# Channel Modeling and Analysis for the Sensor Network Inside Tower Buildings
## 3 Channel Clustering and Performance Analysis
### Equivalent Performance Evaluation  
BER is an important index to evaluate the performance of wireless communication system. Assuming that the transmitting signal is \(s(t)~{}=~{}\text{Re}[\widetilde{s}(t)\text{exp}(j2\pi f_{c}t)]\), the propagation distance of \(n\)th path is \(x_{n}\), the received signal can be represented as  
\[\begin{array}{l}y(t)=\sum\limits_{n=0}^{N}\text{Re}\big{\{}\widetilde{s}(t- \frac{x_{n}}{c})\text{exp}\big{[}j2\pi f_{c}\big{(}t-\frac{x_{n}}{c}\big{)} \big{]}\big{\}}+\widetilde{n}(t)\\ =\text{Re}[\widetilde{y}(t)\text{exp}(j2\pi f_{c}t)]+\widetilde{n}(t)\end{array} \tag{8}\]  
Figure 2: Clustering results of different algorithms: (**a**) Original rays of RT simulation data; (**b**) Traditional k-means algorithm and (**c**) Improved k-means algorithm.  
where \(c\) represents the speed of light and \(\widetilde{n}(t)\) represents the additive environmental noise. The complex envelope of the received signal can be expressed as the superposition of paths with different cluster attenuation and cluster delay, which is expressed as  
\[\widetilde{y}(t) =\sum_{n=0}^{N}\bar{a}_{n}\text{exp}\left(-j2\pi f_{c}\frac{x_{n} }{c}\right)\widetilde{s}\left(t-\frac{x_{n}}{c}\right)=\sum_{n=0}^{N}\bar{a}_{ n}\text{exp}(-j2\pi f_{c}\tau_{n})\widetilde{s}(t-\tau_{n}). \tag{9}\]  
It is assumed that the instantaneous signal to noise ratio (SNR) of received signals is \(r\) and quadrature phase shift keying (QPSK) modulation is adopted in the wireless communication system, the BER of the system \(P_{e}\) at this time can be expressed as  
\[P_{e}(r)=P(0/1)P(1)+P(1/0)P(0). \tag{10}\]  
Assuming the channel noise variance \(\sigma_{n}^{2}=1\), and there are \(k\) terms of inter-symbol interference, which are denoted by \(z_{k}\), the BER of the system can be expressed as  
\[P_{e}=p(y<0)=E_{z}[Q(s_{0}+z_{k})] \tag{11}\]  
where \(E_{z}[\cdot]\) is the expected value of the random variable \(z\). Since it is difficult to implement the above equation directly, we use the trigonometric approximation to solve the performance of QPSK system with inter-symbol interference. Note that with the increase of the modulation code scheme M, the BER will increase, but the amount of information carried by a single symbol will also increase, and the infor-mation rate will be higher. The trigonometric series approximation of Q function can be expressed as  
\[\hat{Q}(x)=\sum_{i=0}^{N_{T}-1}c_{i}\text{e}^{\lambda_{i}x}\cos(\omega_{i}x)= \text{Re}[\sum_{i=0}^{N_{T}-1}c_{i}\text{e}^{\lambda_{i}x+j\omega_{i}x}] \tag{12}\]  
where \(c_{i}\), \(\lambda_{i}\), and \(\omega_{i}\) are undetermined coefficients, \(N_{T}\) denotes order and when \(N_{T}\geq 7\), it meets the requirement of general calculation precision. When \(\sigma_{n}^{2}=1\), the effective SNR can be simplified to \(s_{0}^{2}\). In combination with (11) and (12), it can be deduced  
\[P_{e}\approx\text{Re}\left[\sum_{i=0}^{N_{T}-1}c_{i}\text{e}^{(\lambda_{i}+j \omega_{i})s_{0}}\prod_{k=-\infty,k\
eq 0}\cosh((\lambda_{i}+j\omega_{i})s_{ k})\right]. \tag{13}\]  
The channel capacity is another important evaluation index for the communication system, which represents the maximum amount of information that can be transmitted in the channel. According to [PERSON]'s theorem, when the transmission rate of the transmitted signal is lower than the capacity of the channel, there is a possible way to transmit the signal through the channel without error. Due to the influence of path loss and serious multipath fading inside tower building, the channel capacity would be greatly affected. The definition of channel capacity can be expressed as  
\[C=\frac{1}{2}\text{log}_{2}(1+r) \tag{14}\]  
where the system bandwidth is normalized, and the channel capacity unit is \(\text{bit}/(\text{s}\cdot\text{Hz})\). The instantaneous SNR distribution of the received signal \(r\) can be expressed as  
\[r=10\,\text{lg}\left(\frac{\sum\limits_{n=0}^{N}\bar{P}_{n}}{\sigma_{n}^{2}} \right). \tag{15}\]