Sparse representation based two-dimensional direction of arrival estimation using co-prime array

Abstract

Direction of arrival (DOA) estimation using co-prime array has been attractive for its potential advantages. A co-prime array consists of two uniform linear arrays (ULAs), where one has M elements with $N\lambda /2$ being the inter-element spacing, and the other has N (co-prime to M) elements with $M\lambda /2$ being the inter-element spacing . In this paper, the two ULAs of the co-prime array are placed parallel to each other in the same plane for two-dimensional (2D) DOA estimation, and the uniqueness proof of DOA estimation for this geometry is given. By setting the vectorization of the cross covariance matrix of the two ULAs as an observing vector in sparse representation, MN degrees of freedom (DOF) can be achieved via ($M+N)$ sensors. Then through the enhanced sparse recovery technique, unique and automatically paired 2D DOA estimation can be obtained from the recovery vector via only 1D dictionary. The proposed algorithm can achieve better DOA estimation performance than conventional algorithms. The simulation results verify the effectiveness of the proposed algorithm.

Appendix

As

$$\begin{aligned} E[{\hat{\mathbf{R}}}_C ]= & {} E\left[ \frac{1}{J}\sum _{t=1}^J {\mathbf{x}_M \left( t \right) \mathbf{x}_N^H \left( t \right) } \right] \nonumber \\= & {} \frac{1}{J}\sum _{t=1}^J {E\left[ \mathbf{x}_M \left( t \right) \mathbf{x}_N^H \left( t \right) \right] } \nonumber \\= & {} \mathbf{R}_C \end{aligned}$$

(17)

From Eq. (17), it is shown that the mean of $\Delta \mathbf{r}$ is 0. For the variance matrix of $\Delta \mathbf{r}$, the $i-$th column of ${\hat{\mathbf{R}}}_C$ is

$$\begin{aligned} {\hat{\mathbf{r}}}_{Ci} =\frac{1}{J}\sum _{t=1}^J {\mathbf{x}_M \left( t \right) x_{Ni}^*\left( t \right) } \end{aligned}$$

(18)

where $x_{Ni}\left( t \right)$ is the i-th element of $\mathbf{x}_N \left( t \right)$. According to Ottersten et al. (1998), the signals are independent from sample to sample and circularly Gaussian distributed. Then it can be derived that

$$\begin{aligned} E\left[ {\hat{\mathbf{r}}}_{Ci} {\hat{\mathbf{r}}}_{Cj} ^{H}\right]= & {} \frac{1}{J^{2}}\sum _{t=1}^J {\sum _{p=1}^J {E\left[ \mathbf{x}_M \left( t \right) x_{Ni}^*\left( t \right) \mathbf{x}_M^H \left( p \right) x_{Nj}\left( p \right) \right] }} \nonumber \\= & {} \mathbf{r}_{Ci} \mathbf{r}_{Cj} ^{H}+\frac{1}{J}\hbox {R}_{Nji} \mathbf{R}_M \end{aligned}$$

(19)

where $\hbox {R}_{Nji}$ is the (j,i)th element of covariance matrix $\mathbf{R}_N =E[\mathbf{x}_N \left( t \right) \mathbf{x}_N^H \left( t \right) ]$, and $\mathbf{R}_M =E[\mathbf{x}_M \left( t \right) \mathbf{x}_M^H \left( t \right) ]$. From Eq. (19), it can be derived that the variance matrix of $\Delta \mathbf{r}$ is

$$\begin{aligned} E[\Delta \mathbf{r}\Delta \mathbf{r}^{H}]=\frac{1}{J}(\mathbf{R}_N^T \otimes \mathbf{R}_M ) \end{aligned}$$

(20)