﻿ 威布尔杂波下扩展目标检测

威布尔杂波下扩展目标检测Extended Target Detection under Weibull Clutter

Abstract: Target detection under sea clutter is widely used in military and civilian applications and has broad prospects. As the resolution of the radar increases, the sea clutter exhibits a non-Gaussian amplitude, which brings new challenges to the detection of radar targets. At present, in the research of wide-band radar target detection, the commonly used broadband radar sea clutter amplitude distribution models are: lognormal distribution, Weibull distribution, K distribution and so on. This paper mainly analyzes the detection problem of wide-band targets in the sea clutter background of Weibull distribution, and proposes a new detection algorithm for extended targets under Weibull distribution: firstly, two parameters of Weibull distribution are estimated by moment estimation method. Then a new detection quantity is constructed combined with the amplitude accumulation detector to detect the target. It is found that the Weibull distribution can be converted into a standard exponential distribution by constructing a new detection quantity, and the problem is converted into an exponentially distributed broadband target in the background of the sea clutter. Through detecting problems and greatly simplifying the original problems, finally, the paper proves the effectiveness of the algorithm.

1. 引言

2. 基于矩估计的参数估计

${\mu }_{m}=E\left({X}^{l}\right)={\int }_{-\infty }^{+\infty }{x}^{m}f\left(x;{\theta }_{1},{\theta }_{2},\cdots ,{\theta }_{k}\right)\text{d}x$ ，X连续 (1)

${\mu }_{m}=E\left({X}^{l}\right)=\underset{x\in {R}_{x}}{\sum }{x}^{m}p\left(x;{\theta }_{1},{\theta }_{2},\cdots ,{\theta }_{k}\right)\text{d}x$ ，X离散(2)

${A}_{m}=\frac{1}{n}\underset{i=1}{\overset{n}{\sum }}{X}_{i}^{m}$ (3)

$\left\{\begin{array}{l}{\mu }_{1}={\mu }_{1}\left({\theta }_{1},{\theta }_{2},\cdots ,{\theta }_{k}\right),\\ {\mu }_{2}={\mu }_{2}\left({\theta }_{1},{\theta }_{2},\cdots ,{\theta }_{k}\right),\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}⋮\\ {\mu }_{k}={\mu }_{k}\left({\theta }_{1},{\theta }_{2},\cdots ,{\theta }_{k}\right).\end{array}$ (4)

$A\left({X}^{m}\right)=\frac{1}{n}\underset{i=1}{\overset{n}{\sum }}{X}_{i}^{m}$ (5)

$E\left({X}^{m}\right)=A\left({X}^{m}\right)$ (6)

$f\left(x\right)=\frac{\beta }{\eta }{\left(\frac{x}{\eta }\right)}^{\beta -1}\mathrm{exp}\left(-{\left(\frac{x}{\eta }\right)}^{\beta }\right)$ (7)

$F\left(x\right)=1-\mathrm{exp}\left(-{\left(\frac{x}{\eta }\right)}^{\beta }\right)$ (8)

$E\left({X}^{m}\right)=\underset{0}{\overset{\infty }{\int }}{x}^{m}f\left(x\right)\text{d}x=\underset{0}{\overset{\infty }{\int }}{x}^{m}\frac{\beta }{\eta }{\left(\frac{x}{\eta }\right)}^{\beta -1}\mathrm{exp}\left(-{\left(\frac{x}{\eta }\right)}^{\beta }\right)\text{d}x={\eta }^{m}\Gamma \left(\frac{m}{\beta }+1\right)$ (9)

$\left\{\begin{array}{l}\eta \Gamma \left(\frac{1}{\beta }+1\right)=E\left(X\right)\\ {\eta }^{2}\Gamma \left(\frac{2}{\beta }+1\right)=E\left({X}^{2}\right)\end{array}$ (10)

$\left\{\begin{array}{l}\beta =\frac{\pi E\left({X}^{2}\right)E\left({X}^{-2}\right)/E\left(X\right)E\left({X}^{-1}\right)}{\sqrt{{\left(E\left({X}^{2}\right)E\left({X}^{-2}\right)\right)}^{2}-{\left(E\left(X\right)E\left({X}^{-1}\right)\right)}^{2}}}\\ \eta =\frac{E\left(X\right)}{\Gamma \left(1+\frac{1}{\beta }\right)}\end{array}$ (11)

3. 威布尔分布的非相干积累

${X}_{1},{X}_{2},\cdots ,{X}_{N}$ 为N个杂波样本且独立同分布，经非相干积累后随机变量 $\underset{i=1}{\overset{N}{\sum }}{X}_{i}$ 的PDF为 ${p}_{N}\left(x\right)$${X}_{i}\left(i=1,2,\cdots ,N\right)$ 的矩母函数为

$M\left(s\right)=E\left({\text{e}}^{s{X}_{i}}\right)$ (12)

$M\left(s\right)={\int }_{0}^{+\infty }{\text{e}}^{sx}\frac{c}{\eta }{\left(\frac{x}{\eta }\right)}^{\beta -1}{\text{e}}^{-{\left(\frac{x}{\eta }\right)}^{c}}\text{d}x$ (13)

$M\left(s\right)=\underset{n=0}{\overset{+\infty }{\sum }}\frac{{t}^{n}}{n!}E\left[{X}^{n}\right]$(14)

$M\left(s\right)=\underset{n=0}{\overset{+\infty }{\sum }}\frac{{s}^{n}}{n!}{\eta }^{n}\Gamma \left(\frac{n}{\beta }+1\right)$ (15)

$\underset{i=1}{\overset{N}{\sum }}{X}_{i}$ 的矩母函数为

 (16)

4. 算法设计

4.1. 问题描述

$f\left({s}_{n}\right)=\frac{{s}_{n}^{\left(m-1\right)}{\text{e}}^{-{s}_{n}/2}}{{2}^{m}\Gamma \left(m\right)}$ , ${s}_{n}>0$ (17)

$\left\{\begin{array}{l}{H}_{0}:{z}_{n}={c}_{n};\\ {H}_{1}:{z}_{n}={s}_{n}+{c}_{n};\end{array}n=1,2,3,\cdots ,L$ (18)

4.2. 似然比检验

NP 准则下，假设检验式(16)的最优检测器为似然比检测器 [11] (Likelihood Ratio Test, LRT)，其表达式为：

$\underset{n=1}{\overset{L}{\prod }}\frac{p\left({z}_{n}|{H}_{1}\right)}{p\left({z}_{n}|{H}_{0}\right)}\underset{{H}_{0}}{\overset{{H}_{1}}{\gtrless }}{T}_{h}$ (19)

$\underset{n=1}{\overset{L}{\sum }}f\left({z}_{n}\right)\underset{{H}_{0}}{\overset{{H}_{1}}{\gtrless }}\mathrm{ln}{T}_{h}$ (20)

${H}_{0}$ 假设下，有

$p\left({z}_{n}|{H}_{0}\right)=f\left({z}_{n}\right)=\frac{\beta }{\eta }{\left(\frac{{z}_{n}}{\eta }\right)}^{\beta -1}\mathrm{exp}\left(-{\left(\frac{{z}_{n}}{\eta }\right)}^{\beta }\right)$ (21)

${H}_{1}$ 假设下，有

(22)

4.3. 指数分布与 $Gamma\left(n,\lambda \right)$ 分布的关系

$n=1$ 时，

$Gamma\left(1,\lambda \right)=Exp\left(\lambda \right)$ (23)

${S}_{k+1}={S}_{k}+{X}_{k+1}$ ，其中服从 $Gamma\left(k,\lambda \right)$${X}_{k+1}$ 服从 $Exp\left(\lambda \right)$

$\mathrm{Pr}\left({S}_{k+1}=x\right)={\int }_{0}^{x}\mathrm{Pr}\left({S}_{k}=y\right){P}_{r}\left({X}_{k+1}=x-y\right)\text{d}y={\int }_{0}^{x}\frac{{\lambda }^{k}}{\Gamma \left(k\right)}{y}^{k-1}{\text{e}}^{-\lambda y}×\lambda {\text{e}}^{-\lambda \left(x-y\right)}\text{d}y=\frac{{\lambda }^{k+1}}{\Gamma \left(k+1\right)}{x}^{k}\text{e}$ (24)

4.4. 检测器设计

$p\left({x}_{n}|{H}_{0}\right)=\frac{\beta }{\eta }{\left(\frac{{x}_{n}}{\eta }\right)}^{\beta -1}\mathrm{exp}\left(-{\left(\frac{{x}_{n}}{\eta }\right)}^{\beta }\right)$ (25)

${y}_{i}={\left(\frac{{x}_{i}}{\eta }\right)}^{\beta },\text{\hspace{0.17em}}i=1,2,\cdots ,L$ (26)

$F\left({y}_{i}\right)=1-\mathrm{exp}\left(-{y}_{i}\right)$ (27)

$f\left({y}_{i}\right)=\mathrm{exp}\left(-{y}_{i}\right)$ (28)

$t={y}_{1}+{y}_{2}+\cdots +{y}_{L}$ (29)

$\left\{\begin{array}{l}t\ge T,\text{\hspace{0.17em}}裁决目标存在\\ t

${P}_{\text{fa}}=\underset{T}{\overset{\infty }{\int }}f\left(t\right)\text{d}t$ (30)

1) 根据参考单元估计出威布尔分布的两个参数值；

2) 通过式(30)计算出门限值T；

3) 通过式(26)对检测单元进行变换；

4) 通过式(29)计算检测统计量t的值，将该值与门限值T进行比较得到结果。

5. 威布尔分布与指数分布仿真

Figure 1. PDF curve of standard exponential distribution and distribution after Weibull transformation

6. 算法仿真

6.1. 算法的稳定性

Figure 2. Detection curve under different estimation conditions

6.2. 检测率仿真

6.3. 虚警率仿真

Figure 3. Detection rate simulation curve

7. 结束语

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