1.- From :
An Introduction to Signals and Noise in Electrical Communications
Author : Bruce Carlson
Search for much cheaper copies readily available online.
The sought figure-of-merit is SNR=S/N
S : signal power
N : noise power
S=mean(s^2)
N=mean(n^2)
but receivers always catch S+N , not just S .
2.- Refreshing Random Variables
From this Carnegie Mellon University lecture
in general : mean((x+y)^2)=(mean(x+y))^2+var(x+y)
2.1.- For signals mean(s^2)=(mean(s))^2+var(s)
If the link budget has been correctly designed and since signals and pulses to detect are supposed to be known on the receiver side
(mean(s))^2 >> var(s) and S ~ (mean(s))^2
2.2.- For noise it's the contrary.
It's in the nature of noise var(n) >> mean(n)
3.- AWG n is supposed to be uncorrelated to s
in general mean((x+n)^2)=mean(x^2)+2*mean(x*n)+mean(n^2)
So even if point 1.- takes place, meaning
mean(s^2)>>mean(n^2) and
mean(s^2)+mean(n^2) ~ mean(s^2)
if noise n is is correlated to signal s that the term 2*mean(x*n) cannot be neglected
However, for a satisfatory system design, with Additive White Gausian AWG Noise only, in absence of interference, or any other non-AWG noise correlated to signal, then
S>>N and S+N ~ S
3.- So
(mean(s))^2/(var(n)^2) % SNR linear
same as
20*log10(abs(var(s)))-20*log10(abs(var(n))) % SNR in [dB]
4.- I found this interesting website where SNR is calculated from QAM constellation parameters
QAM webdemo, Institute of Telecommunications, University of Stuttgart, Germany, Jan. 2023.
Author : Michael Bernhard
5.- This Mathworks page shows how to use MATLAB command snr for narrow band signals.
6.- For digital signals you should focus on Eb/N0 rather than SNR, SNR being important, but digital signals quality is about having
enough pulse or bit enery
Ebover noise power spectral density, aka noise power per Hz
N0
rather than signal Watts over Noise Watts (SNR).
Often you will read EbN0 instead of Eb/N0 both being the same, it's just 'simplified' notation.
Eb/N0 101 reading, by Jim Pearce available here.
However
All above said is for narrow band signals only.
For broadband signals more or less complicated expressions or far more simple then do obtain, direct measurements are used.
7.- An example: OFDM signals are considered broadband because several close carriers are often used the overall bandwidth being much larger than the band for each single carrier.
An available detailed explanation how to estimate SNR for OFDM systems is available in this thesis :
Pilot-Based Time Domain SNR Estimation for Broadcasting OFDM Systems
Authors : Abid Muhammad Khan, Varun Jeoti, Muhammad Zaka Ur Rehman, Muhammad Taha Jilani, Omer Chugtai, Mubashir Hussain Rehmani.
Journal of Computer Networks and Communications, vol. 2018, Article ID 9319204, 8 pages, 2018. https://doi.org/10.1155/2018/9319204
Complete text available in Hindawi
Particularly this table shows a couple ways SNR can be estimated for OFDM signals as defined in this text.
And I say it again : Reliable measurements with calibrated instruments are the best way to really know what SNR a signal really has.