5
Wavelet Transform
5.1 INTRODUCTION AND OVERVIEW
This chapter is dedicated to the concepts and applications of the wavelet transform
(WT). The WT has become an essential tool for all types of signal and image processing applications as this transformation provides capabilities that may not be
achieved by other transformations. We start this chapter with justifying the need
for such a transform and then take an intuitive approach toward the definition of
the WT.
5.2 FROM FT TO STFT
Despite the numerous capabilities of the Fourier transform (FT), there exists a serious concert over the use of the FT for certain applications. This concern can be better described using the following examples.
Example 5.1
Consider the two time signals shown in Figure 5.1. Each of the signals is formed of
three sinusoidal components with the same duration. The only difference between
the two signals is the order at which these sinusoidal components appear in the
signal. It is evident that order or relative location of these three components is
indeed an important characteristic that allows differentiating the two signals from
each other.
Next, we calculate the magnitude of the discrete Fourier transform (DFT) for
these two signals, as in Figure 5.2. As can be seen in Figure 5.2, the magnitude
of the DFT for the two signals is exactly the same. In other words, if we limit
ourselves only to the magnitude of the DFT, all the information regarding the order
of the three sinusoidal components is lost.
Before leaving this example, we have to emphasize that the order information
is not truly lost; rather, the order information is contained in the phase of the DFT
of the signals. In other words, if someone observes and interprets the phase of the
two signals, he or she must be able to discover the order of the three sinusoidal
components, even though this may not be an easy task at all. However, as mentioned in the previous chapters, interpreting and dealing with the phase of the DFT
of a signal is often considered as a relatively difficult task and one would rather
focus on the magnitude of DFT only.
In the next example, the shortcomings of relying only on the magnitude of the
FT are further emphasized.
79
Wavelet Transform
5.1 INTRODUCTION AND OVERVIEW
This chapter is dedicated to the concepts and applications of the wavelet transform
(WT). The WT has become an essential tool for all types of signal and image processing applications as this transformation provides capabilities that may not be
achieved by other transformations. We start this chapter with justifying the need
for such a transform and then take an intuitive approach toward the definition of
the WT.
5.2 FROM FT TO STFT
Despite the numerous capabilities of the Fourier transform (FT), there exists a serious concert over the use of the FT for certain applications. This concern can be better described using the following examples.
Example 5.1
Consider the two time signals shown in Figure 5.1. Each of the signals is formed of
three sinusoidal components with the same duration. The only difference between
the two signals is the order at which these sinusoidal components appear in the
signal. It is evident that order or relative location of these three components is
indeed an important characteristic that allows differentiating the two signals from
each other.
Next, we calculate the magnitude of the discrete Fourier transform (DFT) for
these two signals, as in Figure 5.2. As can be seen in Figure 5.2, the magnitude
of the DFT for the two signals is exactly the same. In other words, if we limit
ourselves only to the magnitude of the DFT, all the information regarding the order
of the three sinusoidal components is lost.
Before leaving this example, we have to emphasize that the order information
is not truly lost; rather, the order information is contained in the phase of the DFT
of the signals. In other words, if someone observes and interprets the phase of the
two signals, he or she must be able to discover the order of the three sinusoidal
components, even though this may not be an easy task at all. However, as mentioned in the previous chapters, interpreting and dealing with the phase of the DFT
of a signal is often considered as a relatively difficult task and one would rather
focus on the magnitude of DFT only.
In the next example, the shortcomings of relying only on the magnitude of the
FT are further emphasized.
79