# Rao and Wald Tests for Moving Target Detection in Forward Scatter Radar
## 2 Design of Adaptive Detectors with Secondary Data
### Adaptive Complex Parameter Wald Detector with Secondary Data  
The complex parameter Wald test with secondary data is given by  
\[\left(\hat{\Theta}_{r1}-\Theta_{r0}\right)^{H}\biggl{\{}\left[\mathbf{f}^{-1}\left( \hat{\Theta}_{1}\right)\right]_{\Theta_{r}\Theta_{r}}\biggr{\}}^{-1}\left(\hat{ \Theta}_{r1}-\Theta_{r0}\right)\gtrless\eta \tag{37}\]  
where \(\hat{\Theta}_{1}\) and \(\hat{\Theta}_{r1}\) are the MLEs of \(\Theta\) and \(\Theta_{r}\) under \(H_{1}\), \(\eta\) denotes the detection threshold, and \(\Theta_{r0}\) is the value of \(\Theta_{r}\) under \(H_{0}\).  
From (35), \(\left\{\left[\mathbf{f}^{-1}\left(\Theta\right)\right]_{\Theta_{r}\Theta_{r}} \right\}^{-1}\) is calculated as:  
\[\begin{array}{l}\left\{\left[\mathbf{f}^{-1}\left(\Theta\right)\right]_{\Theta _{r}\Theta_{r}}\right\}^{-1}\\ =\frac{1}{\sigma^{2}}\left[\left\|\mathbf{r}_{1}\right\|^{2}-\frac{r_{1}^{H}r_{d} \mu_{r}^{H}r_{d}}{\left(\left\|\mathbf{r}_{d}\right\|^{2}+\left\|\mathbf{r}_{d}^{H} \right\|^{2}\right)}\right]\mathbf{I}_{2}\end{array} \tag{38}\]  
The value of \(\Theta_{r}\) under \(H_{0}\) is  
\[\Theta_{r0}=\left[0,0\right]^{\mathrm{T}} \tag{39}\]  
The MLE of \(\Theta\) under \(H_{1}\) is:  
\[\begin{array}{l}\hat{\beta}_{1}=\left[\mathbf{r}_{t}^{H}\mathbf{z}-\mathbf{r}_{t}^{H}r_ {d}\left(\mathbf{r}_{d}^{H}\mathbf{z}+\mathbf{r}_{d}^{H}\mathbf{z}^{\prime}\right)\left(\mathbf{r} _{d}^{H}\mathbf{r}_{d}+\mathbf{r}_{d}^{H}\mathbf{r}_{d}\right)^{-1}\right]\\ \times\left[\left\|\mathbf{r}_{t}\right\|^{2}-\left|\mathbf{r}_{d}^{H}\mathbf{r}_{t} \right|^{2}\left(\mathbf{r}_{d}^{H}\mathbf{r}_{d}+\mathbf{r}_{d}^{H}\mathbf{r}_{d}^{H}\right) ^{-1}\right]^{-1}\\ \\ \hat{\alpha}_{1}=\left(\mathbf{r}_{d}^{H}\mathbf{r}_{d}+\mathbf{r}_{d}^{H}\mathbf{r}_{d}^{H} \right)^{-1}\left(\mathbf{r}_{d}^{H}\mathbf{z}+\mathbf{r}_{d}^{H}\mathbf{z}^{\prime}-\hat{ \beta}_{1}\mathbf{r}_{d}^{H}\mathbf{r}_{t}\right)\end{array} \tag{40}\]  
\[\hat{\sigma}_{1}^{2}=\frac{1}{N+K}\Biggl{\{}\left\|\mathbf{z}\right\|^{2}+\left\| \mathbf{z}^{\prime}\right\|^{2}-\frac{\left|\left(\mathbf{r}_{d}^{H}\mathbf{z}+\mathbf{r}_{d}^ {H}\mathbf{z}^{\prime}\right)\right|^{2}}{\left\|\mathbf{r}_{d}\right\|^{2}+\left\| \mathbf{r}_{d}^{H}\right\|^{2}}-\frac{\left|\mathbf{r}_{t}^{H}\left[\mathbf{z}-\frac{r_{ d}\left(\mathbf{r}_{d}^{H}\mathbf{z}+\mathbf{r}_{d}^{H}\mathbf{z}^{\prime}\right)}{\left\|\mathbf{r}_{d} \right\|^{2}+\left\|\mathbf{r}_{d}^{H}\right\|^{2}}\right]^{2}}{\left(\left\|\mathbf{ r}_{t}\right\|^{2}-\frac{\mathbf{r}_{d}^{H}\mathbf{r}_{t}^{H}\mathbf{z}}{\left\|\mathbf{r}_{d} \right\|^{2}+\left\|\mathbf{r}_{d}^{H}\right\|^{2}}\right)}\Biggr{\}} \tag{42}\]  
The complex-parameter Wald test with secondary data based on the FSR (referred to as Wald-SD-FSR) can be derived by substituting (38)-(42) into (37):  
\[\begin{array}{l}T_{Wald-SD-FSR}=\left|\mathbf{r}_{t}^{H}\mathbf{z}-\frac{\mathbf{r}_{t} ^{H}\mathbf{z}\left(\mathbf{r}_{d}^{H}\mathbf{z}+\mathbf{r}_{d}^{H}\mathbf{z}^{\prime}\right)}{ \left\|\mathbf{r}_{d}\right\|^{2}+\left\|\mathbf{r}_{d}^{H}\right\|^{2}}\right|^{2} \\ \times\biggl{(}\left\|\mathbf{r}_{t}\right\|^{2}-\frac{\left|\mathbf{r}_{d}^{H}\mathbf{r} \right|^{2}}{\left\|\mathbf{r}_{d}\right\|^{2}+\left\|\mathbf{r}_{d}^{H}\right\|^{2}} \biggr{)}^{-1}\biggl{[}\left\|\mathbf{z}\right\|^{2}+\left\|\mathbf{z}^{\prime}\right\| ^{2}-\frac{\left|\left(\mathbf{r}_{d}^{H}\mathbf{z}+\mathbf{r}_{d}^{H}\mathbf{z}^{\prime} \right)\right|^{2}}{\left\|\mathbf{r}_{d}\right\|^{2}+\left\|\mathbf{r}_{d}^{H}\right\| ^{2}} \tag{43}\]  
where \(\eta^{\prime}\) denotes the modified detection threshold.