A wavelet is a mathematical function used to divide a given function or continuous-time signal into different frequency components and study each component with a resolution that matches its scale. A wavelet transform is the representation of a function by wavelets. The wavelets are scaled and translated copies (known as "daughter wavelets") of a finite-length or fast-decaying oscillating waveform (known as the "mother wavelet"). Wavelet transforms have advantages over traditional Fourier transforms for representing functions that have discontinuities and sharp peaks, and for accurately deconstructing and reconstructing finite, non-periodic and/or non-stationary signals.
In formal terms, this representation is a wavelet series representation of a square-integrable function with respect to either a complete, orthonormal set of basis functions, or an overcomplete set of Frame of a vector space (also known as a Riesz basis), for the Hilbert space of square integrable functions.
Wavelet transforms are classified into discrete wavelet transforms (DWTs) and continuous wavelet transforms (CWTs). Note that both DWT and CWT are continuous-time (analog) transforms. They can be used to represent continuous-time (analog) signals. CWTs operate over every possible scale and translation whereas DWTs use a specific subset of scale and translation values or representation grid. The word "wavelet" is due to Morlet and Grossmann in the early 1980s. They used the French word "ondelette", meaning "small wave". Soon it was transferred to English by translating "onde" into "wave", giving "wavelet".
Wavelet theory is applicable to several subjects. All wavelet transforms may be considered forms of time-frequency representation for continuous-time (analog) signals and so are related to harmonic analysis. Almost all practically useful discrete wavelet transforms use discrete-time filterbanks. These filter banks are called the wavelet and scaling coefficients in wavelets nomenclature. These filterbanks may contain either finite impulse response (FIR) or infinite impulse response (IIR) filters. The wavelets forming a CWT are subject to the uncertainty principle of Fourier analysis respective sampling theory: Given a signal with some event in it, one cannot assign simultaneously an exact time and frequency response scale to that event. The product of the uncertainties of time and frequency response scale has a lower bound. Thus, in the scaleogram of a continuous wavelet transform of this signal, such an event marks an entire region in the time-scale plane, instead of just one point. This is related to Heisenberg's uncertainty principle of quantum physics and has a similar derivation. Also, discrete wavelet bases may be considered in the context of other forms of the uncertainty principle.
Wavelet transforms are broadly divided into three classes: continuous, discretised and multiresolution-based.
Continuous wavelet transforms (Continuous Shift & Scale Parameters)
In continuous wavelet transforms, a given signal of finite energy is projected on a continuous family of frequency bands (or similar subspaces of the Lp function space ).For instance the signal may be represented on every frequency band of the form for all positive frequencies "f>0". Then, the original signal can be reconstructed by a suitable integration over all the resulting frequency components.
The frequency bands or subspaces (sub-bands) are scaled versions of a subspace at scale "1". This subspace in turn is in most situations generated by the shifts of one generating function , the "mother wavelet". For the example of the scale one frequency band this function is :with the (normalized) sinc function. Other example mother wavelets are:
The subspace of scale "a" or frequency band is generated by the functions (sometimes called "child wavelets"):,where "a" is positive and defines the scale and "b" is any real number and defines the shift. The pair "(a,b)" defines a point in the right halfplane .
The projection of a function "x" onto the subspace of scale "a" then has the form : with "wavelet coefficients":.
See a list of some Continuous wavelets.
For the analysis of the signal "x", one can assemble the wavelet coefficients into a scaleogram of the signal.
Discrete wavelet transforms (Discrete Shift & Scale parameters)
It is computationally impossible to analyze a signal using all wavelet coefficients, so one may wonder if it is sufficient to pick a discrete subset of the upper halfplane to be able to reconstruct a signal from the corresponding wavelet coefficients. One such system is the affine system for some real parameters "a>1", "b>0". The corresponding discrete subset of the halfplane consists of all the points with integers . The corresponding "baby wavelets" are now given as:.
A sufficient condition for the reconstruction of any signal "x" of finite energy by the formula:is that the functions form a tight frame of .
Multiresolution-based discrete wavelet transforms
In any discretised wavelet transform, there are only a finite number of wavelet coefficients for each bounded rectangular region in the upper halfplane. Still, each coefficient requires the evaluation of an integral. To avoid this numerical complexity, one needs one auxiliary function, the "father wavelet" . Further, one has to restrict "a" to be an integer. A typical choice is "a=2" and "b=1". The most famous pair of father and mother wavelets is the Daubechies 4 tap wavelet.
From the mother and father wavelets one constructs the subspaces :, where and :, where .From these one requires that the sequence:forms a multiresolution analysis of and that the subspaces are the orthogonal "differences" of the above sequence, that is, is the orthogonal complement of inside the subspace . In analogy to the sampling theorem one may conclude that the space with sampling distance more or less covers the frequency baseband from "0" to . As orthogonal complement, roughly covers the band .
From those inclusions and orthogonality relations follows the existence of sequences and that satisfy the identities: and and : and .The second identity of the first pair is a refinement equation for the father wavelet . Both pairs of identities form the basis for the algorithm of the fast wavelet transform.
For practical applications, and for efficiency reasons, one prefers continuously differentiable functions with compact support as mother (prototype) wavelet (functions). However, to satisfy analytical requirements (in the continuous WT) and in general for theoretical reasons, one chooses the wavelet functions from a subspace of the space This is the space of measurable functions that are absolutely and square integrable: :