Is the Fourier transf. of an autocorrelation functn always positive?

[jiadong]

I am trying to understand the IR spectra of liquid. I can get the autocorrelation function of atoms' velocity,
<v_{i}(0)v_{i}(t)>
make a Fourier Transformation, the vibrational density of state (VDOS) can be obtained. Does the VDOS always be positive? Or it can also take negative value in some frequency region? Thanks

 

[Steve Drake]

Maybe not directly related to your work, but with what I do, the FT of the autocorrelation function which obtains the power spectral density. Where the peak will be centered at the incident laser wavelength, and the line width relates to the characteristic decay time of the sample. I don't see why it would go negative, but again I am talking about light scattering process, not yours.

 

[Mute]

Since the FT of an auto-correlation function gives the power spectrum, it is supposed to be positive. However, a colleague of mine once found that taking the Fourier transform of a discrete auto-correlation data set he was working with erroneously gave some negative values. His data was simulation data, though, and the problem was that the correlation function he chose and was trying to transform wasn't properly a correlation function for some reason (dimensional problems, maybe).

If you have simulation data, you might need to alter your auto-correlation function or your Fourier transform method somehow. If you have experimental data, I guess you have to improve your Fourier transform method so that you don't get the negatives, or maybe you need more time resolution?

 

[marcusl]

Since your data are real, the autocorrelation function is real and (by definition) symmetric, so its transform is symmetric and real.

 

[jiadong]

Maybe not directly related to your work, but with what I do, the FT of the autocorrelation function which obtains the power spectral density. Where the peak will be centered at the incident laser wavelength, and the line width relates to the characteristic decay time of the sample. I don't see why it would go negative, but again I am talking about light scattering process, not yours.

Thanks, all the same. I also think the FT of the autocorrelation function should give the relative probability of different component, eg the relative probability of different vibritional density of state.Thanks, my reply is too late.

 

[jiadong]

Since the FT of an auto-correlation function gives the power spectrum, it is supposed to be positive. However, a colleague of mine once found that taking the Fourier transform of a discrete auto-correlation data set he was working with erroneously gave some negative values. His data was simulation data, though, and the problem was that the correlation function he chose and was trying to transform wasn't properly a correlation function for some reason (dimensional problems, maybe).

If you have simulation data, you might need to alter your auto-correlation function or your Fourier transform method somehow. If you have experimental data, I guess you have to improve your Fourier transform method so that you don't get the negatives, or maybe you need more time resolution?

My data is simulation data. I also think the problem is in the method of FT. Thanks a lot!

 

[jiadong]

Since your data are real, the autocorrelation function is real and (by definition) symmetric, so its transform is symmetric and real.

Yes.! Finally, I find that someone also define the square of FT of the velocity autocorrelation as the vibrational density of state. now I think I understand it more.