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Sparse representation (SR) algorithms can be implemented for high-resolution direction of arrival (DOA) estimation. Additionally, SR can effectively separate the coherent signal sources because the spectrum estimation is based on the optimization technique, such as the _{1} norm minimization, but not on subspace orthogonality. However, in the actual source localization scenario, an unknown gain/phase error between the array sensors is inevitable. Due to this nonideal factor, the predefined overcomplete basis mismatches the actual array manifold so that the estimation performance is degraded in SR. In this paper, an adaptive SR algorithm is proposed to improve the robustness with respect to the gain/phase error, where the overcomplete basis is dynamically adjusted using multiple snapshots and the sparse solution is adaptively acquired to match with the actual scenario. The simulation results demonstrate the estimation robustness to the gain/phase error using the proposed method.

Direction of arrival (DOA) estimation has long been a useful method for signal detection in sonar, radar and communication applications [^{m×n} is an overcomplete basis, _{1} norm minimization) [

In this paper, an adaptive SR algorithm to dynamically adjust both the overcomplete basis and the sparse solution so that the solution can better match the actual scenario is proposed. The remainder of this paper is organized as follows. Section 2 describes the basic model of the received data. In Section 3, an adaptive SR method to deal with the gain/phase error scenario is illustrated. In Section 4, the performance analysis is implemented to illustrate the robustness of adaptive SR with the simulated data. Section 5 presents our concluding remarks about the proposed algorithm.

Source localization using sensor arrays is a problem with important practical applications including radar, sonar, exploration seismology and many other applications [

As shown in _{i}_{k}

Starting with the ideal model with no gain/phase error, we have _{k}_{s}_{s}N_{s}_{s}_{i}_{i}_{s}_{s}

Recently, the techniques of SR have been illustrated as effective methods for DOA estimation [

With the constraint of sparsity on _{p}_{p}_{2} norm constraint by ɛ guarantees the residual ‖_{2} to be small, whereas the _{1} norm enforces the sparsity of the estimated spectrum _{0} norm. However, this optimization is NP-hard and is unrealizable even for modest data size [_{0} norm, the _{1} norm minimization can be efficiently implemented via convex optimization. The fundamental contribution of SR is to illustrate the equivalence between these two optimizations. It is proven that SR implemented by the _{1} norm minimization can approximate the actual solution as ‖_{0}‖_{2} ≤ Λ · ɛ, where _{0}_{1} norm characteristic is given in [

When multiple measurements are available, the data model is extended as:
^{(1)},⋯,^{(L)}] are multiple snapshots, ^{(1)},⋯,^{(L)}] and ^{(1)},⋯,^{(L)}] are the corresponding noise and spectrum matrixes, respectively. The rows of _{1,2} norm minimization to implement the joint optimization [

The mixed _{1,2} norm minimization is implemented on the solution matrix

Based on this, the _{1,2} norm minimization combines the multiple snapshots using the _{2} norm and the sparsity is only enforced in the spatial dimension via the _{1} norm. Therefore the solution matrix

To decrease the computation load, a simple method is to separate the joint problem in (6) into a series of independent subproblems [

Each subproblem can be solved via the _{1} norm minimization using (5) to obtain the sparse spectrum estimation. Then, the average result of these estimated spectrums ^{(l)}, 1 ≤

This method implements noncoherent average and its main attraction is its simplicity. However, by turning to fully coherent combined processing, as described in the following sections, we expect to achieve greater accuracy and robustness to noise.

A typical coherent sparse representation algorithm using multiple snapshots is the ℓ_{1}-_{SV}_{K}_{K}_{K}_{K}_{SV}_{K}_{SV}_{K}

Then the _{1} norm minimization can be similarly implemented like (5), however, only in the signal subspace. In the ℓ_{1}-_{1}-_{1}-

However, there are some nonideal factors, which is inevitable in a practical radar array system. These factors include gain/phase error, mutual coupling between sensors and so forth [_{i}e^{jΔθi} indicates the gain/phase error at the _{m}_{m}_{1}-

The key feature of adaptive SR is the adaptive adjustment of the overcomplete basis. This process generally learns the uncertainty of the overcomplete basis, which is not available from the prior knowledge, but rather has to be estimated using multiple snapshots. Prior works on basis learning take the strategy that the whole overcomplete basis is optimized to better represent the data of multiple snapshots [_{l}_{1},⋯,_{N}_{i}_{i}_{s}_{1},⋯,_{K}_{n}_{K}_{+1}, ⋯ _{N}_{s}

Once reliable estimations of _{n}_{k}_{n}_{k}_{ij}

Similarly, we need to minimize (20) with respect to ^{H}^{T}

Then the error matrix estimation can be effectively given as

Let ^{(0)} =

Calculate the covariance matrix estimation using (13) and obtain the noise subspace as _{n}_{K+1},⋯_{N}

At the ^{(n)} is estimated by the _{1} norm minimization with the overcomplete basis ^{(n−1)}

Based on the current solution ^{(n)}, only significant peaks (local maxima) are extracted from the spectrum estimation and the manifold estimation is given as _{p1}),⋯, _{pK})], where _{K}

Update the error matrix using the optimization in

In the adaptive SR, the choice of

In our simulations, a ULA with _{1}^{jΔθ1},⋯,Δ_{N}e^{jΔθN}), where the gain error obeys Δ_{i}^{−3}) and the phase error Δ_{i}_{1} norm minimization at each snapshot separately and then averages them to obtain the overall performance. The _{1} – _{1} norm minimization only on the signal subspace [_{1} –

Next, the quantitative results are given to illustrate the advantages of adaptive SR. All the performance comparisons are based on 50 Monte Carlo simulations.

Since SR deals with each snapshot separately, the addition of snapshots provides no obvious benefits for improving the performance. The performance of _{1} –

_{1} – _{1} –

As stated above, compared with traditional SR methods like _{1} – _{1} norm minimization, but not on subspace orthogonality. The following scenario is used to prove the capabilities of dealing with coherent sources of adaptive SR, where subspace-based methods with calibration are ineffective. The array parameters and the gain/phase error keep identical with that in Section 4.1. Two far-field point sources are located at angles −38°, −32°, having a high correlation of _{12} = 0.95. As a performance comparison, MVDR is deployed as the detailed implementation of the subspace-based methods [_{1} norm minimization and the coherent sources can be effectively separated. Therefore, even though effective calibration is deployed in MVDR, it still can not distinguish the coherent sources. On the other hand, adaptive SR can make it because the final spectrum estimation is still based on _{1} norm minimization, but not on subspace orthogonality.

This paper focuses on improving the robustness of sparse representation for the DOA estimation with the gain/phase error. By dynamically calibrating the overcomplete basis and adaptively estimating the sparse solution, the proposed adaptive SR can greatly improve the estimation robustness, and thus, the solution better matches the actual scenario. Additionally, it does separate the coherent sources, which is unrealizable for subspace-based methods with calibration. The following are several considerations for further research: first, the current signal model in SR only considers the far-field point sources, however, the near-field source location is also important and meaningful in the actual scenario. Second, the convergence of the adaptive SR needs to be proved in a strict mathematical way. Finally, more adaptive mechanisms should be added to deal with the mutual coupling scenario.

This work was supported in part by the National Natural Science Foundation of China (No. 40901157) and in part by the National Basic Research Program of China (973 Program, No. 2010CB731901).

An illustration of the array geometry of source localization.

Spectrum estimation result.

DOA MSE against the number of snapshots.

Amplitude MSE against the number of snapshots.