## 3 Methodology: Statistics of the Combined Received SNR  
In this section, we statistically analyze the combined received SNR given in (1). Specifically, our objective is to obtain meaningful analytical results regarding the sum of the \(K\) largest SNRs among \(M\) SNRs the PDF of which is given in (4). If we let \(Z_{1}=\sum_{k=1}^{K-1}S_{k:M}\) and \(Z_{2}=S_{K:M}\), the target statistical result of \(Z^{\prime}=Z_{1}+Z_{2}\) can be obtained from the two-dimensional joint PDF of \(Z_{1}\) and \(Z_{2}\), \(f_{Z}(z_{1},z_{2})\), as  
\[f_{Z^{\prime}}(x)=\int_{0}^{\frac{x}{K}}f_{Z}(x-z_{2},z_{2})dz_{2}, \tag{5}\]  
for \(K\geq 2\). Note that, in general, the conventional MGF-based approach is commonly used for the sum of RVs to be applied to derive the target statistical results for the ordered-RV cases. Hence, the joint PDF, \(f_{Z}(z_{1},z_{2})\), in (5) can be expressed by the inverse Laplace transform (LT) of the joint MGF as  
\[f_{Z}(z_{1},z_{2})=\mathcal{L}_{s_{1},s_{2}}^{-1}\{\text{MGF}_{Z}(-s_{1},-s_{ 2})\}. \tag{6}\]  
However, deriving this two-dimensional statistical result of \(Z_{1}\) and \(Z_{2}\) with the conventional MGF-based approach requires dealing with the computationally intensive and complex original statistical result, the MGF expression of which involves a multiple integral (i.e., \(K\)-fold integrals in this case), as shown in the following equation:  
\[\begin{split}\text{MGF}_{Z}(\lambda_{1},\lambda_{2})& =E\{\exp(\lambda_{1}z_{1}+\lambda_{2}z_{2})\}\\ &=F\int_{0}^{\infty}dS_{1:K}f_{S_{1:K}}(S_{1:K})\exp(\lambda_{1}S _{1:K})\times\\ &\qquad\qquad\qquad\qquad\qquad\qquad\vdots\\ &\times\!\!\!\int_{0}^{S_{K:-2}\!\!\!\!\!\!\!\!\!\!\!\!\!\!where \(F=\frac{M!}{(M-K)!}\), and \(c(S_{K:M})\) is the cumulative distribution function of \(S_{K:M}\) given by  
\[c(S_{K:M})=\int_{0}^{S_{K:M}}dxf_{S_{K:M}}(x). \tag{8}\]  
In this study, by adopting the analytical framework proposed in [35], we obtain a target two-dimensional joint PDF of the arbitrary partial sums of ordered RVs, for example, involving the first \(K\) ordered RVs among \(M\) ones, as a closed-form result. The first step is to derive the analytical expressions of the two-dimensional joint MGF of the partial sums. Then, in step two, we apply the inverse LT to obtain the target two-dimensional joint PDF. Through step one, following the analytical framework proposed in [35], we can obtain a two-dimensional joint MGF of \(Z=[Z_{1},Z_{2}]\) in a single integral expression given by  
\[ \begin{split}\text{MGF}_{Z}(\lambda_{1},\lambda_{2})=& \frac{F}{(K-1)!}\int_{0}^{\infty}ds_{K:M}f_{S_{K:M}}(S_{K:M})\\ &\times\exp(\lambda_{2}S_{K:M})[c(S_{K:M})]^{M-K}[e(S_{K:M}, \lambda_{1})]^{K-1},\end{split}  \tag{9}\]  
where \(e(S_{K:M},\lambda_{1})\) is the mixture of an exceedance distribution function and an MGF defined in [35] and can be obtained as  
\[e(S_{K:M},\lambda_{1})=\int_{S_{K:M}}^{\infty}dxf_{S_{K:M}}(x)\exp(\lambda_{1} x). \tag{10}\]  
Substituting (8) and (10) into (9) and then applying the inverse LT, the target two-dimensional joint PDF can be expressed as  
\[ \begin{split} f_{Z}(z_{1},z_{2})&=\mathcal{L}_{s_{ 1}s_{2}}^{-1}\{\text{MGF}_{Z}(-s_{1},-s_{2})\}\\ &=\frac{F}{(K-1)!}f_{Z_{2}}(z_{2})[c(z_{2})]^{M-K}\mathcal{L}_{s _{1}}^{-1}\Big{\{}[e(z_{2},-s_{1})]^{K-1}\Big{\}},\end{split}  \tag{11}\]  
where \(f_{Z_{2}}(z_{2})\) is expressed by (3). Gamma-distributed RVs, based on (4), (8) and (10) can be expressed by  
\[ \begin{split} c(S_{K:M})&=\int_{0}^{S_{K:M}}dxf_{S_ {K:M}}(x)\\ &=\frac{p_{k}}{\Gamma(m_{k})}\Big{[}\Gamma(m_{k})-\Gamma\bigg{(}m _{k},\frac{S_{K:M}}{\eta_{k}}\bigg{)}\Big{]}\end{split}  \tag{12}\]  
and  
\[