# Interference Avoidance Using TDMA-Beamforming in Location Aware Small Cell Systems
## 2 System Modelling and Problem Formulation
### Evaluation Metrics  
In this section, the parameters used to assess the performance of the proposed algorithm are introduced. 1. Throughput [31]: this describes the competence of the network to deliver data. It also describes the effects of interference on the signals quality. Higher throughput means less interference, high SINR, and efficient resource allocation scheme. Throughput is obtained by the following: \[\mathcal{T}_{s}=\sum_{u\in\mathcal{U}}\mathcal{T}_{s}=\sum_{u\in\mathcal{U}}[ \mathcal{B}^{-}\times log_{2}(1+SINR_{sn})]\] (5) It is noteworthy to mention that Equation (5) is valid for the maximum obtainable throughput. However, the amount of successful packets delivery in real-time scenarios represents the network throughput. 2. [PERSON]'s Fairness model [32]: described by Equation (6); it is used to analyse how fairly the users can access the required throughput. Fairness index varies between 0 and 1 whereby the higher the index, the fairer the system. \[\mathcal{R}_{s}=\frac{\left[\sum\limits_{u\in\mathcal{U}}\mathcal{T}_{sn} \right]^{2}}{\left[\mathcal{U}\times\sum\limits_{u\in\mathcal{U}}\mathcal{T}_ {sn}^{2}\right]}\] (6)
3. SINR: this compares the ratio of the received signal to interference plus noise. **3. Proposed Algorithm**  
As highlighted in Section 2, the synchronisation of both users and base stations increases the complexity of the beamforming system. To avoid this, we introduce the parameter \(\beta\) at the ScMS, which is a matrix of \([s\times u]\) that describes which user is receiving from which base station, \(\beta_{sn}\) is described below:\[\beta_{su}= \begin{cases}1&\theta_{m}-\frac{w_{m}}{2}\leq\theta_{m}\geq\theta_{m}+ \frac{w_{m}}{2}\\ 0&else\end{cases}  \tag{7}\]  
whereby \(w_{m}\) is the beamwidth, and \(\theta_{m}\) is the angle at which the user is located. Algorithm 1 describes how \(\beta_{su}\) and \(\beta_{iu}\forall i\in\mathcal{S}/\{s\}\) are derived from \(\alpha_{su}\). When user \(u\) is associated to small cell \(s\), \(\alpha_{su}\) is said to equal 1. Subsequently, if the angle of user \(u\) is located within angles of the beam of small cell \(s\), \(\beta_{su}\) is set to 1. Similarly, \(\beta_{iu}\) is set to 1 if user \(u\) is located within angles of the beam of small cell \(i\), \(\forall i\in\mathcal{S}/\{s\}\)  
```
1:functionScMS
2:for\(s=1:\mathcal{S}\)do
3:for\(u=1:\mathcal{U}\)do
4: Set \(\beta_{su}=0\)
5:if\(\alpha_{su}=1\)then
6:if\(\theta_{sn}-\frac{w_{m}}{2}\leq\theta_{sn}\geq\theta_{sn}+\frac{w_{m}}{2}\)then
7: Set \(\beta_{su}=1\)
8:endif
9:for\(i=1:\mathcal{S}\)do
10:if\(i\
eq s\)then
11:if\(d_{si}>2\times radius\)then
12: Set \(\beta_{iu}=0\)
13:if\(\theta_{iu}-\frac{w_{m}}{2}\leq\theta_{sn}\geq\theta_{iu}+\frac{w_{m}}{2}\)then
14: Set \(\beta_{iu}=1\)
15:endif
16:endif
17:endif
18:endfor
19:endif
20:endif
21:endfor
22:endfor
23:endfunction
```  
**Algorithm 1** Initialise \(\beta_{su}\)  
Additionally, we introduce the parameter \(\tau\), which is also a matrix of \([s\times u]\). It describes the transmission time slot for user \(u\), and obtained by the following:  
\[\tau_{su}=(u-1)\times[\mathcal{U}\times\mathbb{T}\mathbb{T}]+1 \tag{8}\]  
whereby \(\mathbb{T}\mathbb{T}\mathbb{T}\) is the transmission time interval for a user in LTE. In view of that, the ScMS forces \(s\) to transmit at the angle \(\theta_{su}\) where the user is located. The maximum signal response will result at \(\theta_{su}\) if \(\gamma=-\frac{2\pi}{\lambda}l\sin\theta_{su}\), which is used in obtaining the summation of all antenna elements radiations i.e., array factor [33] as given by Equation (9). \[\mathcal{F}(\theta_{sn})=\sum_{c\in\mathcal{C}}\mathcal{X}_{c}e^{j(\frac{2\pi }{\lambda}d\sin\theta_{su}+c\gamma)} \tag{9}\]  
whereby \(\mathcal{X}_{c}\) is the complex weight of the \(c\)th antenna element, \(\lambda\) is the wavelength, \(l\) is the inter-element spacing, and \(\gamma\) is the phase lead of the \(c\)th element.