# Relationship between Wavelet transform and Fourier Power Spectral Density

Is there anyway to obtain the Fourier Power Spectral Density from a wavelet transform of a time series?

I am particularly interested in this problem because I was wondering if there is any possibility to obtain the local Power Spectral Density from the wavelet transform.

If I am not wrong, according to Torrence and Compo, the average of all the local wavelet spectra tends to approach the Fourier Spectrum of the time series.

However when I compute the Wavelet spectra the results is much widther than the one given by the Fourier Transform.

Here there is an example:

• I made an oscillatory time series with some noise (first panel)
• I compute the Wavelet Continious transform and then I compute the wavelet power spectra (second panel) using a Morlet Transform.
• According to Torrence and Compo, we can compute the Fourier periods with the formulae than they provide, and also we can compute the Fourier frequencies. They said that a vertical cut in any time is the local Fourier Spectrum, and if we average all the local wavelet spectra we will obtain the Fourier Spectra.
• However when I do this the result obtained from the wavelet is much wider than the Power Spectra obtained from Fourier:

The main frequencies seems to be OK, but I don't understand why the spectrum from the wavelets is so wide.

To rephrase the question again, I would like to know if I can approximate the Fourier Spectra with the wavelet transform without this wideness. And again: I am particularly interested in this problem because I was wondering if there is any possibility to obtain the local Power Spectral Density from the wavelet transform.

Thank you very much.

• If you don't get an answer I can work with you. It's been a while since I dug into wavelets seriously. As far as your particular question is concerned it's the time-frequency uncertainty principle. Roughly (without mathematics) the Fourier transform doesn't localize the time of frequencies whereas the wavelet transforms do and this time localization causes the frequency terms to have a broader width. Since both transforms are invertible you can always go from one to the other and back. But note that you will probably be passing through the time representation. – rrogers Mar 29 '17 at 15:03