# Is it possible to approximate given skewness and kurtosis values using an IFFT?

I have time domain data for a signal that behaves randomly but has non-random (non-Gaussian?) skewness and kurtosis values of ~0.9 and ~7.4, respectively.

The FFT of the signal shows that it's not broadband random -- it's generally flat below 100 Hz and tapers off at about -3 dB/oct above 100 Hz up to a Nyquist of 400 Hz.

I would like to create a synthesized signal that is statistically equivalent (or approximately so) to my reference signal, with nearly identical

1. Frequency content
2. RMS
3. Skewness, and
4. Kurtosis.

Additionally, I have extrapolated PSD amplitudes up to 10 kHz that I would like to add into the mix and invert to the time domain.

I have done this in the past satisfying only (1) and (2) by using, for example, MATLAB's randn() function to create random white noise (namely to create random phase angles), applying an FFT to the white noise, scaling its frequency components to match a desired FFT spectrum, and performing an IFFT to convert to time domain. The rand() function approaches a skewness and kurtosis of 0.0 and 3.0, respectively, so my initial thought was to create white noise with a specified skewness and kurtosis.

I have tried this same procedure using MATLAB's pearsnd() function, which allows one to specify skewness and kurtosis values, but the above procedure doesn't seem to preserve the skewness and kurtosis of the scaled time domain signal. I believe the IFFT assumes a Gaussian distribution. Is there a work-around for this?

Thanks!

• Is it possible to provide representative plots? Would it also be possible to mention the physical process that leads to such signals? – A_A Apr 25 at 7:50
• @A_A Thank you for your response. It's aeroacoustic data. What sort of plots are you thinking of? I cannot provide any of the actual data, and I don't think plots would enhance what I've described in my main post. My primary aim is to synthesize new data whose (normalized) histogram closely matches the reference signal's. – gdbb89 Apr 25 at 21:30
• Knowing that it is aeroacoustic data would be enough I guess. I would be interested in a plot of the signal you are trying to approximate (or replicate). And yes, it is very likely that it is generated by a non-linear process. How much would this affect the answer do you think? Do you have reason to believe that the non-stationarity would not be an issue here? – A_A Apr 30 at 15:03