SAMV (algorithm)

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) receives 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, .

The covariance matrix of that contains all information about is

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

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where , , , , and let where is the Kronecker product.

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 .

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 .

A typical application with the SAMV algorithm in SISO radar/sonar range-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.

An open source MATLAB implementation of SAMV algorithm could be downloaded here.

• Array processing – Area of research in signal processing
• Matched filter – Filters used in signal processing that are optimal in some sense
• Periodogram – Estimate of the spectral density of a signal
• Filtered backprojection – Integral transform in mathematics (Radon transform)
• MUltiple SIgnal Classification – Algorithm used for frequency estimation and radio direction finding (MUSIC), a popular parametric superresolution method
• Pulse-Doppler radar – Type of radar system
• Super-resolution imaging – Any technique to improve resolution of an imaging system beyond conventional limits
• Compressed sensing – Signal processing technique
• Inverse problem – Process of calculating the causal factors that produced a set of observations
• Tomographic reconstruction – Estimate object properties from a finite number of projections

1. ^ a b c d e Abeida, 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.
2. ^ a b Glentis, 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. Bibcode:2014SigPr..95...15G. doi:10.1016/j.sigpro.2013.08.003. S2CID 41743051.
3. ^ 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.
4. ^ 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.
5. ^ a b Yardibi, 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.