and $(s'_1,s'_2)$ in $\mathcal{S}_{\text{eff}}$ reduces to zero, *i.e.* if $$\begin{aligned}
\label{eqn:sing}
 \gamma e^{j \theta} = -\frac{(s_1-s'_1)}{(s_2-s'_2)}\end{aligned}$$ then $ d^2_{(s_1,s_2) \leftrightarrow (s_1\prime, s_2\prime) }=0 $. These values of $ (\gamma, \theta) $ are called the *singular fade states* [@Akino], [@Vijay]. Singular fade states can also be defined as follows:

\[def:sing\] A fade state $ (\gamma, \theta) $ is said to be a *singular fade state* if $ \vert \mathcal{S}_{\text{eff}} \vert < M^2 $.

Clearly, $ \gamma e^{j \theta}=0 $ is a singular fade state, for any arbitrary signal set $ \mathcal{S} $. For any input constellation $\mathcal{S}$ the other non-zero singular fade states are obtained using . For a given input constellation $ \mathcal{S} $, let $ \mathcal{H} $ denote the set of all singular fade states.

When $\mathcal{S}_1=\mathcal{S}_2=\mathcal{S}$, where $\mathcal{S}$ is a QPSK constellation, then the non-zero singular fade states are at $$\begin{aligned}
\nonumber
\gamma &= \sqrt{2} , \hspace{5pt} \theta = 45^\circ , 135^\circ ,225^\circ , 315^\circ \\
\nonumber
\gamma &= 1, \hspace{13pt} \theta = 0^\circ , 90^\circ , 180^\circ , 270^\circ\\
\nonumber
\gamma &= \frac{1}{\sqrt{2}}, \,\theta = 45^\circ , 135^\circ ,225^\circ , 315^\circ\end{aligned}$$

Since for an AWGN channel the error performance at the destination is dominated by the minimum 