## 2 System Model
### Multiple Assets in View  
If the sensor device can establish links with multiple HAPs, it is defined as multiple assets in view (MAV) or multiple HAPs in view. The MAV can provide a lot of extra benefits from the users' perspective: Rate and integrity improvements, signal routing flexibility, cooperative relay, and overcoming instability of platform that degrades system performance, _etc._  
It is known that the SNR of receiver is a identical to a filled aperture of the same physical collecting area in the antenna. If there are \(N_{H}\) assets in view, as depicted in Figure 1a, which each have a transmitter antenna aperture diameter of \(D_{H}\), then the receiver antenna measures the radiation field and the signals from each of the different antennas, and combines this to deliver a single waveform to the detector. In the most general case, the input signal power varies across the array. Assuming unit impedance throughout, the average signal power is given by [9]:  
\[\mathrm{E}[S_{i}]=\mathrm{E}\left[\frac{1}{N_{H}}\sum_{j=1}^{N_{H}}s_{ij}^{2}\right] \tag{1}\]  
where \(\mathrm{E}\left[\cdot\right]\) denotes the expectation operator and \(s_{ij}\) are the proportional to the envelope of the RF signal voltage for the \(j\)th transmitter antenna.  
Assuming the signal strength across the antenna is constant, \(s_{ij}=s_{ik}=s_{i}\), and the signal power per transmitter antenna is \(S_{i}=s_{i}^{2}\). If all signals are co-phased by a bank of phase shifters, the output signal voltage after integration is \(N_{H}s_{i}\) and the output signal power of the receiver can be represented as \(S_{0}=N_{H}^{2}S_{i}\). The noise from each antenna is an independent and identically distributed (i.i.d.) complex Gaussian random variable, uncorrelated with the signals, each following a \(\aleph_{C}(0,\sigma_{r}^{2})\) distribution. The average noise power is:  
\[\mathrm{E}[N_{i}]=\mathrm{E}\left[\frac{1}{N_{H}}\sum_{j=1}^{N_{H}}n_{ij}^{2}\right] \tag{2}\]  
Figure 1: System model.  
The square of the noise voltage is the receiving noise power:  
\[N_{0}=\left(n_{0}\right)^{2}=\left(\sum\limits_{j=1}^{N_{H}}n_{ij}\right)^{2}= \sum\limits_{j=1}^{N_{H}}n_{ij}^{2}+\sum\limits_{j=1}^{N_{H}}\sum\limits_{k=1}^ {N_{H}}n_{ij}n_{ik} \tag{3}\]  
where \(n_{ij}\) denote the noise between the \(i\)th receiving antenna and the \(j\)th transmitter antenna.  
Since the noise is uncorrelated, it does not add up in phase and so \(\sum\limits_{j=1}^{N_{H}}\sum\limits_{k=1}^{N_{H}}n_{ij}n_{ik}=0\) and \(\mathrm{E}[N_{i}]=N_{H}\mathrm{E}\left[N_{0}\right]\). The output SNR is therefore equal to:  
\[\gamma_{0}=\frac{S_{0}}{\mathrm{E}\left[N_{0}\right]}=\frac{N_{H}S_{i}}{ \mathrm{E}\left[N_{i}\right]}=N_{H}\gamma_{1} \tag{4}\]  
where \(\gamma_{1}\) denotes the SNR of a single HAP in view. If the QPSK is used and the required BER is \(10^{-6}\), the corresponding threshold of SNR for the sensor device is 5.3 dB. Assuming the value of \(\gamma_{1}\) provided by a single HAP is 1.6 dB, and there are four HAPs and a feasible data transmission can be achieved. However, it will be impossible when there is only one HAP.  
The improvement in SNR of MAV compared with a single asset in view is \(N_{H}\), the number of assets in view, therefore contributing to the rate and integrity improvements.