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Hybrid Digital and Analog Beamforming Design for Large-Scale Antenna Arrays

Foad Sohrabi, Wei Yu

arXiv:1601.06814v1cs.IT

TL;DR

Large-scale mmWave arrays make fully digital beamforming costly because each antenna requires a dedicated RF chain. The paper uses hybrid digital–analog beamforming, proves exact realization with twice as many RF chains as data streams, and proposes heuristic designs that remain close to fully digital performance with fewer chains and finite-resolution phase shifters.

  • Problem

    Fully digital beamforming is impractical for large-scale antenna arrays because it requires one dedicated RF chain per antenna element, imposing prohibitive cost and power consumption at mmWave frequencies.

  • Method

    The paper combines low-dimensional digital beamforming with analog phase-shifter beamforming and develops heuristic designs for point-to-point MIMO and downlink MU-MISO systems.

  • Results

    The hybrid structure can realize any fully digital beamformer exactly with twice the number of data streams in RF chains, while heuristic designs achieve performance close to fully digital beamforming with fewer chains.

  • Takeaways & Limitations

    Hybrid beamforming offers a lower-RF-chain alternative for large-scale arrays, with proposed methods also effective when only finite-resolution phase shifters are available.

  • Takeaways & Limitations

    Exhaustive optimization over finite-resolution RF beamformers is impractical because the feasible set grows exponentially with antenna number and phase-shifter resolution.

Abstract

from arXiv · show

The potential of using of millimeter wave (mmWave) frequency for future wireless cellular communication systems has motivated the study of large-scale antenna arrays for achieving highly directional beamforming. However, the conventional fully digital beamforming methods which require one radio frequency (RF) chain per antenna element is not viable for large-scale antenna arrays due to the high cost and high power consumption of RF chain components in high frequencies. To address the challenge of this hardware limitation, this paper considers a hybrid beamforming architecture in which the overall beamformer consists of a low-dimensional digital beamformer followed by an RF beamformer implemented using analog phase shifters. Our aim is to show that such an architecture can approach the performance of a fully digital scheme with much fewer number of RF chains. Specifically, this paper establishes that if the number of RF chains is twice the total number of data streams, the hybrid beamforming structure can realize any fully digital beamformer exactly, regardless of the number of antenna elements. For cases with fewer number of RF chains, this paper further considers the hybrid beamforming design problem for both the transmission scenario of a point-to-point multipleinput multiple-output (MIMO) system and a downlink multiuser multiple-input single-output (MU-MISO) system. For each scenario, we propose a heuristic hybrid beamforming design that achieves a performance close to the performance of the fully digital beamforming baseline. Finally, the proposed algorithms are modified for the more practical setting in which only finite resolution phase shifters are available. Numerical simulations show that the proposed schemes are effective even when phase shifters with very low resolution are used.

I. INTRODUCTION

The paper addresses the cost and power barriers of fully digital beamforming in large-scale mmWave arrays with a hybrid digital–analog architecture. It establishes an exact RF-chain sufficiency result and develops heuristic designs for constrained MIMO and MU-MISO settings, including finite-resolution phase shifters.

  • Fully digital beamforming is impractical for large-scale mmWave arrays because it requires one dedicated RF chain per antenna element, creating prohibitive cost and power consumption.
  • The proposed architecture concatenates a low-dimensional digital beamformer with an analog RF beamformer implemented using phase shifters.
  • Twice the total number of data streams in RF chains is sufficient to realize any fully digital beamformer exactly, regardless of antenna count.
  • With fewer RF chains, heuristic designs target spectral-efficiency maximization for point-to-point MIMO and downlink MU-MISO systems under total transmit-power constraints.
  • The proposed finite-resolution modification remains effective with very low-resolution phase shifters, whereas straightforward quantization can be ineffective at low resolution.

III. MINIMUM NUMBER OF RF CHAINS TO REALIZE FULLY DIGITAL BEAMFORMERS

The section derives lower and upper bounds on RF chains for realizing fully digital beamformers. At least Ns chains are necessary, while 2Ns chains suffice for any fully digital beamformer.

  • Ns RF chains are necessary to realize a fully digital beamforming matrix because the hybrid beamformer rank cannot exceed the RF-chain count.
  • 2Ns RF chains suffice to realize any fully digital beamforming matrix in the hybrid architecture.
  • For N_RF = 2Ns, the construction selects constant-modulus RF coefficients and digital gains whose product equals each fully digital beamformer entry.
  • The same construction extends to N_RF > 2Ns by setting the extra digital-precoder parameters to zero.
  • Equal digital gains for each data stream allow one realization of a scaled data symbol to be reused twice, reducing the implementation to Ns RF chains and 2NsN phase shifters.
  • For rank-deficient fully digital beamformers, factorization through rank r permits realization using 2r RF chains.

IV. HYBRID BEAMFORMING DESIGN FOR SINGLE-USER LARGE-SCALE MIMO SYSTEMS

This section studies hybrid beamforming for point-to-point large-scale MIMO systems and develops heuristic designs when RF chains are limited. The critical case N_RF = Ns is emphasized because it is the minimum count capable of realizing fully digital beamformers.

  • The point-to-point system has large-scale antenna arrays at both transmitter and receiver, with min(N, M) much greater than Ns.
  • The analysis assumes identical numbers of transmit and receive RF chains to simplify the notation.
  • For N_RF = Ns, the paper proposes a heuristic algorithm intended to achieve a rate close to capacity.
  • The same algorithm can be extended through further approximations to cases with Ns < N_RF < 2Ns.
  • Joint hybrid precoder-combiner rate maximization is difficult because of coupled optimization and non-convex analog-beamformer constraints.

A. Digital Precoder Design for N RF = Ns

For fixed RF precoding, the section derives the digital precoder through the effective channel. In the large-array regime with N_RF = Ns, the solution typically has an approximately equal-gain structure.

  • With a fixed RF precoder, the effective channel is H_eff = H V_RF, and the digital-precoder problem is formulated over this channel.
  • The digital precoder uses the Ns right singular vectors associated with the largest singular values of H_eff Q^-1/2 and allocates power through a diagonal matrix Γ_e.
  • For large-scale MIMO, Q ≈ NI with high probability because diagonal terms equal N while off-diagonal sums are relatively small.
  • When N_RF = Ns, the optimal digital precoder typically satisfies V_D V_D^H ∝ I.
  • Under approximately equal power allocation across streams, the proportionality constant is determined by γ² = P/(N N_RF).

B. RF Precoder Design for N RF = Ns

The RF precoder is designed after approximating the digital Gram matrix by a scaled identity. The resulting non-convex problem is solved with iterative coordinate descent over phase-shifter elements.

  • After assuming V_D V_D^H ≈ γ²I, the RF precoder design automatically satisfies the transmitter power constraint for any RF-precoder design.
  • The RF-precoder objective remains non-concave, but decoupled constraints enable an iterative coordinate-descent algorithm.
  • The objective is rewritten to isolate the contribution of each RF-precoder element while holding all other elements fixed.
  • For each element at row i and column j, the update selects its optimal value under the remaining fixed RF-precoder entries.
  • The algorithm initializes a feasible RF precoder, sequentially updates its elements, and stops when convergence is reached.
  • Each element update does not decrease the objective, so the iterative RF-precoder algorithm is guaranteed to converge to a local optimum.

C. Hybrid Combining Design for N RF = Ns

The receiver design decouples RF and digital combining, using an RF combiner followed by an MMSE digital combiner. The proposed point-to-point MIMO procedure has cubic complexity and achieves rates close to maximum capacity in simulations.

  • Hybrid Combining Design for N_RF = N_s: The RF combiner is designed first, followed by the optimal digital combiner for the fixed RF combiner.For N_RF = N_s, the digital combiner is unconstrained square, enabling this decoupling without loss of optimality.
  • Hybrid Beamforming Design for N_s < N_RF < 2N_s: For N_s < N_RF < 2N_s, the RF precoder is designed using an all-eigenvalue approximation before obtaining the digital precoder for the actual N_s streams.The approximation is described as reasonable when N_RF is of the order of N_s.
  • Hybrid Combining Design for N_RF = N_s: For large M, approximating W_RF^H W_RF as M I makes the decoupled RF-combiner design approximately optimal.The resulting RF-combiner problem has the same form as the RF-precoder design problem, so Algorithm 1 can be reused.
  • Algorithm and Complexity: O(N^3) is the overall complexity of Algorithm 2 when the antenna counts at both ends satisfy M = O(N).This complexity is similar to that of several existing hybrid beamforming designs.
  • Simulation Results: The N_RF = N_s infinite-resolution design achieves a rate very close to maximum capacity in simulations.With finite-resolution phase shifters, additional RF chains can trade off phase-shifter accuracy.

V. HYBRID BEAMFORMING DESIGN FOR MULTI-USER MASSIVE MISO SYSTEMS

The MU-MISO design targets weighted sum rate under inter-user interference and unequal stream priorities. It alternates between ZF digital precoding with power allocation and RF-precoder updates to obtain a good solution.

  • System and Objective: MU-MISO hybrid precoding must account for inter-user interference because receivers are not collocated.Unlike point-to-point MIMO, the receivers cannot cooperate through the rate expression used for the MIMO case.
  • System and Objective: The MU-MISO objective is weighted sum-rate maximization because users or streams may have unequal priorities.Point-to-point MIMO streams are described as having equal priority.
  • Related Work: For N_RF = K and N → ∞, channel-matched RF precoding with low-dimensional ZF achieves a reasonable sum rate relative to fully digital ZF, but finite N leaves a gap to capacity.The cited prior result motivates further hybrid-beamforming design for practical antenna counts.
  • Proposed Design: The proposed method decouples RF and digital precoder design by using ZF beamforming with power allocation for the digital precoder.For a fixed RF precoder, the digital precoder and its power variables can be obtained through the stated ZF formulation and water-filling.
  • Proposed Design: Iterating between fixed-RF digital design and fixed-power RF design yields an approximately local-optimal RF precoder and a good MU-MISO solution.The RF design is based on the power-minimization reformulation of the weighted sum-rate problem.

B. RF Precoder Design

The RF-precoder design alternates between RF phase updates and digital power allocation. Coordinate-wise phase optimization reduces the difficult RF problem to iterative updates that converge to a local minimizer of the approximated objective.

  • RF Precoder Design: The achievable weighted sum rate depends on the RF precoder through the power constraint, motivating RF-precoder power minimization.The resulting objective remains difficult because its dependence on V_RF is complicated.
  • Approximation: The RF design uses the approximation V_RF^H V_RF ≈ N I when N is large.This large-array property simplifies the RF-precoder objective before coordinate descent is applied.
  • Coordinate Update: For fixed RF-precoder entries except one phase, the algorithm finds a stationary phase value and selects the solution minimizing the periodic objective.Only two phase values satisfy the stationarity condition, and periodicity identifies the minimizer.
  • Iterative Algorithm: The RF precoder is updated sequentially entry by entry until convergence to a local minimizer of the approximated objective.The procedure starts from a feasible RF precoder and applies the coordinate-wise phase rule repeatedly.
  • Iterative Algorithm: The complete algorithm alternates RF phase updates with water-filling power allocation until the overall procedure converges.It initializes P = I, updates the RF precoder, then computes the optimal diagonal power allocation for the current RF precoder.

VI. HYBRID BEAMFORMING WITH FINITE RESOLUTION PHASE SHIFTERS

Finite-resolution phase shifters turn RF-beamformer design into a discrete optimization problem. The proposed algorithms optimize directly over feasible phase values, retaining polynomial complexity and improving on simple post-hoc quantization for low-resolution hardware.

  • Finite-Resolution Model: Finite-resolution phase shifters restrict each RF-beamformer element to a finite set of realizable phase angles.The number of realizable angles is typically n_PS = 2^b for b-bit phase shifters.
  • Optimization Challenge: Exhaustive optimization over finite RF beamformers is impractical because the feasible set grows exponentially with antenna count and phase-shifter resolution.Digital-beamformer design is well studied for fixed RF beamformers, but joint RF optimization has combinatorial complexity.
  • Direct Quantized Optimization: Directly incorporating phase quantization into optimization is intended to improve performance over quantizing an infinite-resolution solution afterward.The paper reports that nearest-point quantization is ineffective for low-resolution phase shifters.
  • Complexity: O(N^3) is the proposed single-user MIMO algorithm complexity when M = O(N), versus exponential exhaustive-search complexity.The complexity comparison assumes antenna counts at both ends are in the same asymptotic range.
  • MU-MISO Design: For finite-resolution MU-MISO design, each RF entry is selected by one-dimensional exhaustive search over the feasible phase set.This replaces the continuous phase update used with infinite-resolution phase shifters.

VII. SIMULATIONS

The simulations evaluate proposed hybrid beamforming algorithms under geometric-channel and uniform-linear-array settings across MIMO and MU-MISO systems. Spectral efficiency is averaged over 100 channel realizations as a function of SNR.

  • The simulations use geometric channels with L = 15 scatterers, uniformly random arrival and departure angles, and uniform linear arrays.The antenna spacing is d̃ = λ/2.
  • The evaluation covers proposed and existing hybrid beamforming designs together with optimal or nearly optimal fully digital schemes.The comparison includes both point-to-point MIMO and MU-MISO systems.
  • Average spectral efficiency is plotted against SNR over 100 channel realizations.SNR is defined as P/σ^2.

A. Performance Analysis of a MIMO System with Hybrid Beamforming

The simulations show that the proposed hybrid beamforming methods closely approach fully digital performance and outperform competing designs in point-to-point MIMO and MU-MISO settings. Optimizing finite-resolution phase shifters and increasing RF-chain count reduce performance gaps caused by low phase resolution.

  • A. Performance Analysis of a MIMO System with Hybrid Beamforming: About 1.5 dB and 1 dB gains are achieved over the methods in [27] and, respectively, in a 64 × 16 MIMO system with N_RF = N_s = 6.The proposed algorithm is very close to the optimal fully digital rate.
  • A. Performance Analysis of a MIMO System with Hybrid Beamforming: At least 1.5 dB gain is achieved with 1-bit phase shifters over quantized competing algorithms in a 10 × 10 MIMO system with N_RF = N_s = 2.Performance is very close to the optimal exhaustive-search method.
  • A. Performance Analysis of a MIMO System with Hybrid Beamforming: About 5 dB separates very-low-resolution and infinite-resolution phase shifters in the 64 × 16 MIMO example with N_s = 4.Increasing RF chains and jointly optimizing RF and digital beamformers reduce this gap.
  • B. Performance Analysis of a MU-MISO System with Hybrid Beamforming: In an 8-user MISO system with N = 64, the proposed method uses K + 1 = 9 RF chains, while the compared methods use K = 8 RF chains.The strongest-path matching approach is reported as ineffective for practical N = 64.
  • B. Performance Analysis of a MU-MISO System with Hybrid Beamforming: With b = 1 phase shifters in a 4-user MISO system, the proposed approach improves performance by about 1 dB, 2 dB, and 8 dB over three quantized alternatives.The alternatives are the quantized Section IV method and the methods in and.

APPENDIX A DERIVATION OF (26)

The appendix derives equation (26) through matrix-inverse and trace identities, then simplifies the resulting expressions by expansion. It also reduces the relevant scalar optimization to solving for extrema over one period.

  • The derivation partitions the RF beamformer into a submatrix and expresses the objective in terms of its entries.The notation uses the jth column v^(j) and matrix-entry products involving B_j and D_j.
  • The first equality uses the Sherman–Morrison formula for a full-rank matrix plus a rank-one matrix.The formula expresses (A+B)^−1 using A^−1 and a trace denominator.
  • Trace linearity and cyclic invariance transform the intermediate expression before the remaining terms are expanded.The appendix explicitly uses Tr(AB) = Tr(BA).
  • The scalar trigonometric optimization is reduced to finding maxima and minima by solving the derivative equation.The resulting equation has only two solutions over one 2π period.
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