Page 276 - Introduction to Information Optics

5.4. Wavelet Transform 261

The Fourier transform of the wavelet is

H v r((o) = -j= h ( - — - ] exp( -jct)t) dt = ^s H(so)) (exp - jari), (5. 10)

where H((o) is the Fourier transform of the basic wavelet h(t). In the frequency
domain the wavelet is scaled by 1/s, multiplied by a phase factor exp(— J
l/2
and by a normalization factor s .

5.4.2. TIME-FREQUENCY JOINT REPRESENTATION

According to Eq. (5.9), the wavelet transform of a one-dimensional (ID)
signal is a two-dimensional (2D) function of the scale s and the time shift T.
The wavelet transform is a mapping of the ID time signal to a 2D time-scale
joint representation of the signal. The time-scale joint wavelet representation
is equivalent to the time-frequency joint representation, which is familiar in
the analysis of nonstationary and fast transient signals.
A signal is stationary if its properties do not change during the course of the
signal. Most signals in nature are nonstationary. Examples of nonstationary
signals are speech, radar, sonar, seismic, electrocardiographic signals, music,
and two-dimensional images. The properties, such as the frequency spectrum,
of a nonstationary signal change during the course of the signal.
In the case of the music signal, for instance, the music signal can be
represented by a ID time function of air pressure, or equivalently by ID
Fourier transform of the air pressure function. We know that a music signal
consists of very rich frequency components.
The Fourier spectrum of a time signal is computed by the Fourier trans-
form, which should integrate the signal from minus infinity to plus infinity in
the time axis. However, the Fourier spectrum of the music signal must change
with time. There is a contradiction between the infinity integral limits in the
mathematical definition of the Fourier transform frequency and the non-
stationary nature of the music signal. The solution is to introduce the local
Fourier transform and the local frequency concept. Indeed, nonstationary
signals are in general characterized by their local features rather than by their
global features.
A musician who plays a piece of music uses neither the representation of the
music as a ID time function of the air pressure, nor its ID Fourier transform.
Instead, he prefers to use the music note, which tells him at a given moment
which key of the piano he should play. The music note is, in fact, the
time frequency joint representation of the signal, which better represents a
nonstationary signal by representing the local properties of the signal.