---
title: Hybrid Analog/Digital Beamforming
url: https://www.emergentmind.com/topics/hybrid-analog-digital-beamforming-0b0d0ef9-af39-4536-aa10-8eb0dcd5f375
type: topic
---

# Hybrid Analog/Digital Beamforming

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 $N_t \times N_{\rm RF}$ phase shifters [1711.08408, 1904.10276].
- **Partially-Connected (PC)/Subarray**: Each RF chain connects to a disjoint subarray of $N_t/N_{\rm RF}$ antennas, forming block-diagonal $\mathbf{F}_{\rm RF}$ [1904.10276, 1811.01176].
- **Antenna/Switching**: RF chains are dynamically assigned to antenna subsets via switches [1712.03485].

**Receive architecture** analogously partitions combining into $\mathbf{W}_{\rm RF}$ (analog, phase-shifter or switch network) and $\mathbf{W}_{\rm BB}$ (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 [1902.09511].

## 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**: 
  \[
    \max_{\mathbf{F}_{\rm RF},\,\mathbf{F}_{\rm BB}}\; \log_2\det\left[\mathbf{I} + \frac1{\sigma^2}\mathbf{H}\mathbf{F}_{\rm RF}\mathbf{F}_{\rm BB}\mathbf{F}_{\rm BB}^H\mathbf{F}_{\rm RF}^H\mathbf{H}^H\right]
  \]
  subject to power and modulus constraints [1601.06814, 1711.08408].

- **Joint Sensing-Communication** (ISAC): minimize posterior Cramér–Rao bound (PCRB) for angle estimation under sum rate constraints, e.g., [2406.00689]:
  \[
    \min_{\mathbf{F}_{\rm RF},\mathbf{F}_{\rm BB}}\, \mathrm{PCRB}_\theta(\mathbf{F}_{\rm RF},\mathbf{F}_{\rm BB})
    \;\;\;\text{s.t.}\;\; R(\mathbf{F}_{\rm RF},\mathbf{F}_{\rm BB}) \ge R_0
  \]
- **Radar/Beampattern**: minimize beampattern error, MSE to desired angular profile [2101.06837].
- **Robustness**: Interference suppression/robust Capon beamforming under DOA error via null-space projection (NSP) and diagonal loading [1801.06776].

**Methodologies**:
- **Alternating optimization (AO)**: Outer loop alternates digital and analog updates, as in FPP-SCA for nonconvex QCQP [2406.00689], or Alt-MaG/PE-AltMin for MSE minimization [1712.03485, 2507.02802].
- **Compressed sensing/OMP**: Hybrid precoder approximates digital precoder via a sparse dictionary (overcomplete array steering vectors) plus LS optimization for digital weights [1407.0446, 1711.08408].
- **Coordinate descent**: Phase-only $\mathbf{F}_{\rm RF}$ elements are updated element-wise to maximize rate or MSE objectives with per-element constant modulus [1601.06814, 1711.08408].
- **Softmax neural selection**: For radar, sparse RF/antenna selection is relaxed via softmax networks, trained to match beampatterns [2101.06837].
- **Metaheuristics**: Evolutionary algorithms (e.g., improved bat algorithm) optimize phase-only digital weights under hybrid constraints [1811.01176].

**Fully digital equivalence**: When $N_{\rm RF} \geq 2 N_s$, any digital precoder can be exactly realized by an FC hybrid architecture [1601.06814]; in highly sparse channels or highly correlated eigenstructures, $N_{\rm RF}=N_s$ suffices [1711.08408, 1407.0446]. For ISAC, hybrid arrays with only $N_{\rm RF} \geq 2$ are information-lossless relative to fully digital, as optimal covariance is rank-one [2406.00689].

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

**RF chain count**: Hybrid performance approaches fully digital once $N_{\rm RF}$ exceeds the number of data streams/users; spectral efficiency gap diminishes with $N_{\rm RF}/N_s$ ($\lesssim$1 bps/Hz for $N_{\rm RF} \gtrsim 1.5 N_s$ in i.i.d. channels) [1407.0446, 1711.08408]. 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 ($O(N_t N_{\rm RF})$ hardware).
- PC/switching offers $30-40\%$ complexity/power savings and only small rate loss when beams are properly aligned, and allows PAs to approach higher efficiency [1904.10276].

**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 [2101.06837, 1711.08408].
- Low-res ADCs require modified combining: two-stage analog combining (aggregation + DFT spreading) can achieve optimal MI scaling $N_u\log N_{\rm RF}$ with $b$-bit ADCs [1808.01013].

**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 [1902.09511].
- DAC/ADC and phase-shifter power dissipation scales linearly with $N_{\rm RF}$, 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 [1711.08408].
- Frequency-flat analog stages rely on channel sparsity; for practical arrays ($64\times 32$), FC hybrid with $N_{\rm RF}=4$ is within $1-2$ dB SE of fully digital for $K=64$ 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 [1509.05091, 2107.07622].
- The optimal allocation of training energy follows water-filling over these dominant directions; for highly correlated channels, a small $\tau$ (pilot duration) suffices [2107.07622].
- For $N_t=64$, $N_{\rm RF}=4$, hybrid MMSE achieves 1–2 dB gap to fully digital estimation, with optimal $\tau$ between $4$ and $20$ depending on correlation [1509.05091].

**Beam alignment/training** in MU-MIMO exploits hierarchical analog codebooks (flat-top beams, multi-level DFT) [2101.07106] and non-negative least squares (NNLS) recovery in sparse beamspace [1904.10276]; 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 $N_{\rm RF}\ge2$, and AO/FPP-SCA methods provide efficient joint solutions under rate constraints [2406.00689].
- Low-resolution DACs in ISAC setups can be efficiently modeled (AQNM, Bussgang) and digitally compensated; $b=4-5$ bits recovers almost all the SE, while $b=1$ maintains robust dual-functionality with a modest 2–3 dB loss [2411.02827].

**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 [2605.29481].

**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 [2410.03682].

**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 [2101.06837].

## 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 [1904.10276, 1711.08408] |
| 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 [1808.01013]    |

## 7. Implementation and Practical Recommendations

- **RF chain budget**: $N_{\rm RF} \simeq 1.5 \times N_s$ achieves $>90\%$ of the fully digital sum-rate in most MU-MIMO or radar applications.
- **Codebook sizing**: A few hundred steering angles suffice for $N_t=64$ ULAs [1407.0446].
- **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 $1/N$ and $1/N_{\rm RF}$; 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 [1811.01176].
- **Sensing applications**: When $N_{\rm RF}=1$, convex per-element AO or learning-based selection can approach optimal performance in beam synthesis [2406.00689, 2101.06837].

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 [1601.06814, 1407.0446, 1711.08408, 2406.00689, 1712.03485, 2411.02827, 1904.10276, 2410.03682, 2101.06837, 1808.01013].

Source: https://www.emergentmind.com/topics/hybrid-analog-digital-beamforming-0b0d0ef9-af39-4536-aa10-8eb0dcd5f375