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Massive MIMO 1-Bit DAC Transmission: A Low-Complexity Symbol Scaling Approach

Ang Li, Christos Masouros, Fan Liu, A. L. Swindlehurst

arXiv:1709.08278v3eess.SPcs.ITmath.IT

TL;DR

Massive MIMO downlink with 1-bit DACs presents a hardware-complexity and power challenge, motivating beamforming designs based on constructive interference. The paper proposes quantized linear, non-linear mapping, and low-complexity symbol scaling schemes; in massive MIMO, symbol scaling offers a better performance-complexity tradeoff, while non-linear mapping has a performance advantage.

  • Problem

    How to design constructive-interference beamforming for massive MIMO downlink with 1-bit DACs, given the hardware complexity and power consumption of large DAC arrays.

    Source: §I · pp. 1–2

  • Method

    The paper develops quantized linear and non-linear mapping schemes, then proposes symbol scaling that directly selects quantized transmit signals sequentially across antenna elements.

    Source: §I · p. 2, §VII · p. 13

  • Results

    In massive MIMO, non-linear mapping has a performance advantage, while symbol scaling offers a better performance-complexity tradeoff; in small-scale MIMO, symbol scaling is superior.

    Source: §VII · p. 14

  • Takeaways & Limitations

    The symbol scaling scheme's performance-complexity tradeoff favours its usefulness in practical systems.

    Source: §VII · p. 14

Abstract

from arXiv · show

We study multi-user massive multiple-input single-output (MISO) systems and focus on downlink transmission, where the base station (BS) employs a large antenna array with low-cost 1-bit digital-to-analog converters (DACs). The direct combination of existing beamforming schemes with 1-bit DACs is shown to lead to an error floor at medium-to-high SNR regime, due to the coarse quantization of the DACs with limited precision. In this paper, based on the constructive interference we consider both a quantized linear beamforming scheme where we analytically obtain the optimal beamforming matrix, and a non-linear mapping scheme where we directly design the transmit signal vector. Due to the 1-bit quantization, the formulated optimization for the non-linear mapping scheme is shown to be non-convex. To solve this problem, the non-convex constraints of the 1-bit DACs are firstly relaxed, followed by an element-wise normalization to satisfy the 1-bit DAC transmission. We further propose a low-complexity symbol scaling scheme that consists of three stages, in which the quantized transmit signal on each antenna element is selected sequentially. Numerical results show that the proposed symbol scaling scheme achieves a comparable performance to the optimization-based non-linear mapping approach, while its corresponding complexity is negligible compared to that of the non-linear scheme.

I. INTRODUCTION

The paper targets the hardware cost of massive-MIMO downlink with 1-bit DACs and develops constructive-interference beamforming methods. It compares an analytically characterized linear scheme with nonlinear and sequential symbol-scaling approaches.

I. INTRODUCTION

  • Existing 1-bit downlink schemes either degrade relative to unquantized transmission or require computationally inefficient optimization, motivating symbol-level constructive-interference designs.The paper also motivates 1-bit DACs as a way to reduce RF-chain power consumption and hardware complexity.Source: §I · pp. 1–2
  • In massive MIMO, symbol scaling is comparable to nonlinear mapping, while nonlinear mapping improves over existing schemes and symbol scaling has negligible relative complexity.In small-scale MIMO, symbol scaling achieves the best performance.Source: §I · p. 2

II. SYSTEM MODEL

  • Massive-MIMO downlink is modeled with 1-bit DACs at the BS, K single-antenna users, ideal receiver ADCs, perfect CSI, and uniform transmit power allocation.The symbol vector is assumed to come from a normalized PSK constellation.Source: §II · p. 3

A. Constructive Interference and Constructive Region

  • Constructive interference pushes received signals away from modulation detection thresholds; locating a signal in the constructive region increases its distance from those thresholds.Source: A. Constructive Interference and… · p. 3

B. 1-Bit Transmission Scheme - Linear Beamforming

  • The massive-MIMO quantized linear scheme has optimality conditions yielding real, identical user multipliers, and its quantized output is equivalent to quantized ZF, which has a high-SNR error floor.The linear beamformer is posed as an SOCP optimization.Source: B. 1-Bit Transmission Scheme - Linear… · pp. 4–6

C. 1-Bit Transmission Scheme - Non-linear Mapping

The nonlinear mapping scheme directly designs the transmit signal vector using a constructive-interference formulation. Its optimization is non-convex because of the 1-bit DAC output constraint.

C. 1-Bit Transmission Scheme - Non-linear Mapping

1) Relaxation:

The relaxation step replaces the strict 1-bit transmit-signal constraint with a relaxed constraint. This yields a convex optimization problem that can be solved with convex optimization tools.

1) Relaxation:

  • The relaxed problem P4 retains the user constructive-interference constraints while replacing the strict 1-bit signal constraint.Source: 1) Relaxation · p. 6
  • The relaxed problem P4 is convex and can be solved with convex optimization tools.Source: 1) Relaxation · p. 6
  • The relaxed transmit vector is denoted x̂_T, and x̂_n denotes its n-th entry.Source: 1) Relaxation · p. 6

2) Normalization:

The relaxed solution may not satisfy exact 1-bit transmission, so element-wise normalization sets the DAC outputs to the required levels. The deviation ratio η decreases as antenna count increases, supporting asymptotic optimality of the relaxed problem in the stated setting.

2) Normalization:

  • Element-wise normalization forces the real and imaginary parts of the DAC output to meet the 1-bit transmission constraint.The relaxed solution does not always guarantee equality for both parts.Source: 2) Normalization · p. 6
  • Most entries of the relaxed solution already satisfy the strict 1-bit equality requirement, so only a few require normalization.The deviation ratio η measures the share of entries that do not satisfy 1-bit transmission; η = 0 makes P4 equivalent to P3.Source: 2) Normalization · p. 6
  • η decreases as the number of transmit antennas increases, indicating that the relaxed optimization becomes asymptotically optimal for massive MIMO.The reported study uses K = 4 users and 500 channel realizations.Source: 2) Normalization · p. 6

IV. PROPOSED LOW-COMPLEXITY SYMBOL SCALING APPROACH

The approach transforms constructive-interference constraints into symbol scaling along detection-threshold directions, then uses a three-stage procedure to select 1-bit outputs sequentially. It is proposed as a lower-complexity alternative to nonlinear optimization, with comparable performance and possible gains in small-scale MIMO.

IV. PROPOSED LOW-COMPLEXITY SYMBOL SCALING APPROACH

  • The three-stage symbol scaling scheme offers comparable performance with much lower complexity than nonlinear mapping, and can outperform it in small-scale MIMO.The small-scale advantage is attributed to avoiding relaxation and normalization.Source: §IV · p. 6

A. A New Look at the Constructive Interference Criteria

  • The coordinate transformation decomposes each data symbol along its two corresponding detection thresholds to define scaling bases.For 8-PSK, the text illustrates the decomposition using the two threshold directions for a constellation point.Source: A. A New Look at the Constructive… · p. 7
  • The transformed optimization maximizes the minimum value across its entries, and removes the need for transmit-signal relaxation and normalization.This formulation motivates initialization, allocation, and refinement stages.Source: A. A New Look at the Constructive… · pp. 7–8

IV. PROPOSED LOW-COMPLEXITY SYMBOL SCALING APPROACH

  • The scheme selects each antenna element’s 1-bit DAC output sequentially through initialization, allocation, and refinement.The sequential selection is presented as a response to the high computational complexity of nonlinear optimization in large arrays.Source: A. A New Look at the Constructive… · p. 8

B. Initialization Stage

Initialization allocates some transmit-signal entries before the allocation stage handles the residual entries. It first exploits columns whose entries share a sign, with a fallback that chooses a column having the greatest effect on Λ.

B. Initialization Stage

  • When a column’s entries share the same sign, initialization assigns the corresponding x_i^E that sign, which guarantees an increase in every entry of Λ.The sign-identity condition identifies columns for which this assignment is optimal.Source: B. Initialization Stage · p. 8
  • The set V tracks columns whose corresponding x_i^E values have been allocated; when all 2Nt entries are allocated, t equals Λ.The initialization algorithm outputs t and V.Source: B. Initialization Stage · p. 8
  • If no column satisfies the sign-identity condition, initialization selects one column with the maximum effect on Λ.The initialization stage then assigns the corresponding entry and forms the temporary value t.Source: B. Initialization Stage · p. 8

C. Allocation Stage

The allocation stage assigns values to entries left unallocated during initialization. It considers both Sum-Max and Max-Min allocation criteria.

C. Allocation Stage

  • The allocation stage assigns x_i^E for each residual index i in W.W contains the entries not allocated during initialization.Source: C. Allocation Stage · p. 8
  • W is the set of x_i^E entries whose values remain unallocated after initialization.These are the entries handled in the allocation stage.Source: C. Allocation Stage · p. 8
  • The allocation scheme considers both Sum-Max and Max-Min criteria.The passage names these alternatives without specifying their procedures here.Source: C. Allocation Stage · p. 8

1) Sum-Max:

The allocation stage offers Sum-Max and Max-Min criteria for assigning 1-bit outputs. Sum-Max maximizes the aggregate objective, while Max-Min targets the smallest entry during allocation.

1) Sum-Max:

  • Sum-Max chooses each expanded signal entry's sign to match the corresponding entry of m, maximizing the sum objective.The criterion maximizes the sum of the entries in the objective vector, but does not ensure each individual α_l is large or positive.Source: 1) Sum-Max · p. 8
  • Because Sum-Max does not optimize each α_l individually, one value can be very small or negative, motivating the refinement stage.The passage notes this can lead to performance loss.Source: 1) Sum-Max · p. 8

2) Max-Min:

  • Max-Min repeatedly selects the entry with largest absolute value in the row of t containing its current minimum, then updates V and t until allocation is complete.The minimum entry's row index q determines which row of M guides the next selection.Source: 2) Max-Min · p. 9

D. Refinement Stage

The refinement stage tests sequential sign changes against the current minimum in Λ and retains improvements. The paper then describes the combined algorithm and analyzes the computational costs of exhaustive search and optimization-based mapping.

D. Refinement Stage

  • The refinement stage flips one signal entry at a time and keeps the sign that yields a larger minimum in Λ.It processes each entry sequentially, starting from a vector produced by either allocation criterion.Source: D. Refinement Stage · p. 9

E. Algorithm

V. COMPUTATIONAL COMPLEXITY ANALYSIS

  • The complexity analysis counts real multiplications and additions for the proposed schemes, exhaustive search, and the Pokemon method.The comparison uses floating-point operations as its measure.Source: §V · p. 10

A. Exhaustive Search

  • Exhaustive search requires 8KN_t·2^(2N_t) operations and is inapplicable in massive MIMO because of its overwhelming computational cost.Each antenna element has four possible output values under 1-bit quantization.Source: A. Exhaustive Search · p. 10

B. Optimization-based Non-linear Mapping P4

  • For optimization-based non-linear mapping, the relaxation stage's dominant cost is solving P5 with an interior-point method; its real formulation has M=2N_t variables and N=2K constraints.The normalization stage also requires a one-dimensional search over 2N_t entries to find signals violating the 1-bit output constraint.Source: B. Optimization-based Non-linear… · p. 10

C. Symbol Scaling Scheme

The symbol-scaling complexity analysis accounts for initialization, allocation, and refinement, including additional work for Max-Min. It compares this approach with Pokemon and reports a substantially smaller operation count for symbol scaling.

C. Symbol Scaling Scheme

  • The symbol-scaling stages calculate and update t, with a stated cost of O(8KN_t) for the update work.The analysis also accounts for allocation and refinement, and Max-Min adds operations for iterative residual-entry selection.Source: C. Symbol Scaling Scheme · pp. 10–11

D. Pokemon

  • Pokemon's per-iteration work includes calculating q=Ux and projecting the signal vector x, with total cost depending on its iteration limit.The matrix-vector calculation requires 4N_t^2 multiplications and 4N_t^2 additions per iteration.Source: D. Pokemon · p. 11
  • As transmit antennas increase, with ‘Pokemon’ using nmax = 20 following, symbol scaling requires approximately 12% of Pokemon’s operations, while the proposed non-linear mapping costs more than both symbol scaling and Pokemon.Source: D. Pokemon · p. 11

VI. NUMERICAL RESULTS

The simulations compare the proposed schemes across MIMO scales, modulation settings, computational costs, and imperfect CSI. The conclusion reports that symbol scaling is especially effective in small-scale systems, while in massive MIMO it offers a better performance-complexity tradeoff than the performance-leading non-linear mapping method.

VI. NUMERICAL RESULTS

  • For Nt = 8 and K = 2, Symbol Scaling has the best BER among the 1-bit schemes; relaxation-normalization may worsen detection in this small-scale setting.The authors attribute the proposed scheme's promising performance to selecting each quantized antenna signal directly, without relaxation or quantization.Source: Figure 4 · p. 11, §VI · p. 11
  • In massive MIMO, symbol scaling incurs around 2 dB SNR loss versus Pokemon while using approximately 12% of its computational cost.The comparison is for Nt = 128, K = 16, and QPSK.Source: Figure 5 · p. 12, §VI · p. 12
  • For nmax = 20, Symbol Scaling's execution time is approximately 12% of Pokemon's across 10 channel realizations.The runtime of Symbol Scaling is similar to Max-Min because most allocation-stage computation comes from the Max-Min criterion.Source: Figure 7 · p. 12, §VI · p. 12
  • Pokemon's BER improves as its iteration count rises, with diminishing gains at larger nmax and its best performance around nmax = 25.At nmax = 2, when Pokemon and symbol scaling have similar computational cost, symbol scaling achieves better performance.Source: §VI · pp. 12–13, Figure 8 · p. 13
  • At the same computational cost, Symbol Scaling gains performance over Pokemon with nmax = 2, especially as the antenna count grows.The optimization-based method has the best performance but the highest complexity; Sum-Max is less complex than Pokemon with nmax = 2 and has better BER when Nt is large.Source: Figure 9 · p. 13, §VI · p. 13
  • With imperfect CSI modeled by a noisy channel estimate, non-linear mapping has the best performance among the 1-bit schemes, while symbol scaling achieves comparable performance at greatly reduced computational cost.The generic model uses an error matrix with variance δ = β/ρ; the authors note that an exact imperfect-CSI model for this quantized setting is not yet known.Source: Figure 10 · p. 13, §VI · p. 13

VII. CONCLUSION

  • The conclusion reports that symbol scaling performs best in small-scale MIMO, while in massive MIMO non-linear mapping leads in performance and symbol scaling offers a better performance-complexity tradeoff.The conclusion identifies symbol scaling's practical usefulness through that tradeoff.Source: §VII · p. 14

APPENDIX COORDINATE TRANSFORMATION

The appendix illustrates a coordinate transformation for a selected PSK constellation point. It represents the received vector in a real-imaginary basis and then relates it to the plane expanded by the point's detection thresholds.

APPENDIX COORDINATE TRANSFORMATION

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