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Low-Complexity DOA Estimation Algorithm based on Real-Valued Sparse Bayesian Learning

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Abstract

The present DOA estimation approach based on sparse Bayesian learning has two major shortcomings: high algorithm complexity and large estimate errors. These two flaws are mostly the result of an excessive number of complex-valued matrix inversion operations and incorrect grid partitioning in the EM stage of sparse Bayesian learning (SBL). To overcome the aforementioned issues, we propose a three-procedure, low-complexity DOA estimate technique based on SBL. First, the roots of the estimated covariance are used to complete the real-valued conversion and generate the received signal matrix, which includes the virtual array steering vector. Second, a novel iterative method for immobile spots is devised using the probability distribution function of the noise variance. Finally, iterations are completed utilizing dynamic grid approaches to improve DOA estimation accuracy. The simulation findings reveal that the proposed technique significantly speeds up DOA estimation and, to a lesser extent, improves estimation accuracy.

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Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

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Appendices

Appendix A

Conclusion 1 proof.\({\tilde{\mathbf{R}}}_{y}^{1/2}\) is the Hermitian square root of \({\tilde{\mathbf{R}}}_{y}\), so \({\tilde{\mathbf{W}}}\) can be split into the following equation

$$ \begin{aligned} {\tilde{\mathbf{W}}} = & \frac{1}{T}{\tilde{\mathbf{R}}}_{y}^{{\varvec{T}}} \otimes {\tilde{\mathbf{R}}}_{y} \\ = & \frac{1}{T}\left( {{\tilde{\mathbf{R}}}_{y}^{1/2} {\tilde{\mathbf{R}}}_{y}^{1/2} } \right)^{{\varvec{T}}} \otimes \left( {{\tilde{\mathbf{R}}}_{y}^{1/2} {\tilde{\mathbf{R}}}_{y}^{1/2} } \right) \\ = & \frac{1}{\sqrt T }\left( {{(}{\tilde{\mathbf{R}}}_{y}^{1/2} {)}^{{\varvec{T}}} \otimes {\tilde{\mathbf{R}}}_{y}^{1/2} } \right) \cdot \frac{1}{\sqrt T }\left( {{(}{\tilde{\mathbf{R}}}_{y}^{1/2} {)}^{{\varvec{T}}} \otimes {\tilde{\mathbf{R}}}_{y}^{1/2} } \right) \\ \end{aligned} $$

Based on \(({\mathbf{AC}} \otimes {\mathbf{BD}}) = ({\mathbf{A}} \otimes {\mathbf{B}})({\mathbf{C}} \otimes {\mathbf{D}})\), derive \({\tilde{\mathbf{W}}}^{{ - {1/2}}}\) as follows

$$ {\tilde{\mathbf{W}}}^{{ - {1/2}}} = \sqrt T \left( {{(}{\tilde{\mathbf{R}}}_{y}^{1/2} {)}^{{\varvec{T}}} \otimes {\tilde{\mathbf{R}}}_{y}^{1/2} } \right)^{ - 1} $$
$$ \begin{aligned} {\mathbf{r}}_{{\varvec{W}}} = & \sqrt T \left( {{(}{\tilde{\mathbf{R}}}_{y}^{1/2} {)}^{{\varvec{T}}} \otimes {\tilde{\mathbf{R}}}_{y}^{1/2} } \right)^{ - 1} {\tilde{\text{r}}} \\ { = } & \sqrt T \left( {{(}{\tilde{\mathbf{R}}}_{y}^{1/2} {)}^{{\varvec{T}}} \otimes {\tilde{\mathbf{R}}}_{y}^{1/2} } \right)^{ - 1} vec\left\{ {{\tilde{\mathbf{R}}}_{y}^{1/2} {\mathbf{I}}_{M} {\tilde{\mathbf{R}}}_{y}^{1/2} } \right\} \\ = & \sqrt T \left( {{(}{\tilde{\mathbf{R}}}_{y}^{1/2} {)}^{{\varvec{T}}} \otimes {\tilde{\mathbf{R}}}_{y}^{1/2} } \right)^{ - 1} \left( {{(}{\tilde{\mathbf{R}}}_{y}^{1/2} {)}^{{\varvec{T}}} \otimes {\tilde{\mathbf{R}}}_{y}^{1/2} } \right)vec\left\{ {{\mathbf{I}}_{M} } \right\} \\ \end{aligned} $$

According to \(vec\left\{ {{\mathbf{ABC}}} \right\} = ({\mathbf{C}}^{{\varvec{T}}} \otimes {\mathbf{A}})vec\left\{ {\mathbf{B}} \right\}\), we get.

\({\mathbf{r}}_{{\varvec{W}}} = \sqrt T vec\left\{ {{\mathbf{I}}_{M} } \right\}\).

Conclusion 2 proof. \(i,j \in \left\{ {1,2,...,L} \right\}\),\(L\) is the number of grids and the DOA estimation range \([ - {\varvec{\pi}}/2,{\varvec{\pi}}/2]\) is divided evenly into \(L\) \(\theta\).

Based on \({\tilde{\mathbf{W}}}^{{ - {1/2}}} = \sqrt T \left( {({\tilde{\mathbf{R}}}_{y}^{1/2} )^{{\varvec{T}}} \otimes {\tilde{\mathbf{R}}}_{y}^{1/2} } \right)^{ - 1}\), it follows that

$$ \begin{aligned}& [{\mathbf{A}}_{\theta }^{{\varvec{H}}} {\tilde{\mathbf{W}}}^{ - 1} {\mathbf{A}}_{\theta }^{{}} ]_{i,j} \\ &\quad= \left( {{\tilde{\mathbf{W}}}^{{ - {1/2}}} ({\mathbf{a}}^{*} (\theta_{i} ) \otimes {\mathbf{a}}(\theta_{i} ))} \right)^{{\varvec{H}}} {\tilde{\mathbf{W}}}^{{ - {1/2}}} ({\mathbf{a}}^{*} (\theta_{j} ) \otimes {\mathbf{a}}(\theta_{j} )) \\ &\quad= T \cdot \left( {({\tilde{\mathbf{R}}}_{y}^{ - 1/2} {\mathbf{a}}(\theta_{i} ))^{*} \otimes {\tilde{\mathbf{R}}}_{y}^{ - 1/2} {\mathbf{a}}(\theta_{i} )} \right)^{{\varvec{H}}} \\& \qquad\cdot \left( {({\tilde{\mathbf{R}}}_{y}^{ - 1/2} {\mathbf{a}}(\theta_{j} ))^{*} \otimes {\tilde{\mathbf{R}}}_{y}^{ - 1/2} {\mathbf{a}}(\theta_{j} )} \right) \\ \end{aligned} $$

According to \(({\mathbf{A}} \otimes {\mathbf{B}})^{{\varvec{H}}} = {\mathbf{A}}^{{\varvec{H}}} \otimes {\mathbf{B}}^{{\varvec{H}}}\), it follows that

$$ \begin{aligned}& [{\mathbf{A}}_{\theta }^{{\varvec{H}}} {\tilde{\mathbf{W}}}^{ - 1} {\mathbf{A}}_{\theta }^{{}} ]_{i,j} \\&\quad = T \cdot \left( {({\tilde{\mathbf{R}}}_{y}^{ - 1/2} {\mathbf{a}}(\theta_{i} ))^{{\varvec{T}}} \otimes ({\tilde{\mathbf{R}}}_{y}^{ - 1/2} {\mathbf{a}}(\theta_{i} ))^{{\varvec{H}}} } \right) \\ &\qquad\cdot \left( {({\tilde{\mathbf{R}}}_{y}^{ - 1/2} {\mathbf{a}}(\theta_{j} ))^{*} \otimes {\tilde{\mathbf{R}}}_{y}^{ - 1/2} {\mathbf{a}}(\theta_{j} )} \right) \\ &\quad= T \cdot \left( {({\tilde{\mathbf{R}}}_{y}^{ - 1/2} {\mathbf{a}}(\theta_{i} ))^{{\varvec{T}}} \otimes ({\tilde{\mathbf{R}}}_{y}^{ - 1/2} {\mathbf{a}}(\theta_{i} ))^{*} } \right) \\&\qquad \cdot \left( {({\tilde{\mathbf{R}}}_{y}^{ - 1/2} {\mathbf{a}}(\theta_{j} ))^{{\varvec{H}}} {\tilde{\mathbf{R}}}_{y}^{ - 1/2} {\mathbf{a}}(\theta_{j} )} \right) \\&\quad = T \cdot \left| {{\mathbf{a}}^{{\varvec{H}}} (\theta_{i} ){\tilde{\mathbf{R}}}_{y}^{ - 1} {\mathbf{a}}(\theta_{j} )} \right|^{{\varvec{2}}} . \\ \end{aligned} $$

Appendix B

The results derived from Appendix A \({\mathbf{r}}_{{\varvec{W}}} = \sqrt T vec\left\{ {{\mathbf{I}}_{M} } \right\}\),\(({\tilde{\mathbf{W}}}^{{ - {1/2}}} {\mathbf{A}}_{\theta }^{{}} )^{{\varvec{H}}} {\mathbf{r}}_{{\varvec{W}}}\) are further deduced as follows

$$ \begin{aligned}& [({\tilde{\mathbf{W}}}^{{ - {1/2}}} {\mathbf{A}}_{\theta }^{{}} )^{{\varvec{H}}} {\mathbf{r}}_{{\varvec{W}}} ]_{i}^{{}} \\ &\quad= ({\tilde{\mathbf{W}}}^{{ - {1/2}}} ({\mathbf{a}}^{*} (\theta_{i} ) \otimes {\mathbf{a}}(\theta_{i} )))^{{\varvec{H}}} \cdot \sqrt T vec\left\{ {{\mathbf{I}}_{M} } \right\} \\&\quad = T \cdot \left( {({\tilde{\mathbf{R}}}_{y}^{ - 1/2} {\mathbf{a}}(\theta_{i} ))^{*} \otimes {\tilde{\mathbf{R}}}_{y}^{ - 1/2} {\mathbf{a}}(\theta_{i} )} \right)^{{\varvec{H}}} vec\left\{ {{\mathbf{I}}_{M} } \right\} \\ \end{aligned} $$

According to \(vec\left\{ {{\mathbf{ABC}}} \right\} = ({\mathbf{C}}^{{\varvec{T}}} \otimes {\mathbf{A}})vec\left\{ {\mathbf{B}} \right\}\), we get

$$ \begin{aligned}& [({\tilde{\mathbf{W}}}^{{ - {1/2}}} {\mathbf{A}}_{\theta }^{{}} )^{{\varvec{H}}} {\mathbf{r}}_{{\varvec{W}}} ]_{i} \\&\quad = T \cdot ({\tilde{\mathbf{R}}}_{y}^{ - 1/2} {\mathbf{a}}(\theta_{i} ))^{{\varvec{H}}} {\tilde{\mathbf{R}}}_{y}^{ - 1/2} {\mathbf{a}}(\theta_{i} ) \\&\quad = T \cdot \left\| {{\tilde{\mathbf{R}}}_{y}^{ - 1/2} {\mathbf{a}}(\theta_{i} )} \right\|_{{\varvec{2}}}^{{\varvec{2}}} . \\ \end{aligned} $$

Since the previous equation demonstrates that \(({\tilde{\mathbf{W}}}^{{ - {1/2}}} {\mathbf{A}}_{\theta }^{{}} )^{{\varvec{H}}} {\mathbf{r}}_{{\varvec{W}}}\) is real and that \({\mathbf{V}}\) and \(\Lambda\) are likewise real, it is clear from the following equation that \({\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{r} }}\) is real. We can also demonstrate that \({\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\Psi } }}\) and \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\varepsilon }\) are true values.

\(({\tilde{\mathbf{W}}}^{{ - {1/2}}} {\mathbf{A}}_{\theta }^{{}} )^{{\varvec{H}}} {\mathbf{r}}_{{\varvec{W}}} = ({\mathbf{U}}\Lambda {\mathbf{V}}^{{\varvec{T}}} )^{{\varvec{H}}} {\mathbf{r}}_{{\varvec{W}}} = {\mathbf{V}}\Lambda^{{\varvec{T}}} {\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{r} }}\).

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Wang, G., Kang, Y. & Wang, H. Low-Complexity DOA Estimation Algorithm based on Real-Valued Sparse Bayesian Learning. Circuits Syst Signal Process 43, 4319–4338 (2024). https://doi.org/10.1007/s00034-024-02649-7

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