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A Framework for One-Bit and Constant-Envelope Precoding over Multiuser Massive MISO Channels

Mingjie Shao, Qiang Li, Wing-Kin Ma, Anthony Man-Cho So

arXiv:1810.03159v2cs.ITeess.SPmath.IT

TL;DR

Multiuser precoding with one-bit, continuous CE, or discrete CE transmit constraints is difficult to optimize, despite its motivation of reducing hardware cost and energy consumption. The paper formulates QAM-based minimax SEP design and develops a penalty-based transformation with a custom first-order algorithm. Simulations report better SEP performance than existing designs and competitive runtime.

  • Problem

    Multiuser precoding is difficult under binary and unit-modulus constraints, while coarsely quantized and CE techniques are motivated by hardware-cost and energy-consumption concerns.

    Source: §1 · pp. 2–3

  • Method

    The paper formulates QAM-based minimax SEP precoding and uses a negative square penalty transformation with a custom first-order algorithm.

    Source: §1.1 · p. 4

  • Results

    The framework provides satisfactory SEP performance with competitive runtime, and GEMM and MM outperform SQUID and QZF in the reported one-bit BER simulations.

    Source: §6.1 · p. 18, §7 · p. 21

  • Takeaways & Limitations

    The framework brings one-bit, continuous CE, and discrete CE precoding into one optimization approach for the multiuser MISO downlink.

    Source: §7 · p. 21

    For a unit-modulus example, the penalized problem has an infeasible locally optimal solution for every finite penalty parameter.

    Source: §4 · p. 12

Abstract

from arXiv · show

Consider the following problem: A multi-antenna base station (BS) sends multiple symbol streams to multiple single-antenna users via precoding. However, unlike conventional multiuser precoding, the transmitted signals are subjected to binary, unit-modulus, or even discrete unit-modulus constraints. Such constraints arise in the one-bit and constant-envelope (CE) massive MIMO scenarios, wherein high-resolution digital-to-analog converters (DACs) are replaced by one-bit DACs and phase shifters, respectively, for cutting down hardware cost and energy consumption. Multiuser precoding under one-bit and CE restrictions poses significant design difficulty. In this paper we establish a framework for designing multiuser precoding under the one-bit, continuous CE and discrete CE scenarios---all within one theme. We first formulate a precoding design that focuses on minimizations of symbol-error probabilities (SEPs), assuming quadrature amplitude modulation (QAM) constellations. We then devise an algorithm for our SEP-based design. The algorithm combines i) a novel penalty method for handling binary, unit-modulus and discrete unit-modulus constraints; and ii) a first-order non-convex optimization recipe custom-built for the design. Specifically, the latter is an inexact majorization-minimization method via accelerated projected gradient, which, as shown by simulations, runs very fast and can handle a large number of decision variables. Simulation results indicate that the proposed design offers significantly better bit-error rate performance than the existing designs.

1 Introduction

One-bit and constant-envelope precoding address hardware and energy costs but make multiuser design difficult because of binary, unit-modulus, or discrete constraints. The paper develops a QAM-focused SEP optimization framework spanning these settings, with a penalty transformation and a fast first-order algorithm.

1 Introduction

  • One-bit and CE precoding make multiuser design difficult because binary, M-ary PSK, or unit-modulus restrictions complicate the transmit-signal design.The literature is described as scattered across particular scenarios and constellations.Source: §1 · pp. 2–3

1.1 This Work and Contributions

  • The framework targets multiuser MISO downlink one-bit, continuous CE, and discrete CE precoding through a minimax SEP design for QAM.The paper also states that its optimization methods can be applied to an M-ary PSK formulation studied in.Source: §1.1 · pp. 3–4
  • The negative square penalty method transforms discrete and/or non-convex equality constraints into convex constraints, enabling a shared treatment of one-bit, discrete CE, and continuous CE designs.Its first-order algorithm combines majorization-minimization with accelerated projected gradient and is reported to run fast on many decision variables.Source: §1.1 · p. 4
  • The framework jointly optimizes precoding and users’ QAM inter-point spacings, which the paper identifies as a key factor in SEP performance.The paper contrasts this more general treatment with prior approaches that pre-fix spacings, predict them through analysis, or assume identical spacing across users.Source: §1.1 · p. 4

1.2 Notations and Some Basic Notions

  • The paper defines notation for vector and matrix operations, norms, inner products, and optimization concepts including gradients and stationarity.It states that smoothness on compact sets supplies the Lipschitz continuity conditions used in the problem setting.Source: §1.2 · pp. 4–5

2 Background

The background models a multiuser downlink in which a massive-antenna base station sends streams to single-antenna users over frequency-flat channels. It sets out the received-signal model and the one-bit, continuous CE, and discrete CE transmit constraints used for QAM precoding.

2 Background

  • The received signal is modeled as the channel response to the transmitted vector plus circular complex Gaussian noise with zero mean and variance σ^2_η.The model defines the transmitted vector, channel, received signal, and block and system dimensions.Source: §2 · pp. 5–6
  • One-bit DACs restrict each transmitted signal’s real and imaginary parts to binary values, while phase-shifter CE transmission uses constant-modulus signals.Continuous phase values or sufficiently fine phase resolution are assumed for the continuous CE model.Source: §2 · p. 6
  • The discrete CE model restricts phase shifters to a finite set of uniformly spaced phase values, and the design assumes QAM symbols.The paper describes the precoding task as nonlinear: the transmitted signal need not be a linear combination of the symbols or a linear-precoder output.Source: §2 · p. 6
  • The base station designs transmitted vectors using channel knowledge so each single-antenna user receives its designated symbol stream.The model assumes frequency-flat channels and a transmission block no longer than the channel coherence time.Source: §2 · pp. 5–6

3 Minimum SEP Problem Formulation

The section formulates precoding to minimize the worst symbol-error probability across users and symbol times, while also optimizing QAM inter-point spacings. It characterizes the SEP objective and develops a simpler related formulation for optimization.

3.1 SEP Characterization

  • The base station determines the half inter-point spacings for received QAM symbols, and users are informed of them during training.The spacings apply separately to the real and imaginary parts.Source: §3.1 · p. 7
  • SEP characterization treats the real and imaginary component error probabilities conditioned on the transmitted symbol as the relevant error measures.The paper calls these conditional probabilities SEPs.Source: §3.1 · p. 7

3.2 A Minimum SEP Formulation

  • The design minimizes the worst SEP across users and symbol times while jointly optimizing the precoder and QAM inter-point spacings.The feasible signal set covers one-bit, continuous CE, and discrete CE precoding.Source: §3.2 · p. 8
  • The direct SEP formulation is difficult to handle because its component error probabilities depend on the Q function, which has no closed form.The paper therefore turns to a closely related optimization problem.Source: §3.2 · p. 8
  • Problem (16) has a piecewise-linear objective that is simpler than Problem (14) and provides both upper- and lower-bound approximations to it.The approximation uses the monotonicity of Q.Source: §3.2 · p. 9
  • Problem (16) has an optimal solution with d* bounded above by ρ.The proposition states that such an optimal solution exists.Source: §3.2 · p. 9
  • The final formulation smooths the maximum using log-sum-exponential, whose approximation becomes tighter as σ decreases and is tight as σ approaches zero.This addresses the non-smooth objective.Source: §3.2 · p. 9
  • With σ = (√π/2)ση, Problem (17) tends to suppress the average SEP.The paper presents this as an interpretation of the formulation.Source: §3.2 · pp. 9–10

4 A Negative Square Penalty Method

The negative square penalty (NSP) replaces discrete or non-convex equality constraints with convex-hull constraints and a non-convex penalty. Under stated regularity conditions, sufficiently large penalty parameters preserve global optima, while local guarantees and DCE approximation results have narrower conditions.

4 A Negative Square Penalty Method

  • NSP uses the penalty term −λ∥u∥2 to push variables in the convex hull of U toward its extreme points, while keeping the constraint set convex.The resulting objective is generally non-convex, even when the original objective is convex.Source: §4 · p. 10
  • For Lipschitz-continuous f, NSP and the original problem have the same global optimal solutions when λ exceeds 2L for one-bit, L for CE, or L/sin(π/M) for DCE.Here L is a Lipschitz constant of f on the convex hull; the theorem also states the converse equivalence.Source: §4 · p. 10
  • For continuously differentiable f, sufficiently large λ makes every local optimum of NSP feasible for the original problem; non-feasible stationary points are saddles or local maxima.The local result requires Lipschitz-continuous gradient, with the stated threshold λ > L/2.Source: §4 · p. 11
  • For DCE, rounding an NSP optimum gives an approximate original-problem solution when M is large, with f(ũ) approaching f⋆ as M → ∞.The stated condition is λ ≥ L/cos(π/M), where L is a Lipschitz constant of f on the convex hull.Source: §4 · pp. 11–12
  • The local-optimality guarantee requires at least continuously differentiable f; for f(u)=|u| on the unit circle, u=0 is locally optimal for NSP at every finite λ but infeasible originally.Thus the local result does not extend to non-smooth objectives in general.Source: §4 · p. 12

5 Gradient Extrapolated Majorization-Minimization

The paper applies an inexact majorization-minimization method, GEMM, to the NSP-transformed precoding problem, using accelerated projected-gradient updates. It establishes a stationarity-rate guarantee under stated assumptions and gives projections needed for one-bit, CE, and DCE implementations.

5.1 The Main Idea

  • GEMM replaces each exact MM subproblem solve with one accelerated projected-gradient update, and empirical studies report faster convergence than MM implemented via APG.The authors report satisfactory SEP performance most of the time.Source: §5.1 · p. 14

5.2 Convergence Guarantee of GEMM

  • The convergence analysis addresses GEMM’s extrapolation, which means its objective values need not decrease monotonically at every iteration.The guarantee is presented for differentiable objectives on non-empty, closed convex sets with the stated smoothness and majorant conditions.Source: §5.2 · pp. 14–15
  • GEMM is guaranteed to find an ϵ-stationary point in O(1/ϵ2) iterations under the stated assumptions.The analysis establishes a convergence rate rather than only asymptotic convergence.Source: §5.2 · p. 15

5.3 Implementation Details

  • For DCE, the projection onto the convex hull has a closed-form expression, avoiding the need to call a convex optimization solver for that projection.The implementation also uses element-wise projections for one-bit and continuous CE cases.Source: §5.3 · p. 15

6 Simulation Results

Monte Carlo simulations evaluate BER and runtime for one-bit, continuous CE, and discrete CE precoding. Across the tested settings, GEMM performs competitively against the benchmarks, including at larger problem sizes and with moderate phase resolution.

6.1 One-Bit Precoding

  • For one-bit precoding at (N, K, T) = (128, 16, 10), GEMM and MM outperform SQUID and QZF; their SNR gap to ZF is about 5 dB for 16-QAM and over 10 dB for 64-QAM.Unlike SQUID and QZF, GEMM and MM show no error-floor effects.Source: §6.1 · p. 18, Figure 5 · p. 18, Figure 6 · p. 19
  • GEMM is about three times faster than MM and is fastest among the compared algorithms at N = 256 and N = 512.The runtime comparison uses K = 16, T = 10, and 64-QAM.Source: §6.1 · pp. 18–19, Table 1 · p. 19
  • For N = 256, K = 24, and T = 200, MM and GEMM run on a problem with over 100,000 decision variables; GEMM takes about 4 seconds per trial, while SQUID could not be run.MM and GEMM provide reasonably good BER performance in this larger setting.Source: §6.1 · p. 19, Figure 7 · p. 20

6.2 CE Precoding

  • For CE precoding, GEMM beats MUImin and QZF, stays about 2 dB from ZF at 16-QAM, and remains within 5 dB of ZF at 256-QAM.For 64-QAM at (K, T) = (16, 10), GEMM runs faster than MUImin; GEMM and MM have almost the same BER, with GEMM faster.Source: §6.2 · p. 20, Table 2 · p. 20, Figure 8 · p. 21

6.3 DCE Precoding

  • With 16 phase combinations, DCE approaches CE BER performance; with 8, it is about 2 dB away for 64-QAM at (N, K, T) = (128, 16, 100).The authors suggest moderate phase resolution can achieve near-CE performance.Source: §6.3 · p. 20

7 Conclusion

The paper concludes that its optimization framework covers one-bit, continuous CE, and discrete CE precoding for multiuser MISO downlink. Simulations indicate satisfactory SEP performance and competitive runtime, and the authors identify more challenging settings as future directions.

7 Conclusion

  • The framework addresses one-bit, continuous CE, and discrete CE precoding in the multiuser MISO downlink scenario.The authors hope it will inform work on multiuser MIMO and imperfect channel information.Source: §7 · pp. 21–22
  • Simulations indicate satisfactory SEP performance and competitive runtime for the proposed framework.Source: §7 · p. 21

A Proof of Proposition 1

The proof reduces the optimization over d for a fixed optimal U to separate real- and imaginary-part problems. It then establishes boundedness and optimal-solution properties needed to complete the argument.

A Proof of Proposition 1

  • For fixed U = U⋆, the proof reduces Problem (16) to Problem (38), then decouples it into Problems (35) and (37) for the real and imaginary components.An optimal solution to the decoupled problem yields an optimal solution to Problem (16).Source: A Proof of Proposition 1 · pp. 22–23
  • The proof establishes that the objective in Problem (35) is bounded below, enabling an unconstrained minimizer with the required bound on its component.It applies Lemma 1 after showing boundedness.Source: A Proof of Proposition 1 · p. 23
  • The proof also establishes the corresponding bound for the imaginary component and combines the component bounds to support Proposition 1.Source: A Proof of Proposition 1 · pp. 23–24

B Proof of Lemma 1

The lemma proof rewrites the one-dimensional optimization as a linear objective over a polyhedron. Under the stated assumptions, an optimal extreme point exists, yielding the claimed finite candidate structure.

B Proof of Lemma 1

  • Problem (39) is rewritten as Problem (42) over a polyhedron, with an optimal solution mapping back to an optimal solution of Problem (39).The epigraph variable attains the value of the original objective at the optimum.Source: B Proof of Lemma 1 · p. 24
  • Because the vectors include two distinct a_i values and the objective is bounded below, the polyhedron has an extreme point and an optimal solution can be chosen there.Source: B Proof of Lemma 1 · p. 24
  • At that extreme-point optimum, two active constraint vectors are linearly independent.Source: B Proof of Lemma 1 · p. 24

C Proof of Theorem 1

The proof shows that, for a sufficiently large penalty, the relaxed problem has the same optimal solutions as the constrained problem. It establishes this by showing that optimal solutions of the relaxed problem lie in the feasible set under the penalty condition.

C Proof of Theorem 1

  • For λ ≥ λ̄, every optimal solution of Problem (45) lies in U_n, making the optimal solution sets of Problems (44)–(45) equivalent.The proof establishes feasibility coordinate by coordinate.Source: C Proof of Theorem 1 · p. 25
  • In the continuous CE case, λ > L makes the boundary replacement satisfy the required inequality.The proof replaces an interior point by a unit-modulus point with the same phase.Source: C Proof of Theorem 1 · p. 25
  • For the discrete CE case with even M ≥ 4, including one-bit precoding at M = 4, the proof gives a sufficient threshold λ > L/sin(π/M).The threshold follows from bounds involving cos(π/M) and sin(π/M).Source: C Proof of Theorem 1 · pp. 25–26

D Proof of Theorem 2

The proofs establish properties of the penalized objective: sufficiently large penalties make it strongly concave and force local optima to the feasible set's extreme points. They also give a discrete-CE approximation bound and exhibit an infeasible local optimum for the continuous CE relaxation.

D Proof of Theorem 2

  • When λ > L/2, Fλ is strongly concave on the relaxed feasible set, so any local optimum must be an extreme point and therefore feasible.The proof derives strong concavity with parameter α = L − 2λ.Source: D Proof of Theorem 2 · p. 26

E Proof of Corollary 1

  • The discrete-CE feasible projection has objective gap at most L√n sin(π/M) from the relaxed optimum.The proof bounds each coordinate's projection distance by sin(π/M).Source: E Proof of Corollary 1 · p. 27

F Proof of Fact 3

  • For the continuous CE relaxation, u = 0 is locally optimal when |u| ≤ 1/(4λ), although it is infeasible for the original problem.Here Fλ(u) = |u| − λ|u|² on the unit disk.Source: F Proof of Fact 3 · p. 27

G Proof of Theorem 3

The proof analyzes the GEMM update through a majorization inequality and derives bounds on both iterate changes and first-order stationarity. These bounds use the update's algorithmic constants and the objective gap.

G Proof of Theorem 3

  • The first-order stationarity residual is bounded by terms proportional to successive iterate differences, with constants set by the gradient and majorization bounds.The stated bound uses α_k, L_G, β_k, and L_F.Source: G Proof of Theorem 3 · p. 28
  • Applying Lemma 2 to the GEMM updates yields an inequality relating successive objective values.The derivation uses the majorizing function and the projected-gradient lemma.Source: G Proof of Theorem 3 · p. 28
  • The proof bounds accumulated squared iterate changes using the initial objective gap and algorithm constants.The bound involves the minimum over iterations of successive-step norms.Source: G Proof of Theorem 3 · p. 29
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