# Difference(relation?) between filter banks and wavelet decomposition

I'm implementing multirate system and met concepts of filter banks and polyphase decomposition of filters.

While trying to understand them, I've found that they look very similar to wavelet decomposition, and now I'd like to understand their relation and differences.

For me, the only thing is kernel types of filters (usually we use sinc kernels in filters, and other, like Haar, in wavelets).

If we stick to the linear version and discrete versions of filter banks and wavelets, filter banks represent the generic tool, and wavelets can be implemented as a specific instance of iterated $2$-band filter banks satisfying some additional properties, namely that low-pass spaces are embedded dyadically.

In other words: get a single-level $2$-band perfect reconstruction filter-bank, add some regularity so that iterations converge toward something legit, and you have a concrete Discrete Wavelet Transformation.

Kernel types have almost nothing to do with it, from a theoretical perspective. Kernels are chosen with respect to, mainly, implementation (fast or not) or signal properties (regularity).

That was the simple answer. If you want to dig deeper into it, you can look at continuous wavelets (that remain filter banks, as the admissibility condition impose a wavelet to be, somehow, a bandpass), nonlinear wavelets, morphological or hybrid filter banks, etc.

When I was a student, I was teached filter banks by Maurice Bellanger, one (the?) father of polyphase decomposition (Digital filtering by polyphase network: Application to sample-rate alteration and filter banks, 1976). As a young person, I just heard about wavelets, and asked him about it. His answer was: "they are just some special filter banks".

• Bellanger's Digital Processing of signals, Theory and Practice is my go-to ressource when looking for a clear, concise, correct notation for something. I kinda envy you now for this education! – Marcus Müller Dec 18 '16 at 13:04
• But I fear I was too young and immature to fully benefit from it – Laurent Duval Dec 18 '16 at 13:19
• ha, I hope that feeling never ceases to overwhelm me after learning something new – imagine one would be awe-struck by great ideas or teachers when one could instead take as much from it as one is currently mentally (and maybe emotionally?Physically?As a student, learning from lectures definitely wasn't my only concern…) process and then, later on, build on that when actually using that stuff. Also, I'd do my multirate systems professor wrong if I didn't mention that his book is very good and practical. – Marcus Müller Dec 18 '16 at 13:25
• Nice reference, I had a few papers from Göckler on FBs, never knew there was a book. I hope I can find a copy one day (I can read German a little) – Laurent Duval Dec 18 '16 at 13:50
• Yeah, he was kind of a guest lecturer in Karlsruhe - basically coming back from retirement once a week to hold the lecture that used to be held at his former university in Bochum. Very interesting, since Bochum seems to put more focus on filter theory and design than Karlsruhe does – Marcus Müller Dec 18 '16 at 16:56