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 雷达学报  2015, Vol. 4 Issue (4): 411–417  DOI: 10.12000/JR14129 0

### 引用本文 [复制中英文]

[复制中文]
Zou Kun, Wu De-wei, Zhang Bin, et al. A Tunable Adaptive Detector for Mismatched Signal[J]. . Journal of Radars, 2015, 4(4): 411–417. DOI: 10.12000/JR14129.
[复制英文]

### 文章历史

, , ,

A Tunable Adaptive Detector for Mismatched Signal
, , ,
School of Information and Navigation, Air Force Engineering University, Xi'an 710077, China
Foundation Items: The National Natural Science Foundation of China (61273408, 61302153); The Areospace Tecnnology Innovation Foundation (CSSC020302)
Abstract: The adaptive detection problem for mismatched useful signal is considered. A novel parametric adaptive detector is proposed, and the adaptive matched filter and double normalized adaptive matched filter are the special cases of the tunable detector. The detector parameters can be used to control the detection performance for slight mismatched signal and rejection performance for serious mismatched interference. The probability of false alarm and the probability of detection for mismatched signal are derived. The computer simulation results show thatwe can choose suitable parameter to control the ability to detect mismatched useful signal and the ability to reject interference.
Key words: Adaptive detection     Mismatched signal     Tunable detector     Interference rejection
1 引言

2 可调自适应检测器

 \eqalign{ & {{\rm{H}}_0}:{z} = {n} \cr & {{\rm{H}}_1}:{z} = b{v} + {n} \cr} (1)

 ${t_{{\rm{AMF}}}} = {{{{\left| {{{v}^{\rm{H}}}{{S}^{ - 1}}{z}} \right|}^2}} \over {{{v}^{\rm{H}}}{{S}^{ - 1}}{v}}}$ (2)

 ${t_{{\rm{DN - AMF}}}} = {{{\beta ^2}{t_{{\rm{AMF}}}}} \over {\left( {1 - \beta } \right)\left( {1 - \beta + \beta {t_{{\rm{AMF}}}}} \right)}}$ (3)

 $\beta = {1 \over {1 + {{z}^{\rm{H}}}{{S}^{ - 1}}{z} - {t_{{\rm{AMF}}}}}}$ (4)

 ${t_t}\left( \gamma \right) = {\left( {{{\left( {{\beta ^{ - 1}} - 1} \right)}^{2\gamma }}t_{{\rm{AMF}}}^{ - 1} + {{\left( {{\beta ^{ - 1}} - 1} \right)}^\gamma }} \right)^{ - 1}}$ (5)

 ${t_{{\rm{AMF}}}} = {\beta ^{ - 1}}{t_K}$ (6)

 ${\delta ^2} = {\rm{SNR}}\beta {\cos ^2}\theta$ (7)

 ${\rm{SNR}} = {\left| {b} \right|^2}{{v}^{\rm{H}}}{{R}^{ - 1}}{v}$ (8)

 ${\cos ^2}\theta = {{{{\left| {{{v}^{\rm{H}}}{{R}^{ - 1}}{{v}_m}} \right|}^2}} \over {\left( {{v}_m^{\rm{H}}{{R}^{ - 1}}{{v}_m}} \right)\left( {{{v}^{\rm{H}}}{{\bf{R}}^{ - 1}}{v}} \right)}}$ (9)

 $\delta _\beta ^2 = {\rm{SNR}}{\sin ^2}\theta$ (10)

 ${P_{\rm{d}}} = {\rm{P}}\left\{ {{{\left( {{{\left( {{\beta ^{ - 1}} - 1} \right)}^{2\gamma }}\beta t_K^{ - 1} + {{\left( {{\beta ^{ - 1}} - 1} \right)}^\gamma }} \right)}^{ - 1}} > \eta , \\ \qquad \ \eta > 0;{{\rm{H}}_1}} \right\}$ (11)

 ${P_{\rm{d}}} = {\rm{P}}\left\{ {{t_K} > {\eta _\beta },\beta > {1 \over {{\eta ^{ - 1/\gamma }} \ \ + 1}};{H_1}} \right\}$ (12)

 ${\eta _\beta } = {{\eta \beta {{\left( {{\beta ^{ - 1}} - 1} \right)}^{2\gamma }}} \over {1 - \eta {{\left( {{\beta ^{ - 1}} - 1} \right)}^\gamma }}}$ (13)

 ${P_{\rm{d}}}\left( \gamma \right) = \int\limits_{{1 \over {{\eta ^{ - 1/\gamma }} \ \ + 1}}}^1 {\left( {1 - {P_{{\rm{cf}}}}\left( {{\eta _\beta }} \right)} \right)} {p_\beta }\left( \beta \right){\rm{d}}\beta$ (14)

 ${P_{{\rm{cf}}}}\left( x \right) = {x \over {{{\left( {1 + x} \right)}^{K - N - 1}}}}\sum\limits_{k = 0}^{K - N} {\left( {\matrix{ {K - N + 1} \cr {1 + k} \cr } } \right){x^k} \\ \qquad \qquad \cdot \exp \left( { - {{{\delta ^2}} \over {1 + x}}} \right)\sum\limits_{i = 0}^k {{{\left( {{{{\delta ^2}} \over {1 + x}}} \right)}^i}{1 \over {i!}}} }$ (15)
pβ(x)表示参数为K-N+2和N-1的非中心贝塔分布的概率密度函数：
 ${p_\beta }\left( x \right) = \exp \left( { - \delta _\beta ^2x} \right)\sum\limits_{k = 0}^{K - N + 2} {\left( {\matrix{ {K - N + 2} \cr k \cr } } \right){{K!} \over {\left( {K + k} \right)!}}} \\ \qquad \qquad \cdot \delta _\beta ^{2k}{q_\beta }\left( {\beta ,K - N + 2,N + k - 1} \right)$ (16)

 ${q_\beta }\left( {\beta ,m,n} \right) = {{\left( {n + m - 1} \right)!} \over {\left( {n - 1} \right)!\left( {m - 1} \right)}}{x^{m - 1}}{\left( {1 - x} \right)^{n - 1}}$ (17)

 ${P_{{\rm{fa}}}}\left( \gamma \right) = \int\limits_{{1 \over {{\eta ^{ - 1/\gamma }} \ \ + 1}}}^1 {{1 \over {{{\left( {1 + {\eta _\beta }} \right)}^{K - N + 1}}}} \\ \qquad \qquad \cdot {q_\beta }\left( {\beta ,K - N + 2,N - 1} \right){\rm{d}}\beta }$ (18)
4 计算机仿真分析 4.1 可调检测器性能分析

 图 1 检测器对失配信号检测性能分析 Fig.1 detection performance under mismatched signal model
4.2空时导向矢量失配条件下的检测性能分析

 ${{v}_{\rm{a}}}\left( \phi \right) = \left[ {\matrix{ 1 \cr {\exp \left( {j\cos \phi } \right)} \cr \vdots \cr {\exp \left( {j\left( {{N_{\rm{a}}} - 1} \right)\cos \phi } \right)} \cr } } \right], \\ {{v}_{\rm{c}}}\left( {{f_{\rm{d}}}} \right) = \left[ {\matrix{ 1 \cr {\exp \left( {j2\pi {f_{\rm{d}}}} \right)} \cr \vdots \cr {\exp \left( {j\left( {{N_{\rm{c}}} - 1} \right)2\pi {f_{\rm{d}}}} \right)} \cr } } \right], \\ {v}\left( {{f_{\rm{d}}}, \phi } \right) = {{v}_{\rm{c}}}\left( {{f_d}} \right) \otimes {{v}_{\rm{a}}}\left( \phi \right)$ (19)

 图 2 检测器对失配的空时2维导向矢量的检测性能 Fig.2 detection performances for mismatched space-time steering vector
4.3 可调检测器参数的选择

 ${\gamma ^ * } = {\max _\gamma }\delta {P_{\rm{d}}}\left( {{\rm{SNR}},{\rm{INR}},{{\cos }^2}{\theta _s},{{\cos }^2}{\theta _i};\gamma } \right)$ (20)

 $\delta {P_d}\left( {{\rm{SNR}},{\rm{INR}},{{\cos }^2}{\theta _s},{{\cos }^2}{\theta _i};\gamma } \right) \\ = {P_{\rm{d}}}\left( {{\rm{SNR}},{{\cos }^2}{\theta _s};\gamma } \right) - {P_{\rm{d}}}\left( {{\rm{INR}},{{\cos }^2}{\theta _i};\gamma } \right)$ (21)

 图 3 可调参数值γ与检测性能的关系 Fig.3 relationships between tunable parameter γ and detection performance

 图 4 可调检测器参数γ>1时的检测性能 Fig.4 detection performance when γ>1
5 结论