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

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

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

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

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

7

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 dtj 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 dtj ft

e( , ) ( ) *( )τ τπ

= −−∞

∞−

∫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]

∆fG f df

G f df

f=

∫

∫

2 2

2

( )

( )

9

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]

∆tt 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

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 ψτ

τ ,

/( ) ( )

at 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 aa

x tt

adt( , ) ( ) ( )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]

af

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

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

fQ=

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

tf 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

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: unchanged x(n) y(n)

15

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 22α Ψ

where

α( , ) ( ) ( )/i m x t t m dt

i 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 Wi, 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 Wi. 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 zi= ⊕ ⊕ = ⊕+ +

∈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: halved x(n) y(n) 2

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, Wi 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

Scaling Function and the Wavelet Mother obtained using D4 as the

prototype filter.

(Time)

Figure 8

(Am

pli

tude)

19

Scaling Function and Wavelet Mother using D6 as the prototype filter

Figure 9

20

Scaling Function and Wavelet Mother using Daubecies 10 as the prototype

filter

Figure 10

21

Scaling Function and Wavelet Mothre using Coiflet 15 as prototype filter

Figure 11

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 nn

= −=

∞

∑ 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 nn

= −=

∞

∑ 10

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 nn

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 nn

L

0 20

1

( )=

−

∑ = , h nn

L

0 12

0

1

( ) ==

−

∑

h nn

L

1 00

1

( )=

−

∑ = , h nn

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

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.

2 h0 2

2 h1 2 h1’

x x’

h0’

Figure 12. Analysis and Synthesis Filter Bank

24

Wavelet Packet of the 9th

branch of the 19 level filter bank using Coiflets 15

Figure 13

25

Fre

qu

en

cy

f

0

Time Ƭ

Figure14. Wave packet tiling of the time-frequency plane

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

Input

Signal

28

Table 1

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

31

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(z 2 ) followed by

downsampling [Figure 18]. H(z 2 ) 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 nh L n

− − − −1 0 1( ) , n=0,1,…,L-1

2 2 H(z 2 ) H(z)

33

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

Frequency Responses of the Prototype Filters

Figure 20

(Ga

in)

36

Spectrum of the successive filters using Daubechies 4

Figure 19.a

(Ga

in)

37

Spectrum of the Successive Equivalent Filters using Daubechies 6

Figure 19.6

38

Spectrum of the Successive Equivalent Filters using Daubechies 10

Figure 19.c

39

Spectrum of the Successive Equivalent Filters using Coiflets 6

Figure 19.d

40

Spectrum of successive equivalent filters using Coiflets 15

Figure 19.e

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

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 nn

( ) ( )10

==

∞

∑

The different scaling functions were obtained from finite length prototype

filters that satisfied the condition

ho nn

L

( )=

−

∑ =0

1

2

43

Figure 22

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 ( )26 . Thus, In order to normalize the scaling

function, it has to be multiplied by 22− 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 Ho

j( ) ( )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 nn

0( )∑ ; when the impulse response of the scaling function is

normalized to 1, φ(0)=1

• φ(π)=0

• φ 2 1n

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 ( )nn

=∑ 0

Moreover, the energy in the orthonormal wavelet is equal to 1

ψ 2 1n

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

ψ φ( ). ( )n nn

=∑ 0

ψ ψ1 2 0( ). ( )n nn

=∑

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

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

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

Figure 24

Am

pli

tud

e

Time

49

Figure 25

Ga

in

50

(Time)

Figure 26

(Am

pli

tud

e)

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

Figure 26.b

53

Figure 27

54

Figure 28.a

55

Output of the First Branch using Daubechies 4

Figure 28.b

56

Outputs of the Last Branch using Daubechies 4

Figure 28.c

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

Table 2-1

59

Table 2-2

60

Figure 29

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

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

Table 3

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

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

(Am

pli

tud

e)

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

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

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

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

70

[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

Am

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tud

e

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

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Input Signal

(Time)

(Am

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

Time

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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

(Am

pli

tud

e)

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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.

136

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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138