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Branch-and-Bound Precoding for Multiuser MIMO Systems with 1-Bit Quantization

Lukas T. N. Landau, Rodrigo C. de Lamare

arXiv:1705.00122v2cs.ITmath.IT

TL;DR

The paper addresses precoding for multiuser MIMO systems where low energy consumption and hardware complexity motivate constant-magnitude transmit symbols and 1-bit DACs. The designs maximize minimum distance to receiver decision thresholds, using constant-magnitude symbols or a discrete 1-bit-DAC formulation solved by branch-and-bound with a relaxation-based alternative. The methods show superior BER versus existing 1-bit-DAC methods for M = 10 and below, KL = 2; 1-bit-DAC loss is less than 2 dB versus PoP, and proposed methods are gainful in sum-rate at medium SNR.

  • Problem

    The paper addresses precoding for multiuser MIMO systems where low energy consumption and hardware complexity motivate constant-magnitude transmit symbols and 1-bit DACs.

    Source: §I · p. 1

  • Method

    The designs maximize minimum distance to receiver decision thresholds, using constant-magnitude symbols or a discrete 1-bit-DAC formulation solved by branch-and-bound with a relaxation-based alternative.

    Source: §I · p. 1, §IV · p. 2, B. Concept of Branch-and-Bound Precoding · p. 2

  • Results

    The methods show superior BER versus existing 1-bit-DAC methods for M = 10 and below, KL = 2; 1-bit-DAC loss is less than 2 dB versus PoP, and proposed methods are gainful in sum-rate at medium SNR.

    Source: §V · p. 4, §VI · p. 4

  • Takeaways & Limitations

    The exact and relaxation-based designs provide high-performing 1-bit precoding approaches that can serve as benchmarks for designers.

    Source: §VI · p. 4

Abstract

from arXiv · show

Multiple-antenna systems is a key technique to serve multiple users in future wireless systems. For low energy consumption and hardware complexity we first consider transmit symbols with constant magnitude and then 1-bit digital-to-analog converters. We propose precoding designs which maximize the minimum distance to the decision threshold at the receiver. The precoding design with 1-bit DAC corresponds to a discrete optimization problem, which we solve exactly with a branch-and-bound strategy. We alternatively present an approximation based on relaxation. Our results show that the proposed branch-and-bound approach has polynomial complexity. The proposed methods outperform existing precoding methods with 1-bit DAC in terms of uncoded bit error rate and sum-rate. The performance loss in comparison to infinite DAC resolution is small.

I. INTRODUCTION

The introduction motivates low-resolution DACs in multiuser MIMO and presents precoding designs that maximize the minimum distance to receiver decision thresholds. It proposes constant-magnitude phase-only precoding and a 1-bit-DAC design solved exactly by branch-and-bound, with relaxation as an alternative.

I. INTRODUCTION

  • In multiuser MIMO, DACs at the base station merit attention because the base station commonly has more antennas than the users’ total receive antennas, and converters can affect system cost in short-range links.The introduction identifies 1-bit DACs as promising and assumes pulse shaping can be performed in the analog domain.Source: §I · p. 1
  • Prior work includes linear precoders such as maximal-ratio transmission and zero-forcing, plus nonlinear approaches, including one with complexity linear in the number of antennas.The cited approaches are,,, and, respectively.Source: §I · p. 1
  • The proposed design maximizes the minimum distance to receiver decision thresholds, using constant-magnitude phase-only precoding and a 1-bit-DAC formulation solved by branch-and-bound or approximated by relaxation.The branch-and-bound bounding step uses a linear-program relaxation.Source: §I · p. 1

II. SYSTEM MODEL

The system model is a multiuser MIMO downlink with a base station transmitting to multiple users over a frequency-flat fading channel. Transmitted symbols are received after matched filtering and passed to hard-decision detectors or equivalent 1-bit ADCs.

II. SYSTEM MODEL

  • The downlink has M transmit antennas at the base station and K users with L antennas each; the transmit vector has length M and the symbol vector length KL.The channel matrix has dimensions KL × M and comprises user-specific L × M matrices.Source: §II · pp. 1–2
  • Each user-antenna symbol belongs to {1 + j, 1 − j, −1 + j, −1 − j}, over a frequency-flat channel with zero-mean i.i.d. complex Gaussian coefficients.Thermal noise is described as zero-mean i.i.d. complex Gaussian with variance σ²_n.Source: §II · p. 1
  • After matched filtering, received samples are sent to hard-decision detectors or equivalent 1-bit ADCs, which quantize the output to the four possible complex sign combinations.The model’s receive vector collects KL samples.Source: §II · p. 2

III. PROPOSED PHASE-ONLY PRECODING (POP)

Phase-only precoding chooses constant-magnitude transmit symbols to maximize the minimum distance to receiver decision thresholds. This distance-based criterion is intended to provide robustness to received-signal perturbations.

III. PROPOSED PHASE-ONLY PRECODING (POP)

  • The phase-only precoder maximizes the minimum distance ε to the decision thresholds, a criterion described as providing robustness against received-signal perturbations.The criterion is used instead of BER or achievable rate as the optimization objective.Source: §III · p. 2
  • In phase-only precoding, each transmit entry x_m has constant magnitude; the constraint is expressed using an inequality in the optimization problem.If needed, the equality |x_m| = 1/M is introduced by scaling the optimal vector’s entries.Source: §III · p. 2

IV. PROPOSED PRECODING WITH 1-BIT DAC

The 1-bit-DAC design uses the same decision-threshold-distance objective as the preceding section, while constraining transmit entries to a discrete set. The resulting optimization problem is non-convex.

IV. PROPOSED PRECODING WITH 1-BIT DAC

  • The 1-bit-DAC precoder applies the preceding section’s objective while assuming analog pulse shaping.The optimization problem changes under this DAC constraint.Source: §IV · p. 2
  • Each transmit entry x_m is constrained to the discrete set P, making the optimization problem non-convex.The entries are indexed for m = 1, …, M.Source: §IV · p. 2

A. Proposed Approximate 1-Bit Precoder (1-Bit Approx.)

The approximate 1-bit precoder relaxes the input constraint, then maps the relaxed solution to the nearest discrete input vector. The relaxed optimum provides a bound on the discrete problem’s optimum.

A. Proposed Approximate 1-Bit Precoder (1-Bit Approx.)

B. Concept of Branch-and-Bound Precoding

Branch-and-bound solves the constrained precoding minimization by bounding subproblems after fixing some discrete entries. Bounds can eliminate fixed-vector choices and their extensions before exhaustive search.

B. Concept of Branch-and-Bound Precoding

C. Proposed 1-Bit Branch-and-Bound Algorithm (1-Bit B&B)

The proposed 1-bit branch-and-bound algorithm solves the discrete precoding problem using a real-valued description and bounds from relaxed subproblems. Its breadth-first search prunes candidates until the solution can be obtained by exhaustive search.

C. Proposed 1-Bit Branch-and-Bound Algorithm (1-Bit B&B)

V. NUMERICAL RESULTS

The proposed 1-bit precoders outperform existing 1-bit methods in uncoded BER, while the exact and approximate designs have nearly identical BER for larger arrays. The methods also improve sum rate at medium SNR and the branch-and-bound approach has polynomial complexity.

V. NUMERICAL RESULTS

  • For M = 10 and below with KL = 2, the proposed 1-bit methods outperform existing 1-bit-DAC precoders in BER.For M = 16 and above, the approximation and exact solution have almost identical BER.Source: §V · pp. 3–4, Figure 2 · p. 4
  • The 1-bit-DAC performance loss is no more than 2 dB versus PoP, while 8-PSK-like phase quantization at the transmitter brings only marginal loss.The comparison is to the proposed PoP high-resolution design.Source: §V · pp. 3–4
  • At medium SNR, the proposed methods are gainful in sum rate averaged over varied channel realizations.The right-hand panel of Fig. 3 reports sum rate for M = 10 and KL = 2.Source: §V · p. 4, Figure 3 · p. 4
  • The branch-and-bound method has polynomial complexity, compared with exponential complexity for exhaustive search and O(n^3) for ZF precoders.The complexity estimate assumes LP subproblems solved by interior-point methods; active-set methods can be practically more efficient with warm starts.Source: §V · p. 4, Table I · p. 4

VI. CONCLUSIONS

The conclusion presents exact and approximate 1-bit precoding designs based on maximizing the minimum distance to receiver decision thresholds. Both show strong uncoded BER performance, with less than 2 dB loss from 1-bit DACs.

VI. CONCLUSIONS

  • The proposed 1-bit precoding criterion maximizes the minimum distance to receiver decision thresholds.Source: §VI · p. 4
  • Both the exact solution and the continuous-relaxation approximation followed by mapping show outstanding uncoded BER performance and can serve as benchmarks for 1-bit precoding.Source: §VI · p. 4
  • The simulated performance loss from 1-bit DACs is less than 2 dB.Source: §VI · p. 4
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