1. Jordan University of Science And Technology
Electrical Engineering Department
Graduation Project Two
EE 592
Discrete Wavelet Analysis and Applications to
ECG and PCG Signals
By
Serene Zawaydeh
Supervisor:
Dr. Khaled Mayyas
December 31, 1997

2. 2
Acknowledgment
I dedicate this project
To my parents, brother and sisters
Special thanks go to
Dr. Khaled Mayyas and to Dr. Bassam Al Asir.

3. 3
Table of Contents
Table of Contents ............................................................................................................................................3
Introduction to Wavelets .................................................................................................................................4
Non-stationary Signal Analysis.......................................................................................................................8
STFT Transform: Analysis with Fixed Resolution..........................................................................................8
Properties of the Short Time Fourier Transform ...........................................................................................10
Spectrogram...................................................................................................................................................11
The Continuous Wavelet Transform .............................................................................................................11
Scalograms ....................................................................................................................................................12
Efficiency of the Wavelet Transform ............................................................................................................14
Resolution and Scale of Discrete Signals ......................................................................................................14
Signal Analysis with Multi-resolution...........................................................................................................16
The Discrete Wavelet Transform and Filter Banks .......................................................................................17
Basis of Orthonormal Wavelets Constructed from Filter banks....................................................................17
Wavelet Packets ............................................................................................................................................23
Nineteen level Wavelet Packet Structure ......................................................................................................26
Cutoff frequencies of the Filters in the Wavelet Packet Structure.................................................................26
Implementation of Wavelet Packet Structure................................................................................................32
Coefficients of the Prototype Filters Applied to the Filter Bank ...................................................................32
Frequency Responses of the Prototype Filters:.........................................................................................34
Frequency Responses of The Equivalent Filters of the 19-level Filter Bank.............................................41
Verified Properties of the Scaling Function ..................................................................................................42
Verified Properties of Wavelets ....................................................................................................................44
Electrocardiography ......................................................................................................................................45
The Heart Sounds ..........................................................................................................................................46
Analyzing ECG signals .................................................................................................................................47
ECG signals as inputs to the filter bank....................................................................................................47
Energy in analyzed ECG Using Different Prototype Filters .....................................................................57
A Second ECG signal ................................................................................................................................57
Analyzing Heart Sound Signals.....................................................................................................................61
Effect of the sampling frequency ...............................................................................................................66
Conclusion.....................................................................................................................................................67
References .....................................................................................................................................................69
References to Wavelets on the Web..............................................................................................................70
APPENDIX 1 ................................................................................................................................................71
PART 1: Outputs of Normal and Abonormal ECG Signal with Late Potentials .......................................72
PART 2: Outputs of Late Potentials using Daubechies4, 6, and Coiflets 6...............................................92
Part 3: Outputs of Normal Heart sound, and two Abnormal Cases .........................................................113
APPENDIX 2 ..............................................................................................................................................134
Historical Difficulties Faced in Understanding the Cardiac Cycle ........................................................135
The Cell as a Bioelectric Generator........................................................................................................135
Electro Encephalogram (EEG) ...............................................................................................................136

4. 4
Introduction to Wavelets
The classical Fourier transform is suitable for the analysis of stationary
signals, which have constant statistical mean and variance. The basis
functions in the Fourier transform are the infinite length, periodic sinusoids
with their fixed shape, which makes it efficient for the analysis of signals
with naturally occurring sinusoidal behavior.
The analysis of nonstationary signals and signals with discontinuities,
calls for some other kind of basis functions. Abrupt changes in these signals
would be spread out in the whole frequency range, when the infinite extent
sinusoids are used for their analysis.
Extracting information from biomedical signals has been a difficult
issue. These signals normally have highly complicated time-frequency
characteristics. Frequently, they consist of brief, high frequency components
closely spaced in time, accompanied by long lasting, low frequency
components closely spaced in frequency. Any method for dealing with them
should therefore have good frequency resolution to localize the low
frequency components, along with good time resolution to determine the
high frequency components.
One method of analyzing nonstationary signals is to treat them as
stationary signals by dividing them into short parts whose statistics remain
unchanged for their duration. This method is called the Short Time Fourier
Transform. As will be seen in this report, the resolution in this technique is
fixed since one analysis window is used. If this window is made too short,
the frequency resolution will suffer. On the other hand, extending the
window to capture the low frequencies in the signal, may cancel the
assumption of stationarity within the window.
The problem of fixed resolution can be solved by changing the
window used. In the Wavelet transform, a prolonged window is used at low
frequencies, and therefore, good frequency resolution is obtained at low
frequencies. Moreover, good time resolution is obtained at high frequencies
since a short window is used to capture the fast changes in the signal.
The basis functions in wavelet analysis are created by shifting,
expanding, and contracting the “analyzing” wavelet or “mother wavelet”
whose selection depends on the application at hand.
The Wavelet transform is being applied in different fields such as
biomedical signal processing, medical imaging, digital communications,
radar, remote sensing, astronomy, acoustics, nuclear engineering, optics,

5. 5
earth-quake prediction [Figure 1], human vision, and pure mathematics
applications such as solving differential equations and numerical analysis.
Wavelets are also being used to compress digital signals and images, speed
up fundamental scientific algorithms, and to rid digital signals of noise
[Figure 2]. The approach has proven to be so powerful, that it has become
the main subject of international conferences and new journals, as well as
new books. Some of the web sites that provide information about wavelets
are provided in the corresponding references.
In this report, the differences between the Fourier transform and the
wavelet transform will be elaborated further. Both the redundant,
Continuous Wavelet Transform and the Discrete Wavelet Transform will be
discussed. Filter banks will be used to obtain the coefficients of the Discrete
WT, which analyses the non-stationary signals with no redundancy, such
that they can be reconstructed without any distortion.
In this project, normal and abnormal Electrocardiogram (ECG) and
Phonocardiogram (PCG) or signals of the heart sounds are analyzed using a
19 Level Wavelet Packet structure. This analysis filter bank is an extension
to the one used by Dr. Khaled Mayyas in [2] to analyze heart sound signals.
The program used to apply this structure was a modified version of a
program written by Dr. Mayyas to analyze heart sounds. The ECG signals
and the heart sound signals used were provided by Dr. Bassam Al Asir.
The outputs of the analyzed signals are shown in Appendix 1.
Information about the origin of the bioelectric potential can be found
in Appendix 2, along with a figure of the heart and the action potentials that
form the ECG signal, which is shown along with the heart sounds in another
figure. A historical overview is quoted to denote the difficulties of
understanding the cardiac cycle. Wavelets are recently being used to analyze
EEG or Electroencephalogram signals, therefore, some information is
provided about them. These signals describe the electrical activity of the
brains.

6. 6

7. 7

8. 8
Non-stationary Signal Analysis
The aim of analyzing a signal is to extract information from it. This is
achieved by transforming the signal, or representing the signal in some other
form. Stationary signals are signals whose statistical properties of mean and
variance do not evolve in time. For such signals x(t), the Fourier Transform
is used [1]:
X f x t dt
j ft
e( ) ( )=
−
−∞
∞
∫
2π
Abrupt changes within the signal cannot be captured using this
transformation, since the basis functions used are the infinite length
sinusoids. Therefore, basis functions that are more concentrated in time and
less concentrated in frequency are required.
STFT Transform: Analysis with Fixed Resolution
Frequency dependence on time is introduced in the Short Time Fourier
Transform (STFT). In this transformation, a one dimensional signal x(t) is
mapped into the two-dimensional function of time and frequency STFT(τ,f).
The signal is multiplied by a moving window of limited extent then the
Fourier transform of the modulated window is calculated. The STFT
depends primarily on the window chosen, as seen in the following equation
[1]
STFT f x t g t dt
j ft
e( , ) ( ) *( )τ τ
π
= −
−∞
∞
−
∫
2
The time frequency axis is partitioned to tiles of fixed shape [Figure 3.a].
The signal is filtered at all frequencies using a bandpass filter whose impulse
response is the window function modulated to that frequency. The basis
functions for the STFT care shown in Figure 3.c.
Considering a pair of sinusoids whose frequencies are ∆f Hertz apart,
the minimum value of ∆f that the STFT can resolve is called the frequency
resolution of the STFT, and is defined using the root mean square bandwidth
[1]
∆f
G f df
G f df
f
=
∫
∫
2 2
2
( )
( )

9. 9

10. 10
Where G(f) is the Fourier transform of the window g(t), whose energy is
given in the denominator.
The minimum value of spacing between the pair of short pulses
considered is called the resolution in time where ∆t can be expressed using
the root mean-square duration [1]
∆t
t g t dt
g t
=
∫
∫
2 2
2
| ( )|
| ( )|
In which the denominator is the energy of g(t).
The time bandwidth product imposes a lower bound on time and
frequency resolutions [1]
Time Bandwidth product t f− = ≥_ ∆ ∆
1
4π
This is known as the uncertainty principle or the Heisenberg
inequality. It limits the time and frequency resolutions to the value (1/4π),
which is satisfied when Gaussian windows are used. Since a fixed window is
used in the STFT, either good time resolution or good frequency resolution
can be obtained, but not both. The former is achieved by choosing a short
window and the latter with a filter with narrow bandwidth.
Properties of the Short Time Fourier Transform
• The short time Fourier transform preserves time shifts except for linear
modulation. If STFT(τ,f) is the short time Fourier transform for the signal
x(t), then the STFT(τ,f) of the time shifted signal x(t-to) is given by
exp(-j2πfto) STFT(τ-to, f).
• The short time Fourier transform preserves frequency shifts. If the short
time Fourier transform of the signal x(t) then the STFT of the modulated
signal x(t).exp(-j2πfot) is given by STFT (τ , f-fo).
The disadvantages of the STFT is its fixed resolution in time and
frequency, since the same window is used at all frequencies and times
[Figure 4.a]. This leads to a trade off between the two resolutions, since only
one of them can be obtained.

11. 11
Spectrogram
The squared modulus of the STFT of a signal x(t)is called the
Spectrogram[1]
Spec f STFT f( , ) | ( , )|τ τ= 2
In physical terms, it provides a measure of the signal energy in the
time-frequency plane. The spectrogram is extensively used in the analysis of
speech signals.
The Continuous Wavelet Transform
To overcome the resolution limitation of the STFT, the resolution in time
and frequency, denoted by ∆t and ∆f respectively, are varied in the time
frequency plane to obtain a multi-resolution analysis. This is achieved by the
Continuous Wavelet Transform.
Like the Fourier analysis, the wavelet analysis uses an algorithm to
decompose a signal into simpler elements. However, in contrast to a Fourier
sinusoid, which oscillates forever, a wavelet, is localized in time, and lasts
for only a few cycles.
Given a nonstationary signal x(t), the wavelet transform is defined as
the inner product of x(t) with the two-parameter family of basis functions [1]
ψ ψ
τ
τ ,
/
( ) ( )
a
t a
t
a
=
−−1 2
where (a) is a scale factor, and τ is a time delay. In mathematical terms, the
wavelet transform of x(t) is defined by [2]
WT a
a
x t
t
a
dt( , ) ( ) ( )τ ψ
τ
=
−
−∞
∞
∫
1
The mother wavelet, Ψ(t), is the basis function in the wavelet
Transform. It is an oscillating function so there is no need to use the sines
and cosines (waves) as in Fourier analysis. Wavelets are scaled and shifted
versions of Ψ(t). The scale factor, controls the frequency content of the
wavelet since it satisfies the equality [1]
a
f
f
o
=
If |a|<<1 the wavelet is very concentrated and brief, with frequency
content mostly in the high frequency range. On the other hand, if a>>1 the
wavelet is very much spread out and has mostly low frequencies. Therefore,

12. 12
the scale (a) gives global views of the signal when it is large, and gives
detailed views when it is very small
In wavelet analysis, the filter bank is composed of band pass filters
with constant relative bandwidth [1]
∆f
f
Q=
Where ∆f is the frequency resolution of the wavelet, and Q is a constant.
This equation means that the frequency resolution is linearly proportional to
frequency. Thus, as the midband frequency ( f ) of the wavelet increases, the
bandwidth of the wavelet increases. So, good frequency resolution is
obtained at low frequencies and good time resolution is obtained at high
frequencies. The time frequency plane for the wavelet transform is shown in
figure (3.b), and the wavelets are shown in figure (3.d).
With the Gaussian window applied, the time resolution can be
expressed as [1]
∆
Π∆ Π
t
f Qf
= =
1
4
1
4
The Morlet wavelet is Gaussian shaped , and therefore its time
bandwidth product is 1/4π. However, it is a noncausal filter of infinite
extent.
In the CWT The frequency responses of the analysis filter are
regularly spread in a logarithmic scale [Figure 4.b]. The frequency
resolution ∆f is proportional to f so when the center frequency of the analysis
filter is changed, ∆f and ∆t change.
The Continuous Wavelet Transform is highly redundant because the
scale ‘a’ and the time constant ‘τ’ are continuous. Hence, the corresponding
inverse transform is not unique and the original signal cannot be
reconstructed without being distorted.
Moreover, the CWT is only suitable for off line processing in which
the signal is not processed at the time of operation. On-line processing of the
signals using the CWT is not practical since it requires huge processing
power.
Scalograms
The spectrogram is the square modulus of the STFT. It provides a
distribution of the energy of the signal in the time-frequency plane.
Similarly, the CWT preserves energy. The Wavelet spectrogram, or
Scalogram, is defined as the squared magnitude of the CWT. It is

13. 13

14. 14
distribution of the energy in the signal in the time-scale plane where the
energy is distributed with different resolutions, according to the window
used.
Efficiency of the Wavelet Transform
The linear Fourier Transform represents a signal as a superposition of sum
of sinusoids with different frequencies. The contribution of the sinusoids at
these frequencies is measured by the Fourier coefficients. In a similar
manner, the linear wavelet transform represents a signal as a sum of
wavelets with different locations or positions and scales or duration. The
strength of the contribution of the wavelets at these locations and scales are
quantified by the wavelet coefficients.
An example is a signal in the form of a saw-tooth (ramp) wave. The
signal’s intensity rises steadily with time, then drops abruptly before
ramping up again. This shape can be represented as a sum of wavelets
[Figure5]. Coarse-scale wavelets lasting roughly the duration of the ramp
represent the smooth rising part of the signal, while fine-scale wavelets
capture the discontinuity (jump) in the middle.
The building blocks of the Fourier and Wavelet Transforms, which
are used to decompose the signal uniquely, are the sinusoids and wavelets.
The efficiency of these building blocks differs for a given job . In the
mentioned example, the saw-tooth signal was sampled at 256 observations
per second, and was compactly represented by 16 wavelets. A Fourier
analysis of the same saw-tooth signal would need fully 256 sinusoids
because of the technique’s difficulty in representing the discontinuity in the
middle of the signal.
Resolution and Scale of Discrete Signals
Reducing the resolution of a discrete time signal is achieved by low pass
filtering with a half band low pass filter.
When a signal is lowpass filtered, its scale remains unchanged, while
its resolution is reduced, since the resolution is linked to the signal’s content
of frequency. [Figure 6.a]. Increasing the scale in the analysis of a discrete
time signal involves downsampling, or dropping every other sample, which
automatically reduces the resolution. [Figure 6.b]. Decreasing the scale,
which involves upsampling, or inserting zeros between the samples, doesn’t
change the resolution [Figure 6.c].
a) Halfband
lowpass
Resolution: halved
scale: unchangedx(n) y(n)

15. 15

16. 16
b)
c)
Figure 6.Resolution and Scale changes in Discrete time
Signal Analysis with Multi-resolution
In multiresolution signal analysis, the space of square integrable (finite
energy) signals is built from non-overlapping (orthonormal) signal
subspaces with different resolutions, each subspace with different basis
vector. Therefore, a square integrable signal can be obtained by shifting and
expanding or contracting the wavelet ψ(t) as [2]
x t i m t mi
mi
i
( ) ( , ) ( )/
= −− −
∑∑ 2 22
α Ψ
where
α( , ) ( ) ( )/
i m x t t m dti i
= −− −
−∞
∞
∫2 22
Ψ
i, m ∈ Z where Z is the set of integers numbers.
The wavelet ψ(t) is a band pass filter with central frequency
(ωο), and α(i,m) are the wavelet coefficients. In this equation, the scale (a)
is represented by 2 i
and the time shift is represented by m. The wavelet 2−i/2
ψ(2 −i
t-m) is the basis function for the subspaces W i , and is compressed by
a factor of 2 with respect to the basis function in the subspaces Wi+1
represented by 2− +( )/i 1 2
ψ(2− +( )i 1
t-m). Therefore, the time resolution of the
signal in space W i . is twice better than the time resolution of the signal in
subspace Wi+1 . Each of these subspaces is orthogonal to the other subspaces,
and the summation of these subspaces forms the signal space [2]
signal space W W W Wi i i
i z
i= ⊕ ⊕ = ⊕+ +
∈
1 2...
Since the signal space is represented as a direct sum of the various
resolutions, the signal x(t) can be uniquely expanded into many subband
signals of different time or frequency resolutions
Halfband
lowpass
Resolution: halved
scale: doubled
x(n)
y(n)2
Halfband
lowpass
Resolution: unchanged
scale: halvedx(n) y(n)2

17. 17
The Discrete Wavelet Transform and Filter Banks
The theory of multiresolution is related to filter banks, for the process of
projecting the signal into orthogonal subspaces is achieved using filter
banks. To obtain the coefficients wavelet coefficients of the space, W i or
α(i,m), the tree structure analysis filter bank shown in Figure[7]is used.
signal Space
Figure[7]. A tree structure analysis filter bank
In the tree structure above, h0 and h1 are half band low and high pass
filters. The output of each filter is downsampled to give a full band signal.
The downsampled output of the half band low pass undergoes division into
lowpass and high pass parts and so on. The frequency resolution increases as
the number of iterations of low pass filtering is increased. However, the
number of iterations must be finite.
Basis of Orthonormal Wavelets Constructed from Filter banks
The basis functions in wavelet analysis are the scaling function, which is a
low pass filter, and the wavelets which are band pass filters. The wavelets
are formed by contracting or expanding a bandpass filter called the mother
Wavelet. In figures [8-11], the scale and mother wavelet of different
prototype filters (different coefficients for h0) can been seen. As noticed,
there are different shapes for wavelets.
h1 2
h0 2 h1 2
h0 2 h1 2
h0 2
α(0,m)
α(1,m)
α(2,m)
…

18. 18
Scaling Function and the Wavelet Mother obtained using D4 as the
prototype filter.
(Time)
Figure 8
(Amplitude)

19. 19
Scaling Function and Wavelet Mother using D6 as the prototype filter
Figure 9

20. 20
Scaling Function and Wavelet Mother using Daubecies 10 as the prototype
filter
Figure 10

21. 21
Scaling Function and Wavelet Mothre using Coiflet 15 as prototype filter
Figure 11

22. 22
The scaling function φ(t), can be represented as a summation of its
dilated and shifted versions φ(2t-n) with expansion coefficients h0(n) as
follows [2]
ϕ ϕ( ) ( ) ( )t h n t n
n
= −
=
∞
∑ 0
0
2
Moreover, the mother wavelet can be expressed as a summation of the
shifted scaling function φ(2t-n) of the next higher space [2]
Ψ( ) ( ) ( )t h n t n
n
= −
=
∞
∑ 1
0
2ϕ
The impulse responses ho(n) and h1(n) are the coefficients of finite
length Finite Impulse Response (FIR) filters. Infinite impulse response
filters, which require a large number of coefficients, are associated with
infinite length wavelets, and thus are of no practical importance.
Through filter banks, the discrete wavelet transform expansion is
obtained without computing the wavelet mother nor the scaling function.
Only the lowpass filter coefficients h0(n) are required, since the coefficients
of the complementary half band high pass can be derived from them [1]
h L n h n
n
1 1 1 0( ) ( ) ( )− − = − ,n=0,1,…,L-1
where L is the filter length.
The factor (-1)n
transforms the low pass filter to a high pass filter because it
shifts the frequency response of the low pass by π.
In order for h0(n) and h1(n) to be suitable expansion coefficients for
the scaling function and the wavelet mother, they should satisfy the
following conditions [2,]
h n
n
L
0 2
0
1
( )
=
−
∑ = , h n
n
L
0 12
0
1
( ) =
=
−
∑
h n
n
L
1 0
0
1
( )
=
−
∑ = , h n
n
L
1 12
0
1
( ) =
=
−
∑
The filter h0(n) should also have a maximum number of zeros at half
the sampling frequency, or at ω=π, or the spectrum of H0(z) should be flat at
half the sampling frequency. Such filters are said to be regular. The
regularity order is the number of times that the half band FIR low pass filter
h0 or the half band high pass filter is continuously differentiable. Regularity

23. 23
is important for perfect reconstruction of the signal. Filters that satisfy these
conditions are compact support orthonormal filters.
Orthonormality means that the inner product of two basis functions is
zero unless they are equal, for then it is equal to 1. This means that the
product of two wavelets ,or a wavelet and a scaling functions is zero, and the
energy in the wavelet and the scaling function is 1.
The length of orthonormal wavelets is even, as are the filters from
which they are constructed (the prototype filters, h0 and h1). They are also
not symmetric.
In order to reconstruct the original signal from its wavelet transform,
the analysis filter bank used to compute the wavelet coefficients using
downsampling, has to be followed by a synthesis filter bank which uses
upsampling. Following is the analysis and synthesis of a two channel filter
bank
Wavelet Packets
Wavelet packets and cosine packets are intermediate between wavelets and
sinusoids: they oscillate many times, but are still localized to a segment of
the signal duration, as in figure 13. This wavelet packet was obtained using
Coiflets 15.
Wavelet Packets have a location and duration (like wavelets) as well
as a frequency or oscillation (like sinusoids), and are made up of
orthonormal basis functions. The main feature of wave packets tiling is a
frequency resolution that may be adapted to the signal at hand [Figure 14].
The best frequency resolution is achieved at midband frequencies, and the
frequency resolution at high frequencies is better than that at low
frequencies; the associated time resolutions are proportional in a
corresponding way. As with wavelets and sinusoids, every signal can be
portrayed uniquely as a sum of wavelet packets or cosine packets.
2h0 2
2h1 2 h1’
x
x’
h0’
Figure 12. Analysis and Synthesis Filter Bank

24. 24
Wavelet Packet of the 9th
branch of the 19 level filter bank using Coiflets 15
Figure 13

25. 25
Frequencyf
0 Time Ƭ
Figure14. Wave packet tiling of the time-frequency plane

26. 26
Nineteen level Wavelet Packet Structure
In [2], the tree structure was adapted by branching on the high pass and low
pass outputs, in order to attain the desired resolution for analyzing heart
sound signals. A seven level structure was designed upon a sampling
frequency of 1200 Hz.
When this structure was applied on a normal ECG signal, it was seen
that most of the signal’s energy was concentrated in the frequency band
between (0,75) Hz. Therefore, in order to zoom into the signal at low
frequencies, more iterations of low pass filtering were needed at this low
frequency band.
The sampling frequency for the ECG signal 400 Hz. Using filter
banks with half band low pass and high pass filters, the frequency spectrum
between 0 and half the sampling frequency (200 Hz) was divided into 19
non-overlapping divisions. The lowest frequency band was 0 to 3.125 Hz,
and the highest band from 150 to 200 Hz.
The obtained filter bank is shown in figure 15. The frequency ranges
of the outputs are shown in table [1]. This table shows the discrete frequency
division of the spectrum and the corresponding frequencies in (Hz) for the
sampling frequencies of 400Hz and 1200Hz.
Cutoff frequencies of the Filters in the Wavelet Packet Structure
In order to find the frequency ranges of the output signals of the filter
bank, the frequencies at the output of each filter followed by decimation
should be found. In figure 16, the frequency ranges at some of the outputs of
the filters followed by downsamlping are shown.
Assuming that the sampling frequency is twice the highest frequency
in the signal, the sampled signal is repeated every (ω=2π) in the sampled
frequency domain, and the highest frequency in the signal is equal to fs/2
which is equivalent to π.
The frequency range output of the half band lowpass filter (h0) is
(0,π/2). In figure 17, the operation of a two channel filter bank is described
in the frequency domain. Downsampling by two means taking every other
sample of the signal. This reduces the sampling rate to half its value, and
produces shifted copies of the lowpass filtered outputs, as seen in figure
[17.g]. Thus, the output after downsampling is a full band signal of the low
passed half band signal, with a sampling rate of (fs/2). Therefore, in terms of

27. 27
Input
Signal

28. 28
Table 1

29. 29
the original sampling frequency, the signal at the output of the low pass filter
followed by downsampling by 2 occupies the range (0,π/2). The information
contained in the signal before and after downsampling it is the same.
Similarly, the complementary half band high (h1) passes the range of
frequencies (π/2, π). However, downsampling the high pass signal leads to a
reversion in the frequency of the high pass band and shifting down to the
baseband. Thus, in a filter bank following a high pass output, the low pass
channel corresponds to the upper half, and the high pass channel to the lower
half of the preceding high pass channel.
One doesn’t have to go through the downsampling process to find the
frequency band of every filter followed by downsampling by 2. Upon
filtering a signal with frequency range from 0 to π, the lowpass filter
followed by downsampling by 2 (as one unit), produce the lower range of
frequencies , form 0 to π/2. The high frequency part is passed by the high
pass filter followed by decimation by 2 (as a unit). Filtering after a high pass
filter is different. The low range of frequencies is obtained by filtering with a
high pass filter, while the high range of frequencies is obtained by filtering
with the low pass filter. After two high pass filters, the low frequency range
is obtained from a low pass filter (since the second high pass filter acted as a
low pass filter). Similarly, if a high pass filter is followed by a low pass
filter, then the low frequency range is obtained from a high pass filter
following the low pass filter which acted as a high pass filter.
h0 2
(0,π) (0,π/2)
h1 2
(0,π) (π/2,π)
h0 2
(0,π/4)
h1 2 h1 2
(π/4,π/2
(π/4,π/2.66)
2
…
…
h0
(π/2.66,π/2)
…
…
Input signal
Figure 16. Finding the Frequency ranges of the outputs filters followed by downsampling
Note :
H0(z) 2
LP 2
H1(z) 2
HP 2

30. 30

31. 31

32. 32
Implementation of Wavelet Packet Structure
A program was written by Dr. Mayyas to implement a thirteen level filter
bank without decimation. In this program, no decimation was used, because
the output using decimation has a sampling frequency different than the
original sampling rate. Since the signal should be viewed at different bands,
and at the same sampling frequency of the input signal, there is no need for
downsampling. A similar program was written to implement the 19 level
Wavelet Packet.
The fact used in this implementation is that downsampling by 2 before
a filter H(z) can be written as filtering with H(z2
) followed by
downsampling [Figure 18]. H(z2
)is the interpolated (upsampled) version of
H(z). It means inserting a zero between the every samples of h(n). The
equivalent filter bank to the 19 level filter bank can be seen in figure 19.
This was the filter bank applied to analyze the given signals, but without
decimation at the end of each branch.
The output using decimation gives the wavelet transform coefficients.
Each output has a sampling frequency different than that of the input signal,
according to the decimation rate used at the end.
Figure 18 . Equivalence of downsampling then filtering and interpolation then
downsampling.
Each output in the implemented wavelet packet without decimation is
the convolution of the input signal with an equivalent impulse response,
which is the convolution of the interpolated impulse responses. Since no
decimation is applied, the output has the same sampling frequency as the
input, which is more meaningful for the analysis of the input signal.
Coefficients of the Prototype Filters Applied to the Filter Bank
Following are the coefficients of the prototype lowpass filters h0(n). The
high pass coefficients can be derived from these coefficients using the
equation previously mentioned in another form [2]
h1(n)=(-1)L n
h L n− −
− −1
0 1( ) , n=0,1,…,L-1
22 H(z2
)H(z)

33. 33

34. 34
Daubechies 4 Daubechies 6 Daubechies 10
0.482962913145 0.332670552950 0.160102397974
0.836516303738 0.806891509311 0.603829269797
0.224143868042 0.459877502118 0.724308528438
-0.129409522551 0.135011020010 0.138428145901
-0.085441273882 -0.242294887066
0.035226291882 -0.032244869585
0.077571493840
-0.06241490213
-0.012580751999
0.003335725285
Coiflets 6 Filter close to Coiflets 15
-0.07273261951285 0
0.33789766245781 0
0.85257202021226 0
0.38486484686420 0.01767766952966
-0.07273296511271 -0.04419417382416
-0.01565572813546 -0.07071067811865
0.39774756441743
0.81317279836453
0.39774756441743
-0.07071067811865
-0.04419417382416
0.017677669529660
0
0
0
The scaling function and wavelet mother obtained from these coefficients
were seen in figures [11] previously.
Frequency Responses of the Prototype Filters:
The frequency response of the prototype low pass filters using the
different coefficients can be seen in figure (20). All the filters have the same
amplitude at zero, which is equal to 2 , which is the summation of the low

35. 35
Frequency Responses of the Prototype Filters
Figure 20
(Gain)

36. 36
Spectrum of the successive filters using Daubechies 4
Figure 19.a
(Gain)

37. 37
Spectrum of the Successive Equivalent Filters using Daubechies 6
Figure 19.6

38. 38
Spectrum of the Successive Equivalent Filters using Daubechies 10
Figure 19.c

39. 39
Spectrum of the Successive Equivalent Filters using Coiflets 6
Figure 19.d

40. 40
Spectrum of successive equivalent filters using Coiflets 15
Figure 19.e

41. 41
pass filters’ coefficients. This was one of the constraints that the prototype
filter should satisfy.
Before reducing the sampling frequency of a signal by a factor of M,
it should be band limed by a filter with a cutoff frequency π/M. In our case,
M=2. In other words, downsampling by 2 should be preceded by a band
limiting filter with cutoff frequency π/2. This is why all the prototype filters
had the same cutoff frequency (π/2).
As the length of Daubechies filters increased, the transition band
width decreased, since the order of the filter increased. This also was true for
Coiflets6 and Coiflets 15. However, the It was noticed that the frequency
response of Daubechies 4 was the same as the frequency response of
Coiflets 6, and the transition band width of Coiflets 15 was more than that of
Daubechies 10.
The filters have linear phase, which means that the group delay is
constant. Therefore, the outputs of the system will be shifted by the same
amount. The group delay is the negative of the derivative of the phase of the
filter.
Frequency Responses of The Equivalent Filters of the 19-level Filter Bank
The frequency responses of the 19 equivalent filters of the structure using
different coefficients can be seen in figures [19]. The low pass filter is the
scaling function. It has the highest magnitude, which is equal to the
summation of the coefficients of the scaling function in the time domain.
The first band pass filter (from the left) is the frequency response of the
mother wavelet.
It was noticed that the maximum value of the frequency response of
the mother wavelet using Daubechies10 was the highest, followed by
Daubechies6, then Daubechies4. For Coiflets, the frequency response of the
mother wavelet was the same as that of Daubechies6, while the gain for
Coiflets15 was less than the gain of Daubechies10 and higher than the gain
of D6. From the figures, it can be seen that the side lobes decreased the most
when Daubechies10 was used.
More iterations of filtering leads to increasing the filter’s length, and
this leads to a decrease in the bandwidth of the filter, giving more selectivity
in frequency. The length of the wavelet at the mth scale is given by [10]:
m
m
L L= − − +( )( )2 1 1 11

42. 42
where m is the number of iterations of filtering, and L1 is the length of the
prototype filter. For example, using Daubechies 10, the length of the wavelet
after 6 iterations is L6=568, while the length of the wavelet after 6 iterations
using Daubechies 4 is L6=190.
For long wavelets, the amplitude spectra are narrow and are at low
frequencies. On the other hand, short wavelets have wide amplitude spectra
at high frequencies, as seen in figure [22].
Furthermore, it was noticed that although the number of iterations
were the same in the bands up to the tenth level, frequency response kept
decreasing. However, iterations that ended by filtering with a low pass then
a high pass filter, followed by two iterations of low pass filtering, as in
branches 8,9 had the same maximum value. The same is true for branches
13,14.
The magnitude of the high pass filter of the last branch was the same
using the different coefficients, and was equal to 2 22
= . Since the value
of the frequency response of the scaling function after (6) iterations was
equal to 2 86
= , and the magnitude of the high pass filter after 2 iterations
was 2 22
= , the outputs of the system were normalized by a factor of 1/
2 i
, where (i) is the number of iterations of filtering.
Verified Properties of the Scaling Function
• The scaling function is a low pass filter. All the scaling function obtained
using the different coefficients had the same magnitude at zero. This
value was also equal to the summation of the filter’s coefficients. The
scales were different lengths, since the prototype filters were of different
lengths. The z-transform justifies this.
The frequency response of the low pass filter when ω=0 or z=1 is given by
H ho n
n
( ) ( )1
0
=
=
∞
∑
The different scaling functions were obtained from finite length prototype
filters that satisfied the condition
ho n
n
L
( )
=
−
∑ =
0
1
2

43. 43
Figure 22

44. 44
This is why the different scaling functions had the same magnitude at zero
frequency.
In the 19 level wavelet packet structure, the scaling functions considered
were obtained after 6 iterations. The magnitude of the φ(t) at ω=0 was equal
to 8, which is equal to ( )2 6
. Thus, In order to normalize the scaling
function, it has to be multiplied by 2
2−i/
where (i) is the number of iterations
of lowpass filtering and down sampling by two.
• At ω=π, the magnitude of the frequency response of the scaling function
is equal to 0. This follows from the z transform at ω=π
Ho e Hoj
( ) ( )π
= − =1 0
• The summation of the squared coefficients of an orthonormal scaling
function is equal to 1. In other words, the energy of the low pass filter
scaling function is equal to 1.
Το summarize, the conditions that the scaling function should satisfy are:
• φ(0)= h n
n
0( )∑ ; when the impulse response of the scaling function is
normalized to 1, φ(0)=1
• φ(π)=0
• φ2
1
n
n∑ =( )
These properties were satisfied by the orthonormal scaling functions used.
However, due to the finite precision of the computers, a very small number
was obtained instead of 0 where required.
Verified Properties of Wavelets
• Wavelets and Wavelet packets are band pass filters, therefore, the
magnitude of the wavelet at zero frequency is equal to zero
Ψ(0)=0.
Applying the z transform to the wavelet at zero, the second important
characteristic of the wavelet results:
ψ ( )n
n
=∑ 0
Moreover, the energy in the orthonormal wavelet is equal to 1
ψ 2
1
n
n∑ =( )
For orthonormal wavelets, the product of the scaling function and the
wavelet is equal to zero, and the product of two wavelets is zero since they
are orthogonal basis functions.

45. 45
ψ φ( ). ( )n n
n
=∑ 0
ψ ψ1 2 0( ). ( )n n
n
=∑
Electrocardiography
The heart is the power source which provides the energy to move the
blood through the body and supply cells with nutrients, hormones,
temperature, and gases that they need for cellular function and at the same
time removes waste products- products of the cell’s metabolism- from the
cell.
Electrocardiography is a method used to graphically trace the
electrical activity of the heart muscle during a heart beat. The tracing is
recorded with an electrocardiograph, which is a relatively simple
galvanometer. It provides information on the condition and performance of
the heart.
Electrocardiogams are made by applying electrodes to various parts of
the body to guide the tiny heart current to the recording instrument. The two
arms and, left leg and the chest have become standard sites for applying the
electrodes. The magnitude and shape of the individual waves of ECG waves
vary with the location of the location of the electrodes.
After the electrodes are in place, held with a salt paste, a millivolt
from a source outside the body is introduced so that the instrument can be
calibrated. Standardizing electrocardiograms makes it possible to compare
them as taken from person to person.
The normal electrocardiogram shows typical upward and downward
deflections that reflect the alternate contraction of the two upper chambers
(the atria), and the two lower chambers (the ventricles) of the heart. These
deflections are called the P, QRS and T waves. The first upward deflection,
P, is due to atrial depolarization, the QRS complex is caused by ventricular
depolarization, and the T wave by ventricular repolarization. Atrial
repolarization is normally not seen as it is hidden by the QRS complex. The
U wave is sometimes found after the T wave. In Appendix 2, The
conducting system of the heart is shown, and a typical ECG signal is plotted
along with the action potentials that produce it [figure 23-1].
Any deflection from the normal in a particular electrocardiogram is
indicative of a possible heart disorder. Information that can be obtained from
an electrocardiogram includes whether the heart is enlarged and where the

46. 46
enlargement occurs, whether the heart action is irregular and where the
irregularity originates, whether a coronary vessel is blocked and where the
blockage is located. The presence of high blood pressure and certain types of
malnutrition may also be revealed by the electrocardiogram.
The shape of the ECG signal gives an indication of how serious the
case of a patient is. The appearance of very tall, slender peaked T waves,
with PR and QRS intervals within normal limits is lethal. Moreover, the life
duration of patients with inverted T wave was noticed to be shorter than
other cases.
The Heart Sounds
The graphic recording of heart sounds is achieved by means of a
phonocardiograph, which contains an electronic stethoscope. Therefore,
signals of the heart sounds are called PCG signals or phonocardiograms.
Two sounds can be heard during the cardiac cycle. The first sound
(S1) is of relatively long duration and is soft in quality . The second sound
(S2) is shorter and sharper. These characteristics are best intimated vocally
by the syllables “lub” and “dup” separated by a short pause.
The first heart sound (S1) commences .008 sec before the peak of the
R wave in the electrocardiogram signal. Its duration is about .18 sec, and is
followed by a systolic pause [Figure [2] in Appendix 2].
The second heart sound has two components one preceding the other
by a few milliseconds. However, some diseases lead to the occurrence of a
gap between these two components. In some cases, a low frequency third
heart sound S3 is heard. In late diastole, a fourth heart sound may be heard.
The interval between S1 and S2 is called systole, and the interval between
S2 and S1 in the next cycle is called diastole. These intervals are normally
silent. High frequency noise-like sounds are called murmurs. Murmurs
occurring during the systole are called systolic murmurs (SM), while those
occurring during the diastole are called diastolic murmurs (DM). The
diseases that cause SM are different from diseases which cause DM.
Although murmurs are noise like, certain features aid in distinguishing
between different causes. For example, Aortic stenosis (AS) causes diamond
shaped midsystolic murmurs, while mitral stenosis (MS) is indicated by a
decreasing then increasing type of diastolic presystolic murmurs.
The information above indicates the importance of localization in the
heart sounds. The timing instants of heart sounds and their components,

47. 47
frequency content, location in the cardiac cycle and the envelope shape of
murmurs are of great importance and should be accurately measured.
Analyzing ECG signals
ECG signals are continuous signals. In order to process them digitally,
they were converted into discrete sequences using an A/D converter with a
sampling frequency of 400 Hz.
As previously mentioned, normal ECG signals have most of their
energy concentrated at low frequencies up to 75 Hz. In the following, a
normal ECG will be analyzed using the 19 level wavelet packet structure.
The effect of changing the coefficients of the half band low pass filter
will be discussed. In order to see the band of frequencies that carry the
highest energy among the different bands, the energies of the outputs will be
calculated and compared for the same case using the different coefficients.
Abnormalities will be also added the signal in order to see the changes at the
outputs.
ECG signals as inputs to the filter bank
After designing the wavelet packet structure, an ECG signal was
applied to it. The sampling frequency of the signal was 400 Hz. One cycle of
the signal can be seen in figure [24], in which the power spectral density (in
dB) is plotted to the right. As can be seen, the signal was contaminated with
noise. In this case, noise can be defined as any signal that tends to distort the
original ECG signal. Some causes of noise are muscle noise, respiration,
position of the electrodes, and poor electrode contact.
In order to rid this signal of noise, the chosen filter was elliptic filter
of order 5, (0.7 dB) ripple in the passband, and (20 dB) ripple in the stop
band, and a cutoff frequency of 140 Hz. It can be seen in figure [25].
The zero mean, filtered ECG signal is shown in figure [26]. A (U)
curve can visualized starting with the peak of the P wave, passing through Q,
S and ending with the peak of T wave. Therefore, this signal can be
classified as normal. However, the frequency content of the signal specifies
more accurately whether it is normal or not.
When this signal was analyzed using the wavelet packet structure, the
outputs showed that the activity was mostly concentrated at low frequencies,

48. 48
Figure 24
Amplitude
Time

49. 49
Figure 25
Gain

50. 50
(Time)
Figure 26
(Amplitude)

51. 51
and declined as the frequency increased. Therefore, this signal is a normal
ECG signal.
Different abnormalities were added to the signal in order to see
whether the outputs would alter or not. As discontinuities increased in the
simulated signals, more oscillations at high frequencies occurred. One of the
abnormalities aimed at simulating “Late Potentials”, in which random noise
(mostly in a frequency range 25-80 Hz) occurs between the S and T waves.
The signal can be seen in figure [26.b].
In figure [27], three cycles of an ECG signal can be seen. This signal
is a combination of the normal ECG signal, followed by the abnormal signal,
then the normal signal once again. The outputs of this combination can be
seen in Appendix 1, part 1.
As can be seen, at high frequencies, the normal signal had no activity,
unlike the abnormal signal with late potentials. The outputs showed an
oscillatory behavior at the different ranges of frequencies. This might be due
to respiration.
Different coefficients were applied to the normal case, and the
abnormal case. In Appendix1, part 2, the outputs of the ECG with simulated
late potentials can be viewed using three kinds of prototype filters:
Daubechies4, Daubechies6, and Coiflets6.
As can be seen from there figures, the shape of the output changed
slightly by changing the analyzing wavelet. A phase shift could be observed
between Coiflets6 and Daubechies4.
Inversion of the T wave (which is a serious abnormality) was also
simulated. The signal is shown in figure [28.a] to the left of the normal ECG
signal. The output of the first branch was completely different from the
normal case, since it contained negative values as seen in figure [28.b].
Moreover, the inverted T signal didn’t show oscillations at high frequencies,
and gave an output similar to the normal case at the last band as seen in
figure [28.c]. Therefore, the shape of the output at low frequencies
especially the first branch should be taken into consideration because not all
abnormalities have high frequency components.

52. 52
Figure 26.b

53. 53
Figure 27

54. 54
Figure 28.a

55. 55
Output of the First Branch using Daubechies 4
Figure 28.b

56. 56
Outputs of the Last Branch using Daubechies 4
Figure 28.c

57. 57
Energy in analyzed ECG Using Different Prototype Filters
The energy of a signal is an indication of how much information is in the
signal. Higher energy means more information. The energy of the outputs
for one cycle of the analyzed signal, using the different prototype filters was
computed. The obtained figures are shown in table (2). Daubechies 4
coefficients gave the highest energy, followed by Coiflets 6 then Daubechies
6, Coiflets 15. Daubechies 10 gave the lowest energy. Following are the
figures obtained for the first output of the normal ECG signal
Daubechies4 Coiflets6 Daubechies6 Coiflets15 Daubechies10
79.4758 79.4048 78.7000 78.0756 77.6392
The first output of this normal signal obtained by using the different
coefficients is plotted in figure [29]. The energy in this table is normalized
with respect to the highest energy among the outputs, which was found to be
the lowest frequency range for the normal ECG signal (0 to 3 Hz). It should
be noted that this is not only attributed to the high magnitude of the
frequency branch, but it is also because the energy in ECG signals is
concentrated at low frequencies.
The difference in the energy obtained was due to the difference in the
transition band width of the filters used. However, although Coiflets6 had
the same transition band width of Daubechies4, but also the energy obtained
by using it was less than the energy obtained from Daubechies4. Therefore,
in this application, it is better to use Coiflets4 than Coiflets6 since the latter
gives less energy and more complexity.
It is better to use the shortest filter available because it gives the least
delay. From figure [29], we can see that as the filter’s length increases, the
delay of the output increases. The longest delay Using Coiflets15, the delay
obtained was the same length of the input signal. The length of the longest
wavelet packet used shouldn’t exceed the length of the input signal, which is
400. However, to obtain good frequency resolution, a large number of
coefficients is needed. Therefore, there is a compromise between the length
of the filter and the amount of energy or information obtained.
A Second ECG signal
An ECG signal for another student was compared with the normal ECG
signal previously discussed.

58. 58
Table 2-1

59. 59
Table 2-2

60. 60
Figure 29

61. 61
One cycle of the new ECG signal was taken. The signal and its power
spectral density (in dB) can be seen in figure [30].This signal was filtered
with the same filter used with the previously analyzed normal ECG.
The signal was applied to the 19 Wavelet Packet structure. The
outputs can be seen in the Appendix 1, part 11
. The outputs of the previous
signal ‘sam’ are shown to the left of this ‘new’ signal for fast comparison
between the outputs of the two signals. The prototype filter used is
Daubechies 62
.
As seen in the outputs, noise is spread at all times and at the different
frequencies, unlike the case of Late Potentials where it was concentrated in a
certain time interval. The energy of the signal between 44 Hz and 200 Hz
was higher than the energy of the normal ECG signal “sam”. The energies of
the outputs of the two outputs can be seen in table (3).
Comparing the wavelet analysis of this signal with the normal ECG
signal, it contains some abnormalities. The length of one cycle of this new
ECG signal was less than length of one cycle of the previous normal ECG.
This means that the activity of the heart of this person were faster than
‘sam’. Moreover this signal a high frequency component kept rising the
before the Q wave. This component still existed there after filtering. As can
be seen, noise still contaminated the signal, although it was completely
removed in the case of “sam”.
Lastly, it must be mentioned that this signal was classified by the
physician as normal. Its analysis doesn’t prove that. So, this can be an early
of an abnormality with this subject’s heart. However, such a decision should
be based on further analysis of other cycles. What is good about wavelet
analysis is that it gives an indication of any slight abnormality in the signal.
Analyzing Heart Sound Signals
A normal heart sound signal, and two other abnormal signals were analyzed
using the 19 level wavelet packet structure. The original signals can be seen
in figure [31], where the normal heart sound signal is at the top, at the center
is the first abnormal heart sound called SH (Systemic Hypertension), and at
the bottom VSD or Ventricle Septal Defect can be seen .The sampling
frequency used in these signals was 1200 Hz. The outputs for the different
1
Revised: Part 1. Previously: Part 3
2
Revised: Daubechies 6. Previously: Daubechies 4.

62. 62
bands can be in Appendix 1, part 33
. The analyzing wavelet for these outputs
was Daubechies 4.
Figure 30
3
Revised: Part 3. Previously: Part 4

63. 63
Table 3

64. 64
The normal heart sound signal shows the first heart sound, followed
by a pause and then the second heart sound, which can also be distinguished
in SH. However, in the VSD systolic murmurs (noise) contaminate the
signal.
The outputs of the 19 level structure can be seen in the figures that
follow. The frequencies of the different sub-bands can be seen in the in table
4 which also shows the energies of the output in the three cases. The outputs
were normalized to the highest energy in the normal case, which was found
to be the highest energy among the three cases.
It was observed that Systemic Hypertension shows less activity than
the normal case. It also shows a sinusoidal behavior, during the systole and
diastole in the frequency range 28 to 47 Hz (fourth and fifth band). The
magnitude of this sinusoid was a bit higher at the first and second heart
sound. The activity of the first heart sound diminished in frequency band of
300 to 451 Hz.
In Ventricle Septal Defect murmurs started to appear in the frequency
band 19 to 28 Hz (third output) and the first and second heart sound could
not be distinguished at that band, while these sounds were clear in SH and
the normal case at that band. At the thirteenth frequency band, the activity of
the first heart sound could be separated. The murmurs had an interesting
shape at bands thirteen and fourteen (frequency range 133 to 169 Hz) which
was like an amplitude modulated and frequency modulated sinusoid.
The analysis of SH using wavelet packets shows that the first heart
sound and the second heart sound can be distinguished, although with much
less activity, unlike VSD, where it is difficult to distinguish the sounds due
to noise.
At the eighteenth band, the normal heart sound signal clearly showed
the first heart sound and the second heart sound separated by a pause.
Moreover, in this normal case, the energy was highest at band 7, or in the
frequency range 57 to 66.7 Hz. In Systemic4
Hypertension, the energy was
highest at band 1, or in the frequency range 0 to 9 Hz, and this was higher
than the energy of the normal case at that band, and was only 28% of the
highest energy in the normal case.
For Ventricle Septal Defect, the highest energy was at level 13, or in
the frequency band 44 to 50 Hz, yet it was only 20% of the highest energy in
the normal case.
Other features might be extracted from the outputs of the signals. This
means finding the differences between the analyzed outputs, and not the
4
Revised: Systemic. Previously: Systematic

65. 65
common characters between them. This is because the different cases are
supposed to have similar characteristics, if the two signals were normal.
Input Signals (Heart Sounds)
(Time)
Figure 31
(Amplitude)

66. 66
Effect of the sampling frequency
Another group of heart sound signals were processed. The sampling
frequency for these signals was 10000 Hz. The output of the normal case in
these signals contained frequencies up to the frequency range 1250 Hz.
When an abnormal signal of a sampling frequency 10000 was analyzed, it
was seen that activity still persisted up to range of frequencies 2188 to 2500
Hz. This shows that a sampling frequency of 1200 Hz is too low for heart
sounds, and this leads to a huge loss of information.
A Sampling Frequency of 5000 Hz would give better results. This can
be achieved by downsampling the signal which has a sampling frequency of
10000Hz. With this sampling frequency, the range of high frequencies to be
detected is expanded to 2500Hz instead of 600 Hz.
On the other hand, a sampling frequency of 1000 Hz is more suitable
for analyzing ECG signals. This would also give a wider range of
frequencies to get rid of the noise in the signal before processing it through
the filter bank system, without affecting the components of frequency
founded in the signal itself originally. The sampling frequency of 1000 Hz
was found suitable for analyzing ECG signals in other studies.

67. 67
Conclusion
The wavelet transform can be defined as being the representation of a
discrete signal or image using wavelet functions at different locations and
scales by applying the fast pyramid algorithm. The wavelet itself is an
oscillatory waveform that persists for only one or few cycles, and has both a
location (position) and a scale (duration). Wavelets are most useful for the
representation of nonstationary signals and images with discontinuities.
The wavelet transform has the interesting property of zooming into
the time domain or the frequency domain, according to the frequency of the
signal. This can be imagined as seeing the trees as well as the forest. This is
due to its multiresolution capability. It uses long windows for low
frequencies and short windows for high frequencies. As for the STFT, which
also gives a representation of the signal in the time-frequency domain, it
uses a window of fixed bandwidth. This is inefficient in the analysis of
signals with discontinuities like ECG signals; a signal with low frequency,
for example, would be windowed with the same short window of a signal
with high frequency. This would lead to a huge loss of information.
Wavelet packets are efficient for the analysis of ECG signals and
heart sound signals. The signal could be seen in different bands, and in each
band the duration of the activity could be determined. Instead of having only
one signal that determines whether the signal is normal or not, the 19 level
wavelet packet structure showed 19 other signals. Therefore, this method is
suitable for extracting the features of the different abnormalities associated
with different diseases. The energy is another measure through which the
signal at hand can be classified as normal or not.
The wavelet transform is a good tool for early detection of disease. In
case of the occurrence of abnormality in the signal, the decision that it is
abnormal should not be taken right away. Many cycles should be analyzed,
and at different times. It should also be noted that in many cases the
abnormality occurs at only one cycle amidst normal cycles.
However, analyzing more cycles only means that the input signal to
the wavelet packet should have many cycles, instead of one. The processing
time for the signal depends on the length of the signal and the number of
coefficients used. Therefore, it is best to use the shortest prototype filter
available.
Daubechies filters showed good time and frequency resolution.
However, using different coefficients for the same input signal differed the
output’s shape. This means that the output’s shape is not the only thing that
should be considered. The energy in the output should also be considered.

68. 68
In this application, there was no need to reconstruct the original
signal, since the aim was to view the signal at different frequency bands with
different resolution. The number of iterations was not the same at all bands.
Therefore, when summing the outputs, a distorted signal was obtained due to
the different delay associated with the different lengths of equivalent filters.
The choice of wavelet depends on the application. For example,
longer wavelet packets are used for reproducing the sound of a musical
instrument, and Coiflets are suitable for reconstructing images.
The sampling frequency of the data should be carefully selected;
choosing a low sampling frequency means loss of information. Therefore,
using a high sampling is better. It is better to use it higher than the Nyquist
rate. Abnormal cases have higher frequencies than normal cases, thus it is
better to make abnormal cases the determinants for the sampling frequency.
For heart sounds, it is recommended to use a sampling frequency of 5kHz.
On the other hand, a higher sampling rate should be used for ECG signal. A
sampling frequency of 1KHz is preferred. This figure was seen to be good in
other studies. This gives more freedom in getting rid of the high frequencies
that might exist in the signal before analyzing it.
In the normal heart sound signal discussed, the activity of the first and
second heart sounds still persisted in the last band, which means that
frequencies higher than 600 Hz might still show activity. This contradicts
what was mentioned in [12] where the frequency spectrum of S1 was found
to contain peaks at a low frequency range (10 to 50 Hz) and a medium
frequency range (50 to 400 Hz). As for the frequency spectrum of S2, it was
observed in that paper that it contains peaks in low (10 to 80) , medium (80
to 220 Hz), and high frequency ranges (220 to 400).

69. 69
References
[1] “Wavelets and Signal Processing”, Oliver Rioul and Martin Vetterli,
IEEE Signal Processing magazine, pp.14-37, October 1991
[2] “Discrete Wavelet Analysis of Heart Sounds using Filter Banks”, K.
Mayyas, B. El-Asir, Jordan University of Science and Technology, Irbid-
Jordan, March 1997
[3] “Wavelets and Filter Banks :Theory and Design”, Martin Vitterli, IEEE
Transactions on Signal Processing, Vol.40, NO.9, pp.2207-2232, September
1992
[4] Octave Filter Banks and Wavelets, Chapter 9, pp.239-283
[5] “Wavelet analysis”, Andrew Bruce, David Donoho and Hong-Ye Gao,
IEEE SPECTRM magazine, pp.26-35, October 1996
[6] “Wavelet Applications in Medicine”, Metin Aka, IEEE Spectrum,
pp.50-56, May 1997
[7] “Ten Lectures on Wavelets”, Ingrid Daubechies, Rutgers University and
AT&T Bell Laboratories, Society for industrial and Applied mathematics,
Philadelphia, Pennsylvania, pp.258-285, 1992
[8] “An Introduction into Discrete Finite Frames”, Soo-Chang Pei and Min
Hung Yeh, IEEE Signal Processing Magazine, pp.84-96, November 1997
[9] “Wavelets as Alternative to Short-Time Fourier Transform in Signal-
Averaged Electrocardiography”, B. Gramatikov, I. Georgiev, Medical &
Biological Engineering & Computing, pp. 482-487, May 1995
[10] “Wavelet-Based linear System Modeling and Adaptive Filtering”,
Milos I. Doroslovacki, H.(Howard) Fan, IEEE transactions on signal
processing, Vol.44, No. 5, pp. 1157-1167, May 1996
[11] “Low Bit Rate Transparent Audio Compression using Adapted
Wavelets”, Deepen Sinha and Ahmed Tewfiq, IEEE transactions On Signal
Processing, 1993
[12] “Digital Filters, Analysis, Design, and Applications”, Andreas
Antoniou, Second Edition, McGraw-Hill Inc., pp. 602-605, 1993
[13] “Communication Systems”, S. Haykin, John Wiley and Sons, Canada,
3rd Ed, pp. 781-792, 1994
[14] “Physiological Basis of medical Practice”, Best&Taylor, Williams &
Williams, Fifth edition, pp. 203-235, 1039-1041
[15] “Phonocardiogram Signal analysis: A Review”, Rangaraj M.
Rangayyan, Richard J. Lehner, ‘CRC Critical Reviews in Biomedical
Engineering’, Volume 15, Issue 3, pp 211-234, 1988
[16] “Fundamentals of Medical Instrumentation”, Chapter 3, pp.83-103

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[17] “Origin of the Heart Beat & the Electrical Activity of the Heart”,
Chapter 28, pp.517
[18] “A Comparison of the Template Matching and Feature Extraction of
ECG Analysis”, M.J. Laister &R.J. Riggs, ‘Computers in Cardiology’, pp.
101-104, 1984
[19] “A Single Scan Algorithm for QRS-Detection and Feature Extraction”,
W.A.H. Englese and C. Zeelenberg, ‘Computers in Cardiology’, pp. 37-42,
September 1979
[20] Encyclopedia Britannica, Fifteenth edition, Volume 4, pp. 430, 431
1987
References to Wavelets on the Web
1. New Book: Time-frequency and Wavelets in Biomedical Signal
Processing, edited by Metin Akay, Darmouth College. A volume in the
IEEE Press Series on Biomedical Engineering.
email: makay@northstar.dartmouth.edu
2. Wavelet Digest; a free newsletter sent about once a month and contains
many kinds of information concerning wavelets.
Subscription: email an empty message to: add@wavelet.org
3. Toolsmiths Papers Page, a guide to Papers and other publications on
Wavelet Transforms and WavBox Software
http://www.toolsmiths.com/papers.html
4. Amara’s Wavelet Page
http://www.amara.com/current/wavelet.html
5. http://cm.bell-labs.com/who/wim/
6. http://www.org/wavelet/index.html
7. http://www.wavbox.com/
8. http://www.toolsmiths.com/firwav.html
9. http://math.berkeley.edu/~sethian/level_set.html
10. http://www.wavelet.org/wavelet/digest_06/digest_06.02.html#4
11. http://www.spelman.edu/~jcf
12. http://www.davidson/academic/math/davis/index.html

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APPENDIX 1

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PART 1: Outputs of Normal and Abonormal ECG Signal with Late Potentials

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Figure 27

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Output #1 Using Daubechies 6
Time
Amplitude

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PART 2: Outputs of Late Potentials using Daubechies4, 6, and Coiflets 6

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Input Signal
(Time)
(Amplitude)

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Outputs # 1
Using Daubechies 4 Using Daubechies 6 Using Coiflets 6
Time
Amplitude

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Part 35
: Outputs of Normal Heart sound, and two Abnormal Cases
5
Revised: Part 3. Previously: Part 4

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Input Signals (Heart Sounds)
(Time)
Figure 31
(Amplitude)

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Output # 1, Using Daubechies 4

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APPENDIX 2

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Historical Difficulties Faced in Understanding the Cardiac Cycle
The following is quoted from ‘Physiological Basis of Medical Practice’ by
Best and Taylor.
‘The succession of changes which occurs in the heart and which is
repeated during each beat is referred to as the cardiac cycle. Due to the
rapidity with which the events in the cycle follow one another, it is
impossible to study them by mere inspection. In 1628, William Harvey
remarked upon the difficulties of the problem:
“When I first tried animal experimentation for the purpose of
discovering the motions and functions of the heart by inspection and not by
other people’s books, I found it so truly difficult that I almost believed with
Fracastorious that the motion of the heart was to be understood by God
alone. I could not really tell when systole or diastole took place, or when or
where or constriction occurred, because of the quickness of the movement.
In many animals, this takes place in the twinkling of an eye, like a flash of
lightening. Systole seemed now here, now there; diastole seemed the same;
then all reversed, varied and confused. So, I could reach no decision, neither
about what I might conclude myself nor believe from others.”
These difficulties will hopefully diminish upon using the wavelet transform.
Features about the different diseases can be extracted, and this will hopefully
lead to early prediction of diseases.
The Cell as a Bioelectric Generator.
Surrounding the cells of the body are fluids. These fluids are ionic and
represent a conducting medium for electric potentials. Three main kinds of
ions are involved with the mechanism of producing cell potentials. These are
sodium (Na +), potassium(K+) and chloride (Cl-) ions. Cells, as nerve or
muscle have a cell wall (membrane) that acts as a selective ionic filter to
these ions. The membranes of excitable cells permit the entry of (K+) and
(Cl-) and block the flow of (Na+) even though there might be a very high
concentration of sodium across the cell membrane. The effect of this is that
the concentration of sodium ions inside the cell is less than it is on the
outside of the cell wall. Since sodium is a positive ion, the outside of the cell
becomes more positive than the inside of the cell.
The ions seek equilibrium. Therefore, positive K+ ions tend to move
inside the cell. The chloride ions move with much less impedance than
sodium or potassium across the cell membrane.

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At equilibrium, a potential difference between -50 and -100mV exists
across the cell membrane. When equilibrium is reached, the resulting
potential across the cell membrane is called the resting membrane potential.
When this occurs, the cell is said to be polarized. A decrease in the resting
membrane potential difference is called depolarization, and any increase in
this potential difference is called hyperpolarization.
Electro Encephalogram (EEG)
The wavelet transform is recently being used in analyzing EEG or Electro-
encephalogram. EEG signals give a record of the electrical activity of the
brains. In normal subjects, three wave frequencies may be recorded- the
alpha, beta, delta rhythms. The alpha rhythm consists of rhythmical
oscillations in electrical potential occurring at a rate of 10 per second. The
waves have a voltage of about 50 microvolts on the average. The alpha
waves appear with the eyes shut but are reduced by visual activity or by
mental effort (e.g. mental arithmetic); they return again as the mental work
ceases. The waves are present when the eyes are open, provided that the
visual field is uniform. Any attempt to fix the eyes upon any detail results in
the disappearance of the waves. The beta rhythm is faster than the alpha (25-
50 per second) and of lower voltage (5-10 microvolts). The delta waves have
a frequency from 1 to 5 per second and a voltage from 20 to 200 microvolts.

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