Hi Matthias,
On 2022-05-14 08:58, Matthias Welwarsky wrote:
On Dienstag, 3. Mai 2022 22:08:49 CEST Magnus Danielson via time-nuts wrote:
Dear Matthias,
Notice that 1/f is power-spectrum density, straight filter will give you
1/f^2 in power-spectrum, just as an integration slope.
One approach to flicker filter is an IIR filter with the weighing of
1/sqrt(n+1) where n is tap index, and feed it normal noise. You need to
"flush out" state before you use it so you have a long history to help
shaping. For a 1024 sample series, I do 2048 samples and only use the
last 1024. Efficient? No. Quick-and-dirty? Yes.
I went "window shopping" on Google and found something that would probably fit
my needs here:
https://ccrma.stanford.edu/~jos/sasp/Example_Synthesis_1_F_Noise.html
Matlab code:
Nx = 2^16; % number of samples to synthesize
B = [0.049922035 -0.095993537 0.050612699 -0.004408786];
A = [1 -2.494956002 2.017265875 -0.522189400];
nT60 = round(log(1000)/(1-max(abs(roots(A))))); % T60 est.
v = randn(1,Nx+nT60); % Gaussian white noise: N(0,1)
x = filter(B,A,v); % Apply 1/F roll-off to PSD
x = x(nT60+1:end); % Skip transient response
It looks quite simple and there is no explanation where the filter
coefficients come from, but I checked the PSD and it looks quite reasonable.
This is a variant of the James "Jim" Barnes filter to use lead-lag
filter to approximate 1/f slope. You achieve it within a certain range
of frequency. The first article with this is available as a technical
note from NBS 1965 (available at NIST T&F archive - check for Barnes and
Allan), but there is a more modern PTTI article by Barnes and Greenhall
(also in NIST archive) that uses a more flexible approach where the
spread of pole/zero pairs is parametric rather than fixed. The later
paper is important as it also contains the code to initialize the state
of the filter as if it has been running for ever so the state have
stabilized. A particular interesting thing in that article is the plot
of the filter property aligned to the 1/f slope, it illustrates very
well the useful range of the produced filter. This plot is achieved by
scaling the amplitude responce with sqrt(f).
I recommend using the Barnes & Greenhall variant rather than what you
found. It needs to be adapted to the simulation at hand, so the fixed
setup you have will fit only some needs. One needs to have the filter
covering the full frequency range where used flicker noise is dominant
or near dominant. As one uses flicker shaping for both flicker phase
modulation as well as flicker frequency modulation, there is two
different frequency ranges there they are dominant or near dominant.
There is many other approaches, see Attilas splendid review the other day.
Let me also say that I find myself having this question coming even from
the most senioric researches even the last week: "What is your favorite
flicker noise simulation model?". They keep acknowledge my basic message
of "Flicker noise simulation is hard". None model fit all applications,
so no one solution solves it all. One needs to validate that it fit the
application at hand.
Cheers,
Magnus
The ADEV of a synthesized oscillator, using the above generator to generate 1/
f FM noise is interesting: it's an almost completely flat curve that moves
"sideways" until the drift becomes dominant.
Regards,
Matthias
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