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Direction of Arrival Estimation Using Co-prime Arrays: A Super Resolution Viewpoint

Zhao Tan, Yonina C. Eldar, Arye Nehorai

arXiv:1312.7793v1cs.IT

TL;DR

The paper addresses DOA estimation with co-prime arrays, seeking to exploit their high degrees of freedom without grid-based sparse-recovery errors. It extends continuous super-resolution recovery to this setting and reports improved accuracy, degrees of freedom, and resolution, while theoretically detecting up to MN^2 sources with 2M + N sensors in the noiseless case.

  • Problem

    DOA estimation must exploit co-prime arrays' O(MN) degrees of freedom while avoiding off-grid effects from discretized sparse recovery.

  • Method

    The paper applies continuous super-resolution sparse recovery to co-prime-array covariance data and combines reconstructed spectra with SORTE for source-number detection.

  • Results

    The proposed method improves DOA estimation accuracy, degrees of freedom, and resolution ability over MUSIC with spatial smoothing and discrete sparse recovery.

  • Takeaways & Limitations

    Co-prime arrays can use continuous recovery to exploit O(MN) degrees of freedom and estimate source locations without discretizing the candidate range.

  • Takeaways & Limitations

    The current co-prime-array research assumes that sources are uncorrelated.

Abstract

from arXiv · show

We consider the problem of direction of arrival (DOA) estimation using a newly proposed structure of non-uniform linear arrays, referred to as co-prime arrays, in this paper. By exploiting the second order statistical information of the received signals, co-prime arrays exhibit O(MN) degrees of freedom with only M + N sensors. A sparsity based recovery method is proposed to fully utilize these degrees of freedom. Unlike traditional sparse recovery methods, the proposed method is based on the developing theory of super resolution, which considers a continuous range of possible sources instead of discretizing this range into a discrete grid. With this approach, off-grid effects inherited in traditional sparse recovery can be neglected, thus improving the accuracy of DOA estimation. In this paper we show that in the noiseless case one can theoretically detect up to M N sources with only 2M + N sensors. The noise 2 statistics of co-prime arrays are also analyzed to demonstrate the robustness of the proposed optimization scheme. A source number detection method is presented based on the spectrum reconstructed from the sparse method. By extensive numerical examples, we show the superiority of the proposed method in terms of DOA estimation accuracy, degrees of freedom, and resolution ability compared with previous methods, such as MUSIC with spatial smoothing and the discrete sparse recovery method.

I. INTRODUCTION

Co-prime arrays use second-order statistics and a virtual array to increase degrees of freedom with relatively few sensors. The paper develops continuous super-resolution recovery for more accurate DOA estimation and source detection.

  • I. INTRODUCTION: Co-prime arrays achieve O(MN) degrees of freedom using O(M + N) sensors.Their virtual-array construction exploits cross-difference sensor locations.
  • I. INTRODUCTION: Spatially smoothed MUSIC increases detectable sources but reduces the virtual array aperture.Sparsity-based recovery is introduced to address this limitation of subspace methods.
  • I. INTRODUCTION: Continuous super resolution considers all source locations in the interested range instead of using a discrete grid.This avoids the off-grid issue associated with traditional sparse recovery.
  • I. INTRODUCTION: With 2M + N sensors, co-prime arrays can theoretically detect up to MN^2 sources in the noiseless case.The paper contrasts this analysis with coherence-based identifiability results that are limited to very small source counts.
  • II. DIRECTION OF ARRIVAL ESTIMATION AND CO-PRIME ARRAYS: The co-prime model rearranges covariance data into a virtual ULA with 2MN + 1 sensors.Its steering matrix has the structure of a ULA, enabling MUSIC with spatial smoothing to detect O(MN) sources.

III. DIRECTION OF ARRIVAL ESTIMATION WITH SUPER RESOLUTION THEORY

The paper extends continuous super-resolution theory to co-prime-array DOA estimation. It models DOAs as a sparse continuous signal and recovers them through total-variation minimization under a separation condition.

  • A. The Mathematical Theory of Super Resolution: Super resolution recovers high-frequency signal details from low-frequency measurements.The continuous signal is represented as a weighted sum of spikes.
  • A. The Mathematical Theory of Super Resolution: The measurement model uses a low-frequency operator F to obtain Fourier coefficients r from the continuous signal s.The source locations are encoded by the spike locations τk.
  • A. The Mathematical Theory of Super Resolution: Total variation minimization promotes sparsity for continuous signals, analogously to ℓ1 minimization in discrete spaces.For a spike signal, the total variation equals the sum of the spike magnitudes.
  • A. The Mathematical Theory of Super Resolution: The recovery problem minimizes the total variation norm subject to matching the measured low-frequency coefficients.The formulation is a convex optimization problem solved through semidefinite programming.
  • A. The Mathematical Theory of Super Resolution: When any two spike locations are separated by more than 2/fc, the original sparse signal is the unique solution.This condition provides the stated exact-recovery guarantee for the continuous optimization.

B. DOA estimation with Super Resolution

The paper applies continuous-domain super resolution to co-prime-array DOA estimation, using covariance-derived virtual-array measurements and convex optimization to recover closely spaced sources. It analyzes finite-sample noise robustness and establishes O(MN)-scale resolution and source-detection capability.

  • Continuous super resolution: DOA estimation is transformed into a continuous super-resolution problem through a change of variables from source angles to τ_k.The transformed support is T = {τ_k: 1 ≤ k ≤ K}, with lag indices spanning −MN through MN.
  • Array model: A co-prime array with N and 2M sensors uses coprime spacings Md and Nd, respectively, to support the recovery framework.The sensor spacing satisfies d ≤ λ/2.
  • Resolution guarantee: Exact recovery of source locations is guaranteed by convex optimization when the minimum source separation satisfies the theorem’s constraint.The minimum separation is defined as Δ(θ) = min |sin(θ_i) − sin(θ_j)| over distinct sources.
  • Resolution guarantee: With 2M + N sensors, a co-prime array can detect up to MN sources in the noiseless theoretical setting.This contrasts with the traditional ULA bound stated for the same sensor count and supports O(MN) degrees of freedom.
  • Noisy model: Finite covariance samples are handled by modeling estimation error statistically and solving a noise-aware convex optimization problem.The analysis uses concentration results for complex Gaussian variables and bounds the reconstruction error with probability increasing exponentially in the sample count T.
  • Noisy model: The reconstructed signal recovers high-frequency details with high probability, and this probability approaches one exponentially as T increases.The result uses a Fejér kernel with cut-off frequency f_h > MN and an error bound derived from the co-prime-array noise statistics.

IV. DOA ESTIMATION VIA SEMIDEFINITE PROGRAMMING AND ROOT FINDING

The numerical DOA estimator incorporates unknown noise power into a continuous convex program, converts it to a semidefinite program, and extracts source locations by root finding. A discrete sparse refinement addresses numerical overproduction of candidate roots and possible ill-conditioning.

  • Optimization formulation: Unknown noise power is incorporated directly into a more realistic convex optimization formulation for DOA estimation.This modifies the idealized formulation to account for the fact that σ^2 is normally unavailable.
  • Semidefinite programming: The infinite-dimensional constraint is recast as a semidefinite matrix constraint, yielding an equivalent semidefinite programming problem solvable with CVX.The dual derivation relies on strong duality, with u = 0 providing a feasible solution.
  • Root finding: The support set is estimated by finding roots of the trigonometric polynomial 1 − |F*u(τ)|^2 = 0.The estimated support generates a steering matrix F_est and a corresponding measurement model.
  • Root finding: Numerical root-finding can produce K_est ≥ K candidates, and in some cases K_est ≥ 2MN + 1 makes the resulting linear system ill-conditioned.The method then exploits sparsity through a discrete convex optimization problem.
  • Root finding: The discrete refinement uses a larger noise tolerance because root finding introduces additional error into the measurement model.The tolerance ε_d is normally chosen larger than ε.

V. EXTENSION: SOURCE NUMBER DETECTION

The paper reconstructs a sparse signal from continuous sparse recovery and applies SORTE to estimate the number of sources. When the recovered support has at most two elements, the method directly uses that support size instead.

  • Source Number Detection: Source number detection applies SORTE to the descending squared magnitudes of the reconstructed sparse signal.The element differences are defined as ∇s_est[i] = s_est[i]^2 − s_est[i + 1]^2, followed by a gap measure and SORTE criterion.
  • Source Number Detection: CSORTE uses the reconstructed spectrum from continuous sparse recovery to determine the number of sources.The recovered support is obtained by rooting the continuous sparse-recovery result before applying the SORTE-based rule.
  • Source Number Detection: The estimated source count is the index minimizing the SORTE measure.The criterion is written as K_hat = arg min_i SORTE(i).
  • Source Number Detection: SORTE works only when K_est > 2; otherwise, the method sets the estimated source count directly to K_est.This boundary follows from the definition of SORTE and the support produced by the continuous sparse-recovery rooting process.

VI. NUMERICAL RESULTS

The numerical study evaluates continuous sparse recovery on an 11-sensor co-prime array and compares it with MUSIC and discrete sparse recovery. The setup explicitly examines grid mismatch and uses a fine sin(θ) grid for the discrete method.

  • Experimental Setup: The experiments use an 11-sensor co-prime array with sensor positions [0, 3, 6, 9, 12]d and [0, 5, 10, 15, 20, 25]d.The intersensor reference distance d is set to half the wavelength.
  • Experimental Setup: The study compares continuous sparse recovery with MUSIC and discrete sparse recovery.MUSIC follows the spatial-smoothing approach, while the discrete method is implemented through an equivalent Basis pursuit form of LASSO.
  • Experimental Setup: The discrete sparse-recovery comparison uses a sin(θ) grid from −1 to 1 with step size 0.005.The experiment is designed to consider grid mismatches in DOA estimation.

A. Degrees of Freedom

The first numerical example tests whether continuous sparse recovery realizes the co-prime array’s O(MN) degrees of freedom. Under T = 500 and SNR = −10dB, CSR, DSR, and root MUSIC all achieve O(MN).

  • Degrees of Freedom: O(MN) degrees of freedom are achieved by CSR, DSR, and root MUSIC in the first numerical example.The example uses a co-prime array structure to test the predicted degrees-of-freedom increase.
  • Degrees of Freedom: 0.23%, 0.26%, and 0.42% are the average estimation errors for CSR, DSR, and root MUSIC, respectively.The experiment uses 500 time samples and SNR = −10dB.
  • Degrees of Freedom: 7.30 seconds, 7.82 seconds, and 0.81 seconds are the CPU times for CSR, DSR, and MUSIC, respectively.For MUSIC, the root MUSIC algorithm estimates each source location while assuming that the source count is known.

B. Estimation Accuracy

Monte Carlo experiments compare CSR with DSR and MUSIC for DOA estimation accuracy. CSR performs better than DSR uniformly with less computing time, and both sparse methods outperform MUSIC in accuracy.

  • SNR Variation: CSR performs better than DSR uniformly across changing SNR with less computing time.The comparison averages results over 50 Monte Carlo simulations; the average CPU times are 6.93s for CSR and 9.30s for DSR.
  • SNR Variation: Both sparse recovery methods achieve better DOA estimation accuracy than MUSIC.MUSIC is evaluated as Root MUSIC, with the number of sources assumed known in the simulation.
  • SNR Variation: Finer grids can improve DSR accuracy but further slow its computation.This trade-off is attributed to reducing the grid stepsize in the discrete recovery method.
  • Snapshot Variation: CSR has better estimation accuracy than DSR or MUSIC when the number of snapshots changes.MUSIC and DSR approach CSR’s performance as the number of snapshots approaches 5000; average CPU times are 6.50s, 7.91s, and 1.43s, respectively.
  • Snapshot Variation: A small number of snapshots can let CSR achieve the same estimation accuracy as MUSIC while saving sampling time.This conclusion is reported for the changing-snapshot experiment.

C. Source Number Detection Performance Comparsion

The proposed CSORTE source-number detector is compared with SORTE under co-prime-array conditions, reaching the theoretical detection limit while traditional SORTE fails at higher source counts.

  • The simulation evaluates source-number detection from 11 to 17 sources at 0 dB SNR with 3000 snapshots.
  • 17 is the theoretical maximum number of detectable sources because the co-prime array provides consecutive lags from −17d to 17d.
  • At fewer than 15 sources, CSORTE and SORTE yield comparable detection results.
  • SORTE fails above 15 sources, whereas CSORTE remains stable and achieves perfect detection at the theoretical limit of 17 sources.
  • The results indicate that the sparsity-based method offers more degrees of freedom than the subspace-based method.

D. Resolution Ability

The resolution experiments test CSR against MUSIC and SORTE on closely spaced sources. CSR resolves the targets under conditions where MUSIC or traditional SORTE fails, including lower SNR.

  • At 0 dB SNR and 500 snapshots, MUSIC using traditional SORTE fails because SORTE estimates the source number incorrectly.
  • CSR successfully resolves sources at −32° and −30° without assuming the source number in advance.
  • At −5 dB SNR, MUSIC fails to resolve the closely located sources even when their number is known, while CSR resolves them successfully.
  • CSORTE outperforms traditional SORTE for detecting two sources located at −32° and −30° after 50 Monte Carlo runs.
  • The conclusion reports more powerful resolution ability for the proposed method than traditional MUSIC with spatial smoothing.

APPENDIX

The appendix supplies proof steps for the optimization dual problem and a concentration lemma for sums of independent Gaussian variables.

  • The appendix completes the proofs of Lemmas III.1 and III.2 using the stated intermediate results and referenced lemmas.
  • Lemma A.1 assumes x(t), for t = 1, . . . , T, are i.i.d. zero-mean normal variables with variance σ^2.
  • Introducing z ∈ C^(2MN+1) converts the original primal problem into an optimization with an ℓ2 constraint and equality z = F s − r − σ^2w.
  • The Lagrangian uses multipliers v ∈ R and u ∈ C^(2MN+1), and the dual-domain constraints imply v = ∥u∥2.
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