## 2 Channel Capacity in the Presence of \(\eta\)-\(\mu\) Fading and CCI  
### Derivation of the PDF of the Receiver's Output Signal-to-Co-Channel Interference Ratio  
In this section, we derive the probability density function (PDF) of the output signal-to-co-channel interference-ratio (SIR) for a multi-branch SC receiver within a wireless system disturbed by \(\eta\)-\(\mu\) fading and \(\eta\)-\(\mu\) distributed CCI. The use of appropriate fading and CCI models is valuable for addressing challenges related to channel capacity maximization within wireless systems. From the very name of the distribution, we can deduce that the \(\eta\)-\(\mu\) distribution is characterized by two parameters. The parameter \(\eta\) represents the ratio of the powers of the in-phase and in-quadrature scattered waves in each multipath cluster, while \(\mu\) denotes the number of clusters [2]. This distribution is derived from field measurements and is thus fully characterized in terms of measurable physical parameters [3]. The \(\eta\)-\(\mu\) fading may appear in two different formats corresponding to two physical models. However, mathematically, one format can be obtained from the other using the following relation:  
\[\eta_{\text{Format2}}=\frac{1-\eta_{\text{Format1}}}{1+\eta_{\text{Format1}}} \text{ or }\eta_{\text{Format1}}=\frac{1-\eta_{\text{Format2}}}{1+\eta_{\text{Format2}}}\enspace,\]  
where \(0<\eta_{\text{Format1}}<\infty\) is the \(\eta\) parameter in Format 1, and \(-1<\eta_{\text{Format2}}<1\) is the \(\eta\) parameter in Format 2 [3]. To simplify the notation, \(\eta\) is used in both cases, keeping in mind that they represent different physical phenomena, and that their domains are different. Further simplification is achieved by defining two new parameters, denoted \(h\) and \(H\) and both functions of \(\eta\). These parameters hold distinct meanings and values for the two different formats. The convenience of using \(h\) and \(H\) is that they have a unique representation for both formats [3]. In Format 1, we have \(h=(2+\eta^{-1}+\eta)/4\) and \(H=(\eta^{-1}-\eta)\). When \(0<\eta\leq 1\), then \(H\geq 0\), while for \(0<\eta^{-1}\leq 1\), we have \(H\leq 0\). It is important to note that due to the property \(\text{I}_{\
u}(-z)=(-1)^{\
u}\)\(\text{I}_{\
u}(z)\), the distribution yields identical values within these two intervals, i.e., it is symmetrical around \(\eta=1\). Therefore, it is sufficient to consider \(\eta\) within one of these ranges. Also, in Format 1, we observe that \(H/h=(1-\eta)/\left(1+\eta\right)\)[3]. The Hoyt (Nakagami-\(q\)), one-sided Gaussian, Nakagami-\(m\), and Rayleigh distributions are all special cases of the \(\eta\)-\(\mu\) distribution. The Hoyt (or Nakagami-\(q\)) distribution is obtained by setting \(\mu\) = 0.5, the one-sided Gaussian distribution emerges as \(\eta\) approaches 0 or \(\infty\), whereas the Rayleigh distribution corresponds exactly to \(\mu\) = 0.5 and \(\eta\) = 1. If \(L\) denotes the number of input branches in the multi-branch SC receiver, under the \(\eta\)-\(\mu\) distribution, the signal envelopes at the receiver inputs \(x_{i}\), \(i\) = 1, 2, \(\ldots\), \(L\), are ([9]; (3)):  
\[p_{x_{i}}(x_{i})=\frac{4\sqrt{\pi}\,h_{1}{}^{\mu_{1}}}{\Gamma(\mu_{1})\mathrm{e }^{\frac{2\mu_{1}h_{1}}{\Omega_{i}}x_{i}^{2}}}\sum_{i_{1}=0}^{+\infty}\frac{H_ {1}{}^{2i_{1}}x_{i}^{4i_{1}+4\mu_{1}-1}(\mu_{1}/\Omega_{i})^{2i_{1}+2\mu_{1}}} {i_{1}\Gamma(i_{1}+\mu_{1}+1/2)}. \tag{1}\]  
\(\Omega_{i}\) represents the mean values of the input signals: \(\Omega_{i}\) =\(\overline{x}_{i}^{2}\) and \(\Gamma(\cdot)\) is the Gamma function ([17]; Sec. (8.31)).