# Rao and Wald Tests for Moving Target Detection in Forward Scatter Radar
## 2 Design of Adaptive Detectors with Secondary Data
### Adaptive Complex Parameter Rao Detector with Secondary Data  
The complex parameter Rao test with secondary data for the detection problem (1) is  
\[\frac{\partial\ln f(z,z^{\prime}|H_{1})}{\partial\Theta_{r}}\bigg{|}_{\Theta= \Theta_{0}}^{T}\Big{[}J^{-1}\big{(}\hat{\Theta}_{0}\big{)}\Big{]}_{\Theta_{r} \Theta_{r}}\frac{\partial\ln f(z,z^{\prime}|H_{1})}{\partial\Theta_{r}^{*}} \bigg{|}_{\Theta=\Theta_{0}}\frac{H_{1}}{\gtrless\gamma} \tag{3}\]  
where \(\ln(\cdot)\) denotes the logarithm, \((\cdot)^{T}\) and \((\cdot)^{*}\) are the transpose operator and the complex conjugate operator, \(\partial(\cdot)\) denotes the partial derivative, \(\gamma\) denotes the detection threshold, \(\Theta=\left[\Theta_{r}^{T},\Theta_{s}^{T}\right]^{T}\), \(\Theta_{r}=\left[\beta^{T},\beta^{H}\right]^{T}\) is the vector of the relevant parameter which is related to the complex amplitude of the target signal, \(\Theta_{s}=\left[\alpha^{T},\alpha^{H},\sigma^{2}\right]^{T}\) is the vector of nuisance parameters, which include the complex amplitude of direct signal and the noise variance, \((\cdot)^{H}\) is the complex conjugate transpose operator, \(\hat{\Theta}_{0}\) denotes the MLE of \(\Theta\) under \(H_{0}\), and \(J(\Theta)\) is the FIM:  
\[ \begin{array}{c}\mathbf{J}(\Theta)=E\Big{[}\frac{\partial\ln f(z,z^{\prime}|H_ {1})}{\partial\Theta^{*}}\frac{\partial\ln f(z,z^{\prime}|H_{1})}{\partial \Theta^{T}}\Big{]}\\ =\Big{[}\frac{J_{\Theta_{r}\Theta_{r}}}{J_{\Theta_{r}\Theta_{r}}}\frac{J_{ \Theta_{r}\Theta_{s}}}{J_{\Theta_{r}\Theta_{s}}}\Big{]}\end{array}  \tag{4}\]  
where \(J_{\Theta,\Theta_{r}},J_{\Theta,\Theta_{r}},J_{\Theta,\Theta_{r}}\), and \(J_{\Theta,\Theta_{s}}\) are four blocks of \(J(\Theta)\):  
\[ \begin{array}{c}\mathbf{J}_{\Theta,\Theta_{r}}=E\bigg{[}\frac{\partial\ln f(z,z^ {\prime}|H_{1})}{\partial\Theta_{r}^{*}}\frac{\partial\ln f(z,z^{\prime}|H_{1 })}{\partial\Theta_{r}^{T}}\bigg{]}\end{array}  \tag{5}\]  
\[ \begin{array}{c}\mathbf{J}_{\Theta,\Theta_{s}}=E\bigg{[}\frac{\partial\ln f(z,z^ {\prime}|H_{1})}{\partial\Theta_{r}^{*}}\frac{\partial\ln f(z,z^{\prime}|H_{1 })}{\partial\Theta_{s}^{T}}\bigg{]}\end{array}  \tag{6}\]  
\[ \begin{array}{c}\mathbf{J}_{\Theta,\Theta_{r}}=J_{\Theta,\Theta_{s}}^{H}\\ \mathbf{J}_{\Theta,\Theta_{s}}=E\bigg{[}\frac{\partial\ln f(z,z^{\prime}|H_{1})}{ \partial\Theta_{s}^{*}}\frac{\partial\ln f(z,z^{\prime}|H_{1})}{\partial\Theta _{s}^{T}}\bigg{]}\end{array}  \tag{8}\]  
\(\big{[}\mathbf{J}^{-1}(\Theta)\big{]}_{\Theta,\Theta_{r}}\) is the \((\Theta_{r},\Theta_{r})\) part of the inverse of the matrix \(\mathbf{J}(\Theta)\). According to the inverse formula of partitioned matrices, we have  
\[ \begin{array}{c}\Big{[}\mathbf{J}^{-1}(\Theta)\Big{]}_{\Theta,\Theta_{r}}= \Big{[}J_{\Theta,\Theta_{r}}(\Theta)-\mathbf{J}_{\Theta,\Theta_{s}}(\Theta)\mathbf{J} _{\Theta_{s}\Theta_{s}}^{-1}(\Theta)\mathbf{J}_{\Theta,\Theta_{r}}(\Theta)\Big{]}^{ -1}\end{array}  \tag{9}\]  
To obtain the Rao test, we express the partial derivative of the logarithm of the JPDF \(\ln f(z,z^{\prime}|H_{1})\) to \(\Theta\), as  
\[ \begin{array}{c}\frac{\partial\ln f(z,z^{\prime}|H_{1})}{\partial\Theta_{r} ^{T}}=\bigg{[}\frac{\partial\ln f(z,z^{\prime}|H_{1})}{\partial\Theta_{r}^{* }}\bigg{]}^{H}=\begin{bmatrix}\frac{\partial\ln f(z,z^{\prime}|H_{1})}{ \partial\beta^{*}}\\ \frac{\partial\ln f(z,z^{\prime}|H_{1})}{\partial\beta}\end{bmatrix}^{H}\end{array}  \tag{10}\]  
Substituting (2) into (10), we have:  
\[ \begin{array}{c}\frac{\partial\ln f(z,z^{\prime}|H_{1})}{\partial\beta^{*}} =\frac{1}{\sigma^{2}}\Big{(}\mathbf{r}_{t}^{H}\mathbf{z}-\alpha\mathbf{r}_{t}^{H}\mathbf{r}_{d} -\beta\mathbf{r}_{t}^{H}\mathbf{r}_{t}\Big{)}\end{array}  \tag{11}\]  
\[ \begin{array}{c}\frac{\partial\ln f(z,z^{\prime}|H_{1})}{\partial\beta}=\frac {1}{\sigma^{2}}\Big{(}\mathbf{z}^{H}\mathbf{r}_{t}-\alpha\mathbf{r}_{d}^{H}\mathbf{r}_{t}- \beta\mathbf{r}_{t}^{H}\mathbf{r}_{t}\Big{)}\end{array}