Amazing Mike, your videos are unbelievably good. I'm applying wavelets to detect the closing and opening time of a high-voltage circuit breaker for my thesis. Your understanding of the subject matter and clarity in explanations have helped me grasp mathematical concepts that were previously out of reach. I followed your entire course on signal processing on Udemy. You're a brilliant mind; keep up the excellent work. Wavelets are incredible; the uncertainty principle makes them unique.
Not sure how valid it is but the way I like to think about this is that wavelets with fewer cycles look more like impulses (i.e. delta functions) in the time domain, and therefore have a broadband spectral equivalent. As such, they offer less spectral resolution because they are capturing information across a wide range of frequencies. Wavelets with more cycles are closer to sine waves, and thus have a narrowband spectral equivalent (i.e their Fourier transform is closer to a delta function), and thus offer better spectral resolution. The effects of wavelet cycles in terms of time resolution are as explained in the video (i.e. smudging etc.)
Exactly! That's a great way to think about it. It's often useful to think about what happens as parameters go to extreme values (in this case, either 0 or infinity).
Exactly, this is the Heisenberg uncertainty principle applied to time-frequency analysis. You cannot simultaneously localize a signal in time and in "space" (frequency).
Hi Mike, I would like to ask about the "variable 3-10 cycles" option: do you mean we apply smaller number of cycles to lower frequency, and larger number of cycles to higher frequency? you mentioned the number of cycles is a function of frequency, i.e., dependent on frequency. why is that? I thought the "variable" to be "random" (apply a random number of cycle between 3 and 10 to each of the frequency). considering the relationship between number of cycles and time-frequency trade-off, is there any reason we give priority to temporal precision for lower frequency and to frequency precision for higher frequency?
These parameters are, to some extent, arbitrary and selected by the researcher. The increase in cycle parameter with frequency is done to make higher-frequency bursts a bit more smoothed in time. That helps them be identified in noisy data. It's tricky because you need to think about time in two different ways: ms and cycles. If a low-frequency event lasts for 2 cycles, that might be hundreds of ms. But if a higher-frequency event lasts for 2 cycles, that might be only a few tens of ms. So the lower frequencies don't need as much temporal smoothing to take up visible real estate on the x-axis.
@@mikexcohen1 Wow this is exactly what I was looking for. Our required time resolution gets locked in a particular lower frequency. Extrapolating frequency out in octaves using classic tiling leaves certain higher frequencies representing different physics under-resolved in frequency and more resolved in time than I need. So the proposal here is appealing. Is there a resource to learn more about the details of its implementation and consequences? I assume that doing this affects properties like the frame, inverse, near-orthogonality, leakage etc? What are the properties as the idea approaches a regular/rectangular tiling in time and frequency instead of a pyramid? Is there a citable resource where I could learn more? Thanks so much for your series and for mentioning this important possibility!!
It was a really good lecture. thanks! I had a question. Dose any one know how to use morlet wavelet and stft Simultaneously in eeg signal preprocessing ? (I happened to read a "frontiers in neuroscience" article that used these two as EEG signal preprocessing)
Hi Fatemeh. Wavelet convolution and STFT are both used for time-frequency analysis, so it seems a bit redundant. It's possible they used wavelet convolution for cleaning the data before the STFT.
@@mikexcohen1 thanks for your answer. you mean I should give the output of morlet wavelet as the input of STFT? as far as I know the input of both of them must be in time series form not the time-frequency form!
Amazing Mike, your videos are unbelievably good. I'm applying wavelets to detect the closing and opening time of a high-voltage circuit breaker for my thesis. Your understanding of the subject matter and clarity in explanations have helped me grasp mathematical concepts that were previously out of reach. I followed your entire course on signal processing on Udemy. You're a brilliant mind; keep up the excellent work. Wavelets are incredible; the uncertainty principle makes them unique.
Thank you kindly, Pedro. It's always interesting to see how useful wavelets are in so many different fields.
Not sure how valid it is but the way I like to think about this is that wavelets with fewer cycles look more like impulses (i.e. delta functions) in the time domain, and therefore have a broadband spectral equivalent. As such, they offer less spectral resolution because they are capturing information across a wide range of frequencies. Wavelets with more cycles are closer to sine waves, and thus have a narrowband spectral equivalent (i.e their Fourier transform is closer to a delta function), and thus offer better spectral resolution. The effects of wavelet cycles in terms of time resolution are as explained in the video (i.e. smudging etc.)
Exactly! That's a great way to think about it. It's often useful to think about what happens as parameters go to extreme values (in this case, either 0 or infinity).
perfect explanation, thank you!
P.S. couldn't help thinking about Heisenberg during the whole lesson..
Exactly, this is the Heisenberg uncertainty principle applied to time-frequency analysis. You cannot simultaneously localize a signal in time and in "space" (frequency).
Very clear and informative, thank you!
Hi Mike, I would like to ask about the "variable 3-10 cycles" option: do you mean we apply smaller number of cycles to lower frequency, and larger number of cycles to higher frequency? you mentioned the number of cycles is a function of frequency, i.e., dependent on frequency. why is that? I thought the "variable" to be "random" (apply a random number of cycle between 3 and 10 to each of the frequency). considering the relationship between number of cycles and time-frequency trade-off, is there any reason we give priority to temporal precision for lower frequency and to frequency precision for higher frequency?
These parameters are, to some extent, arbitrary and selected by the researcher. The increase in cycle parameter with frequency is done to make higher-frequency bursts a bit more smoothed in time. That helps them be identified in noisy data.
It's tricky because you need to think about time in two different ways: ms and cycles. If a low-frequency event lasts for 2 cycles, that might be hundreds of ms. But if a higher-frequency event lasts for 2 cycles, that might be only a few tens of ms. So the lower frequencies don't need as much temporal smoothing to take up visible real estate on the x-axis.
@@mikexcohen1 Wow this is exactly what I was looking for. Our required time resolution gets locked in a particular lower frequency. Extrapolating frequency out in octaves using classic tiling leaves certain higher frequencies representing different physics under-resolved in frequency and more resolved in time than I need. So the proposal here is appealing. Is there a resource to learn more about the details of its implementation and consequences? I assume that doing this affects properties like the frame, inverse, near-orthogonality, leakage etc? What are the properties as the idea approaches a regular/rectangular tiling in time and frequency instead of a pyramid? Is there a citable resource where I could learn more? Thanks so much for your series and for mentioning this important possibility!!
Thanks for your great explanation! I wonder how I can reference the figure at 15:17, thanks.
Glad it was helpful! That figure is from my ANTS book (MIT Press, 2014).
Thank you so much! You are a great educator!
Aww thanks! Now you make me blush ;)
Book -> Booklet
More -> Morlet
Thanks. U video really helps. Is the cycle here the same as sub-octaves per octave in R？
I don't know, sorry. I haven't used R in ages, and I never learned it past intro-level.
Your vidoes are great! thank you so much for making this content
Thanks, I'm glad you enjoyed it!
It was a really good lecture. thanks! I had a question. Dose any one know how to use morlet wavelet and stft Simultaneously in eeg signal preprocessing ? (I happened to read a "frontiers in neuroscience" article that used these two as EEG signal preprocessing)
Hi Fatemeh. Wavelet convolution and STFT are both used for time-frequency analysis, so it seems a bit redundant. It's possible they used wavelet convolution for cleaning the data before the STFT.
@@mikexcohen1 thanks for your answer. you mean I should give the output of morlet wavelet as the input of STFT? as far as I know the input of both of them must be in time series form not the time-frequency form!
Thanks, good job!
thanks so much!
My ancestor???