(1,0)$. Once the regions $ \mathcal{R}(\mathcal{C}_{k_i}) $, $ 1\leq i \leq N_W $ are obtained, the region exterior to all these regions, lying within the wedge $ [0, \pi /M] $, is the region where $ d_{min}(\mathcal{S})$ is the minimum distance in $ \mathcal{S}_{\text{eff}} $, *i.e.*, the region $ \mathcal{R}_W(\mathcal{C}_{d_{min}(\mathcal{S})}) $.

4.  The quantization obtained in Step 3, for the wedge $[0,\pi/M]$, can now be extended by the procedure suggested in Section \[sec:quandist\] to cover the entire $ (\Gamma,\Theta) $ plane.

We will illustrate the procedure with two examples.

\[exa:qpskquan\] *Channel Quantization for QPSK signal sets.*\

![Quantization of the wedge $[0,\pi/4]$ for QPSK signal sets. []{data-label="fig:qpsksector"}](qpsksector.eps){width="2.25in"}

Here we consider the scenario where both users use QPSK constellations at input, *i.e.* $M=4$. From Lemma \[periodic\], there are two singular fade states in the wedge $ [0,\pi/4]$, *i.e.* $N_W=2$ and these are at $(1,0)$ and $(\sqrt{2},\frac{\pi}{4})$. The class distance functions in $d_{\bar{\mathcal{C}}}(\gamma, \theta)$ which reduce to zero at these singular fade states are identified. For the singular fade state $(\sqrt{2},\frac{\pi}{4})$ the distance $d_{\mathcal{C}_{k_1}}$ falls to zero, and for $(1,0)$ the distances $d_{\mathcal{C}_{k_2}}$ and $d_{\mathcal{C}_{k_3}}$ both fall to zero, as shown in Fig \[fig:qpskeff\]. These are as follows: $$\begin{aligned}
\nonumber
d_{\mathcal{C}_{k_1}}^2(\gamma, \theta) &= 