## 3 Detector Design
### AE-Rao Detector  
The Rao design criterion can be expressed as [29]  
\[t_{\text{Rao}}=\left.\frac{\partial\ln f_{1}(\mathbf{z},\mathbf{Z})}{\partial \mathbf{\Theta}_{r}}\right|_{\mathbf{\Theta}=\hat{\Theta}_{0}}^{\mathrm{T}} \left[\mathbf{I}^{-1}\left(\hat{\Theta}_{0}\right)\right]_{\mathbf{\Theta}, \hat{\Theta}_{r}}\frac{\partial\ln f_{1}(\mathbf{z},\mathbf{Z})}{\partial \mathbf{\Theta}_{r}^{*}}\bigg{|}_{\mathbf{\Theta}=\hat{\Theta}_{0}} \tag{11}\]  
where \(\left[\mathbf{I}^{-1}\big{(}\hat{\Theta}_{0}\big{)}\right]_{\mathbf{\Theta}, \mathbf{\Theta}}\) is the value at \(\mathbf{\Theta}=\hat{\Theta}_{0}\) of the \((\mathbf{\Theta}_{r},\mathbf{\Theta}_{r})\) block of the inverse of the fisher information matrix (FIM) \(\mathbf{I}(\mathbf{\Theta})\). It is known by the matrix partitioning lemma  
\[\left[\mathbf{I}^{-1}(\mathbf{\Theta})\right]_{\mathbf{\Theta},\hat{\Theta}_{r }}=\left[\mathbf{I}_{\mathbf{\Theta},\hat{\Theta}_{r}}(\mathbf{\Theta})- \mathbf{I}_{\mathbf{\Theta},\hat{\Theta}_{s}}(\mathbf{\Theta})\mathbf{I}_{ \mathbf{\Theta}_{s},\hat{\Theta}_{s}}^{-1}(\mathbf{\Theta})\mathbf{I}_{ \mathbf{\Theta}_{s},\hat{\Theta}_{r}}(\mathbf{\Theta})\right]^{-1} \tag{12}\]  
For the complex parameter \(\mathbf{\Theta}\), its FIM is defined as  
\[\mathbf{I}(\mathbf{\Theta})=\mathrm{E}\left[\frac{\partial\ln f(\mathbf{z}; \mathbf{\Theta})}{\partial\mathbf{\Theta}^{*}}\frac{\partial\ln f(\mathbf{z}; \mathbf{\Theta})}{\partial\mathbf{\Theta}^{\mathrm{T}}}\right] \tag{13}\]which can be divided into  
\[\mathbf{I}(\mathbf{\Theta})= \begin{bmatrix}\mathbf{I}_{\mathbf{\Theta}_{r}, \mathbf{\Theta}_{r}}(\mathbf{\Theta})&\mathbf{I}_{\mathbf{\Theta}_{r},\mathbf{ \Theta}_{s}}(\mathbf{\Theta})\\ \mathbf{I}_{\mathbf{\Theta}_{s},\mathbf{\Theta}_{r}}(\mathbf{\Theta})&\mathbf{I }_{\mathbf{\Theta}_{s},\mathbf{\Theta}_{s}}(\mathbf{\Theta})\end{bmatrix}  \tag{14}\]  
with  
\[\mathbf{\Theta}= \begin{bmatrix}\mathbf{\Theta}_{r}^{\mathrm{T}},\mathbf{ \Theta}_{s}^{\mathrm{T}}\end{bmatrix} ^{\mathrm{T}} \tag{15}\]  
being a \((N^{2}+L+p)\times 1\) dimensional parameter vector. \(\mathbf{\Theta}_{r}=\boldsymbol{\xi}\) denotes the relevant parameter and \(\mathbf{\Theta}_{s}= \begin{bmatrix}\mathbf{\gamma}^{\mathrm{T}},\mathrm{vec} ^{\mathrm{T}}(\mathbf{R})\end{bmatrix} ^{\mathrm{T}}\) represents the irrelevant parameter. \(\mathbf{\hat{\Theta}}_{0}= \begin{bmatrix}\mathbf{\hat{\Theta}}_{r0}^{ \mathrm{T}},\mathbf{\hat{\Theta}}_{s0}^{\mathrm{T}}\end{bmatrix} ^{\mathrm{T}}\) indicates the MLE of parameter \(\mathbf{\Theta}\) under hypothesis \(\mathbf{H}_{0}\) with \(\mathbf{\hat{\Theta}}_{r0}=\mathbf{\hat{\Theta}}_{r\times 1}\) and \(\mathbf{\hat{\Theta}}_{s0}= \begin{bmatrix}\mathbf{\hat{\tau}}_{0}^{\mathrm{T}}, \mathrm{vec}^{\mathrm{T}}(\mathbf{\hat{R}})\end{bmatrix} ^{\mathrm{T}}\). \(\mathbf{\hat{\Theta}}_{1}= \begin{bmatrix}\mathbf{\hat{\Theta}}_{r1}^{\mathrm{ T}},\mathbf{\hat{\Theta}}_{s1}^{\mathrm{T}}\end{bmatrix} ^{\mathrm{T}}\) indicates the MLE of parameter \(\mathbf{\Theta}\) under hypothesis \(\mathbf{H}_{1}\) with \(\mathbf{\hat{\Theta}}_{r1}=\boldsymbol{\hat{\xi}}\) and \(\mathbf{\hat{\Theta}}_{s1}= \begin{bmatrix}\mathbf{\hat{\tau}}_{r1}^{\mathrm{ T}},\mathrm{vec}^{\mathrm{T}}(\mathbf{\hat{R}})\end{bmatrix} ^{\mathrm{T}}\). According to (3), the partial derivatives of \(\ln f_{1}(\mathbf{z},\mathbf{Z})\) with respect to \(\boldsymbol{\xi}\) and \(\boldsymbol{\xi}^{*}\) are  
\[\frac{\partial\ln f_{1}(\mathbf{z},\mathbf{Z})}{\partial\mathbf{\Theta}_{r}} =\frac{\partial\ln f_{1}(\mathbf{z},\mathbf{Z})}{\partial\boldsymbol{\xi}}= \begin{bmatrix}(\mathbf{z}-\mathbf{H}\boldsymbol{\xi})^{\mathrm{H}}\mathbf{R} ^{-1}\mathbf{H}\end{bmatrix} ^{\mathrm{T}} \tag{16}\]  
and  
\[\frac{\partial\ln f_{1}(\mathbf{z},\mathbf{Z})}{\partial\mathbf{\Theta}_{r}^{ *}}=\frac{\partial\ln f_{1}(\mathbf{z},\mathbf{Z})}{\partial\boldsymbol{\xi} ^{*}}=\mathbf{H}^{\mathrm{H}}\mathbf{R}^{-1}(\mathbf{z}-\mathbf{H}\boldsymbol {\xi}) \tag{17}\]  
respectively. According to the definition of (14), substituting (16) and (17) into (13) results in  
\[