Wald tests for direction detection in noise and interference

Abstract

In this paper, we consider the problem of detecting a distributed target in unknown disturbance. The signals reflected by the target are assumed to come from the same direction. However, the exact direction is unknown and yet the signal steering vector lies in a known subspace. The disturbance consists of colored noise and deterministic interference. The interference belongs to a known subspace, linearly independent on the signal subspace. We derive the one-step Wald test and two-step Wald test. Numerical examples show that the proposed detectors can achieve better detection performance than the existing detectors.

Appendix: Equivalence forms of (61) and (62)

Here we show that (61) and (62) can be recast as

$$\begin{aligned} t_{\mathrm{GLRT}} = {\lambda _{\max }}\left[ {{{\tilde{\mathbf {X}}}^H}{{\mathbf {P}}_{{\mathbf {P}}_{\tilde{{\mathbf {J}}}}^ \bot \tilde{\mathbf {H}}}}\tilde{\mathbf {X}}{{\left( {{{\mathbf {I}}_{\mathbf K}} + {{\tilde{\mathbf {X}}}^H}{\mathbf {P}}_{\tilde{{\mathbf {J}}}}^ \bot \tilde{\mathbf {X}}} \right) }^{ - 1}}} \right] \end{aligned}$$

(66)

and

$$\begin{aligned} {t_{\mathrm{2S-GLRT}}} = {\lambda _{\max }}\left( {{{\tilde{\mathbf {X}}}^H}{{\mathbf {P}}_{{\mathbf {P}}_{\tilde{{\mathbf {J}}}}^ \bot \tilde{\mathbf {H}}}}\tilde{\mathbf {X}}} \right) , \end{aligned}$$

(67)

respectively.

Note that if $\lambda _*$ is a non-zero eigenvalue of ${\mathbf {G}}_1{\mathbf {G}}_2$, then $\lambda _*$ is also a non-zero eigenvalue of ${\mathbf {G}}_2 {\mathbf {G}}_1$, with ${\mathbf {G}}_1$ and ${\mathbf {G}}_2$ being two arbitrary conformable matrices. For convenience, this fact is written as

$$\begin{aligned} {\lambda _*}({{\mathbf {G}}_1}{{\mathbf {G}}_2}) = {\lambda _*}({{\mathbf {G}}_2}{{\mathbf {G}}_1}). \end{aligned}$$

(68)

Hence, we can rewrite (61) as

$$\begin{aligned} t_{\mathrm{GLRT}}^{'}={\lambda _{\max }}\left[ {\tilde{\mathbf {X}}{{\left( {{{\mathbf {I}}_{\mathbf K}}+{{\tilde{\mathbf {X}}}^H}{\mathbf {P}}_{\tilde{{\mathbf {J}}}}^\bot \tilde{\mathbf {X}}} \right) }^{-1}}{{\tilde{\mathbf {X}}}^H}{\mathbf {ZP}_{{{\mathbf {Z}}^H}\tilde{\mathbf {H}}}}{{\mathbf {Z}}^H}} \right] . \end{aligned}$$

(69)

Inserting ${{\mathbf {P}}_{{{\mathbf {Z}}^H}\tilde{\mathbf {H}}}} = {{\mathbf {Z}}^H}\tilde{\mathbf {H}}{({\tilde{\mathbf {H}}^H}{\mathbf {P}}_{\tilde{{\mathbf {J}}}}^ \bot \tilde{\mathbf {H}})^{ - 1}}{\tilde{\mathbf {H}}^H}{\mathbf {Z}}$ into ${\mathbf {ZP}_{{{\mathbf {Z}}^H}\tilde{\mathbf {H}}}}{{\mathbf {Z}}^H}$, along with ${\mathbf {P}}_{\tilde{{\mathbf {J}}}}^ \bot = {\mathbf {ZZ}^H}$, we have

$$\begin{aligned} {\mathbf {ZP}_{{{\mathbf {Z}}^H}\tilde{\mathbf {H}}}}{{\mathbf {Z}}^H} = {{\mathbf {P}}_{{\mathbf {P}}_{\tilde{{\mathbf {J}}}}^ \bot \tilde{\mathbf {H}}}}. \end{aligned}$$

(70)

Plugging (70) into (69) yields

$$\begin{aligned} t_{\mathrm{GLRT}}^{'} = {\lambda _{\max }}\left[ {\tilde{\mathbf {X}}{{\left( {{{\mathbf {I}}_{\mathbf K}} + {{\tilde{\mathbf {X}}}^H}{\mathbf {P}}_{\tilde{{\mathbf {J}}}}^ \bot \tilde{\mathbf {X}}} \right) }^{ - 1}}{{\tilde{\mathbf {X}}}^H}{{\mathbf {P}}_{{\mathbf {P}}_{\tilde{{\mathbf {J}}}}^ \bot \tilde{\mathbf {H}}}}} \right] . \end{aligned}$$

(71)

Using (68), we can rewrite (71) as (66). Taking the same method, we can represent (62) by (67)