## 2 Conventional Diagonal Loading Methods
The LNR designs the loading value by the power \(\sigma_{\text{W}}^{2}\) of white noise and a specific parameter \(\zeta_{\text{LNR}}\) (usually set to 10) [10], which aims to reduce the divergence of the eigenvalues that correspond to noise and make a compromise between the noise reduction and interference suppression by loading a value related to the power of white noise. However, the LNR is sensitive to the estimated power of white noise and the specific parameter \(\zeta_{\text{LNR}}\). The SMF uses the output power (\(\overline{\mathbf{a}}_{0}^{H}\mathbf{R}_{\text{XX}}\overline{\mathbf{a}}_{0}\), where \(\overline{\mathbf{a}}_{0}\in\mathbb{C}^{M\times 1}\) is the normalized steering vector of the target source) of the spatially matched filter as the loading value [11]. Since the output power of the SMF may contain the power of the target source, the SMF may cause the loading value to be too large. Therefore, the SMF-based MVDR beamformer is easy to degenerate into a fixed beamformer. Based on the HKB, the BPR introduces a perturbation matrix and a constraint factor to avoid the influence of the number of microphones [12]. However, the approximate solution of the constraint factor \(\delta\) is likely to be too small, which limits the improvement of performance on the CMB. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Methods** & **Diagonal Loading** & **Methods** & **Diagonal Loading** \\ \hline HKB & \(\frac{(M-1)(\Delta\mathbf{A}-\mathbf{b})^{2}}{\|\mathbf{\eta}\|_{2}^{2}}\) & LNR & \(10^{\frac{\zeta_{\text{LNR}}}{\hat{\mathbf{\eta}}}}\sigma_{\text{w}}^{2}\) \\ SMF & \(\overline{\mathbf{a}}_{0}^{H}\mathbf{R}_{\text{XX}}\overline{\mathbf{a}}_{0}\) & BPR & \(\frac{\delta\|\mathbf{A}\hat{\mathbf{\eta}}-\mathbf{b}\|_{2}}{\|\mathbf{\eta}\|_ {2}}\) \\ GLC & \(\min\limits_{a,\beta}\Bigl{\{}\|\mathbf{A}\mathbf{R}_{\text{XX}}+\beta\mathbf{I }-\mathbf{R}_{\text{EX}}\|_{2}^{2}\Bigr{\}}\) & NRP-TMMSE & \(\min\limits_{a,\beta}\Bigl{\{}\|\mathbf{A}\mathbf{R}_{\text{XX}}+\beta\mathbf{I }-\mathbf{R}_{\text{T}}\|_{2}^{2}\Bigr{\}}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Six Diagonal Loading Methods of the CMB. The GLC uses the non-negative shrinkage parameters \(\alpha\) and \(\beta\) to combine the CMOS \(\mathbf{R}_{\mathrm{XX}}\) and the unit matrix \(\mathbf{I}\) linearly, where these two parameters are solved by minimizing the error between the loaded covariance matrix and the covariance matrix \(\mathbf{R}_{\mathrm{EXP}}\)\(\in\)\(\mathbb{C}^{M\times M}\) that we expect [13]. Obviously, the GLC has higher flexibility for designing the loading value, but it is also sensitive to the number of microphones and snapshots. Meanwhile, the NRP-TMMSE uses the denoised CMOS \(\mathbf{R}_{\mathrm{T}}\)\(\in\)\(\mathbb{C}^{M\times M}\) and convex optimization technique to solve the loaded covariance matrix and a more accurate steering vector of the target source [14]. It is an optimized method of the GLC. However, the optimization objectives of the GLC and NRP-TMMSE do not focus on making the loaded covariance matrix approach the CMIN, which limits the improvement of the problem of the self-cancellation of the main lobe. Moreover, the diagonal loading iteration of the PFLI [15] is built as follows:  
\[\left\{\begin{array}{l}\mathbf{w}_{\mathrm{LI}}^{(0)}=0\\ \mathbf{w}_{\mathrm{LI}}^{(i_{\mathrm{iteration}}-1)}=\mathbf{w}_{\mathrm{LI} }^{(i_{\mathrm{iteration}}-1)}+\alpha_{\mathrm{LI}}\Big  
In Module 1, an improved GLC (IGLC) with a full loading matrix is used to correct the steering vector of the target source backward so that the distortion of the target source can be improved. In Module 2, an appropriate loading matrix based on the steered response power of the uncertain sets is used to reconstruct the covariance matrix so that the interferences can be further suppressed.