0.1 Introduction to wavelets  (Page 2/5)

For a Fourier series, the orthogonal basis functions ${\psi }_k\left(t\right)$ are $sin\left(k{\omega }_{0}t\right)$ and $cos\left(k{\omega }_{0}t\right)$ with frequencies of $k{\omega }_0$ . For a Taylor'sseries, the nonorthogonal basis functions are simple monomials $t^k$ , and for many other expansions they are various polynomials. There areexpansions that use splines and even fractals.

For the wavelet expansion , a two-parameter system is constructed such that [link] becomes

where both $j$ and $k$ are integer indices and the ${\psi }_{j,k}\left(t\right)$ are the wavelet expansion functions that usually form an orthogonal basis.

The set of expansion coefficients $a_{j,k}$ are called the discrete wavelet transform (DWT) of $f\left(t\right)$ and [link] is the inverse transform.

What is a wavelet system?

The wavelet expansion set is not unique. There are many different wavelets systems that can be used effectively, but all seem to have the following threegeneral characteristics [link] .

1. A wavelet system is a set of building blocks to construct or represent a signal or function. It is a two-dimensional expansion set(usually a basis) for some class of one- (or higher) dimensional signals. In other words, if the wavelet set is given by ${\psi }_{j,k}\left(t\right)$ for indices of $j,k=1,2,\cdots$ , a linear expansion would be $f\left(t\right)={\sum }_k{\sum }_j{a}_{j,k}{\psi }_{j,k}\left(t\right)$ for some set of coefficients $a_{j,k}$ .
2. The wavelet expansion gives a time-frequency localization of the signal. This means most of the energy of the signal is well representedby a few expansion coefficients, $a_{j,k}$ .
3. The calculation of the coefficients from the signal can be done efficiently . It turns out that many wavelet transforms (the set of expansion coefficients) can be calculated with $O\left(N\right)$ operations. This means the number of floating-point multiplications and additions increaselinearly with the length of the signal. More general wavelet transforms require $O(Nlog(N\left)\right)$ operations, the same as for the fast Fourier transform (FFT) [link] .

Virtually all wavelet systems have these very general characteristics. Where the Fourier series maps a one-dimensional function of a continuousvariable into a one-dimensional sequence of coefficients, the wavelet expansion maps it into a two-dimensional array of coefficients. We willsee that it is this two-dimensional representation that allows localizing the signal in both time and frequency. A Fourier series expansionlocalizes in frequency in that if a Fourier series expansion of a signal has only one large coefficient, then the signal is essentially a singlesinusoid at the frequency determined by the index of the coefficient. The simple time-domain representation of the signal itself gives thelocalization in time. If the signal is a simple pulse, the location of that pulse is the localization in time. A wavelet representation willgive the location in both time and frequency simultaneously. Indeed, a wavelet representation is much like a musical score where the location ofthe notes tells when the tones occur and what their frequencies are.

More specific characteristics of wavelet systems

There are three additional characteristics [link] , [link] that are more specific to wavelet expansions.

1. All so-called first-generation wavelet systems are generated from a single scaling function or wavelet by simple scaling and translation . The two-dimensional parameterization is achieved from the function(sometimes called the generating wavelet or mother wavelet) $\psi \left(t\right)$ by
   $${\psi }_{j,k}\left(t\right)=2^{j/2}\psi (2^{j}t-k)j,k\in \mathbf{Z}$$
   where $\mathbf{Z}$ is the set of all integers and the factor $2^{j/2}$ maintains a constant norm independent of scale $j$ . This parameterization of the time or space location by $k$ and the frequency or scale (actually the logarithm of scale) by $j$ turns out to be extraordinarily effective.
2. Almost all useful wavelet systems also satisfy the multiresolution conditions. This means that if a set of signals can be represented by a weighted sum of $\varphi (t-k)$ , then a larger set (including the original) can be represented by a weighted sum of $\varphi (2t-k)$ . In other words, if the basic expansion signals are made half as wide and translated in steps halfas wide, they will represent a larger class of signals exactly or give a better approximation of any signal.
3. The lower resolution coefficients can be calculated from the higher resolution coefficients by a tree-structured algorithm called a filter bank . This allows a very efficient calculation of the expansion coefficients (also known as the discrete wavelet transform) and relateswavelet transforms to an older area in digital signal processing.