# 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}\mathbf{J}_{\Theta,\Theta_{s}}=E\left[\frac{\partial\ln f(\mathbf{z },\mathbf{z}^{\prime}|H_{1})}{\partial\Theta_{r}^{*}}\frac{\partial\ln f(\mathbf{z}, \mathbf{z}^{\prime}|H_{1})}{\partial\Theta_{r}^{*}}\frac{\partial\ln f(\mathbf{z}, \mathbf{z}^{\prime}|H_{1})}{\partial\Theta_{r}^{*}}\right]\\ =E\left[\begin{array}{cc}\frac{\partial\ln f_{1}}{\partial\beta^{*}}\frac{ \partial\ln f_{1}}{\partial\alpha^{T}}\frac{\partial\ln f_{1}}{\partial\beta^{ *}}\frac{\partial\ln f_{1}}{\partial\beta^{*}}\frac{\partial\ln f_{1}}{ \partial\beta^{*}}\frac{\partial\ln f_{1}}{\partial\beta^{*}}\frac{\partial\ln f _{1}}{\partial\beta^{*}}\frac{\partial\ln f_{1}}{\partial\alpha^{T}}\\ \frac{\partial\ln f_{1}}{\partial\beta}\frac{\partial\ln f_{1}}{\partial\alpha^{ T}}\frac{\partial\ln f_{1}}{\partial\beta}\frac{\partial\ln f_{1}}{\partial\alpha^{ H}}\frac{\partial\ln f_{1}}{\partial\beta}\frac{\partial\ln f_{1}}{\partial\beta^{ *}}\frac{\partial\ln f_{1}}{\partial\alpha^{T}}\end{array}\right]\\ =\left[\begin{array}{cc}\frac{1}{\sigma^{2}}\mathbf{r}_{t}^{H}\mathbf{r}_{t}&0&0\\ 0&\frac{1}{\sigma^{2}}\mathbf{r}_{d}^{H}\mathbf{r}_{t}&0\end{array}\right]\end{array}  \tag{24}\]According to the definition of the block \(J_{\Theta_{i}\Theta_{i}}\), we have the following results:  
\[\frac{\partial^{2}\ln f_{1}}{\partial\alpha^{*}\partial\alpha^{T}}=-\frac{1}{ \sigma^{2}}\Big{(}r_{d}^{H}r_{d}+r_{d}^{JH}r_{d}^{\prime}\Big{)} \tag{25}\]  
\[\frac{\partial^{2}\ln f_{1}}{\partial\alpha^{*}\partial\alpha^{H}}=0 \tag{26}\]  
\[\frac{\partial^{2}\ln f_{1}}{\partial\alpha^{*}\partial\sigma^{2}}=-\bigg{(} \frac{1}{\sigma^{2}}\bigg{)}^{2}\Big{(}r_{d}^{H}z-\alpha r_{d}^{H}r_{d}-\beta r _{d}^{H}r_{t}+r_{d}^{JH}z^{\prime}-\alpha r_{d}^{JH}r_{d}\Big{)} \tag{27}\]  
\[\frac{\partial^{2}\ln f_{1}}{\partial\alpha\partial\alpha^{T}}=0 \tag{28}\]  
\[\frac{\partial^{2}\ln f_{1}}{\partial\alpha\partial\alpha^{H}}=-\frac{1}{ \sigma^{2}}\Big{(}r_{d}^{H}r_{d}+r_{d}^{JH}r_{d}^{\prime}\Big{)} \tag{29}\]  
\[\frac{\partial^{2}\ln f_{1}}{\partial\alpha\partial\sigma^{2}}=-\bigg{(} \frac{1}{\sigma^{2}}\bigg{)}^{2}\Big{(}z^{H}r_{d}-\alpha^{H}r_{d}^{H}r_{d}- \beta^{H}r_{t}^{H}r_{d}+z^{JH}r_{d}^{\prime}-\alpha^{H}r_{d}^{JH}r_{d}^{\prime }\Big{)} \tag{30}\]  
\[\frac{\partial^{2}\ln f_{1}}{\partial\sigma^{2}\partial\alpha^{T}}=-\bigg{(} \frac{1}{\sigma^{2}}\bigg{)}^{2}\Big{(}z^{H}r_{d}-\alpha^{H}r_{d}^{H}r_{d}- \beta^{H}r_{t}^{H}r_{d}+z^{JH}r_{d}^{\prime}-\alpha^{H}r_{d}^{H}r_{d}^{\prime }\Big{)} \tag{31}\]  
\[\frac{\partial^{2}\ln f_{1}}{\partial\sigma^{2}\partial\alpha^{H}}=-\bigg{(} \frac{1}{\sigma^{2}}\bigg{)}^{2}\Big{(}r_{d}^{H}z-\alpha r_{d}^{H}r_{d}-\beta r _{d}^{H}r_{t}+r_{d}^{JH}z^{\prime}-\alpha r_{d}^{JH}r_{d}^{\prime}\Big{)} \tag{32}\]  
\[\frac{\partial^{2}\ln f_{1}}{\partial\sigma^{2}\partial\sigma^{2}}=(N+K) \bigg{(}\frac{1}{\sigma^{2}}\bigg{)}^{2}-2\bigg{(}\frac{1}{\sigma^{2}}\bigg{)} ^{3}\Big{(}\|z-\alpha r_{d}-i\beta r_{t}\|^{2}+\big{\|}z^{\prime}-\alpha r_{d}^ {\prime}\big{\|}^{2}\Big{)} \tag{33}\]  
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