Spectral Transformations - Biomedical Signal Analysis: Contemporary Methods and Applications - page 37

S ( k, n )= 2

N +1 sin ( π ( n +1)( k +1)

1
(2.21)
Its basis sequences in the orthonormal transformation are sine functions.
Both DCT and DST have excellent information concentration properties
since they concentrate most of the energy in a few coecients.
Other important transforms are the Haar, wavelet, Hadamard, and
Walsh transforms [48, 264]. Because of the powerful properties of the
wavelet transform and its extensive application opportunities in biomed-
ical engineering, the next section is dedicated solely to the wavelet trans-
form.

) , , n =0 , 1 ,...,N −

N +1

2.2 The Wavelet Transform

Modern transform techniques such as the wavelet transform are gain-
ing an increasing importance in biomedical signal and image processing.
They provide enhanced processing capabilities compared to the tradi-
tional ones in terms of denoising, compression, enhancement, and edge
and feature extraction. These techniques fall under the categories of mul-
tiresolution analysis, time-frequency analysis, or pyramid algorithms.
The wavelet transform is based on wavelets, which are small waves of
varying frequency and limited duration, and thus represents a devia-
tion from the traditional Fourier transform concept that has sinusoids
as basis functions. In addition to the traditional Fourier transform, they
provide not only frequency but also temporal information on the signal.
In this section, we present the theory and the different types of
wavelet transforms. A wavelet represents a basis function in continuous
time and can serve as an important component in a function represen-
tation: any function f ( t ) can be represented by a linear combination
of basis functions, such as wavelets. The most important aspect of the
wavelet basis is that all wavelet functions are constructed from a single
mother wavelet. This wavelet is a small wave or a pulse.
Wavelet transforms are an alternative to the short-time Fourier
transform. Their most important feature is that they analyze differ-
ent frequency components of a signal with different resolutions. In other
words, they address exactly the concern raised in connection with the