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One-Bit MUSIC

Xiaodong Huang, Bin Liao

arXiv:1901.05109v1eess.SP

TL;DR

One-bit DOA estimation seeks source directions from array measurements while using low-resolution quantization. This paper applies MUSIC directly to the one-bit covariance matrix, and reports performance comparable to covariance reconstruction particularly at mildly low SNRs.

  • Problem

    The paper studies DOA estimation from one-bit array measurements, a setting where prior approaches reconstruct the unquantized covariance matrix or use sparse recovery.

    Source: §I · p. 1

  • Method

    The method uses the eigendecomposition of the one-bit covariance matrix to obtain signal and noise subspaces for MUSIC, without extra preprocessing.

    Source: §I · p. 1, §III · p. 3

  • Results

    For SNR below 0 dB, One-bit MUSIC shows no visible performance difference from Reconstruction One-bit MUSIC; at high SNR and large snapshot counts, approximation error causes a performance gap.

    Source: §IV · pp. 3–4

  • Takeaways & Limitations

    The one-bit covariance matrix can be used directly to obtain DOA signal and noise subspaces, simplifying implementation relative to covariance reconstruction.

    Source: §I · p. 1, §V · p. 4

    The subspace result is stated for identical isotropic sensors and sufficiently small signal SNRs.

    Source: §III · p. 3

Abstract

from arXiv · show

In this letter, we consider the problem of direction-of-arrival (DOA) estimation with one-bit quantized array measurements. With analysis, it is shown that, under mild conditions the one-bit covariance matrix can be approximated by the sum of a scaled unquantized covariance matrix and a scaled identity matrix. Although the scaling parameters unknown because of the extreme quantization, they do not affect the subspace-based DOA estimators. Specifically, the signal and noise subspaces can be straightforwardly determined through the eigendecomposition of the one-bit covariance matrix, without pre-processing such as unquantized covariance matrix reconstruction. With so-obtained subspaces, the most classical multiple signal classification (MUSIC) technique can be applied to determine the signal DOAs. The resulting method is thus termed as one-bit MUSIC. Thanks to the simplicity of this method, it can be very easily implemented in practical applications, whereas the DOA estimation performance is comparable to the case with unquantized covariance matrix reconstruction, as demonstrated by various simulations.

I. INTRODUCTION

Low-resolution one-bit ADCs address hardware cost and power concerns in large antenna arrays. The paper proposes applying subspace-based DOA estimation directly to the one-bit covariance matrix, without covariance reconstruction or other preprocessing.

I. INTRODUCTION

  • One-bit ADCs use simple comparators and consume negligible circuit power, addressing hardware cost and power concerns in large-scale antenna arrays.ADC power consumption grows exponentially with quantization bit number.Source: §I · p. 1
  • Prior one-bit DOA methods reconstruct the unquantized covariance matrix or use sparse recovery formulations.Earlier work also applied covariance reconstruction to sparse arrays.Source: §I · p. 1
  • For uncorrelated signals at mildly low SNR, the one-bit covariance matrix approximates a scaled unquantized covariance matrix, with diagonal errors irrelevant to signal and noise subspaces.This allows MUSIC to use the one-bit covariance matrix directly, without extra preprocessing.Source: §I · p. 1

II. ONE-BIT SIGNAL MODEL

The signal model describes K narrowband far-field sources observed by an M-element array, with uncorrelated Gaussian signals and noise. One-bit measurements are formed using a complex element-wise quantizer composed of sign functions.

II. ONE-BIT SIGNAL MODEL

  • The model considers K narrowband far-field signals arriving from distinct directions at an M-element array, with signal and noise modeled as uncorrelated, independent, zero-mean circular complex Gaussian processes.The steering matrix contains the direction-specific steering vectors, while the signal and noise vectors collect the source and sensor components.Source: §II · p. 1
  • One-bit array outputs are formed with a complex-valued element-wise quantization function composed of two sign functions.The real and imaginary parts of a complex value are treated separately by the sign functions.Source: §II · p. 1
  • The prior arcsine-law approach reconstructs the unquantized covariance matrix from the one-bit covariance matrix, whereas this paper proposes applying subspace methods directly to the latter.The direct approach requires no extra preprocessing.Source: §II · p. 1

III. ONE-BIT MUSIC

For identical isotropic sensors, the paper approximates the one-bit covariance matrix using scaled unquantized covariance and identity terms, with a low-SNR condition and a diagonal-entry exception. Its eigenspaces support direct MUSIC estimation, and experiments report good performance even at high SNR.

III. ONE-BIT MUSIC

  • At sufficiently low SNR, the one-bit covariance is approximated by scaled unquantized covariance plus a scaled identity, but the approximation does not apply to diagonal entries.The analysis assumes identical isotropic sensors and equal-power signals.Source: §III · pp. 2–3
  • For identical isotropic sensors at sufficiently small signal SNRs, eigendecomposition of the one-bit covariance matrix yields the signal and noise subspaces.The paper notes that the covariance matrices have nearly identical eigenvectors despite the unknown scaling parameter.Source: §III · p. 3
  • MUSIC can estimate DOAs from those subspaces using the eigendecomposition of the one-bit covariance matrix.The algorithm outputs the noise subspace from the M − L smallest eigenvalues and selects DOAs at the L highest spectrum peaks.Source: §III · p. 3
  • Experiments report that the subspace method using the approximate covariance can also provide good performance at high SNR.The method is presented as theoretically requiring small SNR, while experiments show good performance at high SNR levels.Source: §III · p. 3

IV. SIMULATION RESULTS

Simulations compare one-bit MUSIC with covariance-reconstruction and unquantized baselines. One-bit MUSIC generally matches reconstruction at mildly low SNR, while approximation error and performance gaps grow in some higher-SNR settings.

IV. SIMULATION RESULTS

  • The one-bit covariance approximation error increases with SNR, consistent with the preceding analysis.The plotted error is computed using the specified covariance expressions while assuming Rx and p are known.Source: Figure 3 · p. 3, §IV · p. 3
  • Below 0 dB, one-bit MUSIC and reconstruction show no visible RMSE difference, while their gap grows as SNR increases.The comparison tests 100, 500, and 1,000 snapshots.Source: §IV · p. 3, Figure 4 · p. 4
  • At mildly low SNR, one-bit MUSIC matches reconstruction across snapshot counts; at high SNR with many snapshots, a performance gap appears but remains practically ignorable.The simulations vary snapshots from 100 to 1,000 at −10, 0, and 10 dB.Source: Figure 5 · p. 4, §IV · p. 4
  • At N = 100, the two approaches perform nearly the same even at high SNR, where limited-snapshot error may dominate approximation error.This result indicates that approximation error need not determine performance when snapshot counts are small.Source: §IV · p. 4
  • Covariance reconstruction provides a resolution-probability improvement only at high SNR.Successful resolution is defined by both DOA-estimate biases being below half the angular separation.Source: §IV · p. 4

V. CONCLUSION

The one-bit measurement covariance can directly provide DOA signal and noise subspaces, especially at relatively low SNRs. Although approximation error grows at high SNRs, the authors consider it generally acceptable in practical applications.

V. CONCLUSION

  • At relatively low SNRs, the one-bit measurement covariance can be used directly as the unquantized covariance to obtain DOA signal and noise subspaces.Source: §V · p. 4
  • At high SNRs, approximation error grows, but the authors judge it generally acceptable in practical applications.Source: §V · p. 4
  • The findings are presented as helping reduce hardware cost and power consumption while simplifying real-time implementation.Source: §V · p. 4
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