Talk:1-f noise

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Revision as of 06:38, 10 February 2007 by imported>Joseph Rushton Wakeling (Little proof that the definition of pink noise implies a 1/f spectrum.)
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Pink noise

In line with the issue a spectrum satisfies the definition of equal power per octave, I put together a little proof of the converse. It's probably inelegant (my maths is rusty these days....). Anyway, here goes. Let be the integral of the power spectrum. Then, since there is equal energy per octave, we must have where is a constant.

If we write , we get , where is constant. Since it's so, and is also a constant, without loss of generality we can write,

So, . Now consider , but also , so , so either (boring, means the signal has no power) or . We are left with,

From which it follows that is a linear function, so we can write with c constant. Substituting into the previous equation, we get,

So, , and so .

It follows that , so

So the differential, which gives us the power spectrum, is .

Someone shout if there's a problem. Like I said, I'm rusty... :-) —Joseph Rushton Wakeling 06:38, 10 February 2007 (CST)