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Wald tests for direction detection in noise and interference

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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.

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Notes

  1. The 1S- and 2S-Wald tests are both based on the criterion of the Wald test. However, the former is obtained by using the joint estimates of all unknown parameters. In contrast, the latter is obtained in two steps. It first obtains the Wald test on the assumption of known noise covariance matrix. Then the covariance matrix is replaced by a suitable estimate, which is usually the sample covariance matrix (SCM).

  2. We find that the GLRT and 2S-GLRT can be written as the forms directly with the original data, which are given in the Appendix.

  3. In Figs. 3 and 4, the noise covariance matrix has the form of (63), while in Fig. 5 the noise covariance matrix has the form \(\mathbf{R}={\mathbf{R}_c}+{\mathbf{R}_n}\), where \(\mathbf{R}_c\) is the clutter covariance matrix and \(\mathbf{R}_n\) is the thermal noise covariance matrix. Moreover, \(\mathbf{R}_c\) and \(\mathbf{R}_n\) have the forms \(\mathbf{R}_c=\mathbf{U}_c\varvec{\Lambda }\mathbf{U}_c^H\) and \(\mathbf{R}_n=\gamma \mathbf{I}_N\), respectively. \(\mathbf{U}_c\) is and \(N\times r\) semi-unitary matrix satisfying \(\mathbf{U}_c^H\mathbf{U}_c=\mathbf{I}_r\). \(\varvec{\Lambda }=\text {diag}(\lambda _1,\lambda _2,\ldots ,\lambda _r)\) is an \(r\times r\) diagonal matrix, with the diagonal elements being the clutter eigenvalues, standing for the clutter power. \(\gamma \) is the noise eigenvalue, denote the thermal noise power. The clutter-to-noise ratio is defined as \(\text {CNR} =\frac{\sum _{i=1}^{r}\lambda _i}{N\gamma }\). Note that the noise model for Fig. 5 is usually encountered by airborne radar when space-time adaptive processing (STAP) is adopted.

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Acknowledgements

This work was supported by the Natural Science Foundation of Hunan Province, China (Grants No. 2015JJ6006), the Research Foundation of Education Bureau of Hunan Province, China (Grant No. 16C0113), and the National Natural Science Foundation of China (Grant No. 61501505).

Author information

Authors and Affiliations

  1. Changsha Normal University, Changsha, 410100, China

    Wenjuan Li

  2. School of Information and Navigation, Air Force Engineering University, Xi’an, 710077, China

    Haibo Tong

  3. School of Electronic Information, Wuhan University, Wuhan, 430072, China

    Kai Li

  4. Military Representative Office Stationed at 720 Factory, Nanjing, 210038, China

    Yang Yang

  5. Wuhan Electronic Information Institute, Wuhan, 430019, China

    Weijian Liu

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  1. Wenjuan Li
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  2. Haibo Tong
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  4. Yang Yang
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Correspondence to Weijian Liu.

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

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)

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Li, W., Tong, H., Li, K. et al. Wald tests for direction detection in noise and interference. Multidim Syst Sign Process 29, 1563–1577 (2018). https://doi.org/10.1007/s11045-017-0517-5

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