A MSWF root-MUSIC based on Pseudo-noise resampling technique

This paper uses the shift-invariance property of uniform linear array in root-MUSIC estimator for obtaining signal and noise subspaces by applying multistage Wiener ﬁlter (MSWF) procedure. Also, the MSWF root-MUSIC based on the pseudo-noise resampling process for estimating the direction of arrival (DOA) of signals is proposed. By this process, a root estimator bank and a corresponding DOA estimator bank are constructed. Then, a hypothesis test is applied to the DOA estimator bank to detect the normal DOA estimators from abnormal DOA estimators called outliers. By averaging the corresponding root estimators of normal DOA estimators, the ﬁnal DOAs can be determined more accurately. When all the DOA estimators fail to pass the hypothesis test, the criterion based on the Gaussian weight average of the root estimator bank is introduced. By applying this criterion, better outlier-free performance of MSWF root-MUSIC can be obtained. Simulations show that our method can improve the DOA estimations, especially in small sample sizes and low signal-to-noise ratios.

This paper uses the shift-invariance property of uniform linear array in root-MUSIC estimator for obtaining signal and noise subspaces by applying multistage Wiener filter (MSWF) procedure. Also, the MSWF root-MUSIC based on the pseudo-noise resampling process for estimating the direction of arrival (DOA) of signals is proposed. By this process, a root estimator bank and a corresponding DOA estimator bank are constructed. Then, a hypothesis test is applied to the DOA estimator bank to detect the normal DOA estimators from abnormal DOA estimators called outliers. By averaging the corresponding root estimators of normal DOA estimators, the final DOAs can be determined more accurately. When all the DOA estimators fail to pass the hypothesis test, the criterion based on the Gaussian weight average of the root estimator bank is introduced. By applying this criterion, better outlier-free performance of MSWF root-MUSIC can be obtained. Simulations show that our method can improve the DOA estimations, especially in small sample sizes and low signal-to-noise ratios.
Introduction: Direction of arrival (DOA) estimation is the main topic in array signal processing for localizing radiating sources and it has a lot of applications in radar and wireless communications [1][2][3][4]. Subspace based methods such as multiple signal classification (MUSIC) [4], root-MUSIC [5], beamspace root-MUSIC [6], and estimation signal parameter via a rotational invariant technique (ESPRIT) [7] are based on eigenvalue decomposition (EVD) or singular value decomposition (SVD) on array output covariance matrix (AOCM). To reduce the complexity, real valued estimators including unitary root-MUSIC [8], unitary root-MUSIC with second forward/backward averaging [9] and unitary ES-PRIT were proposed. These methods use a real-valued EVD/SVD computation on either the real part of AOCM (R-AOCM) or the imaginary part of AOCM (I-AOCM), but they still require O(p 3 ) flops where p denotes the number of sensors. When massive arrays are used [10], p can be a very large number and this term of high complexity is unacceptable. Also, these subspace methods suffer considerable performance degradation when the signal-to-noise ratio or the number of snapshots is small [11,12], which is known as a threshold effect. Numerous authors have attempted to lower this effect by pseudo-noise resampling (PR) technique [11][12][13]. To accurately estimate the DOA of signals with reduced computational burden and reduced SNR threshold, we propose a MSWF root-MUSIC based on PR process. Our method uses a PR process to construct a root estimator bank and a corresponding DOA estimator bank. Then, a hypothesis test [12,13] [14] is applied to the whole DOA estimator bank and only the DOA estimator that has passed this test can be retained. For determining the final DOA estimates, we have proposed a strategy that uses the average of root estimators corresponding to retained DOA estimators. If there is no retained DOA estimator, that is, no DOA estimators have passed the test, we propose a new criterion for determining the final DOAs. This criterion uses the argument of the Gaussian weight average of the root estimator bank. Simulations show that this criterion has a better performance than the median average method or the conventional average method [11][12][13] that uses the information of DOA estimator bank. Also, since in our MSWF root-MUSIC, we do not need to estimate the sample covariance matrix and its EVD, we can implement the PR process more efficiently, especially in a large aperture array. Problem formulation: Assume q narrowband far-field sources are impinging on a ULA with p isotropic sensors. The signals and noises are assumed stationary and uncorrelated random processes. Also, the noises are spatially and temporally white. The output of sensors is as follows: where is the steering matrix, is the source signal vector.
is the steering vector due to qth source where (·) T is the transpose operator, λ is the carrier wavelength and d = λ 2 is the inter-element spacing. Given N i.i.d snapshots, x(1), · · ·, x(N ), the sample covariance matrix is obtained as follows: By eigenvalue decomposition of R x we have: where the columns of p × q and p × (p − q) matrices V s and V n contain the signal and noise eigenvectors.
MSWF root-music: As shown in [15], given a reference signal d 0 (t ), the MSWF can partition the observation data x 0 (t ) step by step to provide i desired signals d i (t ) and their orthogonal components In above formulas, h i is the matched filter and B i = I − h i h i H is the blocking matrix. The matched filter h i can be obtained by the normalization of cross-correlation between x i−1 (t) and d i−1 (t) as follows: where||.||, indicates the vector norm. In [16], the authors use MSWF for subspace decomposition of a linear array consisting of p sensors. They use the output of the first sensor as a reference signal d 0 (t ) and the output of p − 1 remaining sensors as the observation data vector x 0 (t ).
where 1=[1, 1, . . . , 1] T . By partitioning x 0 (t ) similar to that of MSWF procedure, p − 1 desired signals and their orthogonal components can be attained. So, we have p − 1 matched filters h i , i = 1, . . . p − 1 computed from the equation 10. By using the shift-invariance property of the Krylow subspace, the authors have demonstrated that for q uncorrelated narrowband signals impinging upon the array with p sensors are the signal and noise subspaces respectively. So we can express the root-MUSIC polynomial as follows: where a r is obtained by removing the first row of a(θ ) and z i = e ( j2πdsinθi/λ) denotes the roots of (15). After selecting p roots lying inside the unit circle, we can determine the DOA of signals from the q roots that are closest to the unit circle as follows: Pseudo-noise resampling technique: The idea of this technique is to perturb the original measured data matrix X = [x(t 1 ), . . . , x(t N )] by means of artificially generated pseudo random noise as follows: where Y i = [y i (t 1 ), . . . , y i (t N )] is the p × N resampled data matrix, i = 1, . . . , F . Z i is the matrix of independent zero mean circular pseudonoise obtained by a Gaussian random generator such that E(Z i ) = 0, To maintain an acceptable signalto-noise ratio (SNR) in the resampled data, the variance of pseudo-noise (σ ) 2 should be approximately the same as the variance of the original noise σ 2 n . For estimating σ 2 n , we use a diagonal matrix R n . This matrix is constructed from the variances of desired signals of the MSWF σ 2 di , (i = q + 1, . . . , p) after the p-th stage as follows: So As in [16], these d i (t ), (i = q + 1, . . . , p) are uncorrelated with each other, and their variances equal the noise variance.
For each resampling run, the MSWF root-MUSIC is done to obtain a root estimator Z and the corresponding DOA estimator each contains q roots and q DOA estimates, respectively. In some resampling runs, the original noise N = [n(t 1 ), . . . , n(t N )] is permuted in a favourable way by pseudo-noise Z i for the exploited DOA estimator. By applying a hypothesis test H , we can select these successfully resampled estimators in such runs and improve the DOA estimation performance. This hypothesis is defined as follows: H : All the q DOA estimates in a DOA estimator are localized inˆ .
where θ i max , i = 1, . . . , q are the highest peaks of the conventional beamformer output [11] [13] and θ q le f t and θ q right are the left and right boundaries of the qth subinterval. These boundaries are chosen as angular distances between the maximum of the qth peak and the left/right neighbor point with 3 dB drop, respectively.
DOA estimation strategy: After applying the MSWF root-MUSIC for each resampling run, a root estimator Z and the corresponding DOA estimator are obtained. So the jth estimators be where z ( j) q is the root corresponding to θ ( j) q and θ ( j) So after F PR runs, two estimator banks are constructed as follows: By applying the reliability test H to β θ the following two subsets are their corresponding root estimators. In (26) and (27), represent the ith DOA estimator and corresponding root estimator accepted by H . If 0 < J s < F , we calculate the qth DOA by the argument of the following new criterion: If all the member of the DOA estimator bank in (25) do not satisfy the reliability test J s = 0, we propose another criterion that exploits the Guassian weight average (GWA) of root estimator bank. The steps of this strategy are as follows: 1) β z,r is divided into q subsets, with each subset contains F roots corresponding to a DOA. In this case, the ith subset is expressed as follows: 2) The modulus of each z i in(32) is calculated.
3) The final root corresponding to ith DOA is computed by the Gaussian weight average of G i as follows: (17) to obtain the final DOA estimate.

4) Substitute Z i− f inal into
In (34), α is a controlling parameter. In each subset, that is, {G i } F i=1 , some roots are far from the unit circle and some roots are close to the unit circle. By selecting α ≥ 0 in (34), we can increase the effect of roots close to the unit circle and reduce the effect of roots that are far from the unit circle. Thus, all the DOA estimates determined by our method are always closest to the true DOAs. The effectiveness of our method will be further verified in simulation section Simulation: In this section, some simulations have been carried out to show the effectiveness of our method. In all simulations, we assume that the source localization sectors and the number of sources are known or estimated by [16][17]. The array is a ULA with 8 sensors spaced at d = λ/2. The sources are two far-field and uncorrelated narrowband Gaussian signals with the same SNR. The additive noise is assumed to be a white stationary Gaussian random process uncorrelated with signals. The vector denotes the DOAs. In the first simulation, we organize two experimental settings as follows: For RMSE performance comparisons, the unconditional Cramer-Rao lower bound (CRLB) [8] is plotted as a benchmark. Figures 1 and 2 show that the MSWF root-MUSIC outperforms EVD root-MUSIC and RV root-MUSIC. In the next simulation, we set σ to one and vary the signal power from −7 to 7 dB. The number of snapshots is N = 100. According to (23), we use the conventional beamformer to pre-estimate in each independent run. The performance of our method with α = 1 is examined with a different number of the PR process. The RMSEs are shown in Figure 3. As it is obvious, a relatively larger F can lead to better performance for our method, especially in low SNR. At high SNR, since all the DOA estimates are passed by the reliability test H , our proposed method is reduced to be the MSWF root-MUSIC one. Also, to illustrate the effectiveness of our GWA criterion in determining the final DOAs, we add another GWA based algorithm for comparison which is obtained by replacing the median average method [12,13] in PR root-MUSIC with the proposed GWA. For all the PR algorithms, we set F = 10. As Figure 3 shows, the GWA criterion for PR root-MUSIC provides a considerable improvement than the median one at low SNRs. Let us now study the RMSE of the proposed method versus the number of snapshots. For this simulation, we fix the SNR to -2dB. As shown in Figure 4, for N < 100, our method achieves better performance than PR root-MUSIC. Also, the effectiveness of using the GWA criterion instead of the median average method in PR root-MUSIC versus N has been shown. In the last simulation, the comparison of our method with PR unitary ESPRIT versus SNRs and N is made. Figures 3 and 4 show the results of this comparison. As we see, our method can be considered as a good estimator for determining the DOAs of signals.
Conclusion: A PR-based MSWF root-MUSIC algorithm has been derived for DOA estimation. Our method combines the MSWF root-MUSIC algorithm and PR technique to form a root estimator bank and a corresponding DOA estimator bank. Unlike the conventional root-MUSIC, MSWF root-MUSIC does not involve estimating the sample covariance matrix and its EVD. So it is suitable for practical applications where massive arrays are used. Also, two strategies for determining the final DOAs in the PR technique are introduced. Since these strategies use the modulus information of the root estimator bank, we can determine the final DOA estimates more accurately. Simulation results verify the effectiveness of our proposed method. The data that support the findings of this study are available from the corresponding author upon reasonable request.