I saw in Correlation: A Magnitude or Power quantity? post the derivation for $\rho$ (the mean of the normalized correlation coefficient) and its relation to the SNR of the signal.
I'm looking to see how $N$ (the length of the template $x$) is related so I tried to calculate the variance of $\rho$ but I couldn't do it on my own or find a derivation online.
Is there a way to calculate the variance or to see how $N$ effects $\rho$?
$\rho$ for a random process (noise) would itself be a random quantity. If the signal is stationary and the noise is white, then the answer is straight forward in that the longer we can observe, the more accurate an estimate of SNR is that we can make. Specifically the variance of the estimate of $\rho$ would go down at rate $1/N$. When the noise or signal is non-stationary and/or when the noise is not white then the answer depends on those statistics. I find the Allan Variance a good tool to assess a given waveform in terms of determining the length of observation time in which we can optimize the SNR (or our estimate of the mean of the process).
I explain the use of Allan Variance (and square root as Allan Deviation or ADEV) in this context at this post.
