<< Chapter < Page Chapter >> Page >
(Blank Abstract)

Main concepts

The discrete wavelet transform (DWT) is a representation of a signal x t 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 2 using a coefficient set parameterized by two countable indices: d k , n k n .

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

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Digital signal processing (ohio state ee700). OpenStax CNX. Jan 22, 2004 Download for free at http://cnx.org/content/col10144/1.8
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Digital signal processing (ohio state ee700)' conversation and receive update notifications?

Ask