the distance depends on $ (\gamma, \theta) $.

For a given input constellation $ \mathcal{S} $, let $ \bar{\mathcal{C}} $ denote the set of the all distance classes for it.

\[def:classfunc\] Associated with every distance class $ \mathcal{C} $ is a function $ d_{\mathcal{C}}(\gamma ,\theta): (\Gamma,\Theta )\leftrightarrow \mathbb{R} $, called the class distance function, which gives the value of the distance between the two elements of a pair in $ \mathcal{C} $ for any $ (\gamma, \theta) $.

\[def:fadefunc\] For a given fade state $ (\gamma, \theta) $, the function $ d_{\gamma , \theta} (\mathcal{C}): \bar{\mathcal{C}} \rightarrow \mathbb{R} $ gives the value of the distance between the two elements of a pair in $ \mathcal{C} $, for any $ \mathcal{C} \in \bar{\mathcal{C}}$. This is called the fade state distance function.

We use integer $ m $, $ 1\leq m \leq M $ to represent the point $ e^{j\frac{(m-1)2 \pi}{M}} $ in $ \mathcal{S} $ *i.e.* the $ M $-PSK signal set. The integer $ q=m+M(n-1) $, $ 1\leq q \leq M^2 $ denotes the complex point in $ \mathcal{S}_{\text{eff}} $ obtained by combining the points $ m $ and $ n $ of $ \mathcal{S} $ *i.e.* it refers to 