Digital Signal Processing Reference
In-Depth Information
2.1. Wavelet families
The concept of wavelet has emerged and evolved during the last decades. Though new fam‐
ilies of wavelet transforms are rapidly increasing, there are a number of them that have been
established with more strength over time. In most situations, the use of a particular family is
set by the application.
Daubechies wavelets are the most used wavelets, representing the foundations of wavelets
signal processing and founding application in Discrete Wavelet Transform. They are defined
as a family of orthogonal and smooth basis wavelets characterized by a maximum number
of vanishing moments. The degree of smoothness increases as long as the order is higher.
Daubechies wavelets lead to more accurate results in comparison to others wavelet types
and also handle with boundary problems for finite length signals in an easier way [58] [29]
[60] [94]. Wavelets have not an explicit expression except for order 1, which is the Haar
wavelet. The inability to present a wavelet equation by a particular formula will be the gen‐
eral trend for almost all types of wavelet families [76].
As above mentioned, Haar wavelets are Daubechies wavelets when the order is 1. They are
the simplest orthonormal wavelets. The main drawback for Haar wavelets is their disconti‐
nuity as a consequence of not solving breaking points problems for its derivates. The Haar
transform is one of the earliest examples of a wavelet transform and it is supported by a
function is an odd rectangular pulse pair [33]. Haar functions are widely used for applica‐
tions as image coding, edge extraction and binary logic design and are defined as [46] [41]
[34] [30]:
1
ì
1
0
£ <
t
ï
= -
2
1
H(t)
1
£ <
t
1
(5)
í
ï
î
2
0
elsewhere
The main advantages of the Haar wavelet are its accuracy and fast implementation com‐
pared with others methods, its simplicity and small computational costs, and its capacity for
solving boundaries problems [87].
Symlet wavelet transform is an orthogonal wavelet defined by a scaling filter (a low-pass fi‐
nite impulse response filter of length
2N
and sum 1). Symlet wavelet transform is sometimes
called SymletN, where
N
is the order. Symlet wavelets are near symmetric. Furthermore,
they have highest number of vanishing moments for a given width [7].
Coiflet wavelets are a family of wavelets whose main characteristics are similar to the Sym‐
let ones: a high number of vanishing moments and symmetry. Coiflet family is also com‐
pactly supported, orthogonal and capable to give a good accuracy when the original signal
has a distortion. The Coiflet wavelets are defined for 5 orders [18].
