# Rao and Wald Tests for Moving Target Detection in Forward Scatter Radar
## 2 Design of Adaptive Detectors with Secondary Data  
Adaptive moving rectangular-shaped target detection in complex Gaussian noise using the FSR is considered. As shown in Figure 1, the length of the baseline between the receiver and the transmitter of the FSR is \(R\). The transmitter transmits a continuous wave signal with carrier frequency \(f_{c}\). A target with vertical dimension \(L_{v}\) and horizontal dimension \(L_{h}\) moves in a straight line with velocity \(v\) and crosses the baseline at a distance \(R_{0}\) from the receiver when \(t=0\) s.  
When the secondary data are available, we formulate the above detection problem as the binary hypothesis test  
\[\left\{\begin{array}{l}H_{0}:\left\{\begin{array}{l}\mathbf{z}=\alpha\mathbf{r}_{d }+\mathbf{n}\\ \mathbf{z}^{\prime}=\alpha\mathbf{r}_{d}^{\prime}+\mathbf{n}^{\prime}\\ \mathbf{z}=\alpha\mathbf{r}_{d}+\beta\mathbf{r}_{t}+\mathbf{n}\\ \mathbf{z}^{\prime}=\alpha\mathbf{r}_{d}^{\prime}+\mathbf{n}^{\prime}\end{array}\right. \tag{1}\]  
where \(H_{0}\) denotes the null hypothesis indicating the absence of the target signal, \(H_{1}\) denotes the alternative hypothesis indicating the presence of the target signal, \(\mathbf{z}\in\mathbb{C}_{N\times 1}\) denotes the data under test, namely, the primary data, \(\mathbb{C}_{N\times 1}\) stands for the set of \(N\) -dimensional complex vectors, \(\mathbf{z}^{\prime}\in\mathbb{C}_{K\times 1}\) denotes the secondary data which are target-free and are collected from the temporal frames adjacent to the primary data, \(\mathbf{r}_{d}\in\mathbb{C}_{N\times 1}\) and \(\mathbf{r}_{d}^{\prime}\in\mathbb{C}_{K\times 1}\) denote complex vectors containing samples collected from the direct signals, \(\mathbf{r}_{t}\in\mathbb{C}_{N\times 1}\) denotes the \(N\) -dimensional complex vector containing samples collected from the target signal, \(\beta\) and \(\alpha\) are unknown complex amplitudes of the target signal and direct signal, \(\mathbf{n}\in\mathbb{C}_{N\times 1}\) and \(\mathbf{n}^{\prime}\in\mathbb{C}_{K\times 1}\) denote the noise which satisfies \(\mathbf{n}\sim\mathcal{CN}\big{(}\mathbf{0},\sigma^{2}\mathbf{I}_{N}\big{)},\mathbf{n}^{ \prime}\sim\mathcal{CN}\big{(}\mathbf{0},\sigma^{2}\mathbf{I}_{K}\big{)}\), \(\sigma^{2}\) denotes the noise variance, \(\mathbf{I}_{N}\) and \(\mathbf{I}_{K}\) denote the identity matrices, and \(\mathcal{CN}(\cdot)\) denotes complex circular symmetric Gaussian distribution. According to the statistical distributions of the target signal, direct signal, and noise, the joint probability density function (JPDF) of \(\mathbf{z}\) and \(\mathbf{z}^{\prime}\) is  
\[f\big{(}\mathbf{z},\mathbf{z}^{\prime}|H_{i}\big{)}=\frac{1}{\pi^{(N+K)}\sigma^{2(N+K )}}\exp\biggl{[}-\frac{1}{\sigma^{2}}\Bigl{(}\|\mathbf{z}-\alpha\mathbf{r}_{d}-i\beta \mathbf{r}_{t}\|^{2}\ +\bigl{\|}\mathbf{z}^{\prime}-\alpha\mathbf{r}_{d}^{\prime}\bigr{\|}^{2} \Bigr{)}\biggr{]} \tag{2}\]  
where \(i=0,\ 1,\|\cdot\|\) is the Euclidean norm.  
Compared with the GLRT, which is obtained by estimating unknown parameters under both hypothesis \(H_{0}\) and hypothesis \(H_{1}\), only the unknown parameters under hypothesis \(H_{0}\) or the unknown parameters under hypothesis \(H_{1}\) are required to estimate for the Rao test or the Wald test. Meanwhile, considering that the complex-parameter Rao and Wald tests coincide with the real-parameter ones [28], and that the real-parameter Rao and Wald tests may be difficult to handle and have heavy computational complexity, we  
Figure 1: The geometry of the forward scatter radar.  
regard the complex parameter as a whole to design the Rao and Wald detectors for FSR in this section.