d_{\mathcal{C}}(\gamma, \theta) \vert \mathcal{C} \in \bar{\mathcal{C}} \rbrace \\
\nonumber
d_{\Gamma, \Theta}(\mathcal{C})&= \lbrace d_{\gamma , \theta}(\mathcal{C})\vert (\gamma, \theta) \in (\Gamma , \Theta)\, \text{plane} \rbrace .\end{aligned}$$

From Definition \[def:sing\], at a singular fade state the value of at least one of the class distance functions in $ d_{\bar{\mathcal{C}}}(\gamma, \theta) $ will reduce to zero.

\[lemma:ordering\] Among the set of all class distance functions that reduce to zero at the singular fade state $ (\gamma',\theta') $, there is a particular one which is the minimum among that set, for all values of $ (\gamma , \theta) \neq (\gamma',\theta' ) $.

Let $ L $ be the number of class distance functions that reduce to zero at the singular fade state $ (\gamma',\theta') $. Denote these by $ d_{\mathcal{C}_i}(\gamma, \theta) $, $ 1\leq i \leq L $ and let $ \lbrace (s_{1,i}, s_{2,i}),(s'_{1,i}, s'_{2,i})\rbrace $ be the representative element for the distance class $ \mathcal{C}_i $. From and , we have $$\begin{aligned}
\nonumber
d_{\mathcal{C}_i}^2(\gamma,\theta)&= d^2_{(s_{1,i},s_{2,i})\leftrightarrow (s'_{1,i},s'_{2,i})} \\
\nonumber
&= \vert (s_{1,i}-s'_{1,i}) + \gamma e^{j \theta} (s_{2,i}-s'_{2,i})\vert ^2\\
\nonumber
&= \vert s_{2,i}-s'_{2,i}\vert ^2 \vert \gamma e^{j \theta} + \frac{s_{1,i}-s'_{1,i}}{s_{2,i}-s'_{2,i}}\vert ^2 \\
\label{eqn:ordr1}
&= \vert s_{2,i}-s'_{2,i}\vert ^2  \vert \gamma e^{j \theta } - \gamma' e ^{j 