Hi Magnus,
On 14.05.22 08:59, Magnus Danielson via time-nuts wrote:
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.
Thank you for that insight. Duly noted. I'll opt to ignore the residual
correlation. As was pointed out here before, the 5 component power law
noise model is an oversimplification of oscillators, so the remaining
error due to residual correlation is hopefully negligible compared to
the general model error.
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])).
Given the variance of X[k] and E{X[k]} = 0 \forall k, it follows that
X[k] = Var(Re{X[k]})^0.5 * N(0, 1) + 1j * Var(Im{X[k]})^0.5 * N(0, 1)
because the variance is the scaling of a standard Gaussian N(0, 1)
distribution is the square root of its variance.
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.
Can't quite follow on that one. What do you mean by "mirror images"? Do
you mean that my formula for X[k] is missing the complex conjugates for
k = N/2+1 ... N-1? Used with a regular, complex IFFT the previously
posted formula for X[k] would obviously generate complex output, which
is wrong. I missed that one, because my implementation uses a
complex-to-real IFFT, which has the complex conjugate implied. However,
for a the regular, complex (I)FFT given by my derivation, the correct
formula for X[k] should be the following:
{ N^0.5 * \sigma * N(0, 1) , k = 0, N/2
X[k] = { (N/2)^0.5 * \sigma * (N(0, 1) + 1j * N(0, 1)), k = 1 ... N/2 - 1
{ conj(X[N-k]) , k = N/2 + 1 ... N - 1
Best regards,
Carsten
--
M.Sc. Carsten Andrich
Technische Universität Ilmenau
Fachgebiet Elektronische Messtechnik und Signalverarbeitung (EMS)
Helmholtzplatz 2
98693 Ilmenau
T +49 3677 69-4269
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