# Rao and Wald Tests for Moving Target Detection in Forward Scatter Radar
## 3 Design of Adaptive Detectors without Secondary Data
### Adaptive Complex Parameter Rao Detector without Secondary Data  
The complex parameter Rao test without secondary data for the detection problem (44) is  
\[\frac{\partial\ln f\left(\overset{\sim}{\boldsymbol{z}}|H_{1}\right)}{ \partial\overset{\sim}{\Theta}_{r}}\Bigg{|}^{T}\overset{\sim}{\Theta}_{ \Theta}=\left[\overset{\sim}{\boldsymbol{\alpha}}^{T},\overset{\sim}{ \boldsymbol{\alpha}}^{H},\overset{\sim}{\sigma}^{2}\right]^{T},\overset{\sim}{ \Theta}_{0}\text{ is the MLE of }\overset{\sim}{\Theta}\text{ under }H_{0}\text{, and }\varepsilon\text{ denotes the detection threshold.}\]  
According to the PDF of the primary data and the theory of complex-valued matrix differentiation [40], we can obtain the following results:  
\[\frac{\partial\ln f\left(\overset{\sim}{\boldsymbol{z}}|H_{1}\right)}{ \partial\overset{\sim}{\beta}^{*}}=\frac{1}{\overset{\sim}{\sigma}^{2}} \left(\overset{\sim}{\boldsymbol{r}}_{t}^{H}\overset{\sim}{\boldsymbol{z}}- \overset{\sim}{\boldsymbol{\alpha}}^{H}\overset{\sim}{\boldsymbol{r}}_{t}^{H} \overset{\sim}{\boldsymbol{r}}_{d}-\overset{\sim}{\beta}\overset{\sim}{ \boldsymbol{r}}_{t}^{H}\overset{\sim}{\boldsymbol{r}}_{t}\right) \tag{47}\]  
\[\frac{\partial\ln f\left(\overset{\sim}{\boldsymbol{z}}|H_{1}\right)}{ \partial\overset{\sim}{\beta}}=\frac{1}{\overset{\sim}{\sigma}^{2}}\left( \overset{\sim}{\boldsymbol{z}}^{H}\overset{\sim}{\boldsymbol{r}}_{t}- \overset{\sim}{\boldsymbol{\alpha}}^{H}\overset{\sim}{\boldsymbol{r}}_{d}^{H }\overset{\sim}{\boldsymbol{r}}_{t}-\overset{\sim}{\beta}^{H}\overset{\sim} {\boldsymbol{r}}_{t}^{H}\overset{\sim}{\boldsymbol{r}}_{t}\right) \tag{48}\]  
The FIM \(\boldsymbol{J}\left(\overset{\sim}{\Theta}\right)\) when the secondary data are not available is calculated in a similar way as the FIM \(\boldsymbol{J}\left(\Theta\right)\) when the secondary data are available. After some calculation, the four blocks of \(\boldsymbol{J}\left(\overset{\sim}{\Theta}\right)\) are  
\[\boldsymbol{J}_{\overset{\sim}{\Theta},\overset{\sim}{\Theta}_{r}}^{-}=\frac {1}{\overset{\sim}{\sigma}^{2}}\left[\overset{\sim}{\boldsymbol{r}}_{t}^{H} \overset{\sim}{\boldsymbol{r}}_{t}^{H}\overset{\sim}{\boldsymbol{r}}_{t}^{H} \right] \tag{49}\]  
\[\boldsymbol{J}_{\overset{\sim}{\Theta},\overset{\sim}{\Theta}_{s}}^{-}=\frac {1}{\overset{\sim}{\sigma}^{2}}\left[\overset{\sim}{\boldsymbol{r}}_{t}^{H} \overset{\sim}{\boldsymbol{r}}_{d}^{H}\overset{\sim}{\boldsymbol{r}}_{t}^{H} \overset{\sim}{\boldsymbol{r}}_{t}^{H}\overset{\sim}{\boldsymbol{r}}_{t}^{H} \overset{\sim}{\boldsymbol{r}}_{t}^{H}\overset{\sim}{\boldsymbol{r}}_{t}^{H} \overset{\sim}{\boldsymbol{r}}_{t}^{H}\overset{\sim}{\boldsymbol{r}}_{t}^{H} \overset{\sim}{\boldsymbol{r}}_{t}^{H}\overset{\sim}{\boldsymbol{r}}_{t}^{H} \overset{\sim}{\boldsymbol{r}}_{t}^{H}\right] \tag{50}\]  
\[\left\{\left[\boldsymbol{I}^{-1}\left(\overset{\sim}{\Theta}\right)\right]_{ \overset{\sim}{\Theta},\overset{\sim}{\Theta}_{r}}\right\}^{-1}=\frac{1}{ \overset{\sim}{\sigma}^{2}}\left[\left\|\overset{\sim}{\boldsymbol{r}}_{t}^{H} \right\|^{2}-\frac{\left|\overset{\sim}{\boldsymbol{r}}_{t}^{H}\right|^{2}}{ \left\|\overset{\sim}{\boldsymbol{r}}_{t}^{H}\right\|^{2}} \begin{array}{c}0\\ 0\\ \left\|\overset{\sim}{\boldsymbol{r}}_{t}^{H}\right\|^{2}-\frac{\left|\overset{ \sim}{\boldsymbol{r}}_{t}^{H}\right|^{2}}{\left\|\overset{\sim}{\boldsymbol{r}}_{t} ^{H}\right\|^{2}}\end{array} \right. \tag{52}\]The MLEs of the unknown parameters under \(H_{0}\) are: \(\stackrel{{<}}{{\sigma}}_{0}^{2}=\left(\left\|\widetilde{z}\right\|^{2}- \frac{\left|\widetilde{r}_{d}^{-H}\widetilde{z}\right|^{2}}{\left\|\widetilde{r }_{d}\right\|^{2}}\right)/N\), \(\stackrel{{<}}{{\beta}}_{0}^{2}=0\), and \(\stackrel{{<}}{{\alpha}}_{0}^{2}=\left(\widetilde{r}_{d}^{H} \widetilde{r}_{d}\right)^{-1}\left(\widetilde{r}_{d}^{H}\widetilde{z}\right)\).