Noise equ.

[dafter]

For uncorrelated noise, to find the total noise we do the foll.:
(en1 ^2 + en2^2)^1/2
I was wondering why do we do this. I read that for independent signal we must add them and that we must add noise power for noise and take the mean square.
I still want to know the proof. Not convince at all.
I was also told that the above equation is just like adding 2 vectors (yeah but what is the point?).

How do we prove the equation for total correlated noise?
thx

 

[Skogsgurra]

"prove the equation for total correlated noise"

I hope that you meant to say "prove the equation for totally uncorrelated noise". It is only for uncrrelated noise (or signals) that the sqrt(sum of squares) is valid.

I have never seen any proof - but it works.

Gunnar Englund

http://www.gke.org

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100 % recycled posting: Electrons, ideas, finger-tips have been used over and over again...

 

[dafter]

No for correlated noise:
et = (e1 + e2 + 2*C*e1*e2)^1/2
where c is correlation factor

How to derive this?

 

[Skogsgurra]

I think that you are changing your mind mid-way. First you talk about "(en1 ^2 + en2^2)^1/2" and then "(e1 + e2 + 2*C*e1*e2)^1/2".

Gunnar Englund

http://www.gke.org

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100 % recycled posting: Electrons, ideas, finger-tips have been used over and over again...

 

[logbook]

I think you missed the squares of the e1 and e2 terms in your formula for correlated noise.

En= (e1*e1 + e2*e2 + 2*C*e1*e2)^1/2

If they are uncorrelated then C=0 and you go back to the original equation.

The point is that noise powers add. There is no proof it is an assertion that is bourne out by experiment and experience. From this you can say that the noise voltages sum as the root of sum of squares because power is V*V/R.

The second equation is effectively a definition of the correlation coefficient. If two sources are partially correlated their sum will be greater than the root sum of squares would suggest. Clearly the maximum value for total correlation would be sqrt( (E1+E2)*(E1+E2) ) which gives C=1.

 

[pstuckey]

Get a good intro to RADAR book and look under coherent integration. Stimson has a very easy to follow "proof" (actually more of a thought experiment). Barton addresses this in a more mathematical fashion.

The key really is that noise has an amplitude AND phase.

Peter

 

[VisiGoth]

The proof is also in books on statistics and random variables. The equation is correct in the limit.

Also, logbook mentioned proof by experience. You can do this yourself. Generate uncorrelated normal (Gaussian) random variables with zero mean and standard deviations e1 and e2. Find out what the average power is and see if it matches your equation. Otherwise, see Davenport.

Here is an unscientific heuristic (which means it is off the cuff and has not been peer reviewed, so please, no flames!)
We can think of independent as uncorrelated or as orthogonal. If we use the term orthogonal to mean perpendicular (90 degrees in a geometric picture of the problem) then we can say that adding two numbers that are perpendicular gives us a distance from start to finish that is the Pythagorean theorem C^2 = A^ + B^2 or C = sqrt( A^2 + B^2).

now if the noise is correlated, we can say that the angle between them is not 90 degrees. So the formula would give a bigger answer for angles above 90 (but less than 180). We could consider this case, positive correlation. If the noise is negatively correlated the angle is less than 90, so the length of C would be less than the Pythag theorem. This does not tell you how to relate the correlation amount to the angle, but it does give you aa perception of the issue (hopefully).

dafter gave you the modification of the Pythag theorem for non perpendicular or correlated noise. Hope this helps with the concept, but I know it is not a proof. A good book would be a better source for a proof like this.

 

[dafter]

Well I was searching for the proof in books like van der ziel, electronic noise, etc. I guess those were the wrong books. I should have focused more on radar books.
Unfortunately, the library (here) don't have Barton, Davenport and others.
I will better go and search in a radar book for the topic "coherent integration" as I have been told by pstuckey. Thx!"