Hybrid Analog/Digital Beamforming

• Hybrid analog/digital beamforming is a transceiver architecture that splits spatial processing between analog phase control and digital processing, reducing RF-chain count while preserving multiplexing and interference management.
• It utilizes various hardware models, including fully-connected, partially-connected, and switch-based designs, to balance performance, cost, and power consumption in high-frequency and large-scale arrays.
• Optimization techniques such as alternating minimization, compressed sensing, and metaheuristics are employed to achieve near-digital performance in communications, radar, and integrated sensing applications.

Hybrid analog/digital beamforming refers to transceiver architectures that divide spatial precoding and combining tasks between a digitally controlled low-dimensional baseband processor and an analog RF network, typically implemented with variable phase shifters or switches. Such architectures address the prohibitive power, cost, and complexity of full digital MIMO in mmWave, THz, or large-scale arrays, by severely reducing RF-chain count—ideally to the number of spatial streams supported—while retaining the essential spatial processing flexibility required for multiplexing, interference management, and joint sensing-communications integration.

1. System Architectures and Hardware Models

Hybrid beamforming decomposes the high-dimensional beamforming operation as $$\mathbf{F} = \mathbf{F}_{\rm RF}\mathbf{F}_{\rm BB}$$ (precoding) or $$\mathbf{W} = \mathbf{W}_{\rm RF}\mathbf{W}_{\rm BB}$$ (combining), where $\mathbf{F}_{\rm RF}$ is a constant-modulus (phase-only) $N_t \times N_{\rm RF}$ analog matrix, and $\mathbf{F}_{\rm BB}$ is a $N_{\rm RF} \times N_s$ digital matrix, for $N_s$ spatial streams and $N_{\rm RF} \ll N_t$ RF chains.

Hardware connectivity is a defining dimension:

• Fully-Connected (FC): Each of the $N_{\rm RF}$ RF chains controls all $N_t$ antennas, typically via $$\mathbf{W} = \mathbf{W}_{\rm RF}\mathbf{W}_{\rm BB}$$0 phase shifters (Sohrabi et al., 2017, Song et al., 2019).
• Partially-Connected (PC)/Subarray: Each RF chain connects to a disjoint subarray of $$\mathbf{W} = \mathbf{W}_{\rm RF}\mathbf{W}_{\rm BB}$$1 antennas, forming block-diagonal $$\mathbf{W} = \mathbf{W}_{\rm RF}\mathbf{W}_{\rm BB}$$2 (Song et al., 2019, Almagboul et al., 2018).
• Antenna/Switching: RF chains are dynamically assigned to antenna subsets via switches (Ioushua et al., 2017).

Receive architecture analogously partitions combining into $$\mathbf{W} = \mathbf{W}_{\rm RF}\mathbf{W}_{\rm BB}$$3 (analog, phase-shifter or switch network) and $$\mathbf{W} = \mathbf{W}_{\rm RF}\mathbf{W}_{\rm BB}$$4 (digital).

RF Signal Chain Placement: The location of analog variable gain amplifiers (VGAs)—after the DAC (conventional), per-antenna, or per-branch (phase-shifter)—significantly impacts spectral efficiency and DoF, with per-branch placement optimally capturing the full digital beamforming capability when the number of RF chains matches the rank of the channel (Karacora et al., 2019).

2. Beamforming Optimization: Algorithms and Problem Statements

Objective functions are drawn from communication capacity (spectral efficiency), MSE (including WSMSE, MSE gap to digital), or, in ISAC/radar, Cramér–Rao bounds:

• Communication:

$$\mathbf{W} = \mathbf{W}_{\rm RF}\mathbf{W}_{\rm BB}$$5

subject to power and modulus constraints (Sohrabi et al., 2016, Sohrabi et al., 2017).

• Joint Sensing-Communication (ISAC): minimize posterior Cramér–Rao bound (PCRB) for angle estimation under sum rate constraints, e.g., (Wang et al., 2024):

$$\mathbf{W} = \mathbf{W}_{\rm RF}\mathbf{W}_{\rm BB}$$6

• Radar/Beampattern: minimize beampattern error, MSE to desired angular profile (Xu et al., 2021).
• Robustness: Interference suppression/robust Capon beamforming under DOA error via null-space projection (NSP) and diagonal loading (Sun et al., 2018).

Methodologies:

• Alternating optimization (AO): Outer loop alternates digital and analog updates, as in FPP-SCA for nonconvex QCQP (Wang et al., 2024), or Alt-MaG/PE-AltMin for MSE minimization (Ioushua et al., 2017, Li et al., 3 Jul 2025).
• Compressed sensing/OMP: Hybrid precoder approximates digital precoder via a sparse dictionary (overcomplete array steering vectors) plus LS optimization for digital weights (Bogale et al., 2014, Sohrabi et al., 2017).
• Coordinate descent: Phase-only $$\mathbf{W} = \mathbf{W}_{\rm RF}\mathbf{W}_{\rm BB}$$7 elements are updated element-wise to maximize rate or MSE objectives with per-element constant modulus (Sohrabi et al., 2016, Sohrabi et al., 2017).
• Softmax neural selection: For radar, sparse RF/antenna selection is relaxed via softmax networks, trained to match beampatterns (Xu et al., 2021).
• Metaheuristics: Evolutionary algorithms (e.g., improved bat algorithm) optimize phase-only digital weights under hybrid constraints (Almagboul et al., 2018).

Fully digital equivalence: When $$\mathbf{W} = \mathbf{W}_{\rm RF}\mathbf{W}_{\rm BB}$$8, any digital precoder can be exactly realized by an FC hybrid architecture (Sohrabi et al., 2016); in highly sparse channels or highly correlated eigenstructures, $$\mathbf{W} = \mathbf{W}_{\rm RF}\mathbf{W}_{\rm BB}$$9 suffices (Sohrabi et al., 2017, Bogale et al., 2014). For ISAC, hybrid arrays with only $\mathbf{F}_{\rm RF}$0 are information-lossless relative to fully digital, as optimal covariance is rank-one (Wang et al., 2024).

3. Performance Trade-offs, Hardware Constraints, and Practical Guidelines

RF chain count: Hybrid performance approaches fully digital once $\mathbf{F}_{\rm RF}$1 exceeds the number of data streams/users; spectral efficiency gap diminishes with $\mathbf{F}_{\rm RF}$2 ($\mathbf{F}_{\rm RF}$31 bps/Hz for $\mathbf{F}_{\rm RF}$4 in i.i.d. channels) (Bogale et al., 2014, Sohrabi et al., 2017). Further RF chain reductions increase beampattern error (radar), estimation error (channel estimation), or force rank reduction (communication).

Analog network:

• FC maximizes DoF but is costly ($\mathbf{F}_{\rm RF}$5 hardware).
• PC/switching offers $\mathbf{F}_{\rm RF}$6 complexity/power savings and only small rate loss when beams are properly aligned, and allows PAs to approach higher efficiency (Song et al., 2019).

Quantization and Cognition:

• Finite-resolution phase shifters (e.g., 4–6 bits) impose only 10–20% beampattern/SE degradation; quantization-aware design can mitigate most of the loss (Xu et al., 2021, Sohrabi et al., 2017).
• Low-res ADCs require modified combining: two-stage analog combining (aggregation + DFT spreading) can achieve optimal MI scaling $\mathbf{F}_{\rm RF}$7 with $\mathbf{F}_{\rm RF}$8-bit ADCs (Choi et al., 2018).

Hardware nonidealities:

• Analog amplifiers placed per phase-shifter branch allow hybrid to match fully-digital benchmark; per-antenna deployment achieves most of the gain with drastically reduced amplifier count (Karacora et al., 2019).
• DAC/ADC and phase-shifter power dissipation scales linearly with $\mathbf{F}_{\rm RF}$9, dominating total consumption in large arrays.

OFDM and Broadband:

• In wideband systems, analog beamformers are constant over frequency; digital precoders compensate frequency selectivity in each subband (Sohrabi et al., 2017).
• Frequency-flat analog stages rely on channel sparsity; for practical arrays ($N_t \times N_{\rm RF}$0), FC hybrid with $N_t \times N_{\rm RF}$1 is within $N_t \times N_{\rm RF}$2 dB SE of fully digital for $N_t \times N_{\rm RF}$3 subcarriers.

4. Channel Estimation and Training

MMSE channel estimation with hybrid architectures is fundamentally limited by the reduced RF chain count:

• The optimal training beamformer aligns analog/digital stages to the dominant channel correlation eigenvectors (Bogale et al., 2015, Mirzaei et al., 2021).
• The optimal allocation of training energy follows water-filling over these dominant directions; for highly correlated channels, a small $N_t \times N_{\rm RF}$4 (pilot duration) suffices (Mirzaei et al., 2021).
• For $N_t \times N_{\rm RF}$5, $N_t \times N_{\rm RF}$6, hybrid MMSE achieves 1–2 dB gap to fully digital estimation, with optimal $N_t \times N_{\rm RF}$7 between $N_t \times N_{\rm RF}$8 and $N_t \times N_{\rm RF}$9 depending on correlation (Bogale et al., 2015).

Beam alignment/training in MU-MIMO exploits hierarchical analog codebooks (flat-top beams, multi-level DFT) (Alexandropoulos et al., 2021) and non-negative least squares (NNLS) recovery in sparse beamspace (Song et al., 2019); these efficiently balance acquisition latency and SNR.

5. Advanced Topics: Sensing, ISAC, and Near-Field Extensions

Integrated Sensing and Communication:

• Hybrid ISAC architectures share the hardware for dual objectives: communication rate maximization and target parameter estimation (delay, angle, Doppler).
• Posterior Cramér–Rao bounds are adopted as performance metric for sensing; hybrid architectures can exactly meet the optimized digital performance for target estimation provided $\mathbf{F}_{\rm BB}$0, and AO/FPP-SCA methods provide efficient joint solutions under rate constraints (Wang et al., 2024).
• Low-resolution DACs in ISAC setups can be efficiently modeled (AQNM, Bussgang) and digitally compensated; $\mathbf{F}_{\rm BB}$1 bits recovers almost all the SE, while $\mathbf{F}_{\rm BB}$2 maintains robust dual-functionality with a modest 2–3 dB loss (Elbir et al., 2024).

Near-Field Beamforming:

• At THz or large aperture, spherical wavefronts and focal/diffractive beams are exploited; hybrid designs, e.g., Airy beamforming, can overcome blockage and spatial focusing limits by hierarchical selection of analog codewords with digital ZF outer suppression (Zhang et al., 28 May 2026).

Radar/Delay Alignment:

• Hybrid Delay Alignment Modulation (DAM) extends to hybrid arrays by approximating path-based digital beamforming (for both integer and fractional delays) via OMP/greedy codebook approaches for both fully and partially connected architectures with near digital-level performance (Zhang et al., 2024).

Machine Learning for Hardware Reduction:

• Softmax neural selection controls sparsity pattern in hardware: selection of RF chains and antennas is learned to synthesize a desired radar beampattern, trading fidelity for hardware cost (Xu et al., 2021).

6. Summary Table: Representative Hybrid Beamforming Structures

Structure	Key Features	Performance/Hardware Tradeoff
Fully-connected	Each RF chain → all antennas (phase shifters)	Best SE, highest hardware cost and power (Song et al., 2019, Sohrabi et al., 2017)
Partially-connected (subarrays)	RF chain → disjoint subarray	Similar SE (after alignment), 30–40% less hardware, better PA efficiency
Switch-based	Antenna selection/network	Lowest complexity, but reduced DoF and pattern flexibility
Two-stage analog combining	Channel aggregation + DFT spreading	Achieves log-scaling MI-optimality under ADC quantization (Choi et al., 2018)

7. Implementation and Practical Recommendations

• RF chain budget: $\mathbf{F}_{\rm BB}$3 achieves $\mathbf{F}_{\rm BB}$4 of the fully digital sum-rate in most MU-MIMO or radar applications.
• Codebook sizing: A few hundred steering angles suffice for $\mathbf{F}_{\rm BB}$5 ULAs (Bogale et al., 2014).
• Algorithm selection: OMP/PE-AltMin for moderate scales, Alt-MaG or AREE for massive MIMO, FPP-SCA/AO for complex ISAC/radar-objectives, softmax/learning for hardware-constrained radar.
• Parasitic losses: FC architectures incur div/com losses as $\mathbf{F}_{\rm BB}$6 and $\mathbf{F}_{\rm BB}$7; PC and OSPS avoid combiners, operating PAs closer to saturation.
• Quantization: Use 4-6 bit phase shifters/DACs for minimal loss; codebook and design must be quantization-aware.
• Update periodicity: Analog phases are updated less frequently than digital weights in practice, reflecting timescales of channel or scenario changes (Almagboul et al., 2018).
• Sensing applications: When $\mathbf{F}_{\rm BB}$8, convex per-element AO or learning-based selection can approach optimal performance in beam synthesis (Wang et al., 2024, Xu et al., 2021).

In conclusion, hybrid analog/digital beamforming achieves a favorable compromise between array gain, spatial multiplexing, and hardware scalability in large MIMO, mmWave/THz communications, massive MIMO radar, and joint radar-communications systems. With proper architectural choices, algorithmic partitioning, and quantization-aware design, practical hybrid precoding can closely approximate the performance of fully-digital schemes at a fraction of the hardware cost and complexity (Sohrabi et al., 2016, Bogale et al., 2014, Sohrabi et al., 2017, Wang et al., 2024, Ioushua et al., 2017, Elbir et al., 2024, Song et al., 2019, Zhang et al., 2024, Xu et al., 2021, Choi et al., 2018).