distance of the input constellation, it is sufficient to study the minimum distance of $ \mathcal{S}_{\text{eff}} $. Also from Definition \[def:sing\] minimum distance in $ \mathcal{S}_{eff} $ reduces to zero at the singular fade states. The following lemma provides an upper bound on the minimum distance of the effective constellation $ \mathcal{S}_{\text{eff}} $.

\[lem:uppermin\] When both the users use any arbitrary signal set $\mathcal{S}$ (which includes $M$-PSK, $M$-QAM) at the input, then for any fade state $ (\gamma, \theta) $, the minimum distance $ d_{min}(\gamma , \theta) $ between any two points in $ \mathcal{S}_{\text{eff}} $ is upper bounded by the minimum distance in the input constellation $ d_{min} (\mathcal{S}) $.

From the definition of $d_{min}^2(\gamma ,\theta)$, we have, [$$\begin{aligned}
\nonumber
d_{min}^2(\gamma ,\theta) &=\hspace{-0.5cm}\min_{(s_1,s_2) \neq (s'_1,s'_2) \in \mathcal{S}^2} \vert (s_1-s'_1)+\gamma e^{j \theta} (s_2-s'_2)\vert ^2\\
\nonumber
&\leq \hspace{-0.5cm}\min_{(s_1,s_2) \neq (s'_1,s'_2) \in \mathcal{S}^2} \left\lbrace \vert s_1-s'_1\vert ^2 +\gamma ^2 \vert s_2-s'_2\vert ^2 \right\rbrace \end{aligned}$$ $$\begin{aligned}
\label{Lemma1_1}
\text{Now}\hspace{5pt} d_{min}^2(\gamma ,\theta) &\leq \min_{s_1 \neq s'_1 \in \mathcal{S}}\vert s_1-s'_1\vert ^2 =d_{min}^2(\mathcal{S}).\\
\label{Lemma1_2}
\text{Also} \hspace{5pt} d_{min}^2(\gamma ,\theta) &\leq\min_{s_2 \neq s'_2 \in \mathcal{S}}\gamma ^2\vert s_2-s'_2\vert ^2=\gamma ^2 d_{min}^2 (\mathcal{S}). 
%&\leq \min_{s_2 \neq s'_2 \in \mathcal{S}}\vert (s_2-s'_2)\vert ^2 \hspace{5pt} (\because \gamma \geq 1) \tag{B}\end{aligned}$$ $$\begin{aligned}
\nonumber
\text{From \eqref{Lemma1_1} 