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Discrete Wavelet Transform: Main Concepts

Module by: Phil Schniter. E-mail the author

Summary: (Blank Abstract)

Main Concepts

The discrete wavelet transform (DWT) is a representation of a signal xt 2 x t 2 using an orthonormal basis consisting of a countably-infinite set of wavelets. Denoting the wavelet basis as ψ k , n t kZ and nZ ψ k , n t k n , the DWT transform pair is

xt=k=n= d k , n ψ k , n t x t k n d k , n ψ k , n t
(1)
d k , n = ψ k , n t,xt= ψ k , n t*xtdt d k , n ψ k , n t x t t ψ k , n t x t
(2)
where d k , n d k , n are the wavelet coefficients. Note the relationship to Fourier series and to the sampling theorem: in both cases we can perfectly describe a continuous-time signal xt x t using a countably-infinite (i.e., discrete) set of coefficients. Specifically, Fourier series enabled us to describe periodic signals using Fourier coefficients Xk kZ X k k , while the sampling theorem enabled us to describe bandlimited signals using signal samples xn nZ x n n . In both cases, signals within a limited class are represented using a coefficient set with a single countable index. The DWT can describe any signal in 2 2 using a coefficient set parameterized by two countable indices: d k , n kZ and nZ d k , n k n .

Wavelets are orthonormal functions in 2 2 obtained by shifting and stretching a mother wavelet, ψt 2 ψ t 2 . For example,

ψ k , n t=2k2ψ2ktn  ,   k and nZ    k n k n ψ k , n t 2 k 2 ψ 2 k t n
(3)
defines a family of wavelets ψ k , n t kZ and nZ ψ k , n t k n related by power-of-two stretches. As kk increases, the wavelet stretches by a factor of two; as nn increases, the wavelet shifts right.

note:

When ψt=1 ψ t 1 , the normalization ensures that ψ k , n t=1 ψ k , n t 1 for all kZ k , nZ n .
Power-of-two stretching is a convenient, though somewhat arbitrary, choice. In our treatment of the discrete wavelet transform, however, we will focus on this choice. Even with power-of two stretches, there are various possibilities for ψt ψ t , each giving a different flavor of DWT.

Wavelets are constructed so that ψ k , n t nZ ψ k , n t n (i.e., the set of all shifted wavelets at fixed scale kk), describes a particular level of 'detail' in the signal. As kk becomes smaller (i.e., closer to ), the wavelets become more "fine grained" and the level of detail increases. In this way, the DWT can give a multi-resolution description of a signal, very useful in analyzing "real-world" signals. Essentially, the DWT gives us a discrete multi-resolution description of a continuous-time signal in 2 2 .

In the modules that follow, these DWT concepts will be developed "from scratch" using Hilbert space principles. To aid the development, we make use of the so-called scaling function φt 2 φ t 2 , which will be used to approximate the signal up to a particular level of detail. Like with wavelets, a family of scaling functions can be constructed via shifts and power-of-two stretches

φ k , n t=2k2φ2ktn  ,   k and nZ    k n k n φ k , n t 2 k 2 φ 2 k t n
(4)
given mother scaling function φt φ t . The relationships between wavelets and scaling functions will be elaborated upon later via theory and example.

note:

The inner-product expression for d k , n d k , n , Equation 2 is written for the general complex-valued case. In our treatment of the discrete wavelet transform, however, we will assume real-valued signals and wavelets. For this reason, we omit the complex conjugations in the remainder of our DWT discussions

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