90
Biomedical Signal and Image Processing
This relation intuitively means that the energy of the wavelet coefficients for a frame
is bounded on both upper and lower sides by the true energy of the signal. In the case
of a basis (i.e., a minimal frame), the values A and B in the aforementioned inequality
become the same, i.e., A = B. This means that for a basis we have
N −1 M −1
2
E x = ∑
1
j =0
∑ W jk
(5.1 )
k =0
This indicates the energy of the coefficients is exactly the same as the energy of the
signal. This is something we saw in DFT before and reminds us that the complex
exponential basis used in the DFT indeed forms a basis. The next important question
to ask here is: “What are the properties of the functions that allow the function sets to
form a basis?” The most popular basis sets are the orthogonal ones, i.e., the function
sets whose members are orthogonal to each other. There is an extensive literature on
how orthogonal basis function sets for DWT are formed.
Despite the usefulness of DWT computed from continuous signals, this transformation is not very popular. This is due to the fact that the original time signals are
often discrete and not continuous. As a result, the next definition of the WT that is
computed over discrete signals is more popular.
5.4.1 DISCRETE WAVELET TRANSFORM ON DISCRETE SIGNALS
As indicated in our discussion of DFT, in almost all practical applications, signals are
formed of discrete measurements, and therefore in practice we normally deal with
sampled signals. This means that we need to focus on calculating DWT from discrete
signals. At this point, we assume that the discrete signal, if sampled from a continuous
signal, has been sampled according to the Nyquist rate (or faster). This guarantees that
all information of the continuous signal is preserved in the discrete signal. For such a
discrete signal, DWT can be calculated in different ways based on the exact type of
mother wavelets used for transformation. As mentioned in the previous section, the
best types of mother wavelets are the ones that form an orthogonal set.
The question here is how to form such basis sets systematically. The method
described next, called Mallat pyramidal algorithm or quadrature mirror filter (QMF),
allows systematic creation of an unlimited number of orthogonal basis sets for DWT.
The interesting feature of this method is the fact that the method relies only on the
choice of a digital low-pass filter h(n), and once this filter is chosen, the entire algorithm is rather mechanical and straightforward. In reality, the restrictions on h(n) are
so relaxed that many such filters can be easily found, and, therefore, many mother
wavelets can be formed based on different choices of the low-pass filter h(n). This
method can be best described using the schematic diagram of Figure 5.9.
Based on the QMF algorithm, the DWT for a one-dimensional (1-D) signal is
systematically calculated as follows. Assuming a digital filter h(n), we form another
filter g(n) as follows:
g n
( ) = h( 2N −1 − n)
(5.12)
Biomedical Signal and Image Processing
This relation intuitively means that the energy of the wavelet coefficients for a frame
is bounded on both upper and lower sides by the true energy of the signal. In the case
of a basis (i.e., a minimal frame), the values A and B in the aforementioned inequality
become the same, i.e., A = B. This means that for a basis we have
N −1 M −1
2
E x = ∑
1
j =0
∑ W jk
(5.1 )
k =0
This indicates the energy of the coefficients is exactly the same as the energy of the
signal. This is something we saw in DFT before and reminds us that the complex
exponential basis used in the DFT indeed forms a basis. The next important question
to ask here is: “What are the properties of the functions that allow the function sets to
form a basis?” The most popular basis sets are the orthogonal ones, i.e., the function
sets whose members are orthogonal to each other. There is an extensive literature on
how orthogonal basis function sets for DWT are formed.
Despite the usefulness of DWT computed from continuous signals, this transformation is not very popular. This is due to the fact that the original time signals are
often discrete and not continuous. As a result, the next definition of the WT that is
computed over discrete signals is more popular.
5.4.1 DISCRETE WAVELET TRANSFORM ON DISCRETE SIGNALS
As indicated in our discussion of DFT, in almost all practical applications, signals are
formed of discrete measurements, and therefore in practice we normally deal with
sampled signals. This means that we need to focus on calculating DWT from discrete
signals. At this point, we assume that the discrete signal, if sampled from a continuous
signal, has been sampled according to the Nyquist rate (or faster). This guarantees that
all information of the continuous signal is preserved in the discrete signal. For such a
discrete signal, DWT can be calculated in different ways based on the exact type of
mother wavelets used for transformation. As mentioned in the previous section, the
best types of mother wavelets are the ones that form an orthogonal set.
The question here is how to form such basis sets systematically. The method
described next, called Mallat pyramidal algorithm or quadrature mirror filter (QMF),
allows systematic creation of an unlimited number of orthogonal basis sets for DWT.
The interesting feature of this method is the fact that the method relies only on the
choice of a digital low-pass filter h(n), and once this filter is chosen, the entire algorithm is rather mechanical and straightforward. In reality, the restrictions on h(n) are
so relaxed that many such filters can be easily found, and, therefore, many mother
wavelets can be formed based on different choices of the low-pass filter h(n). This
method can be best described using the schematic diagram of Figure 5.9.
Based on the QMF algorithm, the DWT for a one-dimensional (1-D) signal is
systematically calculated as follows. Assuming a digital filter h(n), we form another
filter g(n) as follows:
g n
( ) = h( 2N −1 − n)
(5.12)