cramér–Rao bound (PCRB) for sensing, with strict per-subcarrier constraints for communications, leads to alternately optimized digital/analog beamformers, solved efficiently using convex relaxation, FPP-SCA, or alternating optimization frameworks (Wang et al., 2 Jun 2024, Xu et al., 18 Mar 2024).

3. Channel Rank, RF Chain and Phase Shifter Scalability

The number of radio-frequency (RF) chains required is fundamentally tied to the effective signal subspace: $r_t$ RF chains are necessary and sufficient to achieve full-digital performance when the overall channel (across all subcarriers) has rank $r_t$. Under this configuration, the analog beamforming stage “captures” the entire signal space and the digital stage can realize any desired beamforming operation (Bogale et al., 2014). The phase shifter count can be rigorously reduced—practical systems may require only 20–40 phase shifters per RF chain for negligible performance loss.

User selection and scheduling become critical in cases where the number of available RF chains is less than the channel rank; a robust algorithm for scheduling maximizes coverage of the dominant channel modes per subcarrier (Bogale et al., 2014).

For scenarios such as backhaul or correlated channels, antenna clustering and spreading operations further reduce the number of required phase shifters and can leverage closed-form solutions via approximation of the effective correlation matrix (e.g., tridiagonal Toeplitz structure) (Rajatheva et al., 2016).

4. Performance, Hardware Complexity, and Validation

Performance of frequency-selective hybrid beamforming is typically evaluated in terms of achievable spectral efficiency, bit error rate, or task-oriented metrics (e.g., radar beampattern control for DFRC). The literature reports negligible performance loss (typically within 5% of digital beamforming) under adequately chosen hardware parameters (RF chain and phase shifter counts), even under practical constraints such as finite-precision phase shifters or partially connected RF architectures (Bogale et al., 2014, Zhu et al., 2016, Sohrabi et al., 2017, Majidzadeh et al., 2020).

Simulation results over flat fading and frequency-selective channels, large-scale MIMO backhaul, and broadband OFDM systems confirm both the validity of the hardware scaling laws and the robustness of the proposed algorithms. Multiple works provide computational complexity analysis, demonstrating polynomial complexity ($O(N^3)$) for key subproblems and empirical evidence for rapid convergence of iterative methods.

For dual-function radar-communication (DFRC), novel beampattern metrics such as AISMMR and APSIMR have been introduced, with task-oriented optimization enabling flexible control of space-frequency radiation, subject to communication power/rate constraints (Xu et al., 18 Mar 2024).

5. Practical Considerations and Extensions

Frequency-selective hybrid beamforming design is highly relevant to mmWave and Terahertz MIMO, where practical systems are constrained by hardware cost and energy consumption. Several trends and practical principles are substantiated:

• Finite resolution of phase shifters can be accounted for via quantized codebooks without significant spectral efficiency loss, especially with a moderate surplus of RF chains (Sohrabi et al., 2016, Sohrabi et al., 2017).
• Partially connected RF architectures offer a reduction in hardware complexity with minor penalties in achievable rate, particularly in subarray scenarios or when scheduling is adapted to hardware topology (Majidzadeh et al., 2020, Bychkov et al., 16 Feb 2024).
• Channel estimation overhead is a significant bottleneck. Hybrid beamforming architectures employing deterministic codebooks for compressed sensing or CKMs for location-based beam selection drastically reduce the training burden, enabling practical deployment in large-scale antenna systems (Sung et al., 2019, Wu et al., 2022).
• Deep learning-based hybrid beamforming prediction enables low-complexity, low-latency adaptation in dynamic or limited-CSI regimes, achieving robust near-optimal performance (Elbir et al., 2019).

These advances render frequency-selective hybrid beamforming not only a theoretical construct but a practical enabler for massive MIMO, mmWave cellular, wireless backhaul, and ISAC deployments, balancing the stringent constraints of RF hardware with the demands for near-digital spatial signal processing performance.

6. Joint Sensing and Communication and Task-Oriented Optimization

Recent developments address ISAC systems that must minimize radar sensing error—quantified by the PCRB—while meeting communication rate targets (Wang et al., 2 Jun 2024). For such joint objectives, hybrid beamformers can be designed to closely approach fully digital benchmarks if a sufficient number of RF chains is available, with alternating optimization algorithms adjusting both analog and digital stages subject to non-convex hardware and performance constraints. In multi-task OFDM DFRC, consensus-based ADMM allows simultaneous control of sidelobe/mainlobe characteristics and communication QoS, using closed-form or cyclic coordinate descent updates (Xu et al., 18 Mar 2024).

This area demonstrates that hybrid beamforming is extensible beyond classical communications, supporting a rich set of applications with rigorous task-oriented design and optimization methods.

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In conclusion, frequency-selective hybrid beamforming design encompasses the principled reduction of hardware complexity in large antenna systems, deploying sophisticated analog–digital decomposition, algorithmic optimization adapted to wideband channels, and rigorous validation of performance relative to fully digital architectures. Its significance extends into emerging applications such as ISAC and DFRC, where hardware efficiency, algorithmic advanced scheduling, and robust environment-adaptive techniques converge.