# 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
\begin{array}{l}\boldsymbol{J}_{\Theta_{i}\Theta_{i}}=E\Big{[}\frac{\partial \ln f_{1}}{\partial\alpha^{*}\partial\alpha^{*}}\frac{\partial\ln f_{1}}{ \partial\alpha^{*}}\frac{\partial\ln f_{1}}{\partial\alpha^{*}}\frac{\partial \ln f_{1}}{\partial\alpha^{*}}\frac{\partial\ln f_{1}}{\partial\alpha^{*}} \frac{\partial\ln f_{1}}{\partial\alpha^{*}}\frac{\partial\ln f_{1}}{\partial \alpha^{*}}\Big{]}\\ =E\Bigg{[}\frac{\partial\ln f_{1}}{\partial\alpha^{*}}\frac{\partial\ln f_{1} }{\partial\alpha^{*}}\frac{\partial\ln f_{1}}{\partial\alpha^{*}}\frac{\partial \ln f_{1}}{\partial\alpha^{*}}\frac{\partial\ln f_{1}}{\partial\alpha^{*}} \frac{\partial\ln f_{1}}{\partial\alpha^{*}}\frac{\partial\ln f_{1}}{\partial \alpha^{*}}\Bigg{]}\\ \frac{\partial\ln f_{1}}{\partial\alpha^{*}}\frac{\partial\ln f_{1}}{\partial \alpha^{*}}\frac{\partial\ln f_{1}}{\partial\alpha^{*}}\frac{\partial\ln f_{1 }}{\partial\alpha^{*}}\frac{\partial\ln f_{1}}{\partial\alpha^{*}}\frac{ \partial\ln f_{1}}{\partial\alpha^{*}}\frac{\partial\ln f_{1}}{\partial \alpha^{*}}\frac{\partial\ln f_{1}}{\partial\alpha^{*}}\frac{\partial\ln f_{1 }}{\partial\alpha^{*}}\Bigg{]}\\ =\frac{1}{\sigma^{2}}\left[\begin{array}{ll}\big{(}r_{d}^{H}r_{d}+r_{d}^{JH}r_ {d}^{\prime}\big{)}&0\\ 0&\big{(}r_{d}^{H}r_{d}+r_{d}^{JH}r_{d}^{\prime}\big{)}&0\\ 0&0&\big{(}N+K\big{)}\end{array}\right]\\ \big{[}J^{-1}(\Theta)\big{]}_{\Theta_{i}\Theta_{i}}\text{,}\end{array}  \tag{34}\]  
\[ \begin{array}{l}\big{[}J^{-1}(\Theta)\big{]}_{\Theta_{i}\Theta_{i}}\text{,} \\ =\Big{[}J_{\Theta_{i}\Theta_{i}}(\Theta)-J_{\Theta_{i}\Theta_{i}}(\Theta)J_{ \Theta_{i}\Theta_{i}}^{-1}(\Theta)J_{\Theta_{i}\Theta_{i}}(\Theta)\Big{]}^{-1} \\ =\sigma^{2}\Bigg{[}\|r_{t}\|^{2}-\frac{r_{t}^{H}r_{d}^{H}r_{t}}{\big{(}\|r_{d}\|^{ 2}+\big{\|}r_{d}^{\prime}\big{\|}^{2}\big{)}}\Bigg{]}^{-1}\boldsymbol{I}_{2} \end{array}  \tag{35}\]  
The MLEs of the unknown parameters under \(H_{0}\) is: \(\hat{\beta}_{0}=0\), \(\hat{\alpha}_{0}=\Big{(}\big{\|}r_{d}\big{\|}^{2}+\big{\|}r_{d}^{\prime}\big{\|} ^{2}\Big{)}^{-1}\big{(}r_{d}^{H}z+r_{d}^{JH}z^{\prime}\big{)}\), \(\hat{\sigma}_{0}^{2}=\frac{1}{(N+K)}\bigg{[}\|z\|^{2}+\big{\|}z^{\prime}\big{\|} ^{2}-\frac{\big{(}r_{d}^{H}z+r_{d}^{JH}z^{\prime}\big{)}\big{\|}^{2}}{\|r_{d}\|^{ 2}+\big{\|}r_{d}^{\prime}\big{\|}^{2}}\bigg{]}\). The complex parameter Rao test with secondary data based on the FSR (referred to as Rao-SD-FSR) can be obtained by substituting (10)-(35) and the MLEs of the unknown parameters under \(H_{0}\) into (3):  
\[T_{Rao-SD-FSR}=\frac{\bigg{|}r_{t}^{H}\bigg{[}z-\frac{\tau_{d}\big{(}r_{d}^{H}z +r_{d}^{JH}z^{\prime}\big{)}}{\|r_{d}\|^{2}+\big{\|}r_{d}^{\prime}\big{\|}^{2}} \bigg{]}^{2}}\bigg{/}\bigg{(}\|r_{t}\|^{2}-\frac{r_{t}^{H}r_{d}^{H}r_{t}}{\|r_{d} \|^{2}+\big{\|}r_{d}^{\prime}\big{\|}^{2}}\bigg{)}H_{1}}{\big{\|}z\|^{2}+\big{\|} z^{\prime}\big{\|}^{2}-\frac{\big{(}r_{d}^{H}z+r_{d}^{JH}z^{\prime}\big{)}}{\|r_{d}\|^{ 2}+\big{\|}r_{d}^{\prime}\big{\|}^{2}}\bigg{]}^{2}}\gtrdot\gamma^{\prime} \tag{36}\]where \(\gamma^{\prime}\) denotes the modified detection threshold.