# 1/f noise parameter characterization

I would like to characterize 1/f noise in some time series data. I would like to estimate the 1/f noise corner, and the standard deviation of the 1/f noise component and white noise component. The best thing I could find was this paper: A generalized noise variance analysis model and its application to the characterization of 1/f noise that deals with photon noise arrival rates. That's great, I follow their math and everything. They start of with a random variable that is photon count rates. The problem is I want to do this for an ADC or electronic noise source and I'm not sure what my random variable would be. How do I estimate the 1/f noise corner, and the standard deviation of the 1/f noise component and white noise component with an electronic noise source from time series data?

To show what a 1/f noise spectrum looks like:

Here is what the FFT looks like. I'm hoping for something that is more numerical based than semi-empirically modeled.

• what'sa "$1/f$ noise corner"? and, exactly what do you want to do? i am also not sure what you mean by "characterize $1/f$ noise". what is that? are you analyzing an input as a R.V. or do you want to generate a R.V.? – robert bristow-johnson Apr 1 '16 at 18:19
• @robertbristow-johnson : My reading of it is there is noise being modelled as the addition of white noise and $1/f$ noise. The "corner" frequency is that frequency at which the spectral density of the white noise overtakes that of the $1/f$ noise. OP can correct me if I'm wrong. :-) – Peter K. Apr 1 '16 at 18:42
• yeah @PeterK., i started figuring that out. i guess the OP can FFT several windowed frames of this signal. do that several times and add or average the magnitude (or magnitude-squared). then convert to dB. and fit a bilinear curve of dB to log-frequency to it. it could be something like $$\log \left(e^{\alpha (x-x_0)} + 1 \right) + y_0$$ three parameters to fit. for pink noise on the left, i think that $\alpha$ might be $-\frac12$. $x_0$ would be the corner in log-frequency. – robert bristow-johnson Apr 1 '16 at 19:40
• It would be simple to fit a line to my 1/f noise distribution if it were the continuously sampled version (ie, if I had the whole distribution). From the looks of the FFT this might be difficult to do, and probably not characterize the distribution well. I suppose I could re-sample the distribution if that kind of a method works with flicker noise. I'm hoping for a numerical solution. – laptop2d Apr 1 '16 at 20:59
• first of all, the y-axis of the first plot should be labled "noise density in $20 \log(nV/\sqrt{Hz}) \ (dB)$ because it won't be a straight line on the left for pink or brown or whatever filtered noise. second, you should, even with noise, window each frame with a decent window. Hann or Hamming is good enough. third, you should repeat this several times with the same window, but different noise data, toss phase information, and average the magnitude-squared before applying $10 \log(\text{mean}(|X[k]|^2))$ . that should smooth things out. – robert bristow-johnson Apr 1 '16 at 23:34