# Distributed-Satellite-Clusters-Based Spectrum Sensing with Two-Stage Phase Alignment
## 2 System Model  
We consider a SS_DSC system comprising of one master satellite and \(L-1\) concomitant satellites for spectrum sensing, as shown in Figure 1. After receiving the task from the gateway station, each satellite of the DSC performs local sensing and sends the signal to the master satellite through the ISL for weighted fusion. However, in contrast with the traditional soft fusion methods, this method of fusion is to directly stack the sensing signals in phase, so as to greatly retain the effective information. Finally, the master satellite makes a judgment on the fused signal through energy detection, and transmits the result back to the Earth station to complete the whole sensing task. Figure 1: Proposed spectrum-sensing model comprising of a DSC system and a sensing target. In particular, our purpose is to obtain beamforming gain through signal-weighted fusion between the satellites to improve sensing performance. As mentioned earlier, the sensing target sends the signal \(x\) with \(E(|x|^{2})=1\) to the DSC system through the fading channel. After weighted fusion, the output signal at master satellite can be written as  
\[s(t)=\sqrt{P_{t}}\mathbf{w}^{H}\mathbf{\Phi h}x(t)+\mathbf{w}^{H}\mathbf{n_{0} }(t) \tag{1}\]  
where \(P_{t}\) is the transmit power of the sensing target; \(\mathbf{w}^{H}=[e^{-j\Phi_{1}(t)},e^{-j\Phi_{2}(t)},\ldots,e^{-j\Phi_{L}(t)}]\) is the receive beamforming weight vector; \(\mathbf{\Phi}=diag\big{[}e^{j\Phi_{1}(t)},e^{j\Phi_{2}(t)},\ldots,e^{j\Phi_{L }(t)}\big{]}\) is an \(L\times L\) diagonal matrix where \(\Phi_{l}(t)\) is the phase of the \(ith\) satellite; \(\mathbf{n_{0}}(t)\) denotes the additive white Gaussian noise (AWGN), which is an \(L\times 1\) AWGN vector whose distribution is \(\mathcal{CN}(0,\sigma_{n}^{2}\mathbf{L}_{\times 1})\); \(\mathbf{h}=[h_{1},h_{2},\ldots,h_{L}]^{T}\) is the real-valued channel vector between DSC system and sensing target modeled by the SR fading, which was proposed in [23], which illustrates similar agreements with the experimental data and has been used in most of the literature studies to model satellite channels of various radio frequency bands. The PDF of \(h_{i}\) can be written as  
\[f_{h_{i}}(x)=2a\pi\exp(-\beta x)\ _{1}F_{1}(m_{s};1;\delta x^{2}) \tag{2}\]  
where \(a=(2b_{s}m_{s}/2b_{s}m_{s}+\Omega_{s})^{m_{s}}/2b_{s}\), \(\delta=\Omega_{s}(2b_{s}(2b_{s}m_{s}+\Omega_{s}))^{-1}\), \(\beta=1/2b_{s}\), with \(\Omega_{s}\) represents the average power of the line-of-sight (LoS) component, \(2b_{s}\) is the average power of the scatter component, and \(m_{s}\) is the Nakagami parameter ranging from 0 to \(\infty\). Meanwhile, the function \({}_{1}F_{1}(a,b,z)\) is the confluent hypergeometric function, having expression as  
\[{}_{1}F_{1}(a,b,z)=\sum_{k=0}^{\infty}\frac{(a)_{k}(z)^{k}}{(b)_{k}k!} \tag{3}\]  
where \((x)_{k}=x(x+1)\cdots(x+k-1)\). Thus, the output SNR of DSC can be written as  
\[\gamma=\frac{p_{t}}{\sigma_{n}^{2}}\frac{\mathbf{w}^{H}\mathbf{\Phi h}\mathbf{ h}^{H}\mathbf{\Phi}^{H}\mathbf{w}}{\mathbf{w}^{H}\mathbf{w}}=\frac{\overline{ \gamma}}{L}\big{|}\sum_{i=1}^{L}|h_{i}|e^{j(\Phi_{i}(t)-\Phi_{i}(t))}\big{|}^{2} \tag{4}\]  
where \(\overline{\gamma}=\frac{p_{t}}{\sigma_{n}^{2}}\) is the SNR at sensing target. Obviously, the SNR plays a key role in improving the spectrum-sensing performance. The higher the SNR, the higher the correct detection probability [24].