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.