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Coarrays, MUSIC, and the Cramér Rao Bound

Mianzhi Wang, Arye Nehorai

arXiv:1605.03620v2stat.AP

TL;DR

Coarray-based MUSIC can resolve more sources than physical sensors, but its performance lacks a sufficiently simple theoretical analysis because covariance augmentation changes the relevant statistics. The paper derives a unified asymptotic MSE and sparse-array CRB, finding equal asymptotic errors for two augmentation methods and nonzero high-SNR error when sources outnumber sensors. Numerical results further show that efficiency depends on array type, source count, and SNR.

  • Problem

    Coarray MUSIC extends source resolution beyond the sensor count, but existing physical-array analyses do not directly apply and prior coarray MSE expressions are difficult to analyze.

  • Method

    The paper derives a simplified asymptotic MSE for direct augmentation and spatial smoothing, derives the sparse-array CRB, and evaluates estimator efficiency numerically.

  • Results

    When K ≥ M, coarray-based MUSIC MSE remains strictly positive as SNR approaches infinity, while direct augmentation and spatial smoothing share the same asymptotic MSE expression.

  • Takeaways & Limitations

    The high-SNR saturation of coarray MUSIC is explained theoretically, and sparse-array efficiency varies across source-count regimes and array geometries.

Abstract

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Sparse linear arrays, such as co-prime arrays and nested arrays, have the attractive capability of providing enhanced degrees of freedom. By exploiting the coarray structure, an augmented sample covariance matrix can be constructed and MUSIC (MUtiple SIgnal Classification) can be applied to identify more sources than the number of sensors. While such a MUSIC algorithm works quite well, its performance has not been theoretically analyzed. In this paper, we derive a simplified asymptotic mean square error (MSE) expression for the MUSIC algorithm applied to the coarray model, which is applicable even if the source number exceeds the sensor number. We show that the directly augmented sample covariance matrix and the spatial smoothed sample covariance matrix yield the same asymptotic MSE for MUSIC. We also show that when there are more sources than the number of sensors, the MSE converges to a positive value instead of zero when the signal-to-noise ratio (SNR) goes to infinity. This finding explains the "saturation" behavior of the coarray-based MUSIC algorithms in the high SNR region observed in previous studies. Finally, we derive the Cramér-Rao bound (CRB) for sparse linear arrays, and conduct a numerical study of the statistical efficiency of the coarray-based estimator. Experimental results verify theoretical derivations and reveal the complex efficiency pattern of coarray-based MUSIC algorithms.

I. INTRODUCTION

Sparse-array coarray methods extend MUSIC beyond the source-resolution limits of physical arrays, but their performance requires a model-specific analysis. This paper develops a simpler asymptotic MSE framework and connects it with CRB-based efficiency analysis.

  • I. INTRODUCTION: Traditional MUSIC resolves up to N −1 uncorrelated sources with N ULA sensors, whereas coarray processing can resolve up to O(N^2) sources with N sensors.The enhanced capability comes from constructing an augmented covariance matrix from the coarray structure.
  • I. INTRODUCTION: Existing MUSIC performance analyses rely on the physical array model and original sample-covariance statistics, so they do not directly apply to coarray MUSIC.A prior matrix-transform result is applicable but has an explicit form too complicated for convenient analytical studies.
  • I. INTRODUCTION: The paper derives a simpler asymptotic MSE expression for coarray-based MUSIC and shows that direct augmentation and spatial smoothing have the same asymptotic estimation error.The expression is intended to facilitate performance analysis across sparse linear arrays.
  • I. INTRODUCTION: The analysis covers cases in which the number of sources exceeds the number of physical sensors and also derives a CRB for sparse linear arrays.The paper further studies the statistical efficiency of coarray-based estimators numerically.

II. THE COARRAY SIGNAL MODEL

The coarray model transforms sparse-array covariance information into a virtual ULA with enhanced degrees of freedom, enabling MUSIC to process more sources than the physical array can directly support. Direct augmentation and spatial smoothing construct related augmented covariance matrices, but their altered statistics require dedicated analysis.

  • II. THE COARRAY SIGNAL MODEL: A sparse array with M sensors has integer-spaced locations, receives K far-field narrowband sources, and is modeled through its steering matrix and noisy snapshot vectors.The model assumes uncorrelated Gaussian sources and white Gaussian noise under the stated assumptions.
  • II. THE COARRAY SIGNAL MODEL: Vectorizing the physical covariance produces a coarray model whose steering matrix is A_d = A* ⊙ A.The coarray locations are formed by all pairwise sensor-location differences.
  • II. THE COARRAY SIGNAL MODEL: Selecting rows of the coarray steering matrix constructs a centered virtual ULA with 2M_v−1 positions and enhanced degrees of freedom.The virtual array spans [−M_v + 1, ..., 0, ..., M_v −1]d_0.
  • II. THE COARRAY SIGNAL MODEL: The virtual ULA is partitioned into M_v overlapping subarrays, whose outputs support construction of an augmented covariance matrix.The subarray outputs are formed with selection matrices Γ_i.
  • II. THE COARRAY SIGNAL MODEL: Direct augmentation and spatial smoothing are the two covariance-construction methods, yielding DA-MUSIC and SS-MUSIC, respectively.Their covariance entries arise from different operations on the physical covariance, so traditional MUSIC analysis does not apply directly.
  • II. THE COARRAY SIGNAL MODEL: When M_v > M, applying MUSIC to either augmented covariance matrix provides more degrees of freedom than applying it to the physical covariance matrix.The paper assumes K < M_v so MUSIC remains applicable to the virtual ULA.

III. THE MSE OF COARRAY-BASED MUSIC

The paper derives a unified, explicit asymptotic MSE analysis for coarray-based DA-MUSIC and SS-MUSIC, including cases where the source count exceeds the sensor count. It characterizes how the MSE depends on SNR and array geometry, including its high-SNR limits.

  • Motivation: Sample-covariance estimation errors perturb the noise eigenvectors and produce DOA estimation errors, requiring analysis beyond traditional MUSIC results based on the original covariance matrix.The coarray-derived covariance quantities have different statistical properties because they arise through linear and quadratic operations on the sample covariance.
  • Unified analysis: DA-MUSIC and SS-MUSIC share the same first-order error expression and asymptotic second-order statistics under the theorem assumptions.This equality holds despite the different second-order and fourth-order statistics used to construct their augmented covariance matrices.
  • Geometry dependence: The unified asymptotic MSE depends on both physical-array geometry and coarray geometry, so arrays with identical coarrays can still have different MSEs.The physical geometry is represented through A, while the coarray geometry enters through Av, ξk, and γk.
  • SNR behavior: The MSE depends on source SNRs rather than the absolute source and noise powers, and for equal-power sources it decreases monotonically as SNR increases.The monotonicity statement is given for sufficiently large sample size N.
  • High-SNR limits: At infinite SNR, the limiting MSE is zero for K = 1, not necessarily zero for 2 ≤ K < M, and strictly positive for K ≥ M.The positive limit when K ≥ M explains the high-SNR saturation observed for coarray-based MUSIC; traditional MUSIC can outperform these methods when 2 ≤ K < M.

IV. THE CRAM´ER-RAO BOUND

The paper derives a CRB for the coarray model, where the source count may exceed the physical sensor count, and analyzes its high-SNR behavior and validity conditions.

  • CRB derivation: The coarray CRB is derived because prior unconditional-model results assume fewer sources than sensors and no prior knowledge of source powers.The proposed derivation assumes a diagonal source-power matrix and assumptions A1–A4.
  • CRB derivation: The DOA CRB matrix is obtained by block-wise inversion of the partitioned Fisher information matrix.
  • CRB applicability: The derived CRB applies even when the number of sources exceeds the number of sensors.
  • CRB properties: The CRB depends on SNR ratios rather than the absolute source and noise powers.
  • Validity conditions: CRB validity requires the derivative matrix to have full column rank, which is tied to the coarray steering matrix and can fail with too many sources.
  • High-SNR behavior: When K < M, the high-SNR CRB may approach zero; when K ≥ M, it approaches a positive constant, consistent with the MUSIC MSE result.

V. NUMERICAL ANALYSIS

The numerical section evaluates the analytical MSE and CRB expressions, resolvability prediction, estimator efficiency, and the effect of sensor count for DA-MUSIC and SS-MUSIC.

  • Experimental scope: The experiments analyze DA-MUSIC and SS-MUSIC using the analytical MSE and CRB expressions.They also examine resolvability, asymptotic efficiency, and the effect of the number of sensors on asymptotic MSE.
  • Experimental setup: SNR is defined consistently across the numerical experiments.
  • Array configurations: The simulations use arrays with the same number of sensors but different apertures.

A. Numerical Verification

Monte Carlo experiments verify the analytical MSE expression under many-source conditions and show when its approximation agrees with empirical error for both MUSIC variants.

  • Setup: The verification uses 11 equal-power sources, exceeding the number of sensors, across varying SNRs and snapshot counts.
  • Evaluation: 10,000 trials compare analytical and empirical MSE through their relative error.
  • Results: Analytical and empirical MSE agree well with enough snapshots and sufficiently high SNR.At 0dB SNR, the approximation is reported as accurate with 250 snapshots.
  • Results: DA-MUSIC and SS-MUSIC show no significant difference in relative MSE error.
  • Caveat: In some low-SNR regions, a small relative error with few snapshots is misleading because empirical MSE has saturated.The authors state that this does not establish validity of the analytical expression there.

B. Prediction of Resolvability

The study uses an analytical criterion to predict when SS-MUSIC can resolve two closely spaced sources and compares that prediction across three sparse array geometries.

  • Experimental setup: Two equal-power sources near 30° are varied from 0.3° to 3.0° separation, with resolution assessed over 500 trials at 500 snapshots and 0dB SNR.
  • Prediction method: The analytical criterion predicts resolvability thresholds for the two-source experiment.A more comprehensive criterion is referenced separately.
  • Array comparison: The MRA has the best resolution performance, while the co-prime array has the worst because their apertures differ.
  • Results: Each array’s resolution probability drops to nearly zero at its predicted threshold.This supports using the analytical MSE expression to predict close-source resolvability.

C. Asymptotic Efficiency Study

The study evaluates asymptotic efficiency across SNR, source count, array geometry, and angular separation. Coarray-based MUSIC exhibits geometry- and separation-dependent efficiency, with distinct high-SNR behavior when sources outnumber sensors.

  • κ increases with SNR for K = 1 across all three arrays, but none achieves efficient DOA estimation.
  • At high SNR, the co-prime array achieves higher efficiency than the nested array despite being less efficient in the low-SNR region.
  • For K = 6, κ decreases to zero as SNR increases, indicating statistical inefficiency for both DA-MUSIC and SS-MUSIC.
  • For K = 12, κ decreases with SNR but converges to a positive value instead of zero.
  • Higher degrees of freedom are obtained at the cost of decreased statistical efficiency, motivating direct MUSIC on R when sources are fewer than sensors.
  • With two sources, efficiency decreases from 0dB to 10dB SNR and depends strongly on array geometry and normalized angular separation.

D. MSE vs. Number of Sensors

The sensor-count study compares co-prime, nested, and minimum redundancy arrays under determined and underdetermined source settings. MSE decreases faster than the traditional ULA MUSIC rate, with MRAs and nested arrays outperforming co-prime arrays.

  • The experiments use co-prime pairs, nested-array parameter pairs, and MRAs, with K = 1 or K = M at SNR = 0dB.
  • For K = 1, MSE decreases at approximately O(M^-4.5) for all three sparse array types.
  • For K = M, MSE decreases at approximately O(M^-3.5).
  • MRAs and nested arrays achieve lower MSE than co-prime arrays in both source-count settings.
  • Both tested rates exceed the O(M^-3) asymptotic MSE decay rate reported for traditional MUSIC on an M-sensor ULA.

VI. CONCLUSION

The paper derives unified asymptotic MSE results for DA-MUSIC and SS-MUSIC, extends them to several sparse arrays and source regimes, and derives the sparse-array CRB. The analysis assumes perfect calibration and identifies model errors as future scope.

  • DA-MUSIC and SS-MUSIC share the same asymptotic MSE expression.
  • The MSE expression applies to co-prime arrays, nested arrays, MRAs, and cases where the source number exceeds the sensor number.
  • The paper derives the Cramér-Rao bound for sparse linear arrays and analyzes the statistical efficiency of typical sparse geometries.
  • The results are intended to support performance analysis and optimal design of sparse linear arrays.
  • The derivations assume a perfectly calibrated array, while extending them to model errors remains future work.

APPENDIX B PROOF OF THEOREM 1

The proof derives first-order error expressions for DA-MUSIC and SS-MUSIC by perturbing the augmented covariance matrices and simplifying finite-snapshot covariance expectations. Structural properties of the coarray transform establish the equivalence of the two expressions.

  • The proof begins by deriving the first-order DA-MUSIC error expression from the eigendecomposition of the augmented covariance matrix.
  • Under sufficiently small perturbations, the derivation omits high-order terms to obtain closed-form DOA error expressions.
  • The SS-MUSIC derivation uses that its augmented covariance matrix shares eigenvectors with the direct-augmentation matrix.
  • The coarray transform's conjugate-symmetry and Hermitian-symmetry properties enable compact simplification of the MSE expression.
  • The resulting first-order expression coincides with the corresponding expression for the other MUSIC construction.
  • Finite-snapshot perturbations are not circularly symmetric complex Gaussian, so the proof explicitly computes the required expectations.

APPENDIX D PROOF OF PROPOSITION 2

The appendix proves Proposition 2 by analyzing matrix ranks, Schur complements, and limiting behavior for cases with fewer or at least as many sources as sensors.

  • For K < M, the proof establishes nonsingularity and full-rank properties needed for the relevant Schur complements and CRB expression.It uses the full-column-rank condition on ∂r/∂η and shows A^H E⋆_s A = I is nonsingular.
  • As σ^2_n → 0 with K < M, the CRB for θ converges to zero.
  • When K ≥ M, R is full rank for every positive noise variance, making the FIM and CRBθ positive definite.The argument follows from full column rank of ∂r/∂η and positivity of the relevant Schur complements.
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