[PD] Colored (fractal, 1/f^a) noise generator for PD

[Charles Henry]

On 8/20/06, Mathieu Bouchard <matju at artengine.ca> wrote:
> Doesn't a Gaussian distribution have a spectrum proportional to its own
> density? That is, exp(-s^2/var)/sqrt(2*pi*var). That means that in the dB
> vs octave graph, it would appear as a quadratic instead of a straight
> line, and this doesn't match any of the examples shown in the
> Colors_of_noise article. Right?

There's a difference.  The spectrum of the Gaussian distribution is
only valid for single variables.  Looking at the spectrum of a vector
where each sample is a Gaussian random variable has to be treated
differently.
The Gaussian noise vector has a flat frequency response.

I think we can use linear phase filter theory to describe the way this works.
Our vector is a sum of elementary vectors multiplied by Gaussian
random variables.
noise_vector, N=sum(j=1,N : e-j * X-j ) where e-j's are elementary
vectors and X-j~N(0,var)

The fourier transform is then a sum of linear phase filters
N(f)=sum(j=1,N : X-j * e^(2pi* i * j * f/N) )

but that's where I don't know what to do....it's a big sum of complex
values multiplied by random variables.  I think the expected spectrum
is flat, but I'm not sure about the last part...

Chuck