What happens to gaussian white noise when derived in continuous time?

[lagoule]

Hello,

I've got a problem where a recording signal is a signal + gaussian white noise (quite classic). I derive this signal and while I know the theoretical result of the derivative of the noiseless signal, but I can't figure out what happens to the noise after the operation.

So, basically, what happens to gaussian white noise if you derive it (in continuous time)? Will the result be statically gaussian? something else? What will be the variance and mean?

The goal of the problem is to perform detection of events in white noise, and the derivative is used to increase the SNR of the event.

Thanks for any help,

Jonathan

 

[Antiphon]

True white noise has infinite power spectral density and no maximum frequency. I'm not a mathematician but that's probably not differentiable. Bandlited white noise is probably what you have on the real world an that is differentiable.

 

[Antiphon]

Opps, it's gaussian. You can differentiate in the frequency domain. The phase will continue to be random.

 

[lagoule]

Hello,

Off course, the noise is band-limited, as is the differentiator circuit.

I didn't think of looking at the problem in the frequency domain. If the white noise is flat in frequency domain, then its derivative will be linear. This also confirms that if the noise isn't band-limited, its derivative will have infinite power.

However, this doesn't give me the statistical properties of the derivative, it may hint that they aren't mathematically defined though.

Thanks for your help,

Jonathan

 

[Redbelly98]

For any signal, the spectrum of the derivative is ω times the transform of the signal, i.e. ω·F(ω). So any peaked spectrum gets shifted toward higher frequencies.