How to reduce PAPR with constellations choice?

Question

How to tune constellations mapping M ary-psk or x-qam to reduce PAPR ?

Answer 1

Peak to Average Power Ratio (PAPR) is defined as:

$$ \text{PAPR} = \frac{\text{Peak Power}}{\text{Average Power}} $$

If you're working with a $M$-PSK constellation, then the peak power is $P_s$ and the average power is also $P_s$ assuming each symbol is equally likely. For $M$-PSK, the $\text{PAPR}=1$.

If you working a $M$-QAM constellation, then the peak power is $\text{max}_{i \in \mathbb{C}} |s_i|^2$, where $\mathbb{C}$ is the constellation. Again assuming all the symbols are equally likely, the average power is $\frac{1}{M}\sum_{i \in \mathbb{C}} |s_i|^2$. The specifics should be worked out for a constellation but is hard to write down because in general there are many possible QAM constellations.

One thing that is directly clear is that $\text{PAPR}_{\text{QAM}}>\text{PAPR}_{\text{PSK}}$. In a QAM constellation, you have some symbols with less power, some symbols with more power, and there is an average somewhere in between. The point is that there are some symbols with more power, and this makes the peak power part of PAPR greater than the average power part. This is in contrast to a PSK constellation, where the symbols lie on a circle and no symbol has more or less power than each other.