show that the distance profile is a repetitive structure with period $ 2\pi /M $. Next, it is shown that within the wedge $ [0,2\pi /M] $ the distance profile is symmetric about the bisector of this wedge *i.e.*, the $ \theta = \pi /M $ line. We have from , $$\begin{aligned}
\nonumber
\mathcal{S}_{\text{eff}}= \mathcal{S}+ \gamma e^{j \theta } \mathcal{S} .\end{aligned}$$ For any arbitrary value of $ \theta = \frac{k 2 \pi }{M} +\theta ' $ where $k \in \mathbb{Z},\, 0 \leq \theta ' < \frac{2 \pi}{M}$, $$\begin{aligned}
\nonumber
\mathcal{S}_{\text{eff}}&= \mathcal{S}+ \gamma e^{j \theta ' } (\mathcal{S}e^{j \frac{2 k \pi}{M}})\\
\nonumber
&= \mathcal{S}+ \gamma e^{j \theta ' } \mathcal{S}.\end{aligned}$$ The last equality follows from the fact that rotating a $M$-PSK constellation by an integral multiple of $2\pi /M$ does not alter the distance profile of the constellation. Thus, whatever distance profiles for $ \mathcal{S}_{\text{eff}} $ are obtained for the wedge $ [0,2\pi /M] $, it is exactly repeated for the remaining $M-1$ wedges to cover the entire range of $ \theta $.

To show that the distance profiles are symmetric about $\theta= \pi /M$, we need to show that $\mathcal{S}+\gamma e^ {j (\frac{\pi}{M} + \alpha)} \mathcal{S}$ and $\mathcal{S}+\gamma e^{j (\frac{\pi}{M} 