Reconstructing DOA Estimation in the Second-Order Statistic Domain by Exploiting Matrix Completion

Abstract

It is known that the performance of classical direction-of-arrival (DOA) estimators may be deteriorated considerably in the presence of non-uniform noise and low signal-to-noise ratio (SNR). Focusing on this issue, based on the matrix completion theory, a sparse reconstruction algorithm combining second-order statistical vectors and weighted L1-norm is developed in this paper. In the proposed method, the elastic regularization factor is firstly introduced into the matrix completion model to reconstruct the signal covariance matrix as a noise-free covariance matrix. In what follows, the obtained multi-vector issue associated with the noise-free covariance matrix can be recast as a single vector one by exploiting matrix sum-average operation in the second-order statistical domain. With the constructed single vector, DOA estimation can be complemented by employing the sparse reconstruction weighted L1-norm (WL1) approach. Numerical simulation results show that the proposed algorithm has improved angle estimation accuracy and resolution effectively under low SNR and can suppress the effect of noise non-uniformity significantly.

Appendix: The Selection of η

The vectorization of the estimate error of the noise-free covariance matrix denoted by $\Delta {\mathbf{R}}_{0} \varvec{ = }{\hat{\mathbf{R}}}_{0} \varvec{ - }{\mathbf{R}}_{0}$ is subject to the following distribution [17]:

$$ \text{vec}\left( {\Delta {\mathbf{R}}_{0} } \right) \sim \text{AsN}\left( {0,\frac{1}{L}{\mathbf{R}}_{0}^{T} \otimes {\mathbf{R}}_{0} } \right) $$

(30)

where $\text{AsN}\left( \cdot \right)$ denotes the asymptotically normal distribution and ${\hat{\mathbf{R}}}_{0}$ can be obtained by solving (18).

As illustrated above, the vector ${\varvec{\Upsilon}}$ can be obtained by using sum-average operation on ${\mathbf{R}}_{0}$. Consequently, it indicates a linear relationship between $\Delta {\varvec{\Upsilon}}$ and $\text{vec}\left( {\Delta {\mathbf{R}}_{0} } \right)$, i.e.,

$$ \Delta {\varvec{\Upsilon}} = {\hat{\mathbf{\varvec{\Upsilon}}}}\varvec{ - }{\varvec{\Upsilon}} = {\mathbf{G}} \times \text{vec}\left( {\Delta {\mathbf{R}}_{0} } \right) $$

(31)

where ${\hat{\mathbf{\varvec{\Upsilon}}}}$ is the ${\varvec{\Upsilon}}$ estimate with $L$ snapshots, ${\mathbf{G}} \in \left( {2M - 1} \right) \times M^{2}$ is a linear transformation matrix, and $\text{rank}\left( {\mathbf{G}} \right) = 2M - 1$ [24].

Combining (30) and (31), we have:

$$ {\hat{{\varvec{\Upsilon}}}} - {\tilde{\mathbf{B}}}\left( {\varvec{\Psi}} \right){\tilde{\mathbf{P}}} \sim \text{AsN}\left( {0,\frac{1}{L}{\mathbf{G}}\left( {{\mathbf{R}}_{0}^{T} \otimes {\mathbf{R}}_{0} } \right){\mathbf{G}}^{T} } \right) $$

(32)

that is, ${\hat{{\varvec{\Upsilon}}}} - {\tilde{\mathbf{B}}}\left( {\varvec{\Psi}} \right){\tilde{\mathbf{P}}}$ satisfies the asymptotically normal distribution with mean 0 and variance $\frac{1}{L}{\mathbf{G}}\left( {{\mathbf{R}}_{0}^{T} \otimes {\mathbf{R}}_{0} } \right){\mathbf{G}}^{T}$.

Furthermore, (32) indicates that

$$ {\mathbf{H}}^{{{{ - 1} \mathord{\left/ {\vphantom {{ - 1} 2}} \right. \kern-0pt} 2}}} \left( {{\hat{{\varvec{\Upsilon}}}} - {\tilde{\mathbf{B}}}\left( {\varvec{\Psi}} \right){\tilde{\mathbf{P}}}} \right) \sim \text{AsN}\left( {0,{\mathbf{I}}_{2M - 1} } \right) $$

(33)

where ${\mathbf{H}}^{{{{ - 1} \mathord{\left/ {\vphantom {{ - 1} 2}} \right. \kern-0pt} 2}}}$ is the Hermitian square root of ${\mathbf{H}}^{{{ - }1}}$,

$$ {\mathbf{H}} = \frac{1}{L}{\mathbf{G}}\left( {{\mathbf{R}}_{0}^{T} \otimes {\mathbf{R}}_{0} } \right){\mathbf{G}}^{T} $$

(34)

According to the normal distribution theory, (33) can be reshaped as:

$$ \left\| {{\mathbf{H}}^{{{{ - 1} \mathord{\left/ {\vphantom {{ - 1} 2}} \right. \kern-0pt} 2}}} \left( {{\hat{{\varvec{\Upsilon}}}} - {\tilde{\mathbf{B}}}\left( {\varvec{\Psi}} \right){\tilde{\mathbf{P}}}} \right)} \right\|_{2}^{2} \sim {As\chi }^{2} \left( {2M - 1} \right) $$

(35)

where ${As\chi }^{2} \left( {2M - 1} \right)$ denotes the asymptotic Chi-square distribution with $2M - 1$ degrees of freedom.

Following (35), the error parameter factor $\eta$ can be obtained properly by the Chi-square distribution function with probability $1 - \bar{p}$ and $2M - 1$ degrees of freedom, where $\bar{p}$ is a probability value. In general, $\bar{p}$ is generally determined by simulation experiments [24]. With the analysis in [24], $\bar{p} = 0.01$ would be an acceptable compromise.