Method to determine spectrum of turbulence?
 

I make turbulence measurement by using Pulsed Doppler Ultrasound Velocimetry technic. I calculated the power spectrum of turbulent fluctuations and i showed that energy spectrum decay versus frequency approximately a power law of the form E(f)=f^(-5/3), which is the law predicted by Kolmogorov.

The technic used for to determine the energy spectrum is based on Fast Fourier Transform of Doppler Signal + a technic of Slipping Average which consist of making slip a window of observation on the signal.

I observed that the use of such windows will introduce noise into the signal spectrum signal and the slope of energy spectrum changes with the number of used window and not only for hydronamic or acoustic reasons.

I want to know if my treatement technic is right or not? if not, there is other method of signal treatement which does not present a noise on spectrum?

Thank U

Re: Method to determine spectrum of turbulence?
 

Hi

First I assume that the samples are evenly spaced. Second I also suppose that you want to study steady spectra and that the slipping average you mention is Welch's method : separate a long data run in multiple possibly overlapping shorter segments using a non-square window to reduce leakage, compute FFT over each windowed segment to obtain its power spectrum and then average all these spectra (see for instance numerical recipes for details).

The idea of the method is to reduce the variance of the spectrum estimator : with only 1 segment, the stadard deviation of the estimator is 100%. With more segments, the variance is reduced (by a factor of the sqare of the number of segment if I remember correctly). So if you obtain noisy spectra using this approach that maybe because you do not average enough segments (50 or more for experimental data). However it is quite strange that the slope of the spectrum is altered : maybe the signal is not statistically steady => sample a longer run to obtain longer segments and then damp unsteadyness.

Other methods exist. For instance you can compute the FFT of the whole run yielding a very small delta_f and then average the spectra over some neighbouring frequencies. It's OK for broadband signal but inadequate if peaks are present. You can also try non-FFT approaches such as maximum entropy with a low number of poles to smooth the spectrum (again see numerical recipes for details).

Hope this helps a little

Re: Method to determine spectrum of turbulence?
 

Thank you for your response.

I found the solution to my problem. It is necessary to change the nature of the slipping window.

There are many types of windows, at the beginning I used a rectangular window which is simple but the use of such window will introduce noise into the signal spectrum. therefore when I used a window like " hamming " more complicated but it introduces less parasites into the smoothed signal.