# Connexions

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# Introduction to the Discrete Wavelet Transform

Module by: Phil Schniter. E-mail the author

Summary: (Blank Abstract)

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Definition 1: Wavelet
A wavelet is a family of orthonormal functions obtained by shifting and stretching a mother wavelet, Ψt Ψ t .

## Example

Ψ k , n t=2k2Ψ2ktn   for   kZ nZ Ψ k , n t 2 k 2 Ψ 2 k t n   for   k n
(1)

(Note: k,n:(Ψ=1)( Ψ k , n =1) k n Ψ 1 Ψ k , n 1 )

Definition 2: Discrete Wavelet Transform
The discrete wavelet transform is a representation of ft L 2 f t L 2 in terms of a countable set of wavelets.

## Example

ft= k , n k , n d k , n Ψ k , n t f t k , n k , n d k , n Ψ k , n t
(2)
d k , n = Ψ k , n t,ft= Ψ k , n t¯ftd t d k , n Ψ k , n t f t t Ψ k , n t f t
(3)
Definition 3: Multiresolution
The main idea is to construct Ψ k , n t Ψ k , n t so that larger values of k k correspond to higher levels of "detail" in the signal.

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