Comments on: [Meh-ta] What has CALCULUS done?! http://www.silician.com/anime/2007/12/20/meh-ta-what-has-calculus-done/ A terminal of connections Thu, 10 Dec 2009 04:06:46 +0000 hourly 1 http://wordpress.org/?v=3.0.1 By: introspect http://www.silician.com/anime/2007/12/20/meh-ta-what-has-calculus-done/comment-page-1/#comment-35 introspect Sat, 22 Dec 2007 03:43:10 +0000 http://www.silician.com/anime/2007/12/20/meh-ta-what-has-calculus-done/#comment-35 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! 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!

]]> By: bakaneko http://www.silician.com/anime/2007/12/20/meh-ta-what-has-calculus-done/comment-page-1/#comment-33 bakaneko Thu, 20 Dec 2007 17:42:33 +0000 http://www.silician.com/anime/2007/12/20/meh-ta-what-has-calculus-done/#comment-33 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. :) 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. :)

]]> By: introspect http://www.silician.com/anime/2007/12/20/meh-ta-what-has-calculus-done/comment-page-1/#comment-32 introspect Thu, 20 Dec 2007 16:26:01 +0000 http://www.silician.com/anime/2007/12/20/meh-ta-what-has-calculus-done/#comment-32 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. 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.

]]> By: bakaneko http://www.silician.com/anime/2007/12/20/meh-ta-what-has-calculus-done/comment-page-1/#comment-30 bakaneko Thu, 20 Dec 2007 09:29:13 +0000 http://www.silician.com/anime/2007/12/20/meh-ta-what-has-calculus-done/#comment-30 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. :) 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. :)

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