Euler's formula and extracting power and phase

[@semlikfer]

Impressive! I'm following all the lessons Mike has posted in KZbin and also on his Webpage. That was exactly what I was looking for. Thank you so much. Greetings from Colombia.

[@rblack37]

It's cool that you are using all this for neuro... I'm using it for sound recognition and music analysis/synthesis! Thanks for all the info on wavelets!

[@mikexcohen1]

Nice. That highlights the versatility, ubiquitousness, and importance of these kinds of spectral and time-frequency decomposition methods!

[@Dominus_Ryder]

Mike, I am using your method here to analyze generator current signals from wind and water turbines, to see if you can use the frequency representations of the signal to extract useful fault information. I think it's amazing how similar the generator current signals that I analyze look to your EGG signals. Anyway, quick question, you mentioned that the complex Morlet wavelet allows you to see the power of the signal. Well, in my case, the signals that I want to analyze are already generator power signals, just obtained under different conditions. If I use this complex wavelet, how is the power obtained from it different from the generator power that I am analyzing, (if there is one)?

[@mikexcohen1]

Hi Brit. Interesting! And I think this speaks to the versatility and obsequiousness of time-frequency methods for understanding signal characteristics more generally. The goal of using complex wavelets is to extract the phase and power time series from the frequency component in the signal that matches the frequency spectrum of the wavelet, a.k.a. band-limited phase and power. So it could make sense for your generator power signal if that signal exhibits fluctuations in some frequency. I hope I understand you correctly. You can also feel free to email me with a screenshot of your data and I can give you more specific advice. Mike

[@Dominus_Ryder]

Mike, thanks so much, I would love to do exactly that, I sent you a direct message, any help that you could provide would be very much appreciated.

[@brittnyfreeman3650]

Another question Mike, is the phase angle time series that you obtain from the imaginary axis component of the Morlet wavelet the instantaneous phase angle of the original EEG signal, or the instantaneous phase angle of the complex Morlet wavelet? If it's the latter, is there a way to extract the instantaneous phase angle of the original EEG signal?

[@MattCamp]

Very good videos! Thank you!

[@mikexcohen1]

You're very welcome ;)

[@brittnyfreeman3650]

Hello again Mike, lol. Quick question, You mentioned that you square the magnitude time series data that you get when performing complex Morlet wavelet convolution to get the power. However, if you don't square the magnitude, will that give you the envelope of the signal? If so, what domain is that envelope in (Time, or Frequency)?

[@brittnyfreeman3650]

Also, is that envelope, if for the time domain, is it the envelope of the raw time signal or the envelope of the band pass filtered time domain signal?

[@mikexcohen1]

Without squaring the magnitude, you get the amplitude time series. It's in the time-domain and reflects the estimate of the instantaneous energy at that frequency. Conceptually, the amplitude time series and power time series are basically the same. the power time series amplifies the relatively larger effects. To be honest, I'm not sure why we all look at power instead of amplitude. It's basically just for historical reasons (that is, everyone else does it, so we do it too...).

[@brittnyfreeman3650]

@@mikexcohen1 Thanks for getting back to me so quickly Mike, your response cleared up a lot of confusion. In fact, your response was so clear that I realized that I didn't even ask you the question that I truly meant to. Basically, I am trying to do signal demodulation, and I already know how to extract the Hilbert Envelope of a modulated signal, but how do you create the the Morlet Wavelet equivalent of the Hilbert Envelope from a modulated signal? Is it even possible to do that, and if so, is there any benefits to using one method to generate the envelope over the other (or is the answer to that question the same as why in some cases its better to use Morlets for time frequency analysis as opposed to the HT, and vice-versa).

[@mikexcohen1]

Ah, I see ;) You can (1) create a complex Morlet wavelet and convolve that with the signal, or (2) narrowband filter and then apply the Hilbert transform. The resulting analytic signal from either method is basically the same (minor differences will be due to the shape of the filter in the frequency domain). I have 1-2 videos about comparing those methods (probably titled something like "comparing wavelet, filter-Hilbert, STFFT") where I talk about the subtle differences. But honestly, it's mostly a matter of personal preference. My advice is to use the method that you understand the best, or the one that's easiest to code.

[@fawazsammani5271]

Hi Mike. Thanks for your videos. I'm actually unclear on the differences between applying convolution w/o a complex and w/ a complex wavelet. So w/o a complex wavelet, we perform dot product (element-wise multiply the signal with the wavelet and sum) and then get one point which is at the center of the wavelet. So w/ complex, we perform element-wise multiplication between the signal and the real wavelet, and the signal and the complex wavelet and then sum the results to get the complex point? So for each time-point, we still get one point which has the same position of the center of the wavelet, but this time the point is the addition of the real and imaginary element-wise multiplication . Did i get this correct?

[@mikexcohen1]

Yes, that's correct. If you just want to narrow-band filter the signal, then a real-valued wavelet is fine. You'd need the complex-valued wavelet if you want to extract the power and phase angle time series.

[@fawazsammani5271]

@@mikexcohen1 Thanks a alot for your reply. And Thanks for sharing your knowledge on EEG with us. These signal processing Tutorials are the best online, and your tutorials are the only ones for EEG!