Near-field coherent source localization by planar array design

Abstract

This paper is concerned with near-field source localization for scenarios where coherent narrowband sources exist. In this paper, we propose a new method in which we design a general planar array with a covariance matrix whose rank is not decreased by the coherence between sources. Moreover, conditions for the sensor locations in the designed planar array are derived to reach maximum effective array aperture. The proposed method uses second order statistics and features a separable range-bearing search to reduce the computational complexity. This method localizes near-field sources with a number of one-dimensional searches in two steps. In the first step, ranges of sources is estimated using one 1D search and in the second step, the bearing of each signal source is estimated using the corresponding range estimated in the first step. Simulation results show that the performance of the proposed method is comparable with the Cramer–Rao bound.

Appendix

Elements of spatial covariance matrix in the far-field case, where a ULA is adopted, is derived in Han and Zhang (2005). Here, for the sake of completeness, we derive the covariance matrix elements, where a general planar array is used, with the existence of near-field or far-field sources.

The $\left( {l,k} \right)$th element of covariance matrix is given by

$$\begin{aligned} r\left( {l,k} \right)= & {} E\left\{ {x_l \left( t \right) x_k^*\left( t \right) } \right\} \nonumber \\= & {} E\left\{ {\left[ {\mathop \sum \limits _{v=1}^M s_v \left( t \right) e^{-j\tau _{lv} }+n_l \left( t \right) } \right] \times \left[ {\mathop \sum \limits _{w=1}^M s_w^*\left( t \right) e^{j\tau _{kw} }+n_k^*\left( t \right) } \right] } \right\} \nonumber \\= & {} \mathop \sum \limits _{w=1}^M \mathop \sum \limits _{v=1}^M E\left\{ {s_v \left( t \right) s_w^*\left( t \right) } \right\} e^{-j\tau _{lv} +j\tau _{kw} }+\sigma _n^2 \delta \left( {l,k} \right) \nonumber \\= & {} \mathop \sum \limits _{w=1}^M \mathop \sum \limits _{v=1}^M \sigma _{v,w} e^{-j\tau _{lv} }e^{j\tau _{kw} }+\sigma _n^2 \delta \left( {l,k} \right) ,\quad l,k=1,\ldots ,L \end{aligned}$$

(37)

If we denote

$$\begin{aligned} d_{lw} =\mathop \sum \limits _{v=1}^M \sigma _{v,w} e^{-j\tau _{lv} } \end{aligned}$$

(38)

then the $\left( {l,k} \right)$th element of the covariance matrix can be written as

$$\begin{aligned} r\left( {l,k} \right) =\mathop \sum \limits _{w=1}^M d_{lw} e^{j\tau _{kw} }+\sigma _n^2 \delta \left( {l,k} \right) \end{aligned}$$

(39)

where $\tau _{kw}$, the propagation delay, depends on the array structure and the location of the sources, in general. The $\left( {l,k} \right)$th element of the covariance matrix in (14) can be derived by (39) by using the time delay (3) as $\tau _{kw}$ and $\tau _{lv}$.