GLRT Based Adaptive-Thresholding for CFAR-Detection of Pareto-Target in Pareto-Distributed Clutter

Abstract

Constant false alarm rate (CFAR) detectors are developed to track changes in clutter intensities and to adapt the detection threshold to maintain a constant probability of false alarms. These adaptive thresholding mechanisms are initially intended when both target and clutter are exponentially distributed, and they degrade in performance when applied to newer target and clutter models. So, in application scenarios like Airborne Warning and Control Systems (AWACS) and ship remote sensing, when both the target and clutter are Pareto distributed, instead of the conventional way of tweaking the existing adaptive-thresholding CFAR detector, we pose the detection problem as a two-sample, Pareto vs. Pareto composite hypothesis testing problem. Considering no knowledge of both scale and shape parameters of Pareto distributed clutter, we derive the new adaptive-thresholding detector based on the generalized likelihood ratio test (GLRT) statistic. We further show that our proposed adaptive thresholding detector has a CFAR property and provide extensive simulation results to demonstrate the performance of the proposed detector.

Appendix: Mathematical Simplifications

Simplification of Eq. (21) follows from (43). Further, from Eq. (24), arriving at threshold$- p_{fa}$ relation (25) is as follows:

$$\begin{aligned} \lambda (\varvec{x},y)&=\frac{\hat{\alpha }_{\Theta _0}^{n+1} h^{{\hat{\alpha }_{\Theta _0}}(n+1)} y^{-(\hat{\alpha }_{\Theta _0}+1)} {\left( \prod _{i=1}^{n} x_i\right) ^{-(\hat{\alpha }_{\Theta _0} +1)}} }{\hat{\rho }_{\Theta } \hat{\alpha }_{\Theta }^{n}h^{(\hat{\alpha }_{\Theta }n+\hat{\rho }_{\Theta })} y^{-(\hat{\rho }_{\Theta }+1)} {\left( \prod _{i=1}^{n} x_i\right) ^{-(\hat{\alpha }_{\Theta } +1)} } } \end{aligned}$$

(43)

$$\begin{aligned}&=\frac{\left[ \frac{n+1}{\Lambda (\varvec{x})+\Lambda (y)}\right] ^{n+1}h^{\left( \frac{(n+1)^2}{{\Lambda (\varvec{x})+\Lambda (y)}}\right) } y^{-\left( \frac{n+1}{\Lambda (y)+\Lambda (\varvec{x})}+1\right) }(\prod _{i=1}^{n} x_i)^{-\left( \frac{n+1}{\Lambda (y)+\Lambda (\varvec{x})}+1\right) } }{ \frac{1}{\Lambda (y)}\left[ \frac{n}{\Lambda (\varvec{x})}\right] ^{n} h^{\left( \frac{n^2}{\Lambda (\varvec{x})}+\frac{1}{\Lambda (y)}\right) } y^{-\left( \frac{1}{\Lambda (y)}+1\right) } \left( \prod _{i=1}^{n} x_i\right) ^{-\left( \frac{n}{\Lambda (\varvec{x})}+1\right) } }\end{aligned}$$

(44)

$$\begin{aligned}&=\frac{(n+1)^{n+1}{\Lambda (\textbf{x})}^n\Lambda (y)}{n^n(\Lambda (y)+\Lambda (\textbf{x}))^{n+1}} \frac{\left( \frac{y}{h}\right) ^{-\left( \frac{n+1}{\Lambda (\varvec{x})+\Lambda (y)}\right) } h^{\left( \frac{n+1}{\Lambda (\varvec{x})+\Lambda (y)}\right) n} (\prod _{i=1}^{n} x_i)^{-\left( \frac{n+1}{\Lambda (y)+\Lambda (\varvec{x})}\right) } }{ \left( \frac{y}{h}\right) ^{ -\frac{1}{\Lambda (y)}} h^{\left( \frac{n^2}{\Lambda (\varvec{x})}\right) } \left( \prod _{i=1}^{n} x_i\right) ^{-\left( \frac{n}{\Lambda (\varvec{x})}\right) } } \end{aligned}$$

(45)

$$\begin{aligned}&=\frac{(n+1)^{n+1}}{n^n}\frac{\frac{\Lambda (y)}{\Lambda (\textbf{x})}}{(\frac{\Lambda (y)}{\Lambda (\textbf{x})}+1)^{n+1}}{\left( \frac{y}{h}\right) ^{\left( \frac{1}{\Lambda (y)}-\frac{n+1}{\Lambda (y)+\Lambda (\textbf{x})}\right) }\left( \frac{\prod _{i=1}^{n} x_i}{h}\right) ^{\left( \frac{n}{\Lambda (\varvec{x})}-\frac{n+1}{\Lambda (y)+\Lambda (\textbf{x})}\right) }}\end{aligned}$$

(46)

$$\begin{aligned}&={(n+1)^{n+1}} \frac{n\frac{\Lambda (y)}{\Lambda (\varvec{x})}}{\left( (n\frac{\Lambda (y)}{\Lambda (\varvec{x})} +n)\right) ^{n+1} } \left( e^{\Lambda (y)}\right) ^{\left( \frac{1}{\Lambda (y)}-\frac{n+1}{\Lambda (y)+\Lambda (\textbf{x})}\right) } \left( e^{\Lambda (\varvec{x})}\right) ^{\left( \frac{n}{\Lambda (\varvec{x})}-\frac{n+1}{\Lambda (y)+\Lambda (\textbf{x})}\right) }\end{aligned}$$

(47)

$$\begin{aligned}&={(n+1)^{n+1}} \frac{n\frac{\Lambda (y)}{\Lambda (\varvec{x})}}{\left( (n\frac{\Lambda (y)}{\Lambda (\varvec{x})} +n)\right) ^{n+1} } e^{\left( 0\right) }. \end{aligned}$$

(48)

$$\begin{aligned} p_{fa}&=\sup _{{\alpha } \in {(0,\infty )}}\Pr \left( \frac{B}{C}>\gamma _{1} \right) \end{aligned}$$

(49)

$$\begin{aligned}&=\sup _{{\alpha } \in {(0,\infty )}}\int _{c=0}^{\infty }\int _{b=\gamma _{1} c}^{\infty }f_{BC}\left( b,c\right) \textrm{d}{}b\,\textrm{d}c\end{aligned}$$

(50)

$$\begin{aligned}&{\mathop {=}\limits ^{(a)}}\sup _{{\alpha } \in {(0,\infty )}}\int _{c=0}^{\infty }\left( \int _{b=\gamma _{1}^{} c}^{\infty }f_B\left( b\right) \textrm{d}b\right) f_C(c)\,\textrm{d}c\end{aligned}$$

(51)

$$\begin{aligned}&{\mathop {=}\limits ^{(b)}}\sup _{{\alpha } \in {(0,\infty )}}\textbf{E}_C\left[ \int _{b=\gamma _{1}^{} c}^{\infty }f_B\left( b\right) \textrm{d}b\right] \end{aligned}$$

(52)

$$\begin{aligned}&{\mathop {=}\limits ^{(c)}}\sup _{{\alpha } \in {(0,\infty )}}\textbf{E}_C\left[ e^{(-\alpha \gamma _{1}^{}c)}\right] ,\end{aligned}$$

(53)

$$\begin{aligned}&{\mathop {=}\limits ^{(d)}}\sup _{{\alpha } \in {(0,\infty )}}\left[ 1-\frac{1}{n\alpha }\left( -\alpha \gamma _{1}^{}\right) \right] ^{-n}\end{aligned}$$

(54)

$$\begin{aligned}&=\left[ 1+\frac{\gamma _{1}^{}}{n}\right] ^{-n} \end{aligned}$$

(55)

where the equalities are justified as follows: (a) for $B \text { and } C$ are independent random variables and their joint density products down; (b) by taking expectation with respect to the C random variable; (c) by the complementary cdf formula; (d) by applying moment generating function formula for C, followed by simplification.

For Eq. (27) following the similar lines of above justification, but under $H_1$, the simplification steps are:

$$\begin{aligned} p_d&=\Pr \left( \frac{B}{C}>\gamma _{1} ;H_1\right) \end{aligned}$$

(56)

$$\begin{aligned}&=\int _{c=0}^{\infty }\int _{b=\gamma _{1} c}^{\infty }f_{BC}\left( b,c\right) \textrm{d}b\,\textrm{d}c\end{aligned}$$

(57)

$$\begin{aligned}&=\int _{c=0}^{\infty }\left( \int _{b=\gamma _{1}^{} c}^{\infty }f_B\left( b\right) \textrm{d}b\right) f_C(c)\,\textrm{d}c\end{aligned}$$

(58)

$$\begin{aligned}&=\textbf{E}_C\left[ \int _{b=\gamma _{1}^{} c}^{\infty }f_B\left( b\right) \textrm{d}b\right] \end{aligned}$$

(59)

$$\begin{aligned}&=\textbf{E}_C\left[ e^{(-\rho \gamma _{1}^{}c)}\right] \end{aligned}$$

(60)

$$\begin{aligned}&=\left( 1+\frac{\rho \gamma _1}{\alpha n}\right) ^{-n}. \end{aligned}$$

(61)

Similarly for (case (b)), from Eq. (40) arriving at (41) is as follows:

$$\begin{aligned} p_{fa}&=\sup _{{(\alpha ,h)} \in \Theta _0}\Pr \left( \frac{G}{D}>\gamma \right) \end{aligned}$$

(62)

$$\begin{aligned}&{\mathop {=}\limits ^{}}\sup _{{(\alpha ,h)} \in \Theta _0}\int _{d=0}^{\infty }\int _{g=\gamma d}^{\infty }f_{GD}\,\textrm{d}g\,\textrm{d}d \end{aligned}$$

(63)

$$\begin{aligned}&\qquad {\mathop {=}\limits ^{(a)}}\sup _{{(\alpha ,h)} \in \Theta _0}\int _{d=0}^{\infty }\left( \int _{g=\gamma d}^{\infty }f_G\left( g\right) \,\textrm{d}g\right) f_D(d)\,\textrm{d}d\end{aligned}$$

(64)

$$\begin{aligned}&\qquad {\mathop {=}\limits ^{(b)}}\sup _{{(\alpha ,h)} \in \Theta _0} \int _{d=0}^{\infty } f_D\left( d\right) \int _{g=\gamma d}^{\infty }\frac{n}{n+1}\alpha e^{-\alpha g}\,\textrm{d}g\, \textrm{d}d,\end{aligned}$$

(65)

$$\begin{aligned}&\qquad {\mathop {=}\limits ^{}}\sup _{{(\alpha ,h)} \in \Theta _0}\frac{n}{n+1}\int _{d=0}^{\infty }\left[ {-e^{-\alpha g}}\right] ^{\infty }_{g=\gamma d}f_D\left( d\right) \textrm{d}d\end{aligned}$$

(66)

$$\begin{aligned}&\qquad {\mathop {=}\limits ^{(c)}}\sup _{{(\alpha ,h)} \in \Theta _0}\frac{n}{n+1} \textbf{E}_D \left[ e^{-\alpha \gamma d}\right] \end{aligned}$$

(67)

$$\begin{aligned}&\qquad {\mathop {=}\limits ^{(d)}}\sup _{{(\alpha ,h)} \in \Theta _0}\frac{n}{n+1}\left( 1-\frac{1}{n\alpha }(-\alpha \gamma )\right) ^{-(n-1)}\end{aligned}$$

(68)

$$\begin{aligned}&\qquad =\frac{n}{n+1}\left[ 1+\frac{\gamma }{n}\right] ^{-(n-1)} \end{aligned}$$

(69)

where the equalities are justified as follows: (a) for $G \text { and } D$ are independent random variables; (b) substituting the density function for $g>0$ in (39) as $\gamma d >0$; (c) by taking expectation with respect to the D random variable; (d) by applying moment generating function formula for D, followed by simplification.