## Appendix A  
Define \(\widetilde{c}_{m,k}\) and \(\widetilde{w}\) for computational simplicity,  
\[\widetilde{c}_{m,k}=\left[ \begin{array}{c}\mathrm{Re}\big{\{}c_{m,k}\big{\}} \\ \mathrm{Im}\big{\{}c_{m,k}\big{\}}\end{array} \right]\text{ and }\widetilde{w}=\left[ \begin{array}{c}\mathrm{Re}\{\mathbf{w}\}\\ \mathrm{Im}\{\mathbf{w}\}\end{array} \right] \tag{10}\]  
where \(\mathrm{Re}\{\bullet\}\) and \(\mathrm{Im}\{\bullet\}\) stand for the real and imaginary components of a complex number \(\bullet\). \(\widetilde{c}_{m,k}\) and \(\widetilde{w}\) are \(2N_{\mathrm{R}}\times 1\) dimensional real vectors (\(\widetilde{c}_{m,k}\in\mathbb{R}^{2N_{\mathrm{R}}\times 1}\) and \(\widetilde{w}\in\mathbb{R}^{2N_{\mathrm{R}}\times 1}\)) that satisfy \(||\widetilde{c}_{m,k}||^{2}=||\mathbf{c}_{m,k}||^{2}\) and \(||\widetilde{w}||^{2}=||\mathbf{w}||^{2}\). By (6) and (10), we have  
\[\left(\sum_{k=1}^{N}\sigma_{m,k}(\widetilde{w})\right)^{-q}=\sigma_{m}{}^{-q} (\widetilde{w})=\left[\sum_{k=1}^{N}\exp\left(\frac{||\widetilde{w}-q \widetilde{c}_{m,k}||^{2}}{q\sigma^{2}}-\frac{(q+1)||\widetilde{c}_{m,k}||^{2 }}{\sigma^{2}}\right)\right]^{-q} \tag{11}\]  
Since \(q\) is positive integers, it is easy to verify that  
\[\left\{ \begin{array}{l}\sigma_{m}{}^{-q}(\widetilde{w})=\left[\sum_{k=1}^{N }\exp\left(\frac{||\widetilde{w}-q\widetilde{c}_{m,k}||^{2}}{q\sigma^{2}}- \frac{(q+1)||\widetilde{c}_{m,k}||^{2}}{\sigma^{2}}\right)\right]^{-q}>0\\ \lim_{|\widetilde{w}|_{l}\rightarrow+\infty}\{\sigma_{m}{}^{-q}(\widetilde{w} )\}=\lim_{|\widetilde{w}|_{l}\rightarrow+\infty}\left[\sum_{k=1}^{N}\exp \left(\frac{||\widetilde{w}-q\widetilde{c}_{m,k}||^{2}}{q\sigma^{2}}-\frac{( q+1)||\widetilde{c}_{m,k}||^{2}}{\sigma^{2}}\right)\right]^{-q}\to 0\\ \lim_{|\widetilde{w}|_{l}\rightarrow-\infty}\{\sigma_{m}{}^{-q}(\widetilde{w} )\}=\lim_{|\widetilde{w}|_{l}\rightarrow-\infty}\left[\sum_{k=1}^{N}\exp \left(\frac{||\widetilde{w}-q\widetilde{c}_{m,k}||^{2}}{q\sigma^{2}}-\frac{( q+1)||\widetilde{c}_{m,k}||^{2}}{\sigma^{2}}\right)\right]^{-q}\to 0\\ \end{array} \right. \tag{12}\]  
Therefore, there is a maximum value of \(\sigma_{m}{}^{-q}(\widetilde{w})\), which satisfies the conditions of saddle point approximation calculation. Assuming \(\sigma_{m}{}^{-q}(\widetilde{w})\) achieves the maximum at \(\widetilde{w}=\widetilde{w}_{0}\), \(\widetilde{w}_{0}\) satisfies  
\[\left\{ \begin{array}{l}\mathrm{grad}\{\ln[\sigma_{m}{}^{-q}(\widetilde{w}) \}]\}|_{\widetilde{w}_{0}}=0\\ \mathbf{H}\{\ln[\sigma_{m}{}^{-q}(\widetilde{w})]\}|_{\widetilde{w}_{0}}\prec 0 \end{array} \right. \tag{13}\]  
where \(\mathbf{H}\{\ln[\sigma_{m}{}^{-q}(\widetilde{w})]\}|_{\widetilde{w}_{0}}\) is the Hessian matrix of \(\ln[\sigma_{m}{}^{-q}(\widetilde{w})]\) at \(\widetilde{w}=\widetilde{w}_{0}\). By the fact of \(\sigma_{m}(\widetilde{w})>0\) and \(q>0\), for \(\mathbf{I}=1,2,\ldots,2N_{\mathrm{R}}\), \(\mathrm{grad}\{\ln[\sigma_{m}{}^{-q}(\widetilde{w})]\}|_{\widetilde{w}_{0}}=0\) is equivalent to  
\[\sum_{k=1}^{N}\frac{2(\widetilde{w}-q\widetilde{c}_{m,k})}{q\sigma^{2}}\exp \left(\frac{||\widetilde{w}-q\widetilde{c}_{m,k}||^{2}}{q\sigma^{2}}-\frac{( q+1)||\widetilde{c}_{m,k}||^{2}}{\sigma^{2}}\right)\Bigg{|}_{\widetilde{w}= \widetilde{w}_{0}}=0 \tag{14}\]  
and then the Hessian matrix \(\mathbf{H}\{\ln[\sigma_{m}{}^{-q}(\widetilde{w})]\}_{I,j}(\widetilde{w})|_{ \widetilde{w}_{0}}\) is rewritten as  
\[