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8.3 Discrete wavelet transform: main concepts

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Main concepts

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

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

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

k n k n ψ k , n t 2 k 2 ψ 2 k t n
defines a family of wavelets ψ k , n t k n related by power-of-two stretches. As k increases, the wavelet stretches by a factor of two; as n increases, the wavelet shifts right.
When ψ t 1 , the normalization ensures that ψ k , n t 1 for all k , n .
Power-of-two stretching is a convenient, though somewhat arbitrary, choice. In our treatment of the discrete wavelettransform, however, we will focus on this choice. Even with power-of two stretches, there are various possibilities for ψ t , each giving a different flavor of DWT.

Wavelets are constructed so that ψ k , n t n ( i.e. , the set of all shifted wavelets at fixed scale k ), describes a particular level of 'detail' in the signal. As k 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, theDWT gives us a discrete multi-resolution description of a continuous-time signal in ℒ 2 .

In the modules that follow, these DWT concepts will be developed "from scratch" using Hilbert space principles. Toaid the development, we make use of the so-called scaling function φ t ℒ 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 beconstructed via shifts and power-of-two stretches

k n k n φ k , n t 2 k 2 φ 2 k t n
given mother scaling function φ t . The relationships between wavelets and scaling functions will be elaborated upon later via theory and example .
The inner-product expression for d k , n , 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 theremainder of our DWT discussions
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Source:  OpenStax, Intro to digital signal processing. OpenStax CNX. Jan 22, 2004 Download for free at http://cnx.org/content/col10203/1.4
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