transforms have been one of the important signal processing developments in the
last decade, especially for the applications such as time-frequency analysis,
data compression, segmentation and vision. During the past decade, several efficient
implementations of wavelet transforms have been derived. The theory of wavelets
has roots in quantum mechanics and the theory of functions though a unifying framework
is a recent occurrence. Wavelet analysis is performed using a prototype function
called a wavelet.
functions defined over a finite interval and having an average value of zero.
The basic idea of the wavelet transform is to represent any arbitrary function
f (t) as a superposition of a set of such wavelets or basis functions. These basis
functions or baby wavelets are obtained from a single prototype wavelet called
the mother wavelet, by dilations or contractions (scaling) and translations (shifts).
Efficient implementation of the wavelet transforms has been derived based on the
Fast Fourier transform and short-length 'fast-running FIR algorithms' in order
to reduce the computational complexity per computed coefficient.
of all, why do we need a transform, or what is a transform anyway?
transformations are applied to signals to obtain further information from that
signal that is not readily available in the raw signal. Now, a time-domain signal
is assumed as a raw signal, and a signal that has been transformed by any available
transformations as a processed signal.
are a number of transformations that can be applied such as the Hilbert transform,
short-time Fourier transform, Wigner transform, the Radon transform, among which
the Fourier transform is probably the most popular transform. These mentioned
transforms constitute only a small portion of a huge list of transforms that are
available at engineers and mathematicians disposal. Each transformation technique
has its own area of application, with advantages and disadvantages.
Of The Frequency Information
the information that cannot be readily seen in the time-domain can be seen in
the frequency domain. Most of the signals in practice are time-domain signals
in their raw format. That is, whatever that signal is measuring, is a function
of time. In other words, when we plot the signal one of the axis is time (independent
variable) and the other (dependent variable) is usually the amplitude.
we plot time-domain signals, we obtain a time-amplitude representation of the
signal. This representation is not always the best representation of the signal
for most signal processing related applications.
In many cases, the most distinguished
information is hidden in the frequency content of the signal. The frequency spectrum
of a signal is basically the frequency components (spectral components) of that
signal. The frequency spectrum of a signal shows what frequencies exist in the
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