The discrete wavelet transform (DWT) is a representation of a signal xt∈ ℒ 2 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∈ℤ∧n∈ℤ} ψ k , n t k n , the DWT transform pair is 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|k∈ℤ} X k k , while the sampling theorem enabled us to describe bandlimited signals using signal samples {xn|n∈ℤ} 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 ℒ 2 ℒ 2 using a coefficient set parameterized by two countable indices: { d k , n |k∈ℤ∧n∈ℤ} d k , n k n .

Wavelets are orthonormal functions in ℒ 2 ℒ 2 obtained by shifting and stretching a mother wavelet, ψt∈ ℒ 2 ψ t ℒ 2 . For example, defines a family of wavelets { ψ k , n t|k∈ℤ∧n∈ℤ} 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. 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|n∈ℤ} 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 ℒ 2 ℒ 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∈ ℒ 2 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 be constructed via shifts and power-of-two stretches given mother scaling function φt φ t . The relationships between wavelets and scaling functions will be elaborated upon later via theory and example.