# Probability of similarity for two random white noises

I'm looking for the method to calculate theoretical similarity of two random noises $$x$$ and $$y$$ with mean $$0$$ and standard deviation $$1$$, I've got also $$12500$$ samples of signal, sampled $$2.5$$ Ga/s.

Due to my idea I wanted calculate the probability that the two noises have coherence $$C_{xy}(f)=\frac{|G(f)_{xy}|^2}{G(f)_{xx}G(f)_{yy}}$$ higher than $$0.5$$ in frequency range $$2$$-$$20\mathrm{MHz}$$? I wanted to compare it with experimental result. I found probably the answer to my question: Statistical significance of coherence values. How to do this for my signal? How $$\alpha$$ coefficient in this link was calculated?

• What is $G(f)_{xy}$? Also, your question is only borderline signal processing-related -- can you give more context of your application? – MBaz Jun 12 at 0:38
• @MBaz $G(f)_{xy}$ is cross-spectral density, here is definition: link[en.m.wikipedia.org/wiki/Coherence_(signal_processing)] – Malum Wolfram Jun 12 at 1:47
• @Malum Wolfram: Hi. I may not be understanding but your question seems paradoxical because, if both signals are truly white noise ( well, white noise is imaginary but theoretically speaking ), then, by definition, one cannot be predicted from the other. So, assuming that the coherence is the frequency analogue of cross-correlation, it should be close to zero and not statistically significant. This is because of the definition of white noise. – mark leeds Jun 12 at 11:38
• @markleeds Can we really not have two random white noise signals that are not correlated to each other? Consider the input and output of an amplifier that is amplifying a white noise signal. (Also confusing use of imaginary---which is why I dislike that we decided to call j or i that). – Dan Boschen Jun 12 at 12:56
• @Dan Boschen: I know zero about amplifiers ( and not much about DSP either ) but, if two different RV's are truly white noise, then, by definition, they can't be correlated. Part of their definition is that they are uncorrelated. – mark leeds Jun 12 at 16:30