Hi! Can anyone tell me how to plot wavelet power spectrum in Matlab from the cwt coefficients??

Question

I am dealing with geomagnetic data and want to plot the wavelet power spectrum. I have calculated the coefficients using 'cwt' in Matlab. How to get the power spectra from this, I don't understand?
I'll be thankful if anyone helps me..

Answer 1

Firstly, the wavelet coefs and FT coefs are NOT the same. Secondly, the wavelet PACKET coefs on the coarsets level (full transform, many levels, etc) are CLOSELY related to FT coefs. Thirdly, longer wavelet filters combined with a number of consecutive wavelet packet transform steps (to a given level) is CLOSELY related to a STFT (from which a spectrogram can be made by squaring).

The relation between scale and frequency is as follows: A signal containts frequencies from 0 to fs/2, where fs is the sampling frequency. Any freq higher that fs/2 is mirrored into the interval [0; fs/2] at the time of sampling (this is also called aliasing). After ONE wavelet transform (the signal is now on the next scale) of this signal the low pass part contains APPROXIMATELY the freq [0; fs/4] and the high pass part containts APPROX [fs/4; fs/2]. How well this approximation is to the ideal freq separation depends on the wavelet used. To see the approximation one can plot the freq response of the used wavelet filter (examples of this for Daubechies filters 2, 12, and 22 is shown in "Ripples in Math"). Longer filters IN GENERAL means better approximation to the ideal case. Thus, on a given scale the various parts or elements of the signal represents different freq ranges APPROXIMATELY.

The output of ONE wavelet transform is APPROX and ESSENTIALLY the same as an STFT with window length equal to two samples. The problem is that the STFT is not designed to such short windows and produces rather poor results. Therefore, the WT is a better practical time-frequency representation for THIS PARCITULAR trade-off between time and freq. However, the STFT can be used to understand the freq content of the WT coefs.

Applying the WT once more to BOTH the low pass and high pass parts produces four parts which corresponds in interpretation to the result of an STFT with window length four. Again the WT produces better results than STFT as it is designed to handled this particular TF trade off. As more and more wavelet transforms are performed and the freq resolution of the resulting low and high pass parts increases (as the number of samples in them decreases) the freq content gets more and more messy due to the imperfections (non-idealness) of the freq response of the wavelet filter. BUT, the FT is designed to handled freq separation very well and therefore is superior to WT in this extreme case (after many transforms).

As in all other engineering there is not black and white answer. The WT and STFT gives different outputs, and none is better thatn the other. It is two different things, and useful for different purposes. However, the FREQUENCY INTERPRETATION of the coefs are very similar, and a wavelet scale correspond ESSENTIALLY to a frequency range, but not entirely due to the non-idealness of wavelet filters (no finite ideal filter exists, in fact).

When it comes to freq shifting the FT is a priori a good choice since the goal is to alter frequencies and not time (which inevitably is present in WT coefs). However, as some of you have pointed out, stuff happens as you shift the frequency, mainly because it is difficult to control phase and freq simultaneously (and phase is rather important in audio). The hope might therefore be that wavelets, where phase is sort of hidden in the coefs (rather than explicitely there, as is the case with FT coefs) might be able to somehow easily cope with these unpleasent artifacts that occurs.

BE AWARE that wavelets are not necessarily easier to use that Fourier, it is just different. Nothing comes for free (bitter experience on my part tells me that), and your best hope is that your challenge becomes EASIER (and NOT EASY) by applying wavelets. In the case of pitch shifting the problem of phase is perhaps circumvented, but the problem of "time in the coefs" is now apparent.

Finally, a note on discrete vs continouos WT: These two are essentially the same. NO MORE information is available in the cont case, although it often appears nicer to the human eye! The discrete version is faster (often MUCH faster) and can be implemented very efficiently with the lifting scheme (but the filter impl is somewhat easier). In my opinion the cont case is rarely necessary to employ, in particular because the signals we work with are often discrete of nature. As a sort of in-between thing one can use the stationary wavelet transform (see for instance WaveLab toolbox). But again, I have yet to see a signal which is handled significantly better by this method than by an ordinary wavelet packet.

And finally, finally, if anyone wonder how the TF planes on page 126 in Ripples are made, it is handmade, since at the time of us writing the book there was no (to the best of our knowledge) Matlab software for making a best basis scalogram (as it requires boxes of various dimensions in the same graph). I have not made this Matlab code available as it is just some quickly made "for this purpose only" code. Sry.

Answer 2

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