On 8/20/06, Mathieu Bouchard matju@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