**1.-** From :

[**An Introduction to Signals and Noise in Electrical Communications**][1]

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**][2] 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 **A**dditive **W**hite **G**ausian **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][3] 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**][4] 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** `Eb` 
   
 - over **noise power spectral density**, aka n**oise 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**][5].

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**][6]

Particularly [**this table**][7] 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.


  [1]: https://www.amazon.co.uk/Communication-Systems-Solutions-Introduction-Electrical/dp/0070099588/ref=sr_1_6?crid=3O6YLVNUXE38F&keywords=communication%20systems%20bruce%20carlson&qid=1673637400&s=books&sprefix=communcation%20systems%20bruce%20carlson%2Cstripbooks%2C62&sr=1-6
  [2]: https://www.stat.cmu.edu/~cshalizi/36-220/lecture-6.pdf
  [3]: https://webdemo.inue.uni-stuttgart.de/webdemos/02_lectures/uebertragungstechnik_1/qam_constellation_diagram_from_snr/index.php?id=2#
  [4]: https://uk.mathworks.com/help/signal/ref/snr.html
  [5]: http://www.sss-mag.com/ebn0.html
  [6]: https://www.hindawi.com/journals/jcnc/2018/9319204/
  [7]: https://www.hindawi.com/journals/jcnc/2018/9319204/tab1/