# How does the M-law Synchronizer work?

Relatively new to DSP, I've came across the approach for phase-estimation of PSK-Signals x(k):

My question is:

• What do we get by exponentiating the Baseband-Signal?
• Usually, the books that introduce the M-law synchronizers also have an explanation right next to it! The answer is simple: write your M-PSK symbols in amplitude/phase form (i.e. $x = |x| \cdot e^{j\angle x}$) and exponentiate by M, and don't forget about the $2\pi$-periodicity of $e^{jt}$. Question answered! – Marcus Müller Jun 7 '18 at 14:47
• Ah, OK so the symbols |x| are becoming "1" by exponating with M? – John75 Jun 7 '18 at 14:59
• :) you've got it! And: what will you do if you get something that is not exactly +1 or -1? – Marcus Müller Jun 7 '18 at 15:08
• (by the way, I'm counting on you writing your own answer to your own question if you solve this :) I could write an answer, but I'll be honest: It'd be the same perspective that everyone has who's read the same textbook as me, and thus a bit redundant to similar answers. Your perspective and wording would be valueable!) – Marcus Müller Jun 7 '18 at 15:10
• when the symbols are not exactly +1 and -1, then we have to accumulate over a long stream? dividing by M is normalizing the phase... – John75 Jun 7 '18 at 17:26