## 1 Introduction
Meanwhile, the MMSE-AH criterion gives more complicated computation and only marginally better C/N\({}_{0}\) performance than the PM criterion. However, the MMSE criterion and MMSE-AH criterion are difficult to be carried out in practice as the desired GNSS signal is supposed to be accurately known, due to which we will not take these two MMSE criterions into consideration in the following work. (d) Minimum variance distortionless response constraining single satellite (MVDR-CSS) criterion. In this method, the constraint vector is set to constrain only one GNSS satellite at once, in which beam forms towards the desired GNSS satellite and thus the C/N\({}_{0}\) performance is greatly improved. Nonetheless, the MVDR-CSS criterion also cannot avoid complicated calculations when there are multiple GNSS satellites in the FOV because each GNSS satellite needs a separate adaptive filter. Besides, a very interesting and valuable study is introduced in [31], in which a beamforming algorithm for multiple simultaneous desired signals is proposed. In this work, it emphasizes that different beams should be weighted according to the expected range of signal strengths, which is very useful in practice systems and is closely related to our current work. Since the signal strengths of GNSS signals are almost equal, we can define the weights corresponding to the desired GNSS signals are the same when compared with this work. In this paper, we propose a novel STPAP beamforming algorithm based on MVDR criterion for GNSS receivers, which can achieve a balance between the output C/N\({}_{0}\) performance and computational complexity. The contributions of our work are as follows: (a) A novel constraint vector for the STPAP architecture, which only requires a single adaptive filter or few adaptive filters when multiple GNSS satellites exist in the FOV, is proposed. For simplicity, since the proposed STPAP algorithm based on the MVDR criterion which constrains multiple satellites at once, we can call it MVDR-CMS criterion in this paper. Consequently, it can reduce the computational complexity effectively compared with the traditional STPAP algorithm based on the criteria in [30]; (**b**) The number of the desired GNSS signals processed in a single adaptive filter is discussed, so it will not cause drastic output C/N\({}_{0}\) performance degradation when obtaining a lower computational complexity. Especially, the output C/N\({}_{0}\) performance of the proposed STPAP algorithm based on the MVDR-CMS criterion is close to that of the existing STPAP algorithm based on the MVDR-CSS criterion in [30] if the parameter \(J\), which denotes the number of GNSS satellites that processed by the same adaptive filter, is proper. (c) An adaptive algorithm based on the CLMS method is derived to calculate the weight vector for the proposed STPAP beamforming algorithm. The paper is organized as follows. Section 2 presents the polarization concept and the STPAP architecture. In Section 3, a novel STPAP beamforming algorithm based on the MVDR criterion is proposed. Besides, an adaptive algorithm to calculate the weight vector is derived for the proposed STPAP beamforming algorithm. In Section 4, simulations are carried out to validate the effectiveness of the proposed algorithm. Moreover, the proposed STPAP beamforming algorithm is compared with the existing STPAP algorithms based on the criterions in [29; 30; 31]. Finally, Section 5 concludes this paper.