# Time-Varying Communication Channel High Altitude Platform Station Link Budget and Channel Modeling
## 3 HAPS Link Budget and Channel Modeling  
### HAPS Link Budget  
The main goal of the link budget is to ensure the availability of the HAPS communication link. Considering the cost of space segments and ground station equipment, the HAPS link needs to be carefully designed. To fully optimize and save all available resources. The impairment of the HAPS link performance evaluation can be represented by the link diagram shown in Figure 3.  
Where, \(P_{EIRP}\) indicates the equivalent isotropic radiated power of the transmitter, AWGN is Additive white Gaussian noise. \(l_{FS}\) is the loss of free space, \(G_{AT}\) indicates the gain of acceptance antenna. \(s_{i}(t)\) expresses the input signal and \(s_{o}(t)\) denotes the output signal. \(N\) is the thermal noise power, \(k\) is Boltzmann constant. \(B\) is IF (Intermediate Frequency) effective bandwidth. \(T_{AR}\) indicates antenna noise temperature, \(T_{e}\) indicates the effective input noise temperature of the overall acceptance system, \(P_{TX}\) is the transmit power, \(G_{A}\) denotes the amplifier gain. The uplink interference of the near space HAPS is shown in Figure 4.  
As can be seen from Figure 4, for different users \(i,j,k\), etc., there is interference between them. The smaller the elevation angle is, the longer the distance from the HAPS to its users. The energy per bit \(E_{b}\) of the near space HAPS can be expressed as:  
\[E_{b}=C\ T_{b}\]  
Figure 4: Uplink interference model of HAPS. Figure 3: Sketch map of HAPS link budget. where \(C\) is the carrier wave average power. \(T_{b}\) indicates the time it takes to accept 1 bit(s). The signal to noise ratio can be expressed as:  
\[\frac{C}{N}=\frac{E_{b}T_{b}}{N_{0}B}=\frac{E_{b}R}{N_{0}B} \tag{7}\]  
where \(R=1/T_{b}\) is the bit rate (b/s). \(T_{b}\) indicates the spent time when transmit 1-bit energy. \(N_{0}\) denotes the thermal noise power spectral density, then \(E_{b}/N_{0}\) can be expressed as:  
\[\frac{E_{b}}{N_{0}}=\frac{P_{EIRP}g_{AR}}{kT_{s}R_{E}l_{FS}} \tag{8}\]  
where \(P_{EIRP}\) represents the equivalent isotropic radiated power of the transmitter. \(l_{FS}\) denotes the loss of free space. \(g_{AR}\) indicates the antenna gain. \(k\) is the Boltzmann constant. \(T_{s}\) is the effective temperature in HAPS system. \(R_{E}\) is the radius of earth. As for 2 PSK or QPSK, bit error rate \(P_{e}\) and \(E_{b}/N_{0}\) have the following relationship:  
\[P_{e}=\frac{1}{2}\Bigg{(}1-erfc\sqrt{\frac{E_{b}}{n_{0}}}\Bigg{)} \tag{9}\]  
where \(erfc\) denotes the error function. For example, \(erfc(x)=1-erf(x)=\frac{1}{\sqrt{\pi}}\int_{x}^{\infty}e^{-t^{2}}dt\). When the bit error rate \(P_{e}=10^{-4}\) is required, the normalized ideal threshold SNR can be described by the following equation:  
\[\left[\frac{C}{N}\right]=\left[\frac{E_{b}}{n_{0}}\right]+10\text{lg}k+10 \text{lg}P_{b} \tag{10}\]  
where \(k\) is the Boltzmann constant. \(P_{b}\) is the power when receive 1 bit. The elevation angle of HAPS can be expressed as  
\[A_{avg}\ (\Delta)=tg^{-1}\frac{\cos(\phi_{2}-\phi_{1})-0.5\Delta}{\sqrt{3-2[ \cos(\phi_{2}-\phi_{1})\times\cos\delta]^{2}}} \tag{11}\]  
where \(\delta\) is the central angle of the HAPS. \(\phi_{2},\phi_{1}\) are the elevation angle of two different stations to the HAPS platform, respectively. In the HAPS wireless communication space, the change relationship of the elevation angle \(\phi\) should be considered. The error rate \(P_{e}\) in the un-coded case can be expressed as:  
Figure 4.0  
If PSK (8 phase shift keying) and Hamming code (15, 11) and Golay code (23, 12) are used to encode near space HAPS channels, the bit error rate \(P_{e}\) can be expressed as:  
\[P_{e}=\left\{ \begin{array}{ll}\frac{1}{2\sqrt{\phi^{2}+1}}\Big{(}1-erfc \sqrt{\frac{E_{b}}{n_{0}}}\Big{)};&\text{PSK}\\ \frac{1}{2\sqrt{0.8\phi^{2}+5}}\Big{(}1-erfc\sqrt{\frac{E_{b}}{n_{0}}}\Big{)};& \text{Hamming}\\ \frac{1}{2\sqrt{\phi^{2}+3\phi+1}}\Big{(}1-erfc\sqrt{\frac{E_{b}}{n_{0}}}\Big{)} ;&\text{Golay}\end{array} \right.