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I am confused about the wavelet. I understand that wavelet transform is just a bandpass filter centered at the center frequency. However, in PyWavelets, https://pywavelets.readthedocs.io/en/latest/index.html, a wavelet transform is implemented as a filter bank which can break a signal into low and high components. So my question is: how does wavelet, listed on http://wavelets.pybytes.com/wavelet/bior6.8/, play in this game? If they are used as a bandpass filter, how can the signal be broken into two parts instead of just being bandpassed?

Long WU
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    It's not a single filter. It is repeated application of the same filter, followed each time by subsampling for the next step. In that sense it is similar to a filter bank: https://en.wikipedia.org/wiki/Filter_bank#Multidimensional_filter_banks – alle_meije Oct 05 '21 at 15:13
  • @alle_meije Thanks for answering. I understand that. However that's not what I am asking. Let's say in a single step, a wavelet filter is used. From my understand it's a bandpass filter, my question is how can it be used to separate high and low frequency if it is a bandpass filter? That's something achieved by a high or low pass filter. – Long WU Oct 07 '21 at 01:27

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A wavelet transform uses the scaling function, represented by a lowpass filter, to approximate the signal on the next level, and the wavelet function, represented by a highpass filter, to encode the difference between the current level and the next.

On the page you mention these are the left and right plots of each wavelet basis, respectively. The site also has a short explanation of the algorithm (via the API reference URL). The picture on Wikipedia makes it a bit clearer I think.

So the first-level detail coefficients come from a highpass filter, and the final-level approximation coefficients come from a lowpass filter. The levels in between are repeated lowpass filtering (and usually subsampling) followed by one highpass filter. Comparable to edge detection after repeated smoothing. In other words, the bandpass property does not happen in a single step, see also this earlier answer.

alle_meije
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