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SAMV (algorithm)

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SAMV (iterative sparse asymptotic minimum variance<a href="#cite_note-AbeidaZhang-1">[1]</a><a href="#cite_note-SAR-2">[2]</a>) is a parameter-free <a href="Super-resolution_imaging" title="Super-resolution imaging">superresolution</a> algorithm for the linear <a href="Inverse_problem" title="Inverse problem">inverse problem</a> in <a href="Spectral_density_estimation" title="Spectral density estimation">spectral estimation</a>, <a href="Direction_of_arrival" title="Direction of arrival">direction-of-arrival</a> (DOA) estimation and <a href="Tomographic_reconstruction" title="Tomographic reconstruction">tomographic reconstruction</a> with applications in <a href="Signal_processing" title="Signal processing">signal processing</a>, <a href="Medical_imaging" title="Medical imaging">medical imaging</a> and <a href="Remote_sensing" title="Remote sensing">remote sensing</a>. The name was coined in 2013<a href="#cite_note-AbeidaZhang-1">[1]</a> to emphasize its basis on the asymptotically minimum variance (AMV) criterion. It is a powerful tool for the recovery of both the amplitude and frequency characteristics of multiple highly <a href="Correlation_coefficient" title="Correlation coefficient">correlated</a> sources in challenging environments (e.g., limited number of snapshots and low <a href="Signal-to-noise_ratio" title="Signal-to-noise ratio">signal-to-noise ratio</a>). Applications include <a href="Synthetic-aperture_radar" title="Synthetic-aperture radar">synthetic-aperture radar</a>,<a href="#cite_note-SAR-2">[2]</a><a href="#cite_note-Yang-3">[3]</a> <a href="CT_scan" title="CT scan">computed tomography scan</a>, and <a href="Magnetic_resonance_imaging" title="Magnetic resonance imaging">magnetic resonance imaging (MRI)</a>.

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

The formulation of the SAMV algorithm is given as an <a href="Inverse_problem" title="Inverse problem">inverse problem</a> in the context of DOA estimation. Suppose an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

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</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">-element <a href="Sensor_array" title="Sensor array">uniform linear array</a> (ULA) receive <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

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<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

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</math><img src="./2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> narrow band signals emitted from sources located at locations <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\theta } =\{\theta _{a},\ldots ,\theta _{K}\}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {\theta } =\{\theta _{a},\ldots ,\theta _{K}\}}</annotation>

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</math><img src="./aace31fb8bb4fe4f4318eb2225d732025757bfe2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.668ex; height:2.843ex;" alt="{\displaystyle \mathbf {\theta } =\{\theta _{a},\ldots ,\theta _{K}\}}" loading="lazy">, respectively. The sensors in the ULA accumulates <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

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<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

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</math><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> snapshots over a specific time. The <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\times 1}">

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<annotation encoding="application/x-tex">{\displaystyle M\times 1}</annotation>

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</math><img src="./f641103f28cc9fd98df632ccdba15a4ceb18ec92.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.445ex; height:2.176ex;" alt="{\displaystyle M\times 1}" loading="lazy"> dimensional snapshot vectors are

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {y} (n)=\mathbf {A} \mathbf {x} (n)+\mathbf {e} (n),n=1,\ldots ,N}</annotation>

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</math><img src="./f1ea88179e0a18b1ac42373e124abe7bbb176df7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.549ex; height:2.843ex;" alt="{\displaystyle \mathbf {y} (n)=\mathbf {A} \mathbf {x} (n)+\mathbf {e} (n),n=1,\ldots ,N}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} =[\mathbf {a} (\theta _{1}),\ldots ,\mathbf {a} (\theta _{K})]}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {A} =[\mathbf {a} (\theta _{1}),\ldots ,\mathbf {a} (\theta _{K})]}</annotation>

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</math><img src="./b5fcbe78e2e522636e76b2f6c803b2e7d124990d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.736ex; height:2.843ex;" alt="{\displaystyle \mathbf {A} =[\mathbf {a} (\theta _{1}),\ldots ,\mathbf {a} (\theta _{K})]}" loading="lazy"> is the <a href="Phased_array" title="Phased array">steering matrix</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bf {x}}(n)=[{\bf {x}}_{1}(n),\ldots ,{\bf {x}}_{K}(n)]^{T}}">

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<annotation encoding="application/x-tex">{\displaystyle {\bf {x}}(n)=[{\bf {x}}_{1}(n),\ldots ,{\bf {x}}_{K}(n)]^{T}}</annotation>

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</math><img src="./643ebb9ed7c697b14fe3bc42d854740365f34ba0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.552ex; height:3.176ex;" alt="{\displaystyle {\bf {x}}(n)=[{\bf {x}}_{1}(n),\ldots ,{\bf {x}}_{K}(n)]^{T}}" loading="lazy"> contains the source waveforms, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bf {e}}(n)}">

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<annotation encoding="application/x-tex">{\displaystyle {\bf {e}}(n)}</annotation>

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</math><img src="./b7be919bcf9345b6a0c1d6c11e3b32ec1822f1a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.429ex; height:2.843ex;" alt="{\displaystyle {\bf {e}}(n)}" loading="lazy"> is the noise term. Assume that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {E} \left({\bf {e}}(n){\bf {e}}^{H}({\bar {n}})\right)=\sigma {\bf {I}}_{M}\delta _{n,{\bar {n}}}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {E} \left({\bf {e}}(n){\bf {e}}^{H}({\bar {n}})\right)=\sigma {\bf {I}}_{M}\delta _{n,{\bar {n}}}}</annotation>

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</math><img src="./d60b1e9d998081b97407eac21b067f85b00cd685.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:25.919ex; height:3.509ex;" alt="{\displaystyle \mathbf {E} \left({\bf {e}}(n){\bf {e}}^{H}({\bar {n}})\right)=\sigma {\bf {I}}_{M}\delta _{n,{\bar {n}}}}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{n,{\bar {n}}}}">

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<annotation encoding="application/x-tex">{\displaystyle \delta _{n,{\bar {n}}}}</annotation>

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</math><img src="./6ff14851159690c734216f2cf5e5d390cc63e4c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.694ex; height:3.176ex;" alt="{\displaystyle \delta _{n,{\bar {n}}}}" loading="lazy"> is the <a href="Dirac_delta_function" title="Dirac delta function">Dirac delta</a> and it equals to 1 only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n={\bar {n}}}">

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<annotation encoding="application/x-tex">{\displaystyle n={\bar {n}}}</annotation>

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</math><img src="./1ea357b4dc82580830e5e0bf4716abd3b187be87.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.888ex; height:2.009ex;" alt="{\displaystyle n={\bar {n}}}" loading="lazy"> and 0 otherwise. Also assume that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bf {e}}(n)}">

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<annotation encoding="application/x-tex">{\displaystyle {\bf {e}}(n)}</annotation>

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</math><img src="./b7be919bcf9345b6a0c1d6c11e3b32ec1822f1a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.429ex; height:2.843ex;" alt="{\displaystyle {\bf {e}}(n)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bf {x}}(n)}">

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<annotation encoding="application/x-tex">{\displaystyle {\bf {x}}(n)}</annotation>

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</math><img src="./f4e6470ad24f57c4978c25b8dc4b8cdf3edeeeaf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.615ex; height:2.843ex;" alt="{\displaystyle {\bf {x}}(n)}" loading="lazy"> are independent, and that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {E} \left({\bf {x}}(n){\bf {x}}^{H}({\bar {n}})\right)={\bf {P}}\delta _{n,{\bar {n}}}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {E} \left({\bf {x}}(n){\bf {x}}^{H}({\bar {n}})\right)={\bf {P}}\delta _{n,{\bar {n}}}}</annotation>

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</math><img src="./55028efb5d1f7a53f36a5e0771863b481783f0ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:23.815ex; height:3.509ex;" alt="{\displaystyle \mathbf {E} \left({\bf {x}}(n){\bf {x}}^{H}({\bar {n}})\right)={\bf {P}}\delta _{n,{\bar {n}}}}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bf {P}}=\operatorname {Diag} ({p_{1},\ldots ,p_{K}})}">

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<annotation encoding="application/x-tex">{\displaystyle {\bf {P}}=\operatorname {Diag} ({p_{1},\ldots ,p_{K}})}</annotation>

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</math><img src="./b20645061695b47b82bbab1f7e19aaa21c7304ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.746ex; height:2.843ex;" alt="{\displaystyle {\bf {P}}=\operatorname {Diag} ({p_{1},\ldots ,p_{K}})}" loading="lazy">. Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bf {p}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\bf {p}}}</annotation>

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</math><img src="./d1d1cf8537aa88f160be6de2d463631433c0e788.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.485ex; height:2.009ex;" alt="{\displaystyle {\bf {p}}}" loading="lazy"> be a vector containing the unknown signal powers and noise variance, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bf {p}}=[p_{1},\ldots ,p_{K},\sigma ]^{T}}">

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<annotation encoding="application/x-tex">{\displaystyle {\bf {p}}=[p_{1},\ldots ,p_{K},\sigma ]^{T}}</annotation>

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</math><img src="./12f58e2b0d49c71656f0aeade621a9c868df1066.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.895ex; height:3.176ex;" alt="{\displaystyle {\bf {p}}=[p_{1},\ldots ,p_{K},\sigma ]^{T}}" loading="lazy">.

The <a href="Covariance_matrix" title="Covariance matrix">covariance matrix</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bf {y}}(n)}">

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<annotation encoding="application/x-tex">{\displaystyle {\bf {y}}(n)}</annotation>

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</math><img src="./5a999d26ebd799c0260d71f5c89b4981e0cabf1d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.615ex; height:2.843ex;" alt="{\displaystyle {\bf {y}}(n)}" loading="lazy"> that contains all information about <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\bf {p}}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\bf {p}}}}</annotation>

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</math><img src="./867cc5c4f60021d949608bf4beb17580326ecc00.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.485ex; height:2.009ex;" alt="{\displaystyle {\boldsymbol {\bf {p}}}}" loading="lazy"> is

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<annotation encoding="application/x-tex">{\displaystyle {\bf {R}}={\bf {A}}{\bf {P}}{\bf {A}}^{H}+\sigma {\bf {I}}.}</annotation>

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</math><img src="./3ab220d7f35a3f4e387ae6b66e736aca4ebea921.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:18.49ex; height:2.843ex;" alt="{\displaystyle {\bf {R}}={\bf {A}}{\bf {P}}{\bf {A}}^{H}+\sigma {\bf {I}}.}" loading="lazy"></dd></dl>

This covariance matrix can be traditionally estimated by the sample covariance matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bf {R}}_{N}={\bf {Y}}{\bf {Y}}^{H}/N}">

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<annotation encoding="application/x-tex">{\displaystyle {\bf {R}}_{N}={\bf {Y}}{\bf {Y}}^{H}/N}</annotation>

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</math><img src="./1ffd58858a83707ffc2af677127d72c2d243c4b4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.75ex; height:3.176ex;" alt="{\displaystyle {\bf {R}}_{N}={\bf {Y}}{\bf {Y}}^{H}/N}" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bf {Y}}=[{\bf {y}}(1),\ldots ,{\bf {y}}(N)]}">

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<annotation encoding="application/x-tex">{\displaystyle {\bf {Y}}=[{\bf {y}}(1),\ldots ,{\bf {y}}(N)]}</annotation>

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</math><img src="./88683322a32ac4a60ee717316b02ddc9f019c391.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.256ex; height:2.843ex;" alt="{\displaystyle {\bf {Y}}=[{\bf {y}}(1),\ldots ,{\bf {y}}(N)]}" loading="lazy">. After applying the <a href="Vectorization_(mathematics)" title="Vectorization (mathematics)">vectorization operator</a> to the matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bf {R}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\bf {R}}}</annotation>

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</math><img src="./fd564c86d9f39cd941501ab380742ef459a5d78c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.003ex; height:2.176ex;" alt="{\displaystyle {\bf {R}}}" loading="lazy">, the obtained vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bf {r}}({\boldsymbol {\bf {p}}})=\operatorname {vec} ({\bf {R}})}">

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<annotation encoding="application/x-tex">{\displaystyle {\bf {r}}({\boldsymbol {\bf {p}}})=\operatorname {vec} ({\bf {R}})}</annotation>

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</math><img src="./a38cfd729caf8a4dcd287adc6de03151ac95e6da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.6ex; height:2.843ex;" alt="{\displaystyle {\bf {r}}({\boldsymbol {\bf {p}}})=\operatorname {vec} ({\bf {R}})}" loading="lazy"> is linearly related to the unknown parameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\bf {p}}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\bf {p}}}}</annotation>

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</math><img src="./867cc5c4f60021d949608bf4beb17580326ecc00.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.485ex; height:2.009ex;" alt="{\displaystyle {\boldsymbol {\bf {p}}}}" loading="lazy"> as

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<annotation encoding="application/x-tex">{\displaystyle {\bf {r}}({\boldsymbol {\bf {p}}})=\operatorname {vec} ({\bf {R}})={\bf {S}}{\boldsymbol {\bf {p}}}}</annotation>

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</math><img src="./75cba6adfb6ba4a669199c8da9966d73435673b4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.669ex; height:2.843ex;" alt="{\displaystyle {\bf {r}}({\boldsymbol {\bf {p}}})=\operatorname {vec} ({\bf {R}})={\bf {S}}{\boldsymbol {\bf {p}}}}" loading="lazy">,

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bf {S}}=[{\bf {S}}_{1},{\bar {\bf {a}}}_{K+1}]}">

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<annotation encoding="application/x-tex">{\displaystyle {\bf {S}}=[{\bf {S}}_{1},{\bar {\bf {a}}}_{K+1}]}</annotation>

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</math><img src="./f8700568a80a1b2e7834b6da983e743219d0bd91.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.544ex; height:2.843ex;" alt="{\displaystyle {\bf {S}}=[{\bf {S}}_{1},{\bar {\bf {a}}}_{K+1}]}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bf {S}}_{1}=[{\bar {\bf {a}}}_{1},\ldots ,{\bar {\bf {a}}}_{K}]}">

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<annotation encoding="application/x-tex">{\displaystyle {\bf {S}}_{1}=[{\bar {\bf {a}}}_{1},\ldots ,{\bar {\bf {a}}}_{K}]}</annotation>

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</math><img src="./223d2224ef830e8dbc6fb307a39cb9f1a4c1e31f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.456ex; height:2.843ex;" alt="{\displaystyle {\bf {S}}_{1}=[{\bar {\bf {a}}}_{1},\ldots ,{\bar {\bf {a}}}_{K}]}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {\bf {a}}}_{k}={\bf {a}}_{k}^{*}\otimes {\bf {a}}_{k}}">

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<annotation encoding="application/x-tex">{\displaystyle {\bar {\bf {a}}}_{k}={\bf {a}}_{k}^{*}\otimes {\bf {a}}_{k}}</annotation>

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</math><img src="./17af2c859ffcf8e95f4d71fc22a12180216a662d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.104ex; height:2.843ex;" alt="{\displaystyle {\bar {\bf {a}}}_{k}={\bf {a}}_{k}^{*}\otimes {\bf {a}}_{k}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=1,\ldots ,K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k=1,\ldots ,K}</annotation>

</semantics>

</math><img src="./ef5ac9477a2e5ad3b2348d3f30f7326749ffdecb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.716ex; height:2.509ex;" alt="{\displaystyle k=1,\ldots ,K}" loading="lazy">, and let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {\bf {a}}}_{K+1}=\operatorname {vec} ({\bf {I}})}">

<msub>

<mover>

</mrow>

</mover>

</mrow>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\bar {\bf {a}}}_{K+1}=\operatorname {vec} ({\bf {I}})}</annotation>

</semantics>

</math><img src="./4604ef951dbeaa75e5be9ac62429a77c64b8b197.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.307ex; height:2.843ex;" alt="{\displaystyle {\bar {\bf {a}}}_{K+1}=\operatorname {vec} ({\bf {I}})}" loading="lazy">

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \otimes }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \otimes }</annotation>

</semantics>

</math><img src="./de29098f5a34ee296a505681a0d5e875070f2aea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \otimes }" loading="lazy">

is the Kronecker product.

<div class="mw-heading mw-heading2"><h2 id="SAMV_algorithm">SAMV algorithm</h2></div>

To estimate the parameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\bf {p}}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\bf {p}}}}</annotation>

</semantics>

</math><img src="./867cc5c4f60021d949608bf4beb17580326ecc00.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.485ex; height:2.009ex;" alt="{\displaystyle {\boldsymbol {\bf {p}}}}" loading="lazy"> from the statistic <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bf {r}}_{N}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\bf {r}}_{N}}</annotation>

</semantics>

</math><img src="./d4546a89e8b119a15d3e3f6aa455515ac4bf4473.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.794ex; height:2.009ex;" alt="{\displaystyle {\bf {r}}_{N}}" loading="lazy">, we develop a series of iterative SAMV approaches based on the asymptotically minimum variance criterion. From <a href="#cite_note-AbeidaZhang-1">[1]</a> the covariance matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Cov} _{\boldsymbol {p}}^{\operatorname {Alg} }}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Cov} _{\boldsymbol {p}}^{\operatorname {Alg} }}</annotation>

</semantics>

</math><img src="./1065ffdc5581a49fbde8e6034a9d4b365062f0bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:6.812ex; height:3.676ex;" alt="{\displaystyle \operatorname {Cov} _{\boldsymbol {p}}^{\operatorname {Alg} }}" loading="lazy"> of an arbitrary consistent estimator of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {p}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {p}}}</annotation>

</semantics>

</math><img src="./04cff366782c9fb192fc63992ef75ad59ee77695.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.052ex; width:1.449ex; height:2.009ex;" alt="{\displaystyle {\boldsymbol {p}}}" loading="lazy"> based on the second-order statistic <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bf {r}}_{N}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\bf {r}}_{N}}</annotation>

</semantics>

</math><img src="./d4546a89e8b119a15d3e3f6aa455515ac4bf4473.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.794ex; height:2.009ex;" alt="{\displaystyle {\bf {r}}_{N}}" loading="lazy"> is bounded by the real symmetric positive definite matrix

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Cov} _{\boldsymbol {p}}^{\operatorname {Alg} }\geq [{\bf {S}}_{d}^{H}{\bf {C}}_{r}^{-1}{\bf {S}}_{d}]^{-1},}</annotation>

</semantics>

</math><img src="./b95412087ea6bb7532ba838759115b5a5e1b0e45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:24.202ex; height:3.676ex;" alt="{\displaystyle \operatorname {Cov} _{\boldsymbol {p}}^{\operatorname {Alg} }\geq [{\bf {S}}_{d}^{H}{\bf {C}}_{r}^{-1}{\bf {S}}_{d}]^{-1},}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bf {S}}_{d}={\rm {d}}{\bf {r}}({\boldsymbol {p}})/{\rm {d}}{\boldsymbol {p}}}">

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\bf {S}}_{d}={\rm {d}}{\bf {r}}({\boldsymbol {p}})/{\rm {d}}{\boldsymbol {p}}}</annotation>

</semantics>

</math><img src="./9dd9150bec574af4e8faad8f69cc9543954458e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.129ex; height:2.843ex;" alt="{\displaystyle {\bf {S}}_{d}={\rm {d}}{\bf {r}}({\boldsymbol {p}})/{\rm {d}}{\boldsymbol {p}}}" loading="lazy">. In addition, this lower bound is attained by the covariance matrix of the asymptotic distribution of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\bf {p}}}}">

<mover>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {\bf {p}}}}</annotation>

</semantics>

</math><img src="./0e2bbd36f615a261d69ccb98c2eb096a26654ab7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.485ex; height:2.676ex;" alt="{\displaystyle {\hat {\bf {p}}}}" loading="lazy"> obtained by minimizing

<mover>

</mover>

</mrow>

</mrow>

</munder>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {\boldsymbol {p}}}=\arg \min _{\boldsymbol {p}}f({\boldsymbol {p}}),}</annotation>

</semantics>

</math><img src="./4fcb02e2069215018e519530c23b10f04b047100.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; margin-left: -0.052ex; width:17.671ex; height:4.176ex;" alt="{\displaystyle {\hat {\boldsymbol {p}}}=\arg \min _{\boldsymbol {p}}f({\boldsymbol {p}}),}" loading="lazy"></dd></dl>

where

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f({\boldsymbol {p}})=[{\bf {r}}_{N}-{\bf {r}}({\boldsymbol {p}})]^{H}{\bf {C}}_{r}^{-1}[{\bf {r}}_{N}-{\bf {r}}({\boldsymbol {p}})].}</annotation>

</semantics>

</math><img src="./15df0f7c10b4df3f894bdf0b7733ad03987e86ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.658ex; height:3.176ex;" alt="{\displaystyle f({\boldsymbol {p}})=[{\bf {r}}_{N}-{\bf {r}}({\boldsymbol {p}})]^{H}{\bf {C}}_{r}^{-1}[{\bf {r}}_{N}-{\bf {r}}({\boldsymbol {p}})].}" loading="lazy">

Therefore, the estimate of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\bf {p}}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\bf {p}}}}</annotation>

</semantics>

</math><img src="./867cc5c4f60021d949608bf4beb17580326ecc00.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.485ex; height:2.009ex;" alt="{\displaystyle {\boldsymbol {\bf {p}}}}" loading="lazy"> can be obtained iteratively.

The <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{{\hat {p}}_{k}\}_{k=1}^{K}}">

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{{\hat {p}}_{k}\}_{k=1}^{K}}</annotation>

</semantics>

</math><img src="./e9d1e8d6da7314f083564d76d7897da3ac4063b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.962ex; height:3.176ex;" alt="{\displaystyle \{{\hat {p}}_{k}\}_{k=1}^{K}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\sigma }}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {\sigma }}}</annotation>

</semantics>

</math><img src="./e130d3780e4fc508e843485b3cfbf8347ba288fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.176ex;" alt="{\displaystyle {\hat {\sigma }}}" loading="lazy"> that minimize <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f({\boldsymbol {p}})}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f({\boldsymbol {p}})}</annotation>

</semantics>

</math><img src="./ab367dbb05c44b349d69effd6374493c18348dab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.485ex; height:2.843ex;" alt="{\displaystyle f({\boldsymbol {p}})}" loading="lazy"> can be computed as follows. Assume <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {p}}_{k}^{(i)}}">

<mover>

</mover>

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {p}}_{k}^{(i)}}</annotation>

</semantics>

</math><img src="./dff869db1148661da66d2625098d8901fad5eee5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.089ex; width:3.528ex; height:3.676ex;" alt="{\displaystyle {\hat {p}}_{k}^{(i)}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\sigma }}^{(i)}}">

<msup>

<mover>

</mover>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {\sigma }}^{(i)}}</annotation>

</semantics>

</math><img src="./d2a9d91e3b2aa4e335df6666098872b5a89803fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.409ex; height:2.843ex;" alt="{\displaystyle {\hat {\sigma }}^{(i)}}" loading="lazy"> have been approximated to a certain degree in the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

</semantics>

</math><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy">th iteration, they can be refined at the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (i+1)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (i+1)}</annotation>

</semantics>

</math><img src="./01d1bd3a07f06d94e26759b664525a5eb7570e8f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.615ex; height:2.843ex;" alt="{\displaystyle (i+1)}" loading="lazy">th iteration by

<mover>

</mover>

</mrow>

</mrow>

</mrow>

</msubsup>

<mfrac>

<mrow>

</mrow>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</mrow>

</msup>

<msub>

</mrow>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</mrow>

</msup>

<msub>

</mrow>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

</mrow>

</msubsup>

<mfrac>

<mrow>

</mrow>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</mrow>

</msup>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {p}}_{k}^{(i+1)}={\frac {{\bf {a}}_{k}^{H}{\bf {R}}^{-1{(i)}}{\bf {R}}_{N}{\bf {R}}^{-1{(i)}}{\bf {a}}_{k}}{({\bf {a}}_{k}^{H}{\bf {R}}^{-1{(i)}}{\bf {a}}_{k})^{2}}}+{\hat {p}}_{k}^{(i)}-{\frac {1}{{\bf {a}}_{k}^{H}{\bf {R}}^{-1{(i)}}{\bf {a}}_{k}}},\quad k=1,\ldots ,K}</annotation>

</semantics>

</math><img src="./80c0f98c758f243f8371db6e973c6062404d28d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; margin-left: -0.089ex; width:68.593ex; height:7.843ex;" alt="{\displaystyle {\hat {p}}_{k}^{(i+1)}={\frac {{\bf {a}}_{k}^{H}{\bf {R}}^{-1{(i)}}{\bf {R}}_{N}{\bf {R}}^{-1{(i)}}{\bf {a}}_{k}}{({\bf {a}}_{k}^{H}{\bf {R}}^{-1{(i)}}{\bf {a}}_{k})^{2}}}+{\hat {p}}_{k}^{(i)}-{\frac {1}{{\bf {a}}_{k}^{H}{\bf {R}}^{-1{(i)}}{\bf {a}}_{k}}},\quad k=1,\ldots ,K}" loading="lazy"></dd></dl>

<msup>

<mover>

</mover>

</mrow>

</mrow>

</msup>

<mrow>

<mrow>

<msup>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</msup>

<msub>

</mrow>

</mrow>

</msub>

<msup>

<mover>

</mover>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</msup>

<msup>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

<msup>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {\sigma }}^{(i+1)}=\left(\operatorname {Tr} ({\bf {R}}^{-2^{(i)}}{\bf {R}}_{N})+{\hat {\sigma }}^{(i)}\operatorname {Tr} ({\bf {R}}^{-2^{(i)}})-\operatorname {Tr} ({\bf {R}}^{-1^{(i)}})\right)/{\operatorname {Tr} {({\bf {R}}^{-2^{(i)}})}},}</annotation>

</semantics>

</math><img src="./74a0fa65c7caa2c3994b848d91fe615b4a0257ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:68.732ex; height:4.843ex;" alt="{\displaystyle {\hat {\sigma }}^{(i+1)}=\left(\operatorname {Tr} ({\bf {R}}^{-2^{(i)}}{\bf {R}}_{N})+{\hat {\sigma }}^{(i)}\operatorname {Tr} ({\bf {R}}^{-2^{(i)}})-\operatorname {Tr} ({\bf {R}}^{-1^{(i)}})\right)/{\operatorname {Tr} {({\bf {R}}^{-2^{(i)}})}},}" loading="lazy"></dd></dl>

where the estimate of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bf {R}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\bf {R}}}</annotation>

</semantics>

</math><img src="./fd564c86d9f39cd941501ab380742ef459a5d78c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.003ex; height:2.176ex;" alt="{\displaystyle {\bf {R}}}" loading="lazy"> at the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

</semantics>

</math><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy">th iteration is given by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bf {R}}^{(i)}={\bf {A}}{\bf {P}}^{(i)}{\bf {A}}^{H}+{\hat {\sigma }}^{(i)}{\bf {I}}}">

<msup>

</mrow>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

<msup>

<mover>

</mover>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\bf {R}}^{(i)}={\bf {A}}{\bf {P}}^{(i)}{\bf {A}}^{H}+{\hat {\sigma }}^{(i)}{\bf {I}}}</annotation>

</semantics>

</math><img src="./5208d7f4741581b18f26eaf9471b15002a77319d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:24.08ex; height:3.009ex;" alt="{\displaystyle {\bf {R}}^{(i)}={\bf {A}}{\bf {P}}^{(i)}{\bf {A}}^{H}+{\hat {\sigma }}^{(i)}{\bf {I}}}" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bf {P}}^{(i)}=\operatorname {Diag} ({\hat {p}}_{1}^{(i)},\ldots ,{\hat {p}}_{K}^{(i)})}">

<msup>

</mrow>

</mrow>

</msup>

<mover>

</mover>

</mrow>

</mrow>

</mrow>

</msubsup>

<mover>

</mover>

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\bf {P}}^{(i)}=\operatorname {Diag} ({\hat {p}}_{1}^{(i)},\ldots ,{\hat {p}}_{K}^{(i)})}</annotation>

</semantics>

</math><img src="./aeed583fcd545fe9226178a3ab1f86d651fae523.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.616ex; height:3.676ex;" alt="{\displaystyle {\bf {P}}^{(i)}=\operatorname {Diag} ({\hat {p}}_{1}^{(i)},\ldots ,{\hat {p}}_{K}^{(i)})}" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="Beyond_scanning_grid_accuracy">Beyond scanning grid accuracy</h2></div>

The resolution of most <a href="Compressed_sensing" title="Compressed sensing">compressed sensing</a> based source localization techniques is limited by the fineness of the direction grid that covers the location parameter space.<a href="#cite_note-Malioutov_Cetin_Willsky_2005_pp._3010–3022-4">[4]</a> In the sparse signal recovery model, the sparsity of the truth signal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} (n)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {x} (n)}</annotation>

</semantics>

</math><img src="./baf83e20802b96ec92669173abf5a512d8286251.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.615ex; height:2.843ex;" alt="{\displaystyle \mathbf {x} (n)}" loading="lazy"> is dependent on the distance between the adjacent element in the overcomplete dictionary <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bf {A}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\bf {A}}}</annotation>

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</math><img src="./d0f7b46d371ea1fd658b16a62e2b07022d9f9295.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle {\bf {A}}}" loading="lazy">, therefore, the difficulty of choosing the optimum <a href="Sparse_dictionary_learning#Properties_of_the_dictionary" title="Sparse dictionary learning">overcomplete dictionary</a> 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,<a href="#cite_note-AbeidaZhang-1">[1]</a> which refine the location estimates <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\bf {\theta }}}=(\theta _{1},\ldots ,\theta _{K})^{T}}">

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<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\bf {\theta }}}=(\theta _{1},\ldots ,\theta _{K})^{T}}</annotation>

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</math><img src="./1cf3bfe18222cec41d915661abff66b3ef890d9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.71ex; height:3.176ex;" alt="{\displaystyle {\boldsymbol {\bf {\theta }}}=(\theta _{1},\ldots ,\theta _{K})^{T}}" loading="lazy"> by iteratively minimizing a stochastic <a href="Maximum_likelihood_estimation" title="Maximum likelihood estimation">maximum likelihood</a> cost function with respect to a single scalar parameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta _{k}}">

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</math><img src="./4ae772f574aa637900f7673f6f934bdcf35969ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.179ex; height:2.509ex;" alt="{\displaystyle \theta _{k}}" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="Application_to_range-Doppler_imaging">Application to range-Doppler imaging</h2></div>

A typical application with the SAMV algorithm in <a href="Single-input_single-output_system" title="Single-input single-output system">SISO</a> <a href="Radar" title="Radar">radar</a>/<a href="Sonar" title="Sonar">sonar</a> <a href="Pulse-Doppler_radar" title="Pulse-Doppler radar">range-Doppler imaging</a> problem. This imaging problem is a single-snapshot application, and algorithms compatible with single-snapshot estimation are included, i.e., <a href="Matched_filter" title="Matched filter">matched filter</a> (MF, similar to the <a href="Periodogram" title="Periodogram">periodogram</a> or <a href="Radon_transform" title="Radon transform">backprojection</a>, which is often efficiently implemented as <a href="Fast_Fourier_transform" title="Fast Fourier transform">fast Fourier transform</a> (FFT)), IAA,<a href="#cite_note-Yardibi_Li_Stoica_Xue_2010_pp._425–443-5">[5]</a> and a variant of the SAMV algorithm (SAMV-0). The simulation conditions are identical to:<a href="#cite_note-Yardibi_Li_Stoica_Xue_2010_pp._425–443-5">[5]</a> A <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 30}">

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</math><img src="./5bbb9554d69ffa16547379e6d7dc2f0d76fbf637.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle 30}" loading="lazy">-element polyphase <a href="Pulse_compression" title="Pulse compression">pulse compression</a> 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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5}">

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</math><img src="./29483407999b8763f0ea335cf715a6a5e809f44b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 5}" loading="lazy"> dB power and the rest six are of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 25}">

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</math><img src="./a8acf6937f13156a1301ebb614c5364d16597e42.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle 25}" loading="lazy"> dB power. The received signals are assumed to be contaminated with uniform white Gaussian noise of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">

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</math><img src="./2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy"> dB power.

The <a href="Matched_filter" title="Matched filter">matched filter</a> detection result suffers from severe smearing and <a href="Spectral_leakage" title="Spectral leakage">leakage</a> effects both in the Doppler and range domain, hence it is impossible to distinguish the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5}">

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</math><img src="./29483407999b8763f0ea335cf715a6a5e809f44b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 5}" loading="lazy"> 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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5}">

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</math><img src="./29483407999b8763f0ea335cf715a6a5e809f44b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 5}" loading="lazy"> dB targets.

<div class="mw-heading mw-heading2"><h2 id="Open_source_implementation">Open source implementation</h2></div>

An open source <a href="MATLAB" title="MATLAB">MATLAB</a> implementation of SAMV algorithm could be downloaded <a rel="nofollow" class="external text" href="https://qilin-zhang.github.io/_pages/zips/Iterative_Sparse_Asymptotic_Minimum_Variance_Based_Approach_Matlab_Codes.zip?raw=true">here</a>.

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<ul><li><a href="Array_processing" title="Array processing">Array processing</a> – Area of research in signal processing</li>

<li><a href="Matched_filter" title="Matched filter">Matched filter</a> – Filters used in signal processing that are optimal in some sense</li>

<li><a href="Periodogram" title="Periodogram">Periodogram</a> – Estimate of the spectral density of a signal</li>

<li><a href="Radon_transform" title="Radon transform">Filtered backprojection</a> – Integral transform (Radon transform)</li>

<li><a href="MUSIC_(algorithm)" title="MUSIC (algorithm)">MUltiple SIgnal Classification</a> – Algorithm used for frequency estimation and radio direction finding (MUSIC), a popular parametric <a href="Super-resolution_imaging" title="Super-resolution imaging">superresolution</a> method</li>

<li><a href="Pulse-Doppler_radar" title="Pulse-Doppler radar">Pulse-Doppler radar</a> – Type of radar system</li>

<li><a href="Super-resolution_imaging" title="Super-resolution imaging">Super-resolution imaging</a> – Any technique to improve resolution of an imaging system beyond conventional limits</li>

<li><a href="Compressed_sensing" title="Compressed sensing">Compressed sensing</a> – Signal processing technique</li>

<li><a href="Inverse_problem" title="Inverse problem">Inverse problem</a> – Process of calculating the causal factors that produced a set of observations</li>

<li><a href="Tomographic_reconstruction" title="Tomographic reconstruction">Tomographic reconstruction</a> – Estimate object properties from a finite number of projections</li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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<li id="cite_note-Yardibi_Li_Stoica_Xue_2010_pp._425–443-5">^ <a href="#cite_ref-Yardibi_Li_Stoica_Xue_2010_pp._425–443_5-0">a</a> <a href="#cite_ref-Yardibi_Li_Stoica_Xue_2010_pp._425–443_5-1">b</a> <cite id="CITEREFYardibiLiStoicaXue2010" class="citation journal cs1">Yardibi, Tarik; <a href="Jian_Li_(engineer)" title="Jian Li (engineer)">Li, Jian</a>; 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. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2010ITAES..46..425Y">2010ITAES..46..425Y</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2Ftaes.2010.5417172">10.1109/taes.2010.5417172</a>. <a href="Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/1721.1%2F59588">1721.1/59588</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:18834345">18834345</a>.</cite>

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