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 cases we can perfectly describe a continuous-time signal x t using a countably-infinite (i.e., discrete) set of coefficients. Specifically, Fourier series enabled 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 countable index. 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 wavelet transform, 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, the DWT 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. To aid 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 be constructed 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 the remainder of our DWT discussions