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. By local I mean to obtain the Power Spectral Density as a function of the time.
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, I tried some numerical tests for the signal: $$x(t)=cos(t*2 \pi/10)+cos(t*2 \pi/5) $$ I performed the wavelet transform using different python packages (pywavelets and others) and also I wrote a code myself.
With all the different packages I get similar results; the main frequencies seems to be the same in the Fourier Spectrum and in the averaged wavelet spectrum, but the bandwidth of the two frequencies is much widther for the wavelets average spectrum.
I computed the Wavelet Spectra by:
$$ |W_x(s)|^2 $$ where $s$ are the different scales.
To rephrase the question again, I would like to know if I can approximate the Fourier Spectra with the wavelet transform without this widthness "error". 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.