Tuesday, 5 July 2016

Why is a wavelet transform implemented as a filter bank?


The mother wavelet function ψ(t) must satisfy the following:


+|ψ(ω)|2ωdω<+, ψ(ω)|ω=0=0, and +ψ(t) dt=0


To serve as the wavelet basis for wavelet transform γ(s,τ)=+f(t) ψs,τ(t) dt


where ψs,τ(t)ψ(tτs).


While I understand that the wavelet must be an oscillatory function having no frequency component at ω=0 and effectively have a band pass filter like spectrum, from the equation of wavelet series or wavelet transform can you tell me why is it that the wavelet transform is implemented as a filter bank? What is the intuition behind it? What makes it possible?


I am asking this question since the fact that practically the DWT is implemented as a filter bank means that it is not a DWT anymore, it is just a set of low pass and high pass filters. It is mind bogling.




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