Hybrid Digital and Analog Beamformers

• Hybrid digital and analog beamformers are transceiver architectures that combine phase shifter-based RF processing with flexible digital precoding to balance cost and performance.
• They employ alternating minimization and sparse codebook techniques to optimize the phase-only analog matrices under strict constant-modulus and hardware constraints.
• System evaluations reveal trade-offs in RF chain count, interference suppression, and sum rate performance, guiding practical deployments in large-scale MIMO and mmWave scenarios.

Hybrid digital and analog beamformers refer to transceiver architectures that implement the spatial processing necessary for multi-input multi-output (MIMO) communications across two hardware domains: a digital (baseband) domain and an analog (radio frequency, RF) domain. The analog section, typically realized via phase shifters, enables a significant reduction in the number of radio frequency (RF) chains relative to the antenna count—mitigating the power and cost burdens of fully digital beamforming—while the digital section retains the capability to multiplex data streams and manage interference. This architectural split, common in millimeter-wave (mmWave) and massive MIMO systems, is designed to (a) approach the performance of fully digital architectures in spectral efficiency and sum rate, (b) respect stringent hardware constraints such as constant-modulus responses in the analog domain, and (c) maximize cost and energy efficiency.

1. Architectural Principles and Mathematical Formulation

Hybrid beamforming architectures decompose the overall precoding or combining operation as the product of a high-dimensional analog (RF) matrix and a low-dimensional digital (baseband) processor. At the transmitter, for example, the outgoing symbol vector $s$ is processed as $F_A F_D s$, where $F_A$ is the analog beamforming matrix (constant-modulus, implemented via phase shifters or switches) and $F_D$ is the digital beamforming matrix:

$y = H F_A F_D s$

Subject to:

• $|[F_A]_{i,j}| = 1 \;\; \forall i, j$ (constant-modulus constraint)
• The number of columns of $F_A$ equals the number of available RF chains ($N_{\mathrm{RF}} \ll N_{\mathrm{ant}}$)
• $F_D$ is unconstrained, allowing amplitude and phase adaptation per-stream

System-level optimization for hybrid beamformer design often expresses the performance objective as a minimization of the weighted sum mean square error (WSMSE) between transmit symbols and received estimates:

$J = \| W^{1/2} (D - R) \|_F^2$

where $D$ is the symbol matrix, $R$ is the received symbol matrix, and $W$ is a diagonal matrix of per-stream weights (Bogale et al., 2014). Alternatively, sum-rate maximization under transmit power and hardware constraints is frequently employed (Xue et al., 2015).

The hardware-imposed restrictions on $F_A$ make the optimization highly nonconvex. Consequently, alternating minimization or decoupled, iterative approaches are widely adopted.

2. Digital and Analog Beamformer Design Methods

2.1 Digital Stage: Interference Suppression and Power Allocation

For multiuser MIMO downlink, once analog beamforming has coarsely focused the spatial response, the digital stage provides fine-grained stream control. Techniques such as block diagonalization (BD) are employed whereby each user's digital beamformer $B_k$ is designed to lie in the null space of the aggregate interference channel of other users:

$H_j (F_A B_k) = 0 \quad \forall j \neq k$

where $H_j$ represents the channel matrix for user $j$ (Bogale et al., 2014). Power allocation across streams, e.g., via waterfilling, is then used to maximize the sum rate over the effective multiuser channel.

2.2 Analog Stage: Sparse Codebooks and Phase-Only Optimization

The analog stage, realized by an $N_{\mathrm{ant}} \times N_{\mathrm{RF}}$ matrix of phase shifters (constrained to $|[F_A]_{i,j}| = 1$), is responsible for spatial focusing—especially in the direction(s) of dominant channel paths. With the mmWave channel's high spatial sparsity, compressive sensing algorithms and matching pursuit (e.g., OMP) are often applied to extract principal directions (AoD/AoA), forming the columns of $F_A$ (Xue et al., 2015). For example, with $H \approx D X$, $D$ a codebook of steering vectors, analog beams are aligned with high-magnitude entries in the sparse representation $X$.

Analytic methods for analog-only optimization (in single-user cases or when codebooks are very large) allow the analog beamforming matrix to be constructed as the array response matrix itself, $F_A = A_{\mathrm{BS}}$ (Zou et al., 2017), or as Kronecker products in arrays with specific structure (Zhu et al., 2017). Update rules for analog coefficients, under constant-modulus constraints, often formalize the unconstrained solution then project onto the phase manifold, e.g., $F_A(i,j) = \eta_{ij} / |\eta_{ij}|$ if $\eta_{ij}\neq0$ (Sohrabi et al., 2016).

In practical hardware, phase shifter resolution may be as low as 1–2 bits, necessitating direct quantization in the design process (Sohrabi et al., 2016), or codebook lookups with nearest-angle matching (Zou et al., 2017).

3. Algorithmic Strategies and Alternating Minimization

Ideally, the hybrid beamforming problem is solved as a joint optimization over $F_A$ and $F_D$. In practice, alternating last-step optimization is widely adopted:

• Fix $F_A$, optimize $F_D$: With known analog precoder, the digital stage is solved (frequently in closed form) using standard MIMO digital beamforming methods (ZF, MMSE, BD, or WMMSE) on the reduced effective channel $H_{\mathrm{eff}} = H F_A$ (Sohrabi et al., 2016, Sohrabi et al., 2017).
• Fix $F_D$, optimize $F_A$: With $F_D$ set, $F_A$ is updated via a phase-only algorithm where each element is set to match, as closely as possible, the phase of the corresponding entry in the unconstrained optimal solution. For quantized phase shifters, the update is quantized to the nearest available phase (Sohrabi et al., 2016).

Frameworks such as alternating minimization of approximation gap (Alt-MaG) generalize this process by parametrizing the equivalence class of fully digital solutions (exploiting the fact that the optimal digital beamformer is unique only up to a unitary transformation), thus further reducing the minimum achievable mean-squared error of hybrid approximation (Ioushua et al., 2017).

4. Performance Tradeoffs and System-Level Metrics

Hybrid digital and analog beamformers are benchmarked against fully digital implementations in terms of total system sum rate, interference suppression, and robustness to channel estimation error. Key findings include:

Parameter	Hybrid vs. Digital Performance	Impact
Number of RF chains ($N_{\mathrm{RF}}$)	Increasing $N_{\mathrm{RF}}$ narrows the sum-rate gap (Bogale et al., 2014, Xue et al., 2015, Sohrabi et al., 2016)	Cost-performance tradeoff
Number of multiplexed data streams	More streams increase the rate gap; hybrid is less able to perfectly suppress interference	Throughput vs. interference mitigation
ADC/DAC resolution	Fewer/low-resolution converters further degrade hybrid performance	Hardware cost vs. quantization noise
Phase shifter quantization	Low-resolution phase shifters (1–2 bits) degrade performance, especially if not accounted for in the algorithm (Sohrabi et al., 2016)	Hardware feasibility
Channel estimation accuracy	Performance is sensitive to AoA/AoD estimation error; compressive sensing-based estimation, Kronecker methods, and robust phase alignments mitigate some effects (Xue et al., 2015Zhu et al., 2017)	System robustness

Simulation studies across a variety of scenarios confirm that as $N_{\mathrm{RF}}$ increases and/or fewer streams are simultaneously multiplexed, hybrid designs can closely approach full digital performance (Bogale et al., 2014). Key limitations arise when the number of parallel data streams is increased, or hardware constraints (low $N_{\mathrm{RF}}$ or few-bit phase shifters) are severe.

5. Extensions, Robustness, and Special Cases

Robustification against direction-of-arrival (DOA) estimation errors and array imperfections is a current area of focus. Techniques include null space projection in the analog domain—where the analog combiner is constructed to lie in the null spaces of the estimated interference directions—combined with diagonal loading in the digital domain for enhanced interference suppression under covariance mismatch (Sun et al., 2018). For highly sparse channels (e.g., mmWave), hybrid designs using Kronecker decomposition or hierarchical clustering to generate analog sectors further aid in reducing both computational and hardware complexity by dividing users into spatial clusters (Zhu et al., 2017, Bychkov et al., 16 Feb 2024).

Heuristic and metaheuristic approaches, such as those based on improved bat algorithms, have been utilized for partially-connected architectures, optimizing both analog phase alignment and digital beamformer phases in the presence of interference and practical uncertainties (Almagboul et al., 2018). These stochastic optimization algorithms offer robust performance in uncertain or dynamic environments, trading analytical guarantee for empirical convergence speed and computational simplicity.

6. Practical Implementation Considerations

From a system design perspective, hybrid architectures are favored where:

• The cost or power burden of a full complement of RF chains (with DAC/ADC, mixers, etc.) is excessive for large antenna arrays (massive MIMO, mmWave).
• Performance loss (in terms of sum rate, interference suppression, or robustness) remains subdominant for moderate data stream counts and where the absolute number of RF chains $N_{\mathrm{RF}}$ is at least as large as the number of spatial data streams per user, ideally twice that number (Sohrabi et al., 2016).
• Hardware constraints (such as phase shifter resolution) are carefully accounted for in algorithm design, as enacting phase quantization after digital-dominated design yields suboptimal performance (Sohrabi et al., 2016, Xue et al., 2015).

Industry-standard simulation benchmarks show that with only a fraction of the RF circuitry and power consumption, hybrid digital and analog beamforming methods enable practical near-capacity transmission in massive MIMO and millimeter-wave systems.

7. Ongoing Directions and Future Challenges

Ongoing work includes extending hybrid beamforming to frequency-selective/broadband channels with OFDM, where a common analog network must serve all subcarriers, while the digital stage is adapted per-subcarrier (Sohrabi et al., 2017); improving robustness and performance via hierarchical, Kronecker-structured, or machine learning-based beamformer designs; and integrating hybrid architectures into coordinated multi-point transmission and sensing scenarios.

Unresolved challenges remain around:

• The optimal design of hybrid beamformers in rapidly varying, frequency-selective multipath channels
• Dynamic reconfiguration under mobility or user population changes
• Theoretical bounds on the trade-off space in extreme hardware-constrained regimes (e.g., very low $N_{\mathrm{RF}}$, 1-bit quantization)
• Extending channel estimation efficacy in the face of reduced observability caused by analog pre-processing

Hybrid beamforming offers a rigorous, cost-effective methodology for approaching the performance of fully digital MIMO systems, with careful algorithmic and architectural co-design providing the foundation for future large-scale wireless deployment (Bogale et al., 2014, Xue et al., 2015, Sohrabi et al., 2016, Zhu et al., 2017, Sohrabi et al., 2017, Sun et al., 2018).