Hybrid Analog-Digital Beamforming

• Hybrid analog-digital beamforming is a method that splits signal processing between analog RF and digital baseband domains, achieving nearly digital performance with fewer RF chains.
• Architectural variants—fully connected, partially connected, and two-stage—offer trade-offs between flexibility, hardware cost, and power consumption in massive MIMO and ISAC systems.
• Advanced techniques such as alternating optimization, compressed sensing, and quantized design enable near-optimal beamforming performance despite strict hardware constraints.

Hybrid analog-digital beamforming refers to spatial signal processing architectures that combine analog radio-frequency (RF) and digital baseband (BB) signal processing to realize high-dimensional beamforming with a reduced count of costly and power-hungry RF chains. This paradigm underpins much of modern millimeter-wave (mmWave), sub-6 GHz massive MIMO, and integrated sensing and communications (ISAC) system design. By strategically decomposing the overall precoder or combiner into analog and digital components, hybrid beamforming offers a hardware-efficient yet nearly capacity-achieving solution for large-scale antenna systems, particularly where fully digital architectures are impractical.

1. Hybrid Beamforming Architectures

Hybrid beamforming architectures are typically categorized according to the connectivity between RF chains and antennas:

• Fully Connected: Each RF chain is connected to all antennas via phase shifters, yielding the greatest flexibility and highest possible approximation accuracy of digital beamformers at the cost of more phase shifters and increased power consumption. The analog beamforming matrix (F_RF) in this architecture has ∣F_RF(i, j)∣ = 1, for i, j, reflecting phase-shifter constraints (Bogale et al., 2014, Sohrabi et al., 2016).
• Partially Connected (Subarray): Each RF chain is connected to a subset of antennas. The analog matrix in this case is block-diagonal; while this reduces hardware and energy cost, it generally results in some loss of beamforming gain and performance relative to the fully connected case (Majidzadeh et al., 2020, Song et al., 2019).
• Two-Stage Structures: Some systems split analog processing into cascaded stages for specific goals (e.g., channel gain aggregation and gain spreading in low-resolution ADC scenarios) (Choi et al., 2018). This configuration manages both spatial energy collection and reduces per-ADC quantization error, allowing for optimal mutual information scaling even with coarse quantization.

The overall beamforming matrix is classically expressed as

$F = F_{\mathrm{RF}} F_{\mathrm{BB}},$

where $$F_{\mathrm{RF}} \in \mathbb{C}^{N_t \times N_{\mathrm{RF}}}$$ is the analog beamformer (implemented as phase-shifter network) and $$F_{\mathrm{BB}} \in \mathbb{C}^{N_{\mathrm{RF}} \times N_s}$$ is the digital beamformer.

2. Optimization Objectives and Mathematical Formulation

Design methodologies for hybrid beamforming aim to bridge the performance gap between the hardware-limited hybrid solution and the unconstrained fully digital solution. Typical problem formulations include:

• Sum-rate Maximization: For downlink multiuser MIMO, maximize $$\sum_k \log_2 \left|I + \frac{1}{\sigma^2} H_k F F^H H_k^H \right|$$, subject to hardware constraints and total transmit power (Bogale et al., 2014, Xue et al., 2015).
• Weighted Sum-MSE Minimization: Minimize $\sum_k \omega_k \mathbb{E}|d_k - \hat{y}_k|^2$, where $\omega_k$ are stream priorities, by jointly designing $F_{\mathrm{RF}}$, $F_{\mathrm{BB}}$ (and corresponding combiners at the receiver) (Bogale et al., 2014).
• Posterior/Deterministic Cramér–Rao Bound Minimization: Particularly in ISAC and radar, minimize the (posterior) Cramér–Rao bound (PCRB) of target parameters (e.g., angle) with or without communication rate constraints (Wang et al., 2 Jun 2024).

Solutions proceed by decomposing the unconstrained digital beamforming solution into a feasible hybrid realization:

• Alternating Optimization: Alternate updates of the analog and digital beamformer matrices, often starting from an initial digital solution obtained by block-diagonalization, zero-forcing, or SVD (Ioushua et al., 2017, Sohrabi et al., 2016).
• Compressed Sensing and Dictionary Approaches: For sparse channels (e.g., mmWave), represent analog precoders as sparse combinations of array steering vectors chosen from large codebooks, using methods like orthogonal matching pursuit (OMP) (Bogale et al., 2014, Xue et al., 2015, Zou et al., 2017).
• Tensor or Matrix Factorization: For high-dimensional or frequency-selective MIMO-OFDM systems, shared analog and subcarrier-specific digital beamformers are jointly optimized by tensor decomposition (e.g., constrained Tucker2) (Gherekhloo et al., 2021).
• Convex Relaxation, FPP-SCA: For certain nonconvex formulations involving unit-modulus constraints, feasible point pursuit–successive convex approximation (FPP-SCA) is used within an alternating optimization (AO) framework (Wang et al., 2 Jun 2024).

A key theoretical result is that, for $N_{\mathrm{RF}} \geq 2N_s$, any fully digital (unconstrained) precoder can be exactly realized as $F = F_{\mathrm{RF}} F_{\mathrm{BB}}$ for appropriate phase-shifter settings, regardless of array size (Sohrabi et al., 2016).

3. Performance Limits and Trade-Offs

The gap between hybrid and digital beamforming is fundamentally shaped by RF chain count, ADC/DAC resolution, and architecture:

• For a fixed number of data streams $N_s$, increasing $N_{\mathrm{RF}}$ reduces the performance loss relative to fully digital designs. Asymptotically, with $N_{\mathrm{RF}} = 2N_s$, the fully digital sum-rate and MSE can be matched (Sohrabi et al., 2016, Ioushua et al., 2017).
• If $N_s$ is increased with fixed $N_{\mathrm{RF}}$, the hybrid structure becomes increasingly under-determined; the performance gap widens due to insufficient spatial degrees of freedom (Bogale et al., 2014).
• Hardware constraints such as finite phase-shifter resolution or subarray connectivity can introduce a further gap, but incorporating these limitations in the alternating optimization can mitigate performance losses (e.g., direct quantization during update steps) (Sohrabi et al., 2016).
• For partially connected structures, there is a trade-off: hardware complexity and power decrease, but so does beamforming flexibility and sum-rate, especially at high SNR or in high-rank environments (Majidzadeh et al., 2020, Song et al., 2019).
• Low-resolution ADC/DACs require special treatment: optimal two-stage analog combiners that aggregate and then “spread” channel gains achieve the desired $N_u \log_2 N_{\mathrm{RF}}$ scaling law for mutual information; naive single-stage combiners saturate earlier due to severe quantization distortion (Choi et al., 2018, Elbir et al., 5 Nov 2024).

Numerical results across multiple works confirm that, in scenarios of practical interest (moderate $N_s$, sufficient $N_{\mathrm{RF}}$, moderate phase resolution), hybrid approaches with well-chosen heuristics or alternating optimization nearly achieve the sum-rate and spectral efficiency of fully digital beamforming (Bogale et al., 2014, Sohrabi et al., 2016, Sohrabi et al., 2017, Ioushua et al., 2017).

4. Algorithmic Techniques and Compressed Sensing

Several algorithmic innovations are pivotal to modern hybrid beamforming:

Technique	Key Application	Role in Hybrid Beamforming
Compressed Sensing (CS)	Sparse mmWave channel estimation	Selects dominant beams via OMP/Basis Pursuit
Alternating Minimization	Architecture-agnostic optimization	Joint analog/digital update to minimize WSMSE or loss
Greedy Search/Matching	Analog beamformer selection	Efficiently pick codebook vectors in large arrays
Machine Learning Selection	Antenna/RF chain subset selection	Softmax-based “learn to select” (L2S) for beampatterns
FPP-SCA	Nonconvex constraint handling	Successive convexification for unit-modulus constraints

Compressed sensing (CS) is essential for high-dimensional, sparse settings (e.g., mmWave), where the analog beamformer is selected to match array response vectors corresponding to a few dominant paths. The sparse recovery problem is cast as

$\min_x \|y - D x\|_2^2 + \lambda \|x\|_0,$

with $D$ being the dictionary of candidate beams; OMP or $\ell_1$-relaxation is used to enable tractable solution (Bogale et al., 2014, Xue et al., 2015, Zou et al., 2017). For codebook-based analog beamforming, downselecting from a DFT dictionary or other structured codebooks allows for low-overhead calibration and robust performance (Zou et al., 2017, Zhang et al., 20 Sep 2024).

For systems with reduced RF chains and/or finite-bit phase shifters, hybrid updates directly incorporate quantization in the coordinate descent (e.g., by quantizing updated phases at each iteration) (Sohrabi et al., 2016). In massive MIMO radar, machine learning (“learn to select” softmax networks) can be used to optimize the architecture and hardware selection (e.g., antenna/RF chain selection) jointly with the analog and digital beamformers (Xu et al., 2021).

5. Integration with Communication and Sensing

Hybrid beamforming plays a central role in joint communication and radar (sensing) systems such as ISAC:

• In ISAC, the analog and digital beamformers are jointly optimized to balance communication rate constraints (often, the achievable rate depends on the transmit covariance $R_x$) and sensing precision (typically quantified by posterior Cramér–Rao bound (PCRB) for target parameter estimation) (Wang et al., 2 Jun 2024, Elbir et al., 5 Nov 2024).
• It is analytically demonstrated that, for sensing-only scenarios and $N_{\mathrm{RF}} \geq 2$, the optimal digital beampattern can be exactly replicated by a hybrid implementation (Wang et al., 2 Jun 2024).
• When hardware constraints are tighter (single RF chain, quantized phase, low-resolution DAC), convex relaxation and AO techniques (e.g., FPP-SCA, successive approximation) enable efficient suboptimal designs (Wang et al., 2 Jun 2024, Elbir et al., 5 Nov 2024).

For multi-functional operation, hybrid beamformers can partition their resources (e.g., columns of the analog matrix) between sensing and communication tasks, with trade-offs controlled by design parameters (Elbir et al., 5 Nov 2024). The quantization distortion due to low-resolution DACs/ADCs is accurately incorporated using Bussgang or additive-quantization-noise models, and the digital combiner is designed accordingly for both spectral efficiency and beampattern quality.

6. Hardware Constraints, Complexity, and Implementation Considerations

Realizing hybrid beamforming architectures imposes a set of intricate trade-offs and challenges:

• Phase Shifter Resolution: Limited phase resolution (e.g., 1–2 bit phase shifters) impacts performance; algorithms that optimize for quantized updates yield significant robustness (Sohrabi et al., 2016, Sohrabi et al., 2017).
• Subarray vs. Fully Connected: Partially connected schemes simplify the RF network and save power but restrict the achievable precoding space; full connectivity is optimal when feasible (Majidzadeh et al., 2020, Song et al., 2019).
• ADC/DAC Quantization: For energy efficiency, low-resolution converters are required; optimal two-stage combining or explicit distortion-aware digital design can nearly recover much of the performance of high-resolution systems (Choi et al., 2018, Elbir et al., 5 Nov 2024).
• Codebook Design: The codebook size for analog beams should scale with the number of antennas; for minimal “instantaneous 3 dB” loss per path, codebook cardinality $\gtrsim 1.18 N_{\mathrm{ant}}$, and in practice, twice the array size is often recommended (Zou et al., 2017).
• Power and Cost: Each RF chain represents a significant power and cost penalty; well-designed hybrid systems approach the digital limit with only a fraction of the RF hardware, especially when the number of streams is moderate and the channel is sparse (Bogale et al., 2014, Xue et al., 2015, Sohrabi et al., 2016).

Algorithmic frameworks are often modularized to operate under varying architectural and hardware constraints (e.g., unit-modulus, fixed/flexible subarrays, switches), facilitating adaptation across hardware implementations (Ioushua et al., 2017).

7. Applications, Extensions, and Future Directions

Hybrid analog-digital beamforming is foundational in the following sectors:

• mmWave Massive MIMO Cellular: Enables spatial multiplexing and cell edge coverage in 5G/6G with reasonable hardware complexity (Bogale et al., 2014, Sohrabi et al., 2016, Sohrabi et al., 2017).
• Millimeter Wave/Terahertz Communications: Supports ISI mitigation via delay alignment modulation (DAM) and hybrid beamforming for both integer and fractional path delay channels; enables cost-effective, scalable large-array transceivers (Zhang et al., 20 Sep 2024).
• Integrated Sensing and Communication (ISAC): Facilitates dual-functional radar/communications with prior-aware beamforming and explicit PCRB–rate trade-off optimization (Wang et al., 2 Jun 2024, Elbir et al., 5 Nov 2024).
• Automotive Radar/Joint Radar-Communication: Accommodates wide-beam search and narrow-beam tracking modes with hybrid hardware, using waveform–beamformer co-design (e.g., OTFS radar with hybrid beamforming) (Gaudio et al., 2020).
• Machine Learning in Beam Selection: Data-driven techniques for architecture/hardware selection and beampattern synthesis under resource constraints (Xu et al., 2021).

Ongoing and future directions include:

• Robust designs for highly dynamic or misaligned environments (e.g., flat-top multi-level codebooks for mmWave mobility) (Alexandropoulos et al., 2021).
• Advanced convex–nonconvex optimization and stochastic machine learning techniques for large-scale, low-resolution, and reconfigurable hardware (Xu et al., 2021, Wang et al., 2 Jun 2024, Elbir et al., 5 Nov 2024).
• Integration of hybrid beamforming with intelligent reflective surfaces (IRS), direct localization, and advanced waveform design for ISAC.
• Efficient channel estimation methods for hybrid architectures, leveraging eigen-domain sparsity and prior information (Mirzaei et al., 2021).

In conclusion, hybrid analog-digital beamforming delivers an efficient, scalable, and nearly optimal solution to the high-dimensional spatial processing challenges posed by next-generation wireless, radar, and integrated sensing-communication systems. By judiciously partitioning spatial processing across analog and digital domains, and leveraging advanced optimization and compressed sensing techniques, these architectures meet stringent performance goals under practical hardware constraints.

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Key References:

• "Beamforming for Multiuser Massive MIMO Systems: Digital versus Hybrid Analog-Digital" (Bogale et al., 2014)
• "Hybrid Digital and Analog Beamforming Design for Large-Scale Antenna Arrays" (Sohrabi et al., 2016)
• "Hybrid Analog-Digital Beamforming for Multiuser MIMO Millimeter Wave Relay Systems" (Xue et al., 2015)
• "Hybrid Analog and Digital Beamforming for mmWave OFDM Large-Scale Antenna Arrays" (Sohrabi et al., 2017)
• "Delay Alignment Modulation with Hybrid Analog/Digital Beamforming for Millimeter Wave and Terahertz Communications" (Zhang et al., 20 Sep 2024)
• "Hybrid Beamforming for Integrated Sensing and Communications With Low Resolution DACs" (Elbir et al., 5 Nov 2024)
• "Hybrid Beamforming Design for Integrated Sensing and Communication Exploiting Prior Information" (Wang et al., 2 Jun 2024)