A Sparse Array Direction-Finding Approach Under Impulse Noise

Abstract

To address the issue that existing direction-finding approaches perform poorly under impulse noise and do not work well in underdetermined scenarios, a novel sparse array direction-finding approach on the background of impulse noise is proposed in this work. The approach introduces an infinite norm Gaussian kernel to restrain the impulse noise and obtains accurate estimates via the maximum likelihood algorithm. Meanwhile, a novel quantum transient search optimization (QTSO) algorithm is designed to solve the corresponding cost function. In addition, we prove the convergence of QTSO and derive the Cramér–Rao bound of sparse array direction finding in the presence of impulse noise. Compared with some traditional direction-finding approaches, the proposed approach shows excellent performance through simulation results in different schemes, which can also be a general framework to address other complex direction-finding problems.

Appendix A

To prove the boundness of the complex matrix $\mathbf{{R}}$ with the component ${R_{ij}}$ $(i,j = 1,2, \ldots ,M)$, we can prove the boundness of ${{\text {Re}}} \{ {R_{ij}}\}$ and ${{\text {Im}}} \{ {R_{ij}}\}$ as follows:

$$\begin{aligned} \begin{aligned} {{\text {Re}}} \{ {R_{ij}}\}&=\mathrm{{ Re}}\left\{ {\frac{1}{L}\sum \limits _{l = 1}^L {{z_i}(l)z_j^*(l)\exp \left( { - \eta |{z_i}(l) - \mu z_j^*(l)|} \right) } } \right\} \\&=\frac{1}{L}\sum \limits _{l = 1}^L {{{\text {Re}}} \left\{ {{z_i}(l)z_j^*(l)\exp \left( { - \eta |{z_i}(l) - \mu z_j^*(l)|} \right) } \right\} } \\&\le \frac{1}{L}\sum \limits _{l = 1}^L {\left\{ {\left|{{z_i}(l)z_j^*(l)\exp \left( { - \eta |{z_i}(l) - \mu z_j^*(l)|} \right) } \right|} \right\} } \\&\le \frac{1}{L}\sum \limits _{l = 1}^L {\left\{ {\left|{{z_i}(l)} \right|\left|{{z_j}(l)} \right|\exp \left( { - \eta |{z_i}(l) - \mu z_j^*(l)|} \right) } \right\} } \\&\le \frac{1}{L}\sum \limits _{l = 1}^L {\left\{ {\left|{{z_i}(l)} \right|\left|{{z_j}(l)} \right|} \right\} } \\&=\frac{1}{L}\sum \limits _{l = 1}^L {\left\{ {\frac{{\left|{{y_i}(l)} \right|}}{{\max \{ |{y_1}(l)|,|{y_2}(l)|, \ldots ,|{y_M}(l)|\} }}} \right. } \\&\quad \times \left. {\frac{{\left|{{y_j}(l)} \right|}}{{\max \{ |{y_1}(l)|,|{y_2}(l)|, \ldots ,|{y_M}(l)|\} }}} \right\} \\&\le \frac{1}{L}\sum \limits _{l = 1}^L {\left\{ {\left|{\frac{{{y_i}(l)}}{{{y_i}(l)}}} \right|\times \left|{\frac{{{y_j}(l)}}{{{y_j}(l)}}} \right|} \right\} } = 1 \end{aligned} \end{aligned}$$

(A1)

$$\begin{aligned} \begin{aligned} {{\text {Re}}} \{ {R_{ij}}\}&\ge - \frac{1}{L}\sum \limits _{l = 1}^L {\left\{ {\left|{{z_i}(l)} \right|\left|{z_j^*(l)} \right|\exp \left( { - \eta |{z_i}(l) - \mu z_j^*(l)|} \right) } \right\} } \\&\ge - \frac{1}{L}\sum \limits _{l = 1}^L {\left\{ {\left|{{z_i}(l)} \right|\left|{{z_j}(l)} \right|\exp \left( { - \eta |{z_i}(l) - \mu z_j^*(l)|} \right) } \right\} } \\&\ge - \frac{1}{L}\sum \limits _{l = 1}^L {\left\{ {\left|{{z_i}(l)} \right|\left|{{z_j}(l)} \right|} \right\} } \\&\ge - 1 \end{aligned} \end{aligned}$$

(A2)

Thus, $- 1 \le {{\text {Re}}} \{ {R_{ij}}\} \le 1$. Similarly, $- 1 \le {{\text {Im}}} \{ {R_{ij}}\} \le 1$. In this case, $\mathbf{{R}}$ is also bounded.