## 1 Introduction  
GNSS has been widely applied in both military and civil fields because it can provide all-time, all-weather, and high accuracy position, navigation, and timing service to global users. Although the power of the GNSS signal is 20 dB lower than the ambient noise floor, GNSS receivers can withstand a certain level of interference due to the direct sequence spread spectrum (DSSS) technique used in GNSS [1; 2; 3; 4; 5; 6; 7; 8; 9]. However, GNSS receivers will be interfered when interferences are strong enough. Antenna array processing technique provides an effective method to cancel interferences [10; 11; 12]. The concept is to apply weights on the signals received by different array elements to form nulls towards the arriving directions of incoming interferences while steering the array response towards the desired GNSS signals. Antenna array with the space adaptive processing (SAP) technique performs well in cancelling interferences and preserving GNSS signals [13; 14; 15; 16], while it also has disadvantages. At first, the SAPcan only filter out \(M-1\) interferences with an array antenna consisting of \(M\) elements. Nevertheless, in the multipath interference environment, the number of interferences increases dramatically and thus the performance of the SAP will be greatly deteriorated. Moreover, GNSS signals may be weakened as well when their directions of arrival (DOAs) are close to those of interferences. To overcome these shortcomings of the SAP, the space-time adaptive processing (STAP) technique was proposed in [17; 18; 19; 20; 21]. In the STAP, the temporal filter is adopted by placing a finite impulse response (FIR) filter behind each array element. Consequently, the STAP can cancel \(MK-1\) narrowband interferences or \(M-1\) wideband interferences if array antenna with \(M\) elements that followed by \(K\) order FIR filter is utilized. In addition, interferences incoming from directions close to those of the desired GNSS signals can also be rejected without mitigating GNSS signals if their frequencies are not identical to those of the desired GNSS signals. Although the STAP has combined the space domain and time domain to cancel interferences, the polarization domain is another field that can be utilized to distinguish and suppress interferences. In [22; 23; 24; 25; 26; 27], the space-polarization adaptive processing (SPAP) technique was introduced to filter out interferences in the joint space-polarization domain while the time domain is ignored. Furthermore, the STPAP technique was proposed in [28; 29], which can mitigate interferences for GNSS receivers in the joint space-time-polarization domain. Using DPSA with \(M\) elements that followed by \(K\) order FIR filter, the STPAP can cancel up to \(2 MK-1\) narrowband interferences or \(2M-1\) wideband interferences. It indicates that the STPAP can cancel more interferences than the SAP, STAP, and SPAP with the same size array antenna, which is of great significance for platforms that can only vacate small place for equipping GNSS array antennas. To implement the STPAP technique, several criterions to derive weight vector for DPSA have been proposed in [29; 30], which can be summarized as follows: (a) power minimization (PM) criterion. The PM criterion is easily put into practice as it just keeps the reference channel undistorted and does not need any prior knowledge, but its C/N\({}_{0}\) performance is worse than the other three criterions. (b) Minimum mean square error (MMSE) criterion. The MMSE criterion performs better than the PM criterion, while it makes the process computationally intensive since each GNSS satellite in the field of view (FOV) requires a separate adaptive filter. (c) Minimum mean square error averaged over hemisphere (MMSE-AH) criterion. Compared with the MMSE criterion, the MMSE-AH criterion achieves less computational complexity but withstands C/N\({}_{0}\) performance degradation.