What happens when you take the derivative of white noise and set it zero?
The PBF is strong in this one.
While not as doomed as our hapless hero, I still think that numerous LOL moments will be had by my exam markers. While there wasn’t any cake at the finish line, music was fine too.
By way of apology, a semi-relevant entry is coming up. In fact, it may even be up as you read this.
4 Comments
What’s the derivative of white noise? That’s easy. The Fourier transform of white noise is the constant 1. When you differentiate a function, you multiply its Fourier transform by (i pi f). So, the Fourier transform of white noise is just (i pi f). Take the inverse transform, and you have the derivative of white noise. :)
Two entries down I recount the catch-all concept of taking the derivative (of anything) and setting it to zero. It’s just that differentiating noise and setting it to zero makes, er, zero sense. Kind of like the sketch.
But indeed, it’s possible to straight up take the derivative by using power spectral density. I’m a j*omega type person myself, though.
BTW, my last post was a joke. You cannot take the Fourier transform of white noise. It will not be well-defined. In engineering applications, what you do is that you take the Fourier transform of its second moment, a.k.a. power spectral density. Unfortunately, the derivative of the PSD has nothing to do with the derivative of the stochastic process itself. So, the multiplying j*omega trick does not shed any light here.
Furthermore, in general, it makes no sense to take the derivative of a function that is not of bounded variation, which the white noise is certainly not of. Its derivative is simply nowhere defined. So, you are right that it makes no sense to talk about differentiating noise.
The largest class of objects that you can take the derivative of (that I know of) is the class of tempered distributions. But unless you’re into partial differential equations or quantum mechanics, that stuff is of little use.
Just want to clarify things a bit so people won’t take my last post as anything serious. :)
True true. I shouldn’t have said straight up. When you said FT of noise I automatically switched into autocorrelation/PSD mode. Thank you for your insight nonetheless!