## 3 Performance Analysis
### System Capacity  
The system capacity is an important indicator to measure the effectiveness of the wireless communication network. Unlike the AWGN channel, due to the random characteristic of the wireless fading channel, there is no uniform definition of system capacity applicable to all fading situations. In a wireless fading environment, indicators that evaluate the channel capacity include ergodic capacity and outage capacity.  
The ergodic capacity \(C_{erg}\) reflects the average transmission rate of the channel, is mainly used for communication services that are insensitive to delay, e.g., non-real-time multi-media data in multi-media WSNs. The outage capacity is defined as the average transmission rate that channels can provide under the condition of a specified outage probability value \(\varepsilon\). In this paper, we will mainly investigate the ergodic capacity. We assume that the adaptive transmission scheme is adopted, _i.e._, each sub-channel dynamically transmits signal at the optimal data rate and power, based on the characteristics of the channel at that point in time. Various data bits are allocated to each sub-channel during each transmission period, and power allocation obeys the water-filling theorem. The principle of this algorithm is that the HAPs adaptively allocate transmitted power according to known or perfect CSI based on a predetermined guideline. Generally, more power is allocated to better sub-channels and less power is allocated to poorer sub-channels. Therefore, the ergodic capacity is the expectation of the total capacity of all sub-channels, which is given by:  
\[C_{erg}=\mathrm{E}\left(\sum_{i=1}^{r}\log_{2}(1+\frac{P_{i}}{N_{0}}\uplambda _{i}^{2})\right) \tag{29}\]where \(r\leq\min(N_{H},N_{r})\) is the rank of the transmitting channel, _i.e._, the conversion of the channel matrix of the multiple HAP V-MIMO network to \(r\) mutually independent parallel sub-channels. \(P_{i}\) denote the allocated power to the \(i\)th sub-channel and \(P_{s}=\sum\limits_{i=1}^{r}P_{i}\). The optimal power can be given by \(P_{i}^{opt}=\max\left\{\left(\mu-N_{0}/\lambda_{i}^{2}\right),0\right\}\) according to the water-filling algorithm, where \(\lambda_{i}\) is the gain of the \(i\)th sub-channel.  
However, the CSI may be unknown to HAPs in general, so the average energy allocation scheme will then be used, _i.e._, each sub-channel has equal power allocation \(P_{i}=P_{s}/N_{H}\)[31]. By using the Shannon channel capacity formula, the capacity in our constellation of HAPs network can be written as:  
\[C_{erg}=\mathrm{E}\left(W\sum\limits_{i=1}^{r}\log_{2}(1+\frac{\lambda_{i}P_{ s}}{N_{H}\sigma^{2}})\right)=\mathrm{E}\left(W\mathrm{log}_{2}\prod\limits_{i= 1}^{r}\left(1+\frac{\lambda_{i}P_{s}}{N_{H}\sigma^{2}}\right)\right)=\mathrm{ E}\left(W\mathrm{log}_{2}\left|I_{N_{rH}}+\frac{P_{s}}{N_{H}\sigma^{2}}\mathbf{Q} \right|\right) \tag{30}\]  
where \(W\) denotes the bandwidth of the sub-channel, and \(\mathbf{Q}\) is the Wishart matrix, defined as:  
\[\mathbf{Q}=\left\{\begin{array}{l}\mathbf{H}\mathbf{H}^{H},N_{H}>N_{r}\\ \mathbf{H}^{H}\mathbf{H},N_{H}\leq N_{r}\end{array}\right. \tag{31}\]