I'm wondering why noise sources are added such a way their RMS values are added in square root of sum of squares fashion.
For instance if the RMS noise values from three sources are X,Y and Z, then the total noise is given as \$N = \sqrt{X^2+Y^2+Z^2}\$.
Is there a way to show this relation mathematically?
If the noise sources are uncorrelated, then the power (variance) of its sum equals the sum of their powers (variances), because all the co-variances between them are zero, and only the variances remain.
An example with two noise sources (with zero mean):
\$\sigma_N^2 = E[N^2] = E[(X+Y)^2] = E[X^2]+E[Y^2]+2E[XY] = \sigma_X^2 + \sigma_Y^2 +2\sigma_{XY}\$
If \$X\$ and \$Y\$ are uncorrelated then \$\sigma_{XY}=0\$ and \$\sigma_N^2 = \sigma_X^2 + \sigma_Y^2 \$, thus:
\$\sigma_N = \sqrt{\sigma_X^2 + \sigma_Y^2} \$, where \$\sigma_N\$, \$\sigma_X\$ and \$\sigma_Y\$ are the RMS values of \$N\$, \$X\$ and \$Y\$.
It can be demonstrated for any number of sources, if required. The result ends up being the same: power of the sum equals the sum of the powers.
Why are different RMS noise sources added in this way?
Take two waveforms having RMS values x and y. To find the total RMS, you square (x+y) thus: -
$$x^2 + 2xy +y^2$$
That accomplishes the squaring as in RMS (root of the mean of the square).
Then, take the mean (as in mean of the square); \$2xy\$ will have a mean of zero because x and y are uncorrelated noise sources. If they correlated then \$xy\$ would not have a zero mean value.
So, the mean of the squares is \$x^2+y^2\$ and, to get the total RMS you take the square root to get: -
$$\sqrt{x^2+y^2}$$
RMS means root of the mean of the square(s).
You could turn this on its head and say that a defining feature of noise is that it adds as power to any signal, that is, as RMS.
Two uncorrelated signals add as power. You can determine the degree of correlation by using the amount the power of their sum differs from the sum of their powers.
