The formulation of the SAMV algorithm is given as an inverse problem in the context of DOA estimation. Suppose an -element uniform linear array (ULA) receive narrow band signals emitted from sources located at locations , respectively. The sensors in the ULA accumulates snapshots over a specific time. The dimensional snapshot vectors are
where is the steering matrix, contains the source waveforms, and is the noise term. Assume that , where is the Dirac delta and it equals to 1 only if and 0 otherwise. Also assume that and are independent, and that , where . Let be a vector containing the unknown signal powers and noise variance, .
This covariance matrix can be traditionally estimated by the sample covariance matrix where . After applying the vectorization operator to the matrix , the obtained vector is linearly related to the unknown parameter as
,
where , , , , and let
where
is the Kronecker product.
SAMV algorithm
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To estimate the parameter from the statistic , we develop a series of iterative SAMV approaches based on the asymptotically minimum variance criterion. From,[1] the covariance matrix of an arbitrary consistent estimator of based on the second-order statistic is bounded by the real symmetric positive definite matrix
where . In addition, this lower bound is attained by the covariance matrix of the asymptotic distribution of obtained by minimizing,
where
Therefore, the estimate of can be obtained iteratively.
The and that minimize can be computed as follows. Assume and have been approximated to a certain degree in the th iteration, they can be refined at the th iteration by,
where the estimate of at the th iteration is given by with .
Beyond scanning grid accuracy
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The resolution of most compressed sensing based source localization techniques is limited by the fineness of the direction grid that covers the location parameter space.[4] In the sparse signal recovery model, the sparsity of the truth signal is dependent on the distance between the adjacent element in the overcomplete dictionary , therefore, the difficulty of choosing the optimum overcomplete dictionary arises. The computational complexity is directly proportional to the fineness of the direction grid, a highly dense grid is not computational practical. To overcome this resolution limitation imposed by the grid, the grid-free SAMV-SML (iterative Sparse Asymptotic Minimum Variance - Stochastic Maximum Likelihood) is proposed,[1] which refine the location estimates by iteratively minimizing a stochastic maximum likelihood cost function with respect to a single scalar parameter .
Application to range-Doppler imaging
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A typical application with the SAMV algorithm in SISOradar/sonarrange-Doppler imaging problem. This imaging problem is a single-snapshot application, and algorithms compatible with single-snapshot estimation are included, i.e., matched filter (MF, similar to the periodogram or backprojection, which is often efficiently implemented as fast Fourier transform (FFT)), IAA,[5] and a variant of the SAMV algorithm (SAMV-0). The simulation conditions are identical to:[5] A -element polyphase pulse compression P3 code is employed as the transmitted pulse, and a total of nine moving targets are simulated. Of all the moving targets, three are of dB power and the rest six are of dB power. The received signals are assumed to be contaminated with uniform white Gaussian noise of dB power.
The matched filter detection result suffers from severe smearing and leakage effects both in the Doppler and range domain, hence it is impossible to distinguish the dB targets. On contrary, the IAA algorithm offers enhanced imaging results with observable target range estimates and Doppler frequencies. The SAMV-0 approach provides highly sparse result and eliminates the smearing effects completely, but it misses the weak dB targets.
Open source implementation
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An open source MATLAB implementation of SAMV algorithm could be downloaded here.
^ abcdeAbeida, Habti; Zhang, Qilin; Li, Jian; Merabtine, Nadjim (2013). "Iterative Sparse Asymptotic Minimum Variance Based Approaches for Array Processing" (PDF). IEEE Transactions on Signal Processing. 61 (4): 933–944. arXiv:1802.03070. Bibcode:2013ITSP...61..933A. doi:10.1109/tsp.2012.2231676. ISSN 1053-587X. S2CID 16276001.
^ abGlentis, George-Othon; Zhao, Kexin; Jakobsson, Andreas; Abeida, Habti; Li, Jian (2014). "SAR imaging via efficient implementations of sparse ML approaches" (PDF). Signal Processing. 95: 15–26. doi:10.1016/j.sigpro.2013.08.003. S2CID 41743051.
^Yang, Xuemin; Li, Guangjun; Zheng, Zhi (2015-02-03). "DOA Estimation of Noncircular Signal Based on Sparse Representation". Wireless Personal Communications. 82 (4): 2363–2375. doi:10.1007/s11277-015-2352-z. S2CID 33008200.
^Malioutov, D.; Cetin, M.; Willsky, A.S. (2005). "A sparse signal reconstruction perspective for source localization with sensor arrays". IEEE Transactions on Signal Processing. 53 (8): 3010–3022. Bibcode:2005ITSP...53.3010M. doi:10.1109/tsp.2005.850882. hdl:1721.1/87445. S2CID 6876056.
^ abYardibi, Tarik; Li, Jian; Stoica, Petre; Xue, Ming; Baggeroer, Arthur B. (2010). "Source Localization and Sensing: A Nonparametric Iterative Adaptive Approach Based on Weighted Least Squares". IEEE Transactions on Aerospace and Electronic Systems. 46 (1): 425–443. Bibcode:2010ITAES..46..425Y. doi:10.1109/taes.2010.5417172. hdl:1721.1/59588. S2CID 18834345.