88
Biomedical Signal and Image Processing
belonging to parameterized family of functions that have different levels of complexity. For example, while all dbX functions look more or less similar, db2 is a simpler
mother wavelet than db3 or db4. As a rule of thumb, more complex mother wavelet
may be needed to analyze more complex signals. For instance, in order to process
medical signals such as ECG, one would limit X to 10 or 15, while, in processing of
complex signals such as speech signals, much more complex mother wavelets such
as db30 or even higher provide better performances.
Other popular mother wavelets are Mexican hat or sombrero (as shown previously), Coiflets, Symlets, Morlet, and Meyer. The details about the exact mathematical definition of these functions are outside the scope of this book, and the interested
readers are referred to the introduced references for further studies. In this book, we
will focus on the applications of the wavelets for biomedical signal processing.
The most logical question at this point is how to choose a mother wavelet for a
particular application. This question, to the most part, is an open problem, and it does
not appear to have a definite answer. However, two intuitive rules of thumb are widely
followed when choosing a mother wavelet: (1) complex mother wavelets are needed for
complex signals (as discussed earlier) and (2) the mother wavelet that resembles the
general shape of the signal to be analyzed would be a more suitable choice.
As in the CFT, there are some major issues with the CWT that encourages the invention and use of a discrete version of the CWT. While some of the concerns are the same
as the concerns applicable to the CFT (such as the dominance of the digital processing
and storage systems), the CWT suffers from a more computationally serious problem.
A closer look at the CWT reveals that this transformation requires the calculations based
on all continuous shifts and all continuous scales. This obviously makes the computational complexity of the CWT and the ICWT unsuitable for many practically important
applications. This leads us to the discrete version of this transform.
5.4 ONE-DIMENSIONAL DISCRETE WAVELET TRANSFORM
Discrete wavelet transform (DWT) accepts continuous signals and applies only discrete shifts and scales to form the transform. This means that if the original signal
is sampled with a suitable set of scaling and shifting, the entire continuous signal
can be reconstructed from the DWT. In order to see how this is done, we start with
providing the equations for the DWT. Define
a
a
j , b jk = k
j
jk = 0
a 0 T
(5.5)
where
T is the sampling time
a 0 is a positive nonzero constant
Also define
1
⎛ t b
−
Ψ t
jk ⎞
jk ( ) =
Ψ
=
j
⎜
⎟ a
− 2
0
Ψ a
− j
0 t − kT
(5.6)
a jk ⎝ a jk ⎠
(
)
Biomedical Signal and Image Processing
belonging to parameterized family of functions that have different levels of complexity. For example, while all dbX functions look more or less similar, db2 is a simpler
mother wavelet than db3 or db4. As a rule of thumb, more complex mother wavelet
may be needed to analyze more complex signals. For instance, in order to process
medical signals such as ECG, one would limit X to 10 or 15, while, in processing of
complex signals such as speech signals, much more complex mother wavelets such
as db30 or even higher provide better performances.
Other popular mother wavelets are Mexican hat or sombrero (as shown previously), Coiflets, Symlets, Morlet, and Meyer. The details about the exact mathematical definition of these functions are outside the scope of this book, and the interested
readers are referred to the introduced references for further studies. In this book, we
will focus on the applications of the wavelets for biomedical signal processing.
The most logical question at this point is how to choose a mother wavelet for a
particular application. This question, to the most part, is an open problem, and it does
not appear to have a definite answer. However, two intuitive rules of thumb are widely
followed when choosing a mother wavelet: (1) complex mother wavelets are needed for
complex signals (as discussed earlier) and (2) the mother wavelet that resembles the
general shape of the signal to be analyzed would be a more suitable choice.
As in the CFT, there are some major issues with the CWT that encourages the invention and use of a discrete version of the CWT. While some of the concerns are the same
as the concerns applicable to the CFT (such as the dominance of the digital processing
and storage systems), the CWT suffers from a more computationally serious problem.
A closer look at the CWT reveals that this transformation requires the calculations based
on all continuous shifts and all continuous scales. This obviously makes the computational complexity of the CWT and the ICWT unsuitable for many practically important
applications. This leads us to the discrete version of this transform.
5.4 ONE-DIMENSIONAL DISCRETE WAVELET TRANSFORM
Discrete wavelet transform (DWT) accepts continuous signals and applies only discrete shifts and scales to form the transform. This means that if the original signal
is sampled with a suitable set of scaling and shifting, the entire continuous signal
can be reconstructed from the DWT. In order to see how this is done, we start with
providing the equations for the DWT. Define
a
a
j , b jk = k
j
jk = 0
a 0 T
(5.5)
where
T is the sampling time
a 0 is a positive nonzero constant
Also define
1
⎛ t b
−
Ψ t
jk ⎞
jk ( ) =
Ψ
=
j
⎜
⎟ a
− 2
0
Ψ a
− j
0 t − kT
(5.6)
a jk ⎝ a jk ⎠
(
)