I am trying to understand how the CDF of the equation (A) is derived and represented as eq.(B), where eq.(A), eq.(B) are given as $$\gamma_{eq}=\frac{\gamma_1\gamma_2}{\gamma_{inf}(\gamma_2+1)+\gamma_1}\tag{A},$$ where $\gamma_1=P_s|h_{sr}|^2$, $\gamma_2=\frac{P_r}{\sigma^2_d}|h_{rd}|^2$ and $\gamma_{inf}=\sum_{j=1}^{N} P_j|h_j|^2$. Also $h_j,h_{sr}$ and $h_{rd}$ are rayleigh random variables.

The CDF is $$F_{\gamma_{eq}}(\gamma_{th}) = \int_0^{\infty}Pr\left(\gamma_1\leq\frac{\gamma_{th}(y+1)z}{y-\gamma_{th}}\right)\times f_{\gamma_2}(y)f_{inf}(z) dydz\tag{B}.$$ I had reached to eq.(B) to some extent but not getting completely. Any help in this regard is highly appreciated.

  • $\begingroup$ Can somebody tell why PDF of $\gamma_2$ and $\gamma_{inf}$ are multiplied. $\endgroup$ – Pranu Feb 15 at 1:00

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