## 3 Detector Design
### AE-GLRT Detector  
The GLRT design criterion can be formulated as [28]  
\[\frac{\max_{\mathbf{z},\mathbf{R},\gamma_{f}}f_{1}(\mathbf{z},\mathbf{Z}; \mathbf{\xi},\mathbf{R},\boldsymbol{\gamma})}{\max_{\mathbf{R},\gamma_{f}}f_{ 0}(\mathbf{z},\mathbf{Z};\mathbf{R},\boldsymbol{\gamma})}\overset{\mathrm{H}_ {1}}{\underset{\mathrm{H}_{0}}{\gtrless}}\sigma \tag{5}\]  
where \(\sigma\) denotes the detection threshold set based on the predesigned PFA. Under hypothesis \(\mathrm{H}_{1}\), taking the partial derivative of (3) with respect to \(\mathbf{R}\) to be zero, we can obtain the maximum likelihood estimation (MLE) of \(\mathbf{R}\) for a given \(\gamma_{l}\) as  
\[\mathbf{\hat{R}}_{1}=\frac{1}{L+1}\left[(\mathbf{z}-\mathbf{H}\mathbf{\xi})( \mathbf{z}-\mathbf{H}\mathbf{\xi})^{\mathrm{H}}+\sum\limits_{l=1}^{L}\frac{ \mathbf{z}_{l}\mathbf{z}_{l}^{\mathrm{H}}}{\gamma_{l}}\right] \tag{6}\]  
Utilizing the same method, we can derive the MLE of \(\mathbf{R}\) under hypothesis \(\mathrm{H}_{0}\) as  
\[\mathbf{\hat{R}}_{0}=\frac{1}{L+1}\left(\mathbf{z}\mathbf{z}^{\mathrm{H}}+\sum \limits_{l=1}^{L}\frac{\mathbf{z}_{l}\mathbf{z}_{l}^{\mathrm{H}}}{\gamma_{l}}\right) \tag{7}\]  
Substituting (6) and (7) into (3) and (4), respectively, we can obtain  
\[l_{1}(\mathbf{z},\mathbf{Z};\mathbf{\xi},\mathbf{\hat{R}}_{1},\boldsymbol{ \gamma})=\frac{\left(\frac{L+1}{e\pi}\right)^{N(L+1)}}{\prod\limits_{l=1}^{L} \gamma_{l}^{N}\left|(\mathbf{z}-\mathbf{H}\mathbf{\xi})(\mathbf{z}-\mathbf{ H}\mathbf{\xi})^{\mathrm{H}}+\sum\limits_{l=1}^{L}\frac{\mathbf{z}_{l} \mathbf{z}_{l}^{\mathrm{H}}}{\gamma_{l}}\right|^{(L+1)}} \tag{8}\]  
and  
\[l_{0}(\mathbf{z},\mathbf{Z};\mathbf{\hat{R}}_{0},\boldsymbol{\gamma})=\frac{ \left(\frac{L+1}{e\pi}\right)^{N(L+1)}}{\prod\limits_{l=1}^{L}\gamma_{l}^{N} \left|\mathbf{z}\mathbf{z}^{\mathrm{H}}+\sum\limits_{l=1}^{L}\frac{\mathbf{z} _{l}\mathbf{z}_{l}^{\mathrm{H}}}{\gamma_{l}}\right|^{(L+1)}} \tag{9}\]Directly solving for the MLE of \(\xi\) and \(\gamma_{l},\ l=1,2,\cdots,L\) does not yield a closed-form solution. Therefore, we use the alternate estimation algorithm to address this problem. For ease of expression, we specify the following symbols: \(\hat{\gamma}_{l}^{1,(0)}\) and \(\hat{\gamma}_{l}^{1,(t_{\max})}\) represent the initial value and final estimation of \(\gamma_{l}\) under hypothesis H\({}_{1}\), respectively. \(\hat{\xi}^{(t_{\max})}\) denotes the final estimation of \(\xi\) under hypothesis H\({}_{1}\); \(\hat{\gamma}_{l}^{0,(0)}\) and \(\hat{\gamma}_{l}^{0,(t_{\max})}\) represent the initial value and final estimation of \(\gamma_{l}\) under hypothesis H\({}_{0}\), respectively. The estimation process is as follows: First, we let the initial value of \(\gamma_{l}\) under hypothesis H\({}_{1}\) be \(\hat{\gamma}_{l}^{1,(0)},\ l=1,2,\cdots,L\)[22], and \(\hat{\xi}^{(1)}\) can be obtained by (A4); then, we obtain \(\hat{\gamma}_{l}^{1,(1)},\ l=1,2,\cdots,L\) based on (A8). The process is repeated in a loop until the maximum number of iterations \(t_{\max}\) is reached. In this paper, we set it to 15, whose validity will be shown in the fourth section. Algorithm 1 details the steps of the alternate estimation iterative algorithm.