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Quantized Precoding for Massive MU-MIMO

Sven Jacobsson, Giuseppe Durisi, Mikael Coldrey, Tom Goldstein, Christoph Studer

arXiv:1610.07564v3cs.IT

TL;DR

The paper addresses downlink precoding for massive MU-MIMO systems whose BS uses low-resolution DACs, motivated by hardware power and fronthaul constraints. It develops linear-quantized and nonlinear precoding approaches with analytical approximations and algorithms, finding near-infinite-resolution performance with 3–4 bits and a smaller 1-bit penalty for nonlinear precoding than for linear precoding. Practical assessment remains bounded by neglected out-of-band emissions and intersymbol interference.

  • Problem

    The paper studies how to perform massive MU-MIMO downlink precoding with low-resolution BS DACs while addressing the associated hardware power and fronthaul constraints.

  • Method

    The paper develops linear-quantized precoders and 1-bit nonlinear precoders, using Bussgang’s theorem for closed-form rate approximations and optimization-based algorithms for approximate solutions.

  • Results

    3 to 4 DAC bits close the linear-quantized precoding gap to infinite resolution, while at BER 10^-3 nonlinear 1-bit precoding has about a 3 dB gap versus about 8 dB for linear precoding.

  • Takeaways & Limitations

    Massive MU-MIMO can use low-resolution DACs without significant error-rate or information-theoretic-rate loss, while nonlinear 1-bit precoding improves performance at higher complexity.

  • Takeaways & Limitations

    The analysis and algorithms use symbol-rate low-resolution DACs while ignoring their out-of-band emissions and intersymbol interference; frequency-selective extensions remain future work.

Abstract

from arXiv · show

Massive multiuser (MU) multiple-input multiple-output (MIMO) is foreseen to be one of the key technologies in fifth-generation wireless communication systems. In this paper, we investigate the problem of downlink precoding for a narrowband massive MU-MIMO system with low-resolution digital-to-analog converters (DACs) at the base station (BS). We analyze the performance of linear precoders, such as maximal-ratio transmission and zero-forcing, subject to coarse quantization. Using Bussgang's theorem, we derive a closed-form approximation on the rate achievable under such coarse quantization. Our results reveal that the performance attainable with infinite-resolution DACs can be approached using DACs having only 3 to 4 bits of resolution, depending on the number of BS antennas and the number of user equipments (UEs). For the case of 1-bit DACs, we also propose novel nonlinear precoding algorithms that significantly outperform linear precoders at the cost of an increased computational complexity. Specifically, we show that nonlinear precoding incurs only a 3 dB penalty compared to the infinite-resolution case for an uncoded bit error rate of 10^-3, in a system with 128 BS antennas that uses 1-bit DACs and serves 16 single-antenna UEs. In contrast, the penalty for linear precoders is about 8 dB.

I. INTRODUCTION

Massive MU-MIMO can improve wireless-system performance, but large antenna arrays make RF hardware, DAC power, and fronthaul costly. This paper studies frequency-flat downlink precoding with low-resolution DACs, developing linear and nonlinear methods and showing that coarse quantization can approach infinite-resolution performance.

  • Hundreds of BS antennas promise higher spectral efficiency, energy efficiency, reliability, and coverage, but increase hardware complexity, cost, and circuit power.
  • DAC power consumption grows exponentially with resolution and linearly with bandwidth, making low-resolution DACs attractive for massive arrays and fronthaul reduction.
  • The paper considers frequency-flat massive MU-MIMO downlink precoding with low-resolution DACs, where multiple independent UEs share the same time-frequency resource.
  • Linear-quantized precoding applies a finite-resolution DAC after linear processing, whereas nonlinear precoding directly generates quantized outputs from the data vector and CSI.
  • Bussgang-based closed-form rate approximations show that 3 to 4 DAC bits can close the performance gap to infinite resolution for linear-quantized precoding.
  • For 1-bit DACs, nonlinear precoding outperforms linear-quantized precoding, while requiring increased signal-processing complexity and specialized approximate algorithms.

B. Precoding

The paper formulates MMSE quantized precoding under finite-resolution DACs and contrasts linear-quantized processing with nonlinear precoders. Linear methods quantize a matrix-vector product, whereas nonlinear methods directly compute the transmit vector from symbols and CSI, trading lower error rates for higher complexity.

  • Quantized precoding problem: Finite-resolution DACs constrain the precoder outputs and introduce distortion relative to infinite-resolution precoding.The MMSE objective minimizes the error between the received signal and transmitted symbols under a power constraint.
  • Linear-quantized precoding: Linear-quantized precoders apply matrix-vector multiplication followed by quantization.Their transmit vector is x = Q(Ps), where Q is the quantizer-mapping function.
  • Nonlinear precoding: Nonlinear precoders use the transmit symbols and available CSI to directly compute a quantized transmit vector.They implement x = P(s,H) rather than quantizing the output of a linear precoder.
  • Performance trade-off: Nonlinear precoders outperform linear-quantized precoders in error-rate performance at the cost of higher computational complexity.This comparison is reported for the proposed quantized precoding approaches.
  • Motivation: Linear precoding remains attractive for massive MU-MIMO because of its relatively low computational complexity.In the infinite-resolution setting, MRT approaches virtually optimal performance in the large-antenna limit.

1) WF precoding:

This section describes Wiener-filter, zero-forcing, and maximal-ratio-transmission precoders alongside the finite-resolution DAC quantizer model. The quantizer uses uniform levels and thresholds, with scaling chosen to satisfy the transmit-power constraint.

  • WF precoding: Infinite-resolution Wiener-filter precoding provides the solution to the MMSE problem.
  • ZF precoding: Zero-forcing precoding uses the channel pseudoinverse to null multiuser interference.Its matrix follows from the Wiener-filter solution by setting the noise variance N0 to zero.
  • MRT precoding: Maximal-ratio transmission maximizes power directed toward each user while ignoring multiuser interference.Its matrix follows from the Wiener-filter solution as the noise variance N0 tends to infinity.
  • DAC quantization: The DAC model uses symmetric uniform quantizers with step size ∆ and scales their output to satisfy the power constraint.The quantizer is specified by quantization labels and thresholds, and the step size is selected under a Gaussian per-antenna input assumption.
  • 1-bit DACs: For 1-bit DACs, the possible complex-valued output per antenna is restricted to four scaled sign combinations.The chosen outcomes ensure that the power constraint is satisfied with equality.

C. Signal Decomposition using Bussgang’s Theorem

Bussgang’s theorem decomposes the quantized transmit signal into a linear component and distortion uncorrelated with the input symbols. This decomposition enables achievable-rate analysis, including a closed-form treatment for 1-bit DACs.

  • Bussgang decomposition: Bussgang’s theorem represents the quantized signal as a linear function of the DAC input plus an uncorrelated distortion term.The result applies for Gaussian inputs and quantization acting independently on real and imaginary components.
  • Multi-bit formulation: For general Q-bit uniform DACs, the analysis establishes a decomposition using a diagonal gain matrix determined by the quantizer parameters.The theorem assumes s ∼ CN(0,I_U), and the distortion is uncorrelated with s.
  • Received signal: The received-signal decomposition separates the desired signal, multiuser interference, DAC distortion, and receiver noise.The distortion term captures both multiuser interference and finite-resolution DAC effects.
  • Achievable-rate analysis: The nonlinearity of the DACs prevents closed-form characterization of the error distribution, motivating an auxiliary-channel lower bound.The bound corresponds to mismatched nearest-neighbor decoding at the user equipments.
  • 1-bit analysis: For 1-bit DACs, the covariance matrix of the quantized signal has a closed-form expression.This permits a closed-form SINDR and a lower bound on the per-user achievable rate with Gaussian signaling.

E. Achievable Rate Approximation for Multi-Bit DACs

For multi-bit DACs, the paper approximates achievable rates using large-system random matrix arguments and a Gaussian error assumption. Simulations verify that the approximation is accurate for realistic antenna and user dimensions.

  • Approximation assumptions: The multi-bit rate approximation assumes both B and U are large and models the error term as Gaussian.The derivation relies on standard random matrix theory arguments.
  • SINDR approximation: The approximation covers the SINDR of WF, ZF, and MRT linear-quantized precoders in the large-B and large-U regime.These are the three precoders introduced in the preceding section.
  • Achievable rates: Substituting the approximated SINDRs yields achievable-rate expressions for Gaussian signaling with nearest-neighbor decoding.The resulting approximation is stated to be valid for large B and U.
  • Validation: Numerical simulations verify that the rate approximation is accurate already for realistic values of B and U.

IV. NONLINEAR PRECODERS FOR 1-BIT DACS

The 1-bit DAC problem simplifies because all DAC outputs have equal amplitude, enabling nonlinear precoders that approximate the MMSE-optimal solution. Exact exhaustive search is computationally infeasible for large antenna arrays, motivating lower-complexity alternatives.

  • The paper develops approximate nonlinear methods for 1-bit DACs, including semidefinite relaxation, squared-ℓ∞-norm relaxation, and sphere decoding.
  • 1-bit DACs produce equal-amplitude outputs, simplifying the quantized precoding problem.
  • The 1-bit quantized precoding problem minimizes noise-averaged MSE for a given symbol vector and channel, with β depending on the instantaneous symbol vector.
  • The resulting optimization resembles an ℓ2-norm-regularized closest-vector problem whose discrete vectors are parameterized by the continuous precoding factor β.
  • 4^B objective evaluations are required by straightforward exhaustive search, making exact optimization impractical as the number of BS antennas grows.
  • For B = 128, exhaustive search would require evaluating the objective more than 10^77 times, and the closest-vector formulation is NP-hard.

A. Semidefinite Relaxation

Semidefinite relaxation converts the rank-constrained formulation of 1-bit quantized precoding into a convex semidefinite program. It provides polynomial-time approximate solutions, but its lifted dimension and solver complexity limit applicability to small or moderately sized systems.

  • A. Semidefinite Relaxation: Semidefinite relaxation transforms the real-valued 1-bit precoding problem into a semidefinite program by omitting the nonconvex rank-1 constraint.
  • A. Semidefinite Relaxation: The original formulation imposes equal diagonal constraints, positive semidefiniteness, and rank(B_R) = 1.
  • A. Semidefinite Relaxation: If the relaxed solution has rank one, SDR recovers the exact solution; otherwise, a discrete precoding vector is extracted from the leading eigenvector.
  • A. Semidefinite Relaxation: The SDR1 method quantizes the leading eigenvector, while SDRr uses randomized procedures to obtain a vector in the discrete precoding set.
  • A. Semidefinite Relaxation: B^4.5 worst-case complexity makes SDR polynomial-time but computationally unsuitable for massive arrays with hundreds of antennas.
  • A. Semidefinite Relaxation: SDR lifts the problem from 2B dimensions to (2B + 1)^2 dimensions and is considered suitable only for systems with roughly 16 BS antennas or fewer.

B. Squared ℓ∞-Norm Relaxation

The squared-ℓ∞-norm relaxation rewrites the 1-bit quantized precoding problem as a convex relaxation that avoids lifting and can be solved efficiently. SQUID uses Douglas-Rachford splitting and a specialized proximal operator to obtain computationally efficient solutions with performance comparable to SDR.

  • B. Squared ℓ∞-Norm Relaxation: The proposed method rewrites the real-valued 1-bit precoding problem as a convex relaxation without lifting to a higher dimension.The relaxation is obtained by dropping nonconvex constraints and produces a problem that can be solved efficiently.
  • B. Squared ℓ∞-Norm Relaxation: The relaxed problem is solved using squared-infinity norm Douglas-Rachford splitting, called SQUID.SQUID is based on two convex functions and iteratively applies their proximal operators.
  • B. Squared ℓ∞-Norm Relaxation: The proximal operator for the squared ℓ∞-norm is presented as a novel component needed by the SQUID method.The paper contrasts it with the previously known proximal operator for the ordinary ℓ∞-norm.
  • B. Squared ℓ∞-Norm Relaxation: The proximal operator for the squared ℓ∞-norm can be evaluated through an efficient procedure based on sorting and thresholding.Algorithm 1 sorts the absolute input values and returns thresholded signed entries.
  • B. Squared ℓ∞-Norm Relaxation: Each SQUID iteration uses simple matrix and vector operations plus the proximal-operator evaluation, yielding performance comparable to SDR at lower computational complexity.The paper also notes that matrix-inverse evaluation can be accelerated using the Woodbury matrix identity and precomputation.

C. Sphere Precoding

Sphere precoding adapts sphere decoding to the 1-bit quantized precoding problem by exploiting a transformed triangular structure. Because the optimal precoding factor is unknown, the method alternates between updating the factor and solving the sphere-precoding problem.

  • C. Sphere Precoding: Sphere decoding solves the closest-vector problem exactly while often reducing average complexity relative to exhaustive search.Its tree-search formulation prunes candidate branches that exceed a prescribed hypersphere radius.
  • C. Sphere Precoding: Sphere precoding adapts sphere decoding to 1-bit quantized precoding by assuming a known optimal precoding factor and reformulating the objective.The resulting precoding problem can be solved using sphere decoding.
  • C. Sphere Precoding: A QR factorization transforms the problem into an equivalent form with an upper-triangular matrix, enabling standard sphere-decoding methods.The triangular structure supports the resulting tree-search procedure.
  • C. Sphere Precoding: Because the optimal precoding factor is unknown in practice, the proposed approach alternates between updating the factor and solving the sphere-precoding problem.Initialization uses the factor obtained from Wiener-filter precoding.
  • C. Sphere Precoding: The alternating procedure usually converges in 1 to 3 iterations and achieves near-optimal performance for small to moderately sized MIMO systems.The paper reports numerical results for a system with B = 8 antennas, while noting that sphere decoding has exponential complexity in the number of variables.

D. Decoding at the UEs

The simulations compare uncoded QPSK BER for linear and nonlinear precoders under coarse DAC quantization. Nonlinear precoding substantially outperforms linear-quantized precoding in the 1-bit massive-MIMO setting, while a few additional DAC levels nearly recover infinite-resolution performance.

  • 1-bit DACs: 4 dB separates optimal nonlinear and infinite-resolution WF precoding at BER 10^-3 for B = 8 and U = 2.SP and SDRr achieve near-optimal performance in this moderately sized system.
  • 1-bit DACs: 1-bit linear-quantized precoders saturate at BER 10^-2 or above in the B = 8, U = 2 comparison.This degradation motivates nonlinear precoding for coarse quantization.
  • Massive MIMO: 3 dB is the SQUID gap to infinite-resolution BER at BER 10^-3, versus 8 dB for WFQ, with B = 128 and U = 16.The larger antenna array lets ZF, WF, and WFQ support error probabilities below 10^-3, but nonlinear precoders remain better.
  • Massive MIMO: At low SNR, all precoders have comparable error-rate performance.Linear-quantized precoders may therefore offer a better performance-complexity trade-off in this regime.

B. Robustness to Channel-Estimation Errors

The paper evaluates precoding with imperfect CSI and validates achievable-rate approximations through simulation. Nonlinear precoders remain advantageous for moderate estimation errors, while low-resolution DACs can still support high sum-rate throughput.

  • Robustness to Channel-Estimation Errors: ε ≤ 0.5 is the range in which nonlinear precoders outperform linear-quantized precoders for 1-bit DACs, B = 128, and U = 16.Here ε = 0 denotes perfect CSI and ε = 1 denotes no CSI.
  • Achievable rate: The rate approximation from (32) matches numerical results well for WF precoding across DAC levels and SNR.The comparison includes the 1-bit lower bound (25) and numerically estimated achievable rates.
  • Achievable rate: High sum-rate throughputs can be achieved despite low-resolution DACs at the BS.This conclusion follows from the simulated validation of the asymptotic approximation.
  • Scope and future work: The analysis assumes frequency-flat channels and DACs operating at symbol-rate sampling frequency.Extending the analysis to OFDM and frequency-selective channels remains future work.

APPENDIX A PROOF OF THEOREM 2

The appendix proves the Bussgang-based covariance relations and derives the quantizer distortion representation used in the paper. It then solves the associated clipping proximal operator by characterizing active and inactive entries and selecting an optimal threshold.

  • Covariance relations: x = Gz + d represents the quantized signal as a gain-scaled input plus distortion uncorrelated with z.This relation follows from the covariance result used in the theorem proof.
  • Covariance relations: The uniform DAC quantizer-mapping function supplies the moments needed to evaluate the covariance expression.The proof uses the independent, identically distributed real and imaginary components of the precoded vector.
  • Proximal operator: The KKT derivation begins from the Lagrangian and its stationarity conditions for the optimization problem.The stationarity equation gives x_k = y_k/(1 + 2u_k).
  • Proximal operator: The proximal operator clips entries with magnitude above α to sgn(y_k)α and leaves smaller-magnitude entries unchanged.This follows from stationarity and complementary slackness in the KKT conditions.
  • Proximal operator: α* is obtained by sorting |y_k|, computing candidate thresholds α_ℓ, and selecting the largest valid candidate.Algorithm 1 implements this threshold-selection and element-wise clipping procedure.
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