## 1 Introduction  
The minimum variance distortionless response (MVDR) beamformer has proved to be effective for suppressing interferences and noise by using an effective covariance matrix of the interference-plus-noise (CMIN) and an accurate steering vector of the target source [1; 2; 3; 4; 5]. Since the CMIN and the steering vector of the target source are unknown in the conventional MVDR beamformer (CMB), their applications are greatly limited. Thus, many improved MVDR beamforming methods [6; 7; 8; 9] have been proposed successively. In the typical examples, the diagonal loading-based MVDR beamformers [9; 10; 11; 12; 13; 14; 15; 16] have been paid more attention to recently. The purpose of the diagonal loading operation is to reduce the diffusion of the eigenvalues of the noise by loading the values on the diagonal elements of the covariance matrix of the observed signal (CMOS), which can reduce the error of the CMOS and the susceptibility to mismatch of the steering vector [9; 10]. However, a larger loading value lets the CMB degrade to a general fixed beamformer, on the contrary, a smaller loading value can hardly improve the performance of the CMB [11; 12]. In order to find the appropriate loading value, many diagonal loading methods have been proposed in recent decades. They can be divided into three categories. The first one is the single loading parameter-based diagonal loading method, whereby the loading matrix is the product of the loading value and unit matrix and loaded on the CMOS, such as the Hoerl-Kennard-Baldwin (HKB) method [9] and the load-to-white-noise ratio (LNR) method [10]. Although these two methods can reduce the diffusion of the eigenvalues of the noise with the appropriate loading value, they are susceptible to specific parameters. Hence, some specific parameter-free methods have been proposed, such as the spatially matched filter (SMF) method [11] and the bounded perturbation regularization (BPR) method [12]. Nevertheless, these methods usually depend on a priori assumptions or constraints, which can easily lead the loading value to be too large or too small. The second category of diagonal loading methods is the multiple loading parameters-based diagonal loading methods, which is usually a linear combination of the CMOS and the loading matrix, such as the general linear combination (GLC) method [13] and the noise reduction preprocessing into a truncated minimum mean square error (NRP-TMMSE) method [14]. Although this category of methods has a higher flexibility for designing the loading value, they are also sensitive to the number of microphones and snapshots. Moreover, the optimal objectives of these methods do not focus on making the loaded covariance matrix approach to the CMIN, which limits the improvement on the self-cancellation of the main lobe. The third category is the iterative diagonal loading methods, which tries to improve the performance of the CMB by performing diagonal loading iteratively, such as the parameter-free Landweber iteration (PFLI) method [15] and the iterative diagonally loaded sample matrix inverse (IDL-SMI) method [16]. This category of methods utilizes iteration to compensate for the limitation of a single diagonal loading operation but is sensitive to the loading value or the weighting value used for the iterative update. Currently, the available diagonal loading-based MVDR beamforming methods are suitable for cases in which the steering vector of the target source does not mismatch or only has a small mismatch [17]. However, the studies in recent years have still mainly been focused on designing the loading value [18; 19; 20; 21], which is intent on obtaining an effective compromise between the noise reduction and interference suppression, whereas the essential problem of the mismatch of the steering vector is ignored. In addition, the diagonal loading-based methods also ignore the influence of the off-diagonal elements.