# Discrete wavelet transform: main concepts

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## Main concepts

The discrete wavelet transform (DWT) is a representation of a signal $x(t)\in {ℒ}_{2}$ using an orthonormal basis consisting of a countably-infinite set of wavelets . Denoting the wavelet basis as $\{{\psi }_{k,n}(t)\colon k\in \mathbb{Z}\land n\in \mathbb{Z}\}$ , the DWT transform pair is

$x(t)=\sum_{k=()}$ n d k , n ψ k , n t
${d}_{k,n}={\psi }_{k,n}(t)\dot x(t)=\int_{()} \,d t$ 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 $\{X(k)\colon k\in \mathbb{Z}\}$ , while the sampling theorem enabled us to describe bandlimited signals using signal samples $\{x(n)\colon n\in \mathbb{Z}\}$ . 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: $\{{d}_{k,n}\colon k\in \mathbb{Z}\land n\in \mathbb{Z}\}$ .

Wavelets are orthonormal functions in ${ℒ}_{2}$ obtained by shifting and stretching a mother wavelet , $\psi (t)\in {ℒ}_{2}$ . For example,

$\forall k, n, (k\land n)\in \mathbb{Z}\colon {\psi }_{k,n}(t)=2^{-\left(\frac{k}{2}\right)}\psi (2^{-k}t-n)$
defines a family of wavelets $\{{\psi }_{k,n}(t)\colon k\in \mathbb{Z}\land n\in \mathbb{Z}\}$ 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 $(\psi (t))=1$ , the normalization ensures that $({\psi }_{k,n}(t))=1$ for all $k\in \mathbb{Z}$ , $n\in \mathbb{Z}$ .
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 $\psi (t)$ , each giving a different flavor of DWT.

Wavelets are constructed so that $\{{\psi }_{k,n}(t)\colon n\in \mathbb{Z}\}$ ( 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 $\phi (t)\in {ℒ}_{2}$ , 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

$\forall k, n, (k\land n)\in \mathbb{Z}\colon {\phi }_{k,n}(t)=2^{-\left(\frac{k}{2}\right)}\phi (2^{-k}t-n)$
given mother scaling function $\phi (t)$ . 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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