## 2 GPS SDR Architecture for CRPA Processing and Computational Challenges  
Adaptive array processing starts from phase calibration between antenna elements for each satellite tracking channel. Once the carrier phases of all antenna elements are aligned, signal-to-noise-density-ratio (C/N\({}_{0}\)) of the corresponding satellite channel can be enhanced by combining all phase-aligned signals (_i.e._, high antenna gain, or beam, would be formed toward the satellite). One method of phase alignment is to use array synthesis techniques that consider array orientation, baseline geometry, satellite constellation, line biases, filter delays, etc. Furthermore, antenna anisotropy and mutual electronic coupling between antenna elements should be considered for precise phase alignment. In our experience this method is considerably more difficult than our method illustrated in Figure 2.  
Figure 2: Inter-channel phase alignments using independent satellite tracking channels for each antenna element. Different array steering vectors are obtained at each epoch for different satellites (_i.e._, 12 different steering vectors for our GPS SDR supporting 12 beamsteering channels). Figure 2 shows phase alignment of one satellite channel as an example. Four antenna elements independently track the same satellite. Then, a carrier phase of each antenna for the satellite (_i.e.,_\(\phi_{i}\), \(\phi_{2}\), \(\phi_{3}\), and \(\phi_{4}\)) is obtained from its tracking loop. Now phase alignment can be done by simple phase rotation with phase difference between each antenna and a reference antenna (_i.e.,_\(\Delta\phi_{i}=\phi_{i}-\phi_{i}=0\), \(\Delta\phi_{2}=\phi_{2}-\phi_{1}\), \(\Delta\phi_{3}=\phi_{3}-\phi_{1}\), and \(\Delta\phi_{4}=\phi_{4}-\phi_{1}\) ). This phase alignment method is conceptually straightforward, but it has several limitations. For each satellite tracking, five independent tracking channels are required for a four-element array system (_i.e.,_ four tracking channels for four antenna elements and one tracking channel for a synthesized beamsteering signal). Thus, five-times more computational power than for a conventional receiver is required. This method is applicable for mobile GPS receivers with CRPA to reduce multipath effects by forming high antenna gains in the directions of direct signal paths from satellites. However, this method may not be useful for mobile users under strong RFI because tracking channels for a single antenna element can lose lock under RFI, and thus phase differences between antenna elements cannot be calculated on-the-fly. As mentioned in Section 1, the applications of our anti-jamming GPS SDR for CRPA are GPS ground station receivers such as time transfer receivers for GBTs, GBAS ground facility receivers, and DGPS reference station receivers. Since these receivers are stationary, the simple phase alignment method is still applicable even under RFI. When RFI does not exist, which is a usual condition, an array steering vector, \(\mathbf{T}\)*, for each satellite can be obtained by independently tracking each antenna element as in Figure 2. Then, the array steering vectors are stored as a lookup table with time stamps. Since GPS satellite ground tracks repeat in a known basis, the stored array steering vectors can be used for a stationary GPS receiver when RFI makes the on-the-fly phase alignment impossible. Again, this method is not adequate if a GPS receiver with CRPA is not stationary. Once array steering vectors are given, an adaptive RFI cancellation algorithm is applied as in Figure 3. Among two algorithms reviewed in [37], the minimum-variance distortionless-response (MVDR) algorithm [38] is applied in this paper. The optimal weight vector, \(\mathbf{W}\), in MVDR processing is estimated by Equation (1).