Suppose you have 1,000 numbers chosen at random with a uniform 
distribution [-1, +1]. Now suppose you take the discrete Fourier 
transform of this sequence of numbers. Since the input is random, the 
output is also random. But what distribution does it follow?

I've searched all over the Internet, and I can't seem to find the answer 
to this trivial question.

My best guess is that the Central Limit Theorem would apply, resulting 
in both the sine and cosine coefficients being normally distributed. 
(But with what parameters?) The CLT definitely applies to independent 
/identically distributed/ random variables. I'm fuzzy about what happens 
when the variables are not identically distributed, however. In this 
case, each variable follows a uniform distribution with the same 
midpoint, but the width of the distribution varies for each variable.

My simulation results seem to indicate a normal distribution also. But 
that doesn't really mean anything; there could just be a bug in my 
program which causes it to spit out dud data. Or perhaps the 
distribution /looks/ normal but is actually subtly different.

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On 07/05/11 12:48, Invisible wrote:
> Suppose you have 1,000 numbers chosen at random with a uniform
> distribution [-1, +1]. Now suppose you take the discrete Fourier
> transform of this sequence of numbers. Since the input is random, the
> output is also random. But what distribution does it follow?

Hum... isn't it white noise?

http://en.wikipedia.org/wiki/White_noise


			Lars R.

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On 05/07/2011 01:25 PM, Lars R. wrote:
> On 07/05/11 12:48, Invisible wrote:
>> Suppose you have 1,000 numbers chosen at random with a uniform
>> distribution [-1, +1]. Now suppose you take the discrete Fourier
>> transform of this sequence of numbers. Since the input is random, the
>> output is also random. But what distribution does it follow?
>
> Hum... isn't it white noise?

Yes it is. But it's /uniform/ white noise rather than the more usual 
/Gaussian/ white noise. It's also a real-world realisation of the 
theoretical concept of white noise. Theoretical noise has a perfectly 
flat spectrum, but any real-world sample of it does not.

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On 05/07/2011 11:48 AM, Invisible wrote:
> Suppose you have 1,000 numbers chosen at random with a uniform
> distribution [-1, +1]. Now suppose you take the discrete Fourier
> transform of this sequence of numbers. Since the input is random, the
> output is also random. But what distribution does it follow?
>
> I've searched all over the Internet, and I can't seem to find the answer
> to this trivial question.
>
> My best guess is that the Central Limit Theorem would apply, resulting
> in both the sine and cosine coefficients being normally distributed.
> (But with what parameters?) The CLT definitely applies to independent
> /identically distributed/ random variables. I'm fuzzy about what happens
> when the variables are not identically distributed, however.

Apparently the sum of random variables is distributed as the 
/convolution/ of the distributions of each variable.

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