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

6.2. BASIC IDEAS OF THE WAVELET TRANSFORM 50
For a function f(t) and a mother wavelet ψ(t), the corresponding discrete form of the
continuous wavelet transform is then:
Wψf(j, n) = 2 −j/2
�
f(t)ψ ∗
� j t − 2 n
2j �
dt (6.10)
In such a way, the discretization of the continuous wavelet transform is done. Since
only the scale and the translation are discrete in the power of two and the time variable
is continuous, the discrete transform in Equation 6.10 is called the continuous time
wavelet series or the dyadic wavelet transform.
A discrete form of the continuous wavelet transform is specified by the sampling
choice of the scale and the translation in the time-scale plane and the choice of the
mother wavelet. It is clear that there might be infinite such choices but certainly that
the choices cannot be made arbitrarily. The limit is that such the discretization should
satisfy a basic property, namely, invertibility.
This very practical filtering algorithm yields a fast wavelet transform - a box into which
a signal passes, and out of which wavelet coefficients quickly emerge. Let’s examine
this in more depth.
One-Stage Filtering: Approximations and Details
For many signals, the low-frequency content is the most important part. It is what gives
the signal its identity. The high-frequency content, on the other hand, imparts nuance.
It is like the human voice. Removing of high-frequency components, causes the voice
sounding different, but it can be understand. However, enough removing of the low-
frequency components, causes the voice not to be understood. In wavelet analysis, this
parts are called approximations and details. The approximations are the high-scale,
low-frequency components of the signal. The details are the low-scale, high-frequency
components.
The one stage filtering process, at its most basic level, looks like this:
To get as much samples as the input signals, the approximated and the detailed part