Hi Carsten,
On 2022-05-13 09:25, Carsten Andrich wrote:
On 11.05.22 08:15, Carsten Andrich wrote:
Also, any reason to do this via forward and inverse FFT? AFAIK the
Fourier transform of white noise is white noise, [...]
I had the same question when I first saw this. Unfortunately I don't
have a good
answer, besides that forward + inverse ensures that the noise looks
like it is
supposed to do, while I'm not 100% whether there is an easy way to
generate
time-domain Gauss i.i.d. noise in the frequency domain.
If you know how, please let me know.
Got an idea on that, will report back.
Disclaimer: I'm an electrical engineer, not a mathematician, so
someone trained in the latter craft should verify my train of though.
Feedback is much appreciated.
Turns out the derivation of the DFT of time-domain white noise is
straightforward. The DFT formula
X[k] = \Sigma_{n=0}^{N-1} x[n] * e^{-2j*\pi*k*n/N}
illustrates that a single frequency-domain sample is just a sum of
scaled time-domain samples. Now let x[n] be N normally distributed
samples with zero mean and variance \sigma^2, thus each X[k] is a sum
of scaled i.i.d. random variables. According to the central limit
theorem, the sum of these random variables is normally distributed.
Do note that the model of no correlation is not correct model of
reality. There is several effects which make "white noise" slightly
correlated, even if this for most pratical uses is very small
correlation. Not that it significantly changes your conclusions, but you
should remember that the model only go so far. To avoid aliasing, you
need an anti-aliasing filter that causes correlation between samples.
Also, the noise has inherent bandwidth limitations and futher, thermal
noise is convergent because of the power-distribution of thermal noise
as established by Max Planck, and is really the existence of photons.
The physics of it cannot be fully ignored as one goes into the math
field, but rather, one should be aware that the simplified models may
fool yourself in the mathematical exercise.
To ascertain the variance of X[k], rely on the linearity of variance
[1], i.e., Var(a*X+b*Y) = a^2*Var(X) + b^2*Var(Y) + 2ab*Cov(X,Y), and
the fact that the covariance of uncorrelated variables is zero, so,
separating into real and imaginary components, one gets:
Var(Re{X[k]}) = \Sum_{n=0}^{N-1} Var(x[n]) * Re{e^{-2j*\pi*k*n/N})}^2
= \Sum_{n=0}^{N-1} \sigma^2 * cos(2*\pi*k*n/N)^2
= \sigma^2 * \Sum_{n=0}^{N-1} cos(2*\pi*k*n/N)^2
Var(Im{X[k]}) = \Sum_{n=0}^{N-1} Var(x[n]) * Im{e^{-2j*\pi*k*n/N})}^2
= ...
= \sigma^2 * \Sum_{n=0}^{N-1} sin(2*\pi*k*n/N)^2
The sum over squared sin(…)/cos(…) is always N/2, except for k=0 and
k=N/2, where cos(…) is N and sin(…) is 0, resulting in X[k] with real
DC and Nyquist components as is to be expected for a real x[n].
Finally, for an x[n] ~ N(0, \sigma^2), the DFT's X[k] has the
following variance:
{ N * \sigma^2, k = 0
Var(Re{X[k]}) = { N * \sigma^2, k = N/2
{ N/2 * \sigma^2, else
{ 0 , k = 0
Var(Im{X[k]}) = { 0 , k = N/2
{ N/2 * \sigma^2, else
Therefore, a normally distributed time domain-sequence x[n] ~ N(0,
\sigma^2) with N samples has the following DFT (note: N is the number
of samples and N(0, 1) is a normally distributed random variable with
mean 0 and variance 1):
{ N^0.5 * \sigma * N(0, 1) , k = 0
X[k] = { N^0.5 * \sigma * N(0, 1) , k = N/2
{ (N/2)^0.5 * \sigma * (N(0, 1) + 1j * N(0, 1)), else
Here you skipped a few steps compared to your other derivation. You
should explain how X[k] comes out of Var(Re(X[k])) and Var(Im(X[k])).
Greenhall has the same results, with two noteworthy differences [2].
First, normalization with the sample count occurs after the IFFT.
Second, his FFT is of size 2N, resulting in a N^0.5 factor between his
results and the above. Finally, Greenhall discards half (minus one) of
the samples returned by the IFFT to realize linear convolution instead
of circular convolution, fundamentally implementing a single iteration
of overlap-save fast convolution [3]. If I didn't miss anything
skimming over the bruiteur source code, it seems to skip that very
step and therefore generates periodic output [4].
This is a result of using real-only values in the complex Fourier
transform. It creates mirror images. Greenhall uses one method to
circumvent the issue.
Cheers,
Magnus
Best regards,
Carsten
[1] https://en.wikipedia.org/wiki/Variance#Basic_properties
[2] https://apps.dtic.mil/sti/pdfs/ADA485683.pdf#page=5
[3] https://en.wikipedia.org/wiki/Overlap%E2%80%93save_method
[4]
https://github.com/euldulle/SigmaTheta/blob/master/source/filtre.c#L130
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