## 3 Performance Analysis
### Path Diversity Analysis  
Path diversity makes use of two or more statistically independent radio paths to improve the transmission reliability [7]. The probability that all radio paths exhibit deep fading is very low, so if all received signals are properly combined, the network performance can be significantly improved. The statistically independent paths can be obtained by [27]: Spatial diversity, frequency diversity, temporal diversity, angle diversity, and polarization diversity. Clearly, in order to increase the transmission reliability, the diversity techniques reduce the capacity of the communication network by sacrificing radio resources. In the case of diversity reception, the best received signal should be selected (selection diversity) or all received signals should be combined into one signal for data estimation (combination diversity). Due to the predominant LOS channel conditions in a HAP operating environment, propagation channels are highly correlated and most diversity techniques are not applicable. The exception may be spatial diversity on the ground or use of multiple HAPs [11]. The diversity gain here is that the multiple antennas in the system utilize the non-correlation of the received signal in each antenna, reduce the amplitude of fading by combining signals, and then achieve the performance gain. The principle is that when each channel is affected by independent fading, the HAPs transmit multiple copies of the same signal in different channels, which can significantly reduce probability of the deep fading in all copies at the same time. Thus, the wireless channel fading can be overcome and the transmission performance is also improved. Additionally, different incoherent signals paths are diversely processed by the receiver, which can greatly reduce the bit error rate (BER) of the information, and effectively resist blocking interference. We assume that the number of HAPs is \(N_{H}\), the number of receiving antennas for the sensor device is \(N_{r}\), and the transmitting signal vector is \(\mathbf{s}\left(t\right)\in\mathbb{C}^{N_{H}\times 1}\) which has a power expectation of \(\mathrm{E}\left[\left|\mathbf{s}\left(t\right)\right|^{2}\right]=1\). The beamforming vector in the V-MIMO transmitter is \(\mathbf{w}\left(N_{H}\times 1\right)\). The signal is transmitted through a channel gains matrix of complex fading \(\mathbf{H}(t)=[h_{ij}(t)]_{N_{r}\times N_{H}}\in\mathbb{C}^{N_{r}\times N_{H}}\), and is received using the \(N_{r}\times 1\) beamforming vector \(\mathbf{v}\) in the receiver. Ultimately, the received signals \(\mathbf{y}\left(t\right)\in\mathbb{C}^{N_{r}\times 1}\) can then be represented as:  
\[\mathbf{y}\left(t\right)=\mathbf{v}^{H}\left[\mathbf{H}\mathbf{w}\sqrt{P_{s}} \mathbf{s}\left(t\right)+\mathbf{n}\left(t\right)\right]=\mathbf{v}^{H} \mathbf{H}\mathbf{w}\sqrt{P_{s}}\mathbf{s}\left(t\right)+\mathbf{v}^{H} \mathbf{n}\left(t\right) \tag{12}\]  
where the superscript \(H\) denotes the hermitian conjugate transpose operator and \(P_{s}\) is the transmitted power. The channel matrix \(\mathbf{H}(N_{r}\times N_{H})\) is shadowed Rician fading, whose NLOS element \(\left[\mathbf{H}_{NLOS}\right]{}_{i,j}\) is the i.i.d. complex Gaussian distribution \(\sqrt{1/(\kappa+1)}\times\aleph_{C}(0,1)\), and the amplitude of LOS element \(\left[\mathbf{H}_{LOS}\right]{}_{i,j}\) is \(\sqrt{\kappa/(\kappa+1)}\), which obeys the lognormal distribution, as given by Equation (7). \(\mathbf{n}(t)\ \in\ \mathbb{C}^{N_{r}\times 1}\) denotes the vector of additive white Gaussian noise (AWGN), whose elements follow the i.i.d. \(\aleph_{c}(0,\sigma_{r}^{2})\) distribution, and \(\mathrm{E}\left[\mathbf{n}\left(t\right)\mathbf{n}^{H}\left(t\right)\right]=N_ {0}\mathbf{I}_{N_{r}}\). Then, the SNR of receiver is therefore given by:  
(13)  
where \(\gamma_{s}=P_{s}/N_{0}\) is the SNR of the virtual transmitter.