Development of a wavelet-based algorithm to detect and determine ...

6.2. BASIC IDEAS OF THE WAVELET TRANSFORM 36
Suppose that a sequence {ϕn(t)} is a base for a space, and then any functions f(t) in
the space can be represented with this base. Namely, there is Equation 6.1.
f(t) = � Inϕn(t) n ∈ Z (6.1)
Where In is a scalar sequence generated through the inner product of f(t) and ϕn(t), as
shown in equation 6.2.
{In} = 〈f(t), ϕn(t)〉 =
Where ϕ ∗ n(t) is the complex conjugation of the function ϕn(t).
�
+∞
−∞
f(t)ϕ ∗ n(t)dt (6.2)
Equations 6.1 and equation 6.2 form a pair of transforms. Equation 6.2 is called the (for-
ward) transform or the decomposition because it breaks a function into pieces through
the base for the space. Equation 6.1 is called the inverse transform or the reconstruction
because it puts the pieces back together to retrieve the original signal. It is clear that
the property of a base is determinate for the transform.
A transform theory always involves how to break a function apart (transform) and
then to put it back together (inverse transform). The reason for doing this is that sig-
nificant insights, clear patterns, and efficiencies can be obtained through the operation
on the pieces rather than the original function. Electrical engineers are familiar to many
transforms: Fourier transform, Laplace transform, z transform, symmetric components
transform and so on. These transforms all have own well working areas.
The wavelet transform is nothing else but another transform. It represents signals
through the space base that have the expected characteristics.
Representation of signals through the wavelet transform
Wavelets and the admissible condition: A wavelet belongs to one kind of functions
which have the following characteristics:
• The amplitude quickly decays to zero in both directions.