[time-nuts] Simulation of oscillator noise
Jim Lux
jimlux at earthlink.net
Fri Nov 29 18:27:37 UTC 2013
On 11/29/13 8:50 AM, Magnus Danielson wrote:
> On 11/29/2013 04:11 PM, Jim Lux wrote:
>> On 11/29/13 5:56 AM, Azelio Boriani wrote:
>>> Unfortunately that was a contribution from Magnus in 2010
>>>
>>> (see www.febo.com/pipermail/time-nuts/2010-April/046932.html )
>>>
>>> that I have simply reported without verifying the link and found that
>>> link unusable after sending the message. My best guess is this:
>>>
>>> http://www.crya.unam.mx/radiolab/recursos/Allan/Kasdin-Walter.pdf
>>>
>>> based on a search on FLFM (flicker of frequency).
>>
>>
>>
>> one limitation of the Kasdin-Walter method is that it is "batch mode",
>> and doesn't lend itself to an implementation which is continuous.
>>
>> The paper does have a nice discussion of why the "white noise into a
>> filter" technique doesn't work very well if the slopes you need aren't
>> integer powers of frequency. Integer powers in frequency correspond to
>> rational functions in filter characteristics, which are
>> straightforward, but how do you make a 1.5th order filter section or
>> half a pole or zero?
>>
>> The fractal literature, though, may provide mechanisms that might be
>> useful.
> Actually, NIST (or actually this was in it's NBS days) did a few good
> articles, comparing the Mandelbrot simulation method with their filter
> method. Turns out that you need to dimension the filter to the
> simulation length, as the number of lead-lag sections needs to cover the
> range where 1/f slope is needed and then the density of them (lead-lag
> pole/zeros per decade) will control how close it will approximate, that
> is, how little "pass-band" ripple there is from the ideal. Also, you
> need to apply the corrections to start the filter up in the correct state.
>
That's essentially what the Kasdin-Walter paper talks about. The number
of taps/sections is adjusted to approximate whatever curve you want
"well enough".
Then, they sort of shunt all that with an FFT based method.. Generate
white noise, filter it with a FFT convolution scheme where you've loaded
the bins of the FFT with the desired power spectrum.
> It's non-trivial to do well.
And, I suspect, non-trivial to do with low computational complexity.
>
> There are many many methods to do this. Everyone has a favorite.
No doubt about it.
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