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Beamforming for Multiuser Massive MIMO Systems: Digital versus Hybrid Analog-Digital

Tadilo Endeshaw Bogale, Long Bao Le

arXiv:1407.0446v2math.OC

TL;DR

The paper addresses how to design hybrid beamforming for downlink multiuser massive MIMO while approaching digital-beamforming performance under lower-dimensional analog-digital hardware constraints. It uses a WSMSE formulation tied to block-diagonalization digital beamforming and solves the resulting problem with compressed sensing. Simulations show that fewer multiplexed symbols narrow the digital–hybrid gap, while more RF chains and ADCs increase hybrid sum rate.

  • Problem

    Digital beamforming offers high performance but requires costly RF chains and ADCs, motivating hybrid beamforming for massive MIMO systems.

  • Method

    The paper designs hybrid beamforming by minimizing a WSMSE objective incorporating block-diagonalization digital beamforming, then solves it using compressed sensing.

  • Results

    For fixed RF chains and ADCs, fewer multiplexed symbols reduce the digital–hybrid performance gap; for fixed symbols, more RF chains and ADCs increase hybrid sum rate.

  • Takeaways & Limitations

    Hybrid beamforming can approach digital performance, especially at low to moderate SNR, while its sum rate improves with additional RF chains and ADCs.

Abstract

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This paper designs a novel hybrid (a mixture of analog and digital) beamforming and examines the relation between the hybrid and digital beamformings for downlink multiuser massive multiple input multiple output (MIMO) systems. We assume that perfect channel state information is available only at the transmitter and we consider the total sum rate maximization problem. For this problem, the hybrid beamforming is designed indirectly by considering a weighed sum mean square error (WSMSE) minimization problem incorporating the solution of digital beamforming which is obtained from the block diagonalization technique. The resulting WSMSE problem is solved by applying the theory of compressed sensing. The relation between the hybrid and digital beamformings is studied numerically by varying different parameters, such as the number of radio frequency (RF) chains, analog to digital converters (ADCs) and multiplexed symbols. Computer simulations reveal that for the given number of RF chains and ADCs, the performance gap between digital and hybrid beamformings can be decreased by decreasing the number of multiplexed symbols. Moreover, for the given number of multiplexed symbols, increasing the number of RF chains and ADCs will increase the total sum rate of the hybrid beamforming which is expected.

I. INTRODUCTION

The paper motivates hybrid beamforming for downlink multiuser massive MIMO by balancing digital beamforming’s performance against the cost of RF chains and ADCs. It designs the hybrid approach through WSMSE minimization and evaluates how multiplexed symbols, RF chains, and ADCs affect the digital–hybrid relationship.

  • I. INTRODUCTION: Digital beamforming determines signal phase and amplitude at baseband but requires one expensive RF chain per antenna and as many ADCs as receiver antennas.Analog beamforming uses low-cost phase shifters but has constant amplitude, making its performance inferior to digital beamforming.
  • I. INTRODUCTION: The paper designs a novel hybrid beamforming scheme for downlink multiuser massive MIMO with perfect channel state information available only at the transmitter.The assumed transmitter-side channel knowledge can be obtained through time division duplex training.
  • I. INTRODUCTION: Hybrid beamforming is designed indirectly through a WSMSE minimization problem that incorporates digital beamforming designed by block diagonalization for total sum rate maximization.The abstracted introduction states that the WSMSE weights depend on the design criteria and digital beamforming approach.
  • I. INTRODUCTION: For fixed RF-chain and ADC counts, decreasing the number of multiplexed symbols reduces the performance gap between hybrid and digital beamforming.For fixed multiplexed-symbol counts, increasing RF chains and ADCs increases hybrid-beamforming total sum rate.
  • II. SYSTEM MODEL: The paper models multiuser transmission with K users, where user k has M_k antennas and multiplexes S_k symbols.The total numbers of receiver antennas and symbols are defined across users.

A. Digital Downlink MIMO System Model

The digital downlink model uses a transmitter precoder and user-specific linear receivers, while the hybrid model factors transmission and reception into analog RF and baseband digital matrices. RF-chain and ADC dimensions determine the hybrid representation’s available processing resources.

  • A. Digital Downlink MIMO System Model: The digital transmitter precodes the aggregate data vector with B=[B_1,…,B_K], while receiver k uses W_k to recover d_k from its received signal.B_k is the kth user’s digital precoder and W_k is its linear receiver.
  • A. Digital Downlink MIMO System Model: The received signal includes the kth user’s MIMO channel H_k, digital precoders, linear receiver, and additive Gaussian noise n_k.The channel dimensions and roles of B_k and W_k are specified in the model.
  • A. Digital Downlink MIMO System Model: Hybrid beamforming combines RF transmitter and receiver matrices with baseband transmitter and receiver matrices, and every analog-matrix element has constant modulus.The RF matrices are A and F_k, while the baseband matrices are the corresponding tilded precoders and receivers.
  • A. Digital Downlink MIMO System Model: The hybrid output equals the digital output when all RF dimensions equal antenna dimensions and the RF matrices are identity matrices.Specifically, P_t=N, P_rk=M_k, A=I_N, and F_k=I_Mk recover the digital representation.
  • A. Digital Downlink MIMO System Model: P_t denotes transmitter RF chains and P_rk denotes ADCs at receiver k, so hybrid performance depends on selecting these dimensions.The model also reduces to analog beamforming under the stated identity baseband settings.

III. CHANNEL MODEL

The paper adopts a geometric multiuser channel model with user-specific scatterers, path gains, path loss, and angular array responses. Transmit and receive arrays are modeled as uniform linear arrays.

  • III. CHANNEL MODEL: The channel between the transmitter and user k is modeled geometrically with L_k scatterers.This is the paper’s stated channel-model assumption.
  • III. CHANNEL MODEL: Each propagation path has complex gain g_ki, path loss ρ_k, and transmit and receive angles θ_t(i) and θ_rk(i).The path-gain normalization is E{|g_ki|^2}=1, and the angles lie in [0,2π].
  • III. CHANNEL MODEL: The antenna response vectors describe the transmitter and receiver arrays, which the paper models using uniform linear arrays.The response depends on the transmission wavelength and antenna spacing.

IV. HYBRID BEAMFORMING DESIGN

The proposed hybrid design approximates a digital block-diagonalization reference by minimizing weighted MSE under hybrid-structure constraints. The optimization uses symbol-dependent weights and is solved through alternating subproblems, including compressed sensing.

  • IV. HYBRID BEAMFORMING DESIGN: Because hybrid matrices are more constrained than digital matrices, the digital beamforming solution serves as the reference for design.The paper quantifies hybrid quality using the Euclidean distance, expressed as MSE, between hybrid and digital received signals.
  • IV. HYBRID BEAMFORMING DESIGN: The digital reference uses block diagonalization, yielding zero inter-user terms and per-user diagonal signal terms involving Z_k and Q_k.The resulting digital approach is used to maximize the downlink total sum rate.
  • IV. HYBRID BEAMFORMING DESIGN: The proposed objective is WSMSE minimization with weights chosen per user and symbol to make the hybrid–digital performance gap approximately constant across symbols.Symbols whose digital contribution is approximately zero are ignored because digital beamforming switches them off.
  • IV. HYBRID BEAMFORMING DESIGN: The optimization alternates between jointly optimizing each user’s baseband receiver and RF receiver, then jointly optimizing the transmitter RF matrix and baseband precoder.The second step holds the receiver-side variables fixed.

A. Step 1

The first design step reformulates hybrid receive-side beamforming as a constrained dimension-reduction problem and approximates its solution with compressed sensing.

  • A. Step 1: The kth user's receive-side variables are jointly optimized through a coupled objective with a nonconvex constraint.The formulation seeks a suboptimal, close-to-optimal solution rather than an anticipated global optimum.
  • A. Step 1: The left singular eigenvectors of the channel are used to motivate a dictionary built from analog beamforming vectors and null-space vectors.Each column of the receive dictionary is normalized and selected as a candidate column for Fk.
  • A. Step 1: Introducing ¯Fk simplifies the receive-side formulation while preserving a dimension-matching equality constraint for ¯Wk and ˜˜Wk.The equality constraint ensures that the dimensions of the two digital matrices agree.
  • A. Step 1: Orthogonal matching pursuit solves the resulting nonconvex dimension-reduction problem by selecting matching projections from the over-complete dictionary.For this problem, ¯Fk serves as the kth user's dictionary matrix.

B. Step 2

The second design step applies the same matching-pursuit strategy to construct the transmit analog beamformer and recover the associated digital precoder.

  • B. Step 2: The transmit analog beamformer A is constructed from channel right-singular-vector structure, using analog steering vectors and null-space vectors as a normalized dictionary.Each column of A is assumed to be selected from the columns of the resulting dictionary.
  • B. Step 2: The transmit-side optimization problem is solved with an orthogonal matching pursuit procedure analogous to the receive-side algorithm.The algorithm iteratively selects dictionary columns and updates the residual before producing the digital precoder.
  • B. Step 2: The transmitter solves the optimization problems, while each receiver requires its selected Fk and ˜˜Wk to decode its own data.Because Fk columns are selected from steering-vector columns, the kth receiver can construct Fk from θrk(i).

V. SIMULATION RESULTS

The simulations evaluate sum rate under a four-user massive-MIMO setup, with rates averaged over random channel realizations and SNR varied through the noise variance.

  • V. SIMULATION RESULTS: N = 128, Mk = 32, d = 0.5λ, K = 4, and Lk = 16 define the simulation configuration.The transmit power is set as Pt = KPrk.
  • V. SIMULATION RESULTS: SNR is varied by changing σ2 while keeping the total transmitted power Pmax = KSkmw and Pav = Pmax Sk.The simulation defines SNR through the average transmitted power and noise variance.
  • V. SIMULATION RESULTS: The total sum rate is Rt = ∑K i=1 Rki, with each symbol rate calculated as Rki = log2(1 + γki).γki is the achieved SINR of the kth user's ith symbol.
  • V. SIMULATION RESULTS: All plots average results over 1000 realizations of gk for every user.

A. Comparison of Digital and Hybrid Beamformings

With Sk = 8 and Prk = 16 for every user, hybrid beamforming nearly matches digital beamforming at low-to-moderate SNR, with a small gap at high SNR.

  • A. Comparison of Digital and Hybrid Beamformings: The hybrid beamforming sum rate is almost the same as the digital beamforming sum rate from low to moderate SNR regions.The comparison uses Sk = 8 and Prk = 16, ∀k.
  • A. Comparison of Digital and Hybrid Beamformings: A small performance gap appears between hybrid and digital beamforming at high SNR regions.

B. Effect of Prk on Hybrid Beamforming

Increasing Prk, the number of RF chains and ADCs, improves the hybrid beamforming performance.

  • Hybrid beamforming performance improves as Prk, representing the number of RF chains and ADCs, increases.The comparison uses SNR values of −4 dB and 6 dB with Sk = 8.

C. Joint effects of Sk and Prk on Digital and Hybrid Beamformings

The digital and hybrid beamforming sum rates increase with multiplexed symbols, while reducing multiplexed symbols narrows their performance gap under limited RF-chain and ADC resources. Increasing RF chains and ADCs raises hybrid-beamforming sum rate.

  • C. Joint effects of Sk and Prk on Digital and Hybrid Beamformings: The achieved sum rates of both digital and hybrid beamformings increase as Sk increases.The passage attributes this to Pmax = KSk increasing with Sk.
  • C. Joint effects of Sk and Prk on Digital and Hybrid Beamformings: For limited Prk, reducing the number of multiplexed symbols decreases the performance gap between hybrid and digital beamformings.Here, Prk denotes the number of RF chains and ADCs.
  • C. Joint effects of Sk and Prk on Digital and Hybrid Beamformings: Increasing the number of RF chains and ADCs increases the achievable sum rate of hybrid beamforming for a fixed number of multiplexed symbols.

APPENDIX A

Appendix A summarizes digital block diagonalization beamforming: it suppresses inter-user and self-interference, then optimizes per-symbol powers through a sum-rate problem solved by water filling.

  • APPENDIX A: Block diagonalization first eliminates other-user interference, either partially or completely when possible.
  • APPENDIX A: It then cancels each user’s self-interference so the desired symbols become parallel.
  • APPENDIX A: The appendix defines transformed channel and SVD-related matrices used to construct the block diagonalization beamformers.The definitions include H̃k, Ũhk, Ṽhk, and diagonal matrix Zk.
  • APPENDIX A: The third step optimizes symbol powers by solving a sum-rate maximization problem.The power allocation matrix Qk is real, diagonal, and non-negative.
  • APPENDIX A: The global optimum of the power-allocation problem is obtained using a water-filling algorithm.qki is the ith diagonal element of Qk, and Pmax is the transmitter’s maximum available power.
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