#### hntko

##### Member level 1

I just wonder if anyone can show me how to get probability distribution (or standard deviation) of square of amplitude of M-PSK signals (QPSK for example).

I think with QPSK signal, there is two components, in-phase (X) and quadrature (Y)

X = S_I + n_I

Y = S_Q + n_Q

where n_I and n_Q present the in-phase and quadrature components of noise, being zero-mean i.i.d Gaussian random variables with variance of sigma^2.

Now, the square of amplitude can be taken as

A = X^2 + Y^2;

Assume that SNR = 30 dB, Signal Power = 0 dBW, by doing MATLAB simulation, I can get the standard deviation of A is about 0.0448 (see the code below).

My question is:

How to calculate the standard deviation theoretically? I mean how to know probability distribution function (and the standard deviation as well) of square of amplitude of the signals? Please help!!!!

%--------------------

clear;

clc;

M = 4;

Len = 10^6;

SNR = 30;

hMod = comm.PSKModulator(M,'BitInput',true);

data = logical(randi([0 1], Len, 1)); % Generate data

modData = step(hMod, data); % Modulate

tx = awgn(modData,SNR,0);

squareamptx = (abs(tx)).^2;

std(squareamptx)

%--------------------------

Thank you very much.

hntko