## 2 Conventional Diagonal Loading Methods  
Considering a free far-field case including one target source, \(G\) interference sources and environmental noise, a uniform linear array (ULA) with \(M\) (\(G<M\)) microphones was used to pick up the multi-channel speech signals. Assuming that the target source, interference sources and noise are mutually independent, by minimizing the output power of the interference plus noise with the constraint that the target source is distortionless, the MVDR beamformer at the \(k\)th frequency bin of the \(l\)th frame was built as follows [1; 2; 3; 4; 5]:  
\[\underset{\mathbf{w}}{\text{min}}\mathbf{w}^{H}\mathbf{R}_{\text{IN}}\mathbf{w }\quad\text{s.t.}\ \mathbf{w}^{H}\mathbf{a}_{0}=1 \tag{1}\]  
where \(\mathbf{w}\)\(\in\)\(\mathbb{C}^{M\times 1}\) is the\(\in\) weighting vector of the MVDR beamformer, \(\mathbf{R}_{\text{IN}}\)\(\in\)\(\mathbb{C}^{M\times M}\) is the real CMIN, \(\mathbf{a}_{0}\)\(\in\)\(\mathbb{C}^{M\times 1}\) is the real steering vector of the target source, the superscript \({}^{\text{\tiny{$H$\prime$}}}\) indicates the conjugate transpose and the symbol \(``\mathbb{C}\"\) indicates the complex space. Equation (1) is commonly solved by the Lagrange multiplier method [3; 5] as follows:  
\[\mathbf{w}=\frac{\mathbf{R}_{\text{IN}}^{-1}\mathbf{a}_{0}}{\mathbf{a}_{0}^{H }\mathbf{R}_{\text{IN}}^{-1}\mathbf{a}_{0}} \tag{2}\]Since \(\mathbf{R}_{\text{IN}}\) is unknown in practice, it is usually replaced by the CMOS \(\mathbf{R}_{\text{XX}}\in\mathbb{C}^{M\times M}\), i.e. :  
\[\mathbf{R}_{\text{IN}}\rightarrow\mathbf{R}_{\text{XX}}=\frac{1}{J}\sum_{j=l+1 -J}^{l}\kappa(j)\mathbf{x}^{H}(j) \tag{3}\]  
where \(J\) is the number of the snapshots and \(\mathbf{x}(j)\in\mathbb{C}^{M\times 1}\) is the \(j\)th snapshot of the observed signal. Substituting Equation (3) into Equation (2), the weighting vector of the CMB can be expressed as follows by using the estimated steering vector \(\hat{\mathbf{a}}_{0}\in\mathbb{C}^{M\times 1}\) of the target source:  
\[\mathbf{w}_{\text{CMB}}=\frac{\mathbf{R}_{\text{XX}}^{-1}\hat{\mathbf{a}}_{0} }{\hat{\mathbf{a}}_{0}^{H}\mathbf{R}_{\text{XX}}^{-1}\hat{\mathbf{a}}_{0}} \tag{4}\]  
The purpose of diagonal loading methods is to improve the performance of the CMB by loading values on the diagonal elements of the \(\mathbf{R}_{\text{XX}}\) before the inverse operation [8; 9; 11], so that the general form of the weighting vector \(\mathbf{w}_{\text{DL}}\in\mathbb{C}^{M\times 1}\) of the MVDR beamformer realized by the single loading parameter-based diagonal loading method can be expressed as follows:  
\[\mathbf{w}_{\text{DL}}=\frac{(\mathbf{R}_{\text{XX}}+\xi\mathbf{I})^{-1}\hat{ \mathbf{a}}_{0}}{\hat{\mathbf{a}}_{0}^{H}(\mathbf{R}_{\text{XX}}+\xi\mathbf{I })^{-1}\hat{\mathbf{a}}_{0}} \tag{5}\]  
where \(\xi\) is the loading value and \(\mathbf{I}\in\mathbb{C}^{M\times M}\) is a unit matrix. Obviously, \(\xi\) is the key parameter for improving the CMB. When \(\xi=0\), Equation (5) degenerates to the CMB. When \(\xi\) is much larger than the powers of the sound sources, Equation (5) degenerates to the delay and sum (DAS) beamformer [9; 10; 11; 12]. Table 1 gives six diagonal loading methods of the CMB including HKB, LNR, SMF, BPR, GLC and NRP-TMMSE, respectively, where the symbol \" \(\mid\mid\mid\mid_{2}\)\" indicates the 2-norm operation. From Table 1, we can see that the HKB uses the difference between the output \(\mathbf{b}\in\mathbb{C}^{M\times 1}\) of the fixed filter and the output \(\mathbf{A}\hat{\mathbf{\eta}}\) of the blocking filter to solve the loading values [9], where \(\mathbf{A}\in\mathbb{C}^{M\times(M-1)}\) and \(\hat{\mathbf{\eta}}\in\mathbb{C}^{(M-1)\times 1}\) are the outputs of the blocking matrix and weighting vector, respectively. The performance of the HKB is greatly affected by the number of microphones.