## 2 System Model
### Channel Models  
In this section, we model the LEO satellite downlink channel [32]. \(h_{l,k}(n)\) denotes the channel of satellite \(l\) and antenna \(k\), which can be represented as  
\[h_{l,k}(n)=h_{l,k}^{LOS}(n)+h_{l,k}^{NLOS}(n) \tag{8}\]  
where \(h_{l,k}^{LOS}(n)\) represents the LOS part and \(h_{l,k}^{NLOS}(n)\) is the none-line-of-sight (NLOS) part of the channel. They can be expressed as  
\[h_{l,k}^{LOS}(n)=\sqrt{\frac{K_{l,k}}{K_{l,k}+1}}\exp[j2\pi(\varepsilon_{l,k}^ {LOS}\frac{n}{N}-j2\pi f_{c}\tau_{l,k}^{LOS})]v_{l,k}^{LOS} \tag{9}\]  
\[h_{l,k}^{NLOS}(n)=\sqrt{\frac{1}{K_{l,k}+1}}\sum_{m=1}^{M_{l,k}}g_{l,k,m}\exp[ j2\pi(\varepsilon_{l,k,m}^{NLOS}\frac{n}{N}-j2\pi f_{c}\tau_{l,k,m}^{NLOS})]v_{l,k}^{ NLOS} \tag{10}\]  
where \(K_{l,k}\) denotes the Rician factor. \(\varepsilon_{l,k}^{LOS}=f_{l,k}^{LOS}/\Delta f\), \(\varepsilon_{l,k,m}^{NLOS}=f_{l,k,m}^{NLOS}/\Delta f\) denote the normalized Doppler shift of the LOS and NLOS, respectively. \(\Delta f\) represents the subcarrier spacing. \(N\) denotes the Fourier transform points. \(f_{c}\) is the carrier frequency. \(\tau_{l,k}^{LOS}\) and \(\tau_{l,k,m}^{NLOS}\) represent the propagation delay of the LOS and NLOS. \(M_{l,k}\) denotes the number of channel propagation paths of NLOS. \(v_{x,y}\stackrel{{\Delta}}{{=}}\{v_{l,k}^{LOS},v_{l,k}^{NLOS}\}\) denote the array response array. As the satellite is very high relative to the terrestrial terminal, the array response \(v_{l,k}^{LOS}=v_{l,k,m}^{NLOS}\), which can be expressed as  
\[v_{x,y}=v^{x}\otimes v^{y}\in\mathbb{C}^{1\times M} \tag{11}\]  
Figure 3: SS/PBCH resource mapping.  
\[v^{x}= \frac{1}{\sqrt{N_{l,x}}}[1,\exp(-j2\pi\frac{d_{v}}{\lambda}\sin\theta _{1}^{y}\cos\theta_{1}^{x}),\ldots, \tag{12}\] \[\exp(-j2\pi(N_{l,x}-1)\frac{d_{v}}{\lambda}\sin\theta_{N_{l,x}-1}^ {y}\cos\theta_{N_{l,x}-1}^{x})]\in\mathbb{C}^{1\times N_{l,x}}\] \[v^{y}= \frac{1}{\sqrt{N_{t,y}}}[1,\exp(-j2\pi\frac{d_{v}}{\lambda}\cos \theta_{1}^{y}),\ldots,\] (13) \[\exp(-j2\pi(N_{l,y}-1)\frac{d_{v}}{\lambda}\cos\theta_{N_{l,y}-1} ^{y})]\in\mathbb{C}^{1\times N_{l,y}}\]  
where \(\lambda\) denotes the wavelength, \(d_{v}\) represents the element space of the antenna array. \(M\) represents the number of antenna elements. \(N_{l,x}\), \(N_{t,y}\) denote the number of antenna elements along \(x\) axis and \(y\) axis, respectively.  
Doppler frequency shift includes the frequency shift brought by satellite and terrestrial terminals. Assuming that the IoT UT is not moving, the frequency shift \(f=f_{sat}\). According to [33], the frequency shift of different paths can be assumed the same values, \(\varepsilon_{l,k}^{LOS}=\varepsilon_{l,k,m}^{NLOS}\). Because the distance between the two receiving antennas of the IoT terminal is far less than the height of the satellite, the frequency shift caused by the same satellite to the two antennas of the IoT terminal can be regarded as the same value, \(\varepsilon_{l}=\varepsilon_{l}^{LOS}=\varepsilon_{l,k}^{LOS}\). Therefore, Formulas (9) and (10) can be rewritten as  
\[h_{l,k}^{LOS}(n)=\sqrt{\frac{K_{l,k}}{K_{l,k}+1}}\exp[j2\pi(\varepsilon_{l} \frac{n}{N}-j2\pi f_{c}\tau_{l,k}^{LOS})]v_{l,k}^{LOS} \tag{14}\]  
\[h_{l,k}^{NLOS}(n)=\sqrt{\frac{1}{K_{l,k}+1}}\sum_{m=1}^{M_{l,k}}g_{l,k,m}\exp [j2\pi(\varepsilon_{l}\frac{n}{N}-j2\pi f_{c}\tau_{l})]v_{l,k}^{LOS} \tag{15}\]