# Rao and Wald Tests for Moving Target Detection in Forward Scatter Radar
## 4 The Equivalence of the Wald Test, Rao Test, and GLRT  
### The Equivalence of the Wald Test, Rao Test, and GLRT with Secondary Data  
In this part, the equivalence of the Wald test, Rao test, and GLRT is validated when the secondary data are available. We rewrite the test statistics of the Rao-SD-FSR (36) and the Wald-SD-FSR (43) as  
\[\begin{split} T_{Rao-SD-FSR}&=\ \frac{\left|\mathbf{r}_{t}^{H} \left[\mathbf{z}-\frac{r_{d}\left(\mathbf{r}_{d}^{H}z+\mu_{d}^{H}z^{  
\[T_{Rao-FSR}=2N\left|\widetilde{\mathbf{r}}_{t}^{H}\left(\widetilde{ \mathbf{z}}-\frac{\mathbf{r}_{d}^{H}\widetilde{\mathbf{z}}}{\left\|\widetilde{\mathbf{r}}_{d} \right\|^{2}}\right)\right|^{2}\] \[\times\left[\left(\left\|\widetilde{\mathbf{r}}_{t}\right\|^{2}- \frac{\widetilde{\mathbf{r}}_{t}^{H}\widetilde{\mathbf{r}}_{t}\widetilde{\mathbf{r}}_{d} ^{H}\widetilde{\mathbf{r}}_{t}}{\left\|\widetilde{\mathbf{r}}_{d}\right\|^{2}}\right) \left(\left\|\widetilde{\mathbf{z}}\right\|^{2}-\frac{\left|\widetilde{\mathbf{r}}_{d} ^{H}\widetilde{\mathbf{z}}\right|^{2}}{\left\|\widetilde{\mathbf{r}}_{d}\right\|^{2}} \right)\right]^{-1}\] \[=2N\left|\widetilde{\mathbf{r}}_{t}^{H}\mathbf{P}_{t}^{\perp}\widetilde{ \mathbf{z}}\right|^{2}/\left[\left(\widetilde{\mathbf{r}}_{t}^{H}\widetilde{\mathbf{r}}_{ t}^{\perp}\widetilde{\mathbf{r}}_{t}\right)\left(\widetilde{\mathbf{z}}^{H}\mathbf{P}_{ rd}^{\perp}\widetilde{\mathbf{z}}\right)\right] \tag{59}\] \[=2N\widetilde{\mathbf{z}}^{H}\widetilde{\mathbf{r}}_{t}\widetilde{\mathbf{z}} /\left(\widetilde{\mathbf{z}}^{H}\widetilde{\mathbf{r}}_{t}\widetilde{\mathbf{z}}+ \widetilde{\mathbf{z}}^{H}\mathbf{P}_{\widetilde{\mathbf{r}}}\widetilde{\mathbf{z}}\right)\] \[=\frac{2N\left\|\mathbf{P}_{\widetilde{\mathbf{r}}}\widetilde{\mathbf{z}} \right\|^{2}/\left\|\widetilde{\mathbf{P}}_{\widetilde{\mathbf{r}}}^{\perp}\widetilde {\mathbf{z}}\right\|^{2}}{1+\left\|\mathbf{P}_{\widetilde{\mathbf{r}}}\widetilde{\mathbf{z}} \right\|^{2}/\left\|\mathbf{P}_{\widetilde{\mathbf{r}}}^{\perp}\widetilde{\mathbf{z}} \right\|^{2}}=\frac{2N\Gamma_{GIRT-FSR}/(N-2)}{1+\Gamma_{GIRT-FSR}/(N-2)}\] \[T_{Wald-FSR}=2N\left\|\mathbf{P}_{\widetilde{\mathbf{r}}}\widetilde{\mathbf{z }}\right\|^{2}/\left\|\mathbf{P}_{\widetilde{\mathbf{r}}}^{\perp}\widetilde{\mathbf{z}} \right\|^{2}=2 NT_{GIRT-FSR}/(N-2) \tag{60}\]  
where \(T_{GIRT-FSR}=(N-2)\left\|\mathbf{P}_{\widetilde{\mathbf{r}}}\widetilde{\mathbf{z}}\right\| ^{2}/\left\|\mathbf{P}_{\widetilde{\mathbf{r}}}^{\perp}\widetilde{\mathbf{z}}\right\|^{2}\) is the GLRT without secondary data based on the FSR (referred to as GLRT-FSR) [22]. Equations (59) and (60) indicate that the GLRT and the complex parameter Rao and Wald detectors are identical for adaptive target detection in complex white Gaussian noise using the FSR when the secondary data are not available.  
For the problem of adaptive detection of a signal using the monostatic radar in a homogeneous environment wherein the primary data and the secondary data share the same noise covariance matrix, the GLRT, Wald and Rao detectors are demonstrated to be three different detectors [41; 42; 43]. For cases involving partially homogeneous environment wherein the primary data and the secondary data have different noise variance, the GLRT, Wald, and Rao tests coincide [44]. Different from monostatic radar, Equations (57)-(60) indicate that the three tests for FSR coincide with each other in the complex white Gaussian noise wherein the primary data and the secondary data have equal variance. A possible reason for the difference is that the noise follows the complex white Gaussian distribution for the problem of adaptive moving target detection using the FSR.