## Appendix A  
In order to find the final value of the alternate estimation of \(\xi\) under hypothesis H\({}_{1}\), we set  
\[f(\xi)=\left|(\mathbf{z}-\mathbf{H}\xi)(\mathbf{z}-\mathbf{H}\xi)^{\mathrm{H} }+\mathbf{A}^{(t)}\right| \tag{1}\]  
where \(\mathbf{A}^{(t)}=\sum\limits_{l=1}^{L}\frac{\mathbf{z}_{l}\mathbf{z}_{l}^{ \mathrm{H}}}{\hat{\gamma}_{1}^{\frac{1}{2}(t)}}\) and the superscript \(t\) indicates the \(t\)-th estimation of the matrix \(\mathbf{A}\). Figure 9: PD versus \(\cos^{2}\!\phi\) with signal mismatch with IPIX radar data. \(\mathrm{SCR}=25\) dB. By the matrix determinant lemma, for matrices \(\mathbf{A},\mathbf{B},\mathbf{C}\), and \(\mathbf{D}\) of compatible dimensions, it holds that  
\[\left|\mathbf{A}+\mathbf{B}\mathbf{D}^{\mathrm{H}}\right|=\left|\mathbf{A} \right|\left(\mathbf{I}+\mathbf{D}^{\mathrm{H}}\mathbf{A}^{-1}\mathbf{B}\right) \tag{10}\]  
so (10) can be simplified to  
\[f(\boldsymbol{\xi})=\left|\mathbf{A}^{(t)}\right|\left\{1+(\mathbf{z}-\mathbf{ H}\boldsymbol{\xi})^{\mathrm{H}}\Big{[}\mathbf{A}^{(t)}\Big{]}^{-1}(\mathbf{z}- \mathbf{H}\boldsymbol{\xi})\right\} \tag{11}\]  
Calculating the partial derivative of (11) with respect to \(\boldsymbol{\xi}\) and setting it to zero result in  
\[\boldsymbol{\xi}^{(t+1)}=\left\{\mathbf{H}^{\mathrm{H}}\Big{[}\mathbf{A}^{(t)} \Big{]}^{-1}\mathbf{H}\right\}^{-1}\mathbf{H}^{\mathrm{H}}\Big{[}\mathbf{A}^{( t)}\Big{]}^{-1}\mathbf{z} \tag{12}\]  
where the superscript \(t+1\) indicates the \((t+1)\)-th estimation of \(\boldsymbol{\xi}\). Next, solve for \(\hat{\gamma}_{h}\), \(h=1,2,\cdots,L\). From (8), we define  
\[f(\gamma_{h})= \gamma_{h}^{N}\Bigg{|}(\mathbf{z}-\mathbf{H}\boldsymbol{\xi}^{(t+ 1)})(\mathbf{z}-\mathbf{H}\boldsymbol{\xi}^{(t+1)})^{\mathrm{H}}+\sum_{l=1}^{ h-1}\frac{\mathbf{z}_{l}\mathbf{z}_{l}^{\mathrm{H}}}{\gamma_{l}^{1,(l+1)}}+\sum_{l=h+1 }^{L}\frac{\mathbf{z}_{l}\mathbf{z}_{l}^{\mathrm{H}}}{\gamma_{l}^{1,(l)}}+ \frac{\mathbf{z}_{h}\mathbf{z}_{h}^{\mathrm{H}}}{\gamma_{h}}\Bigg{|}^{L+1} \tag{13}\] \[= \gamma_{h}^{N}\Bigg{|}\mathbf{B}_{h}^{1,(t+1)}+\frac{\mathbf{z}_{ h}\mathbf{z}_{h}^{\mathrm{H}}}{\gamma_{h}}\Bigg{|}^{L+1}\] \[= \gamma_{h}^{N}\Big{|}\mathbf{B}_{h}^{1,(t+1)}\Big{|}^{L+1}\! \left(1+\frac{\mathbf{z}_{h}^{\mathrm{H}}\Big{[}\mathbf{B}_{h}^{1,(t+1)} \Big{]}^{-1}\mathbf{z}_{h}}{\gamma_{h}}\right)^{L+1}\]  
where  
\[\mathbf{B}_{h}^{1,(t+1)}= (\mathbf{z}-\mathbf{H}\boldsymbol{\xi}^{(t+1)})(\mathbf{z}-\mathbf{ H}\boldsymbol{\xi}^{(t+1)})^{\mathrm{H}}+\sum_{l=1}^{h-1}\frac{\mathbf{z}_{l} \mathbf{z}_{l}^{\mathrm{H}}}{\gamma_{l}^{1,(l+1)}}+\sum_{l=h+1}^{L}\frac{ \mathbf{z}_{l}\mathbf{z}_{l}^{\mathrm{H}}}{\gamma_{l}^{1,(l)}} \tag{14}\]  
The above simplification used (10). Upon analyzing (13), it can be seen that when \(\gamma_{h}\to 0\), \(f(\gamma_{h})\to\infty\); when \(\gamma_{h}\to\infty\), \(f(\gamma_{h})\to\infty\). Therefore, a certain value within \((0,\infty)\) exists that minimizes \(f(\gamma_{h})\). Taking the partial derivative of (13) with respect to \(\gamma_{h}\) results in  
\[\frac{\partial f(\gamma_{h})}{\partial\gamma_{h}}= N\gamma_{h}^{\mathrm{\,N-1}}\Big{|}\mathbf{B}_{h}^{1,(t+1)}\Big{|}^{L+1} \bigg{(}1+\frac{1}{\gamma_{h}}\mathbf{z}_{h}^{\mathrm{H}}\Big{[}\mathbf{B}_{ h}^{1,(t+1)}\Big{]}^{-1}\mathbf{z}_{h}\bigg{)}^{L+1} \tag{15}\] Figure 9:0 Figure 9:1  
We can further obtain that  
Figure 9:2  
where \(h=1,2,\cdots,L\). To date, we can give an initial value of \(\hat{\gamma}_{l}^{1,(0)}\), \(l=1,2,\cdots,L\) and loop from (12) and (16) to the termination condition, and we can obtain the final estimation of \(\boldsymbol{\xi}^{(t_{\mathrm{max}})}\) and \(\gamma_{l}^{1,(t_{\mathrm{max}})}\), \(l=1,2,\cdots,L\) under hypothesis H\({}_{1}\). Obviously, we can set \(\xi=0_{p\times 1}\) under hypothesis H\({}_{0}\). Then, we also need to calculate the estimations of \(\gamma_{l},~{}l=1,2,\cdots,L\) under hypothesis H\({}_{0}\), which is similar to the derivation under hypothesis H\({}_{1}\), so we directly give the following expressions  
Figure 9:3  
and  
Figure 9:4