[time-nuts] Re: Simple simulation model for an OCXO?

[Carsten Andrich]

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