# Performance Analysis of Downlink 5G Networks in Realistic Environments
## 4 Capacity Bounds Analysis
### Multiservice Capacity Bounds
The SNR value is service-dependent on the modulation technique, BER, and the forward error correction scheme considered. Therefore, the SNR can be considered as the ratio of received power after dispreading and the interference caused by other users' signals and can be expressed as follows:  
\[\frac{C_{i}}{I_{i}} \tag{21}\]  
From (19)-(21), the following relation is considered:  
\[\frac{C_{i}}{I_{i}}=\frac{SF_{i}\cdot P_{Ri}}{N_{t}+\chi(\mathbf{n})+\sum_{j=1,\ j \
eq i}^{n_{1}}V_{j}\cdot P_{Rj}\cdot\varepsilon_{ij}}\geq\left(\frac{E_{b}}{ N_{0}}\right)_{i} \tag{22}\]  
If there are \(n_{1}\) active users in a given cell, (22) can be written as follows:  
\[\left(\frac{E_{b}}{N_{0}}\right)_{i}=\frac{SF_{i}\cdot P_{Ri}}{N_{t}+\chi(\bm {n})+\sum_{j=1,\ j\
eq i}^{n_{1}}V_{j}\cdot P_{Rj}\cdot\varepsilon_{ij}}\geq \left(\frac{E_{b}}{N_{0}}\right)_{i} \tag{23}\]  
For \(i\) and \(j\), they are from \(1,\ldots,n_{1}\)  
The service factor S can be defined as follows:  
\[S_{i}=\frac{SF_{i}}{\left(\frac{E_{b}}{N_{0}}\right)_{i}} \tag{24}\]  
we also introduce the ratio of the interference from other cells (i.e., inter-cell interference) to the interference of the visiting cell (intra-cell interference) as follows [31; 32; 33]:  
\[IR(n)=\frac{\text{inter-cell interference}}{\text{intra-cell interference}}=\frac{\chi(\mathbf{n})}{\varphi(n_{1})} \tag{25}\]where \(\varphi(n_{1})=\sum_{k=1}^{n_{k}-1}V_{k}\cdot\varepsilon_{ij}\cdot P_{Rj}\) is for single-service case and \(\mathbf{n}=(n_{2},\ldots,n_{m})\) is the number of connections among all neighboring cells, and \(m\) is the number of neighboring cells. The inter-cell interference factor \(F\) is defined [34; 35] as  
\[F=\frac{1}{IR+1} \tag{26}\]  
Equation (25) can be rewritten as follows:  
\[\chi(\mathbf{n})=IR(n)\cdot\varphi(n_{1}) \tag{27}\]  
By substituting \(\chi(\mathbf{n})\) of (27) and \(\varphi(n_{1})\) of (25) into (23), we obtain the following:  
\[\left(\frac{E_{b}}{N_{0}}\right)_{i}=\frac{SF_{i}\cdot P_{Ri}}{N_{t}+IR(n) \cdot\varphi(n_{1})+\varphi(n_{1})} \tag{28}\]  
To simplify the derivation, we consider \(n_{1}=n\), and (28) can be rewritten as follows:  
\[C_{i}=SF_{i}\cdot P_{Ri} \tag{20}\]0  
By substituting the value of \(\varphi(n)\) from (25) into (29), we obtain the following:  
\[C_{i}=SF_{i}\cdot P_{Ri} \tag{20}\]1  
Using matrix notation, the power conditions for all user signals arriving at Node B can be expressed as follows:  
\[C_{i}=SF_{i}\cdot P_{Ri} \tag{20}\]2  
where  
\(\mathbf{N}=(N_{t}\ldots N_{t})^{T}\) is a noise vector of dimension \(n\);  
\(\mathbf{S}=\text{diag}(S_{1}\ldots S_{n})\) is a diagonal matrix of dimension \((n\times n)\);  
\(\mathbf{P_{R}}=(P_{R1}\ldots P_{Rn})^{T}\) is the power vector;  
\(\mathbf{V}=\text{diag}(V_{1}\ldots V_{n})\) is the service activity factor;  
\(\mathbf{E}\) is the non-orthogonality factor in the following structure:  
\[\mathbf{E}= \begin{bmatrix}0&\varepsilon_{12}&\cdots&\varepsilon_{1n}\\ \varepsilon_{21}&0&\cdots&\varepsilon_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ \varepsilon_{n1}&\varepsilon_{n2}&\cdots&0\end{bmatrix}  \tag{32}\]  
Solving the above linear system of equations, user \(i\) can obtain the received power at Node B. By (25), the intra-cell interference for a user of class \(k\) can be redefined in a multiservice case as follows:  
\[\varphi_{k}(\mathbf{n})=\sum_{m=1}^{n_{k}-1}V_{k}\cdot\varepsilon_{k}\cdot P_{Rk} +\sum_{l\
eq k,\ \ l=1}^{s}\sum_{i=1}^{n_{l}}V_{l}\cdot\varepsilon_{lk}\cdot P_{Rl} \tag{33}\]  
Equation (33) can be simplified as follows:  
\[\varphi_{k}(\mathbf{n})=(n_{k}-1)V_{k}\cdot\varepsilon_{k}\cdot P_{Rk}+\sum_{l \
eq k,\ \ l=1}^{s}n_{l}\cdot V_{l}\cdot\varepsilon_{lk}\cdot P_{Rl} \tag{34}\]for \(k,l=1\ldots s\), where \(\mathbf{n}=(n_{1},\ n_{2},\ldots,n_{s})\); \(s\) represents several different service classes; \(\varepsilon_{k}\) is the interference factor between users in the same type of service \(k\) (i.e., intra-class interference factor).