# Distributed-Satellite-Clusters-Based Spectrum Sensing with Two-Stage Phase Alignment
## 3 The Proposed Two-Stage Phase Alignment Frame
### Two-Bit Feedback Algorithm  
Note that the iterative speed and accuracy are the most important indicators for an iterative-type algorithm. Herein, the size of the disturbance is the key to controlling the convergence speed and stabilizing the performance of the phase compensation algorithm. Specifically, the perturbation size should be sufficiently large to improve the convergence speed during the initial stage. However, when close to the optimal point, the perturbation size should be reduced to achieve steady performance. Therefore, an excellent algorithm should have faster convergence speed and stability to adapt to the high dynamic characteristics of satellites. In view of this, an RPC_TF algorithm is proposed which introduces an additional direction correction factor \(\tau_{i}(t)\) and a step control factor \(q\) to improve the convergence speed by changing the iteration step while ensuring the accuracy of the disturbance direction. Without loss of generality, one slot in RPC_TF algorithm is set to start from the computation of the transmitting weight and end at the receiving of the two-bit feedback message from the receiver. During the convergence process, the master satellite sends two bits of information, \(b_{0}\) and \(b_{1}\), to the concomitant satellite in a time slot, and each concomitant synchronizes its phase based on these two bits. Specifically, at time slot \(t\), based on the weight fusion of the signals sent by all concomitant satellites, the master satellite compares the strength of the sensing signal \(S_{S}(t)\) with its previously measured best strength \(S_{S}^{best}(t-1)\). If \(S_{S}(t)>S_{S}^{best}(t-1)\), the master satellite makes the first feedback bit \(b_{0}=1\), updates the value \(S_{S}^{best}(t)\) with the \(S_{S}(t)\), and calculates the times of positive feedback, that is, the cumulative times of \(b_{0}=1\). In addition, the second bit \(b_{1}\) is set to 1 if the cumulative number of \(b_{0}=1\) equals the preset threshold \(\Gamma_{1}\) and to 0 otherwise. On the other hand, if \(S_{S}(t)\leqslant S_{S}^{best}(t-1)\), the receiver sets \(b_{0}=0\), maintains the previously measured best \(S_{S}^{best}\), and also calculates the times of negative feedback, namely, the cumulative times of \(b_{0}=0\). Similarly, the second bit \(b_{1}\) is set to 1 if the cumulative number equals the preset threshold \(\Gamma_{2}\) and to 0 otherwise. Upon receiving the feedback from the master satellite, each concomitant satellite checks the two-bit information and makes a series of corresponding responses. The basic idea behind the proposed algorithm is that the concomitant satellites change the best phase compensation when the first bit \(b_{0}=1\), and change the size of random perturbation when the second bit \(b_{1}=1\), and vice versa. The proposed RPC_TF algorithm is summarized in Algorithm 1. ```
A: At the master satellite Input: the sensing signal from the concomitant satellites \(s_{i}(t)\), Initial value \(c=0\), \(d=0\) Two-bit information generation process: The master satellite measures the strength of the sensing signal and updates the best strength and the bits. A1. If \(S_{S}(t)>S_{S}^{best}(t-1)\),Then\(S_{S}^{best}(t)=S_{S}(t)\), \(b_{0}=1\), \(c=c+1\); A2. if \(c=\Gamma_{1}\), then \(b_{1}=1\), \(c=0\) A3. else \(b_{1}=0\) A4. end if A5. Else\(S_{S}^{best}(t)=S_{S}^{best}(t-1)\), \(b_{0}=0\), \(d=d+1\); A6. if \(d=\Gamma_{2}\), then \(b_{1}=1\), \(d=0\) A7. else \(b_{1}=0\) A8. end if A9. End If Output: two-bit feedback information \(b_{0}b_{1}\), the best strength \(S_{S}^{best}(t)\). The master satellite sends \(b_{0}b_{1}\) to the concomitant satellites. B: At the concomitant satellites Input: direction correction factor \(\tau_{i}(0)=0\), initial phase compensation \(\overline{u_{i}(0)}=0\), step control factor \(q,q>1\), random perturbation \(\pm\delta_{0}\), two-bit feedback information \(b_{0}b_{1}\). Phase compensation process: The transmitter obtains the feedback \(b_{0}b_{1}\) from the receiver and updates its best transmission angle B1.