Upload
others
View
6
Download
0
Embed Size (px)
Citation preview
27
CHAPTER 3
ELECTROCARDIOGRAM, PHOTOPLETHYSMOGRAM
AND WAVELET TRANSFORM
This chapter presents the fundamentals of ECG, PPG and wavelet
transform. In the first part of this chapter the basics of ECG signal
measurement, P, QRS complex, ST segments and T wave are discussed.
Using these morphological changes how the abnormalities in cardiac activity
can occur is discussed. The second part of this chapter presents the
fundamentals of PPG signal measurement and its characteristic features. The
third part of this chapter presents the basics of wavelet transform and the use
of wavelet transform in signal processing. At the end of the chapter the
overview of the proposed work is presented.
3.1 ELECTROCARDIOGRAM
An electrocardiogram (ECG) is an electrical recording of the heart
and is used in the diagnosis of heart disease. These impulses are recorded as
waves called P-QRS-T deflections. Each cardiac cell is surrounded by and
filled with a solution that contains sodium (Na+), potassium (K+), and
calcium (Ca++). In its resting condition the interior of the cell membrane is
considered negatively charged, with respect to the outside. When an electrical
impulse is initiated in the heart, the inside of a cardiac cell rapidly becomes
positive in relation to the outside of the cell.
28
The electrical impulse causes this excited state and this change of
polarity, is called depolarization. Immediately after depolarization, the
stimulated cardiac cell returns to its resting state, which is called
repolarization. The resting state is maintained until the arrival of the next
wave of depolarization. This change in cell potential from negative to positive
and back to negative is called an action potential. That action potential
initiates a cardiac muscle contraction. Figure 3.1 shows that the components
of ECG signal.
The ECG is a measurement of the effect of this depolarization and
repolarization for the entire heart on the skin surface, and is also an indirect
indicator of heart muscle contraction, because the depolarization of the heart
leads to the contraction of the heart muscles (Jin Yinbin et al 1998). Although
the phases of the ECG are due to action potentials traveling through the heart
muscle, the ECG is not simply a recording of an action potential. During each
heartbeat, cells fire action potentials at different times, and the ECG reflects
patterns of that electrical activity.
Figure 3.1 Components of an ECG signal
29
3.1.1 ECG Lead Configuration
To record an electrocardiogram, a number of electrodes, 3, 6, 10, 12
or 16 can be affixed to the body of the patient. The electrodes are connected
to the ECG machine by the same number of electrical wires. These electrodes
are called as leads.
There are three types of electrode systems:
Bipolar limb leads (or) standard leads
Augmented unipolar limb leads
Chest leads (or) precordial leads
3.1.2 Bipolar Limb Leads (OR) Standard Leads
By convention, lead I have the positive electrode on the left arm and
the negative electrode on the right arm and therefore measure the potential
difference between the two arms. Figure 3.2 shows the Einthoven’s triangle.
In this and the other two limb leads, an electrode on the right leg serves as a
reference electrode for recording purposes. In the lead II configuration, the
positive electrode is on the left leg and the negative electrode is on the right
arm. Lead III has the positive electrode on the left leg and the negative
electrode on the left arm.
Figure 3.2 Einthoven’s Triangle
30
These three bipolar limb leads roughly form an equilateral triangle
(with the heart at the center) that is called Einthoven’s triangle in honor of
Willem Einthoven who developed the electrocardiogram in 1901. Figure 3.3
and 3.4 shows the bipolar lead configuration and the output waveform of it.
Figure 3.3 Bipolar Lead Configurations
Figure 3.4 Output Waveforms for bipolar Lead configuration
Whether the limb leads are attached to the end of the limb (wrists
and ankles) or at the origin of the limb (shoulder or upper thigh) makes no
difference in the recording because the limb can simply be viewed as a long
wire conductor originating from a point on the trunk of the body.
3.1.3 Augmented Unipolar Limb Leads
These are termed unipolar leads because there is a single positive
electrode that is referenced against a combination of the other limb
31
electrodes. Figure 3.5 and 3.6 shows the augmented unipolar limb leads. The
positive electrodes for these augmented leads are located on the left arm
(aVL), the right arm (aVR), and the left leg (aVF). In practice, these are the
same electrodes used for leads I, II and III. (The ECG machine does the
actual switching and rearranging of the electrode designations).
Figure 3.5 Augmented lead configuration
Figure 3.6 Output waveforms for Augmented Lead configuration
These leads are unipolar in that they measure the electric potential at
one point with respect to a null point (one which doesn't register any
significant variation in electric potential during contraction of the heart). This
null point is obtained for each lead by adding the potential from the other two
leads. For example, in lead aVR, the electric potential of the right arm is
compared to a null point, which is obtained by adding together the potential of
lead aVL and lead aVF.
32
3.1.4 The Chest Leads
In addition to the four limb leads, a 12-lead ECG includes six chest
leads. The chest leads sample the electrical activity over small areas of the
heart. The chest leads look at the heart’s electrical activity in a slightly off-
horizontal plane around the front of the chest. This detects problems that
might not be obvious from the standard limb leads, which measure electricity
in a vertical plane. The chest leads are often called V-leads. Figure 3.7 shows
the chest lead. The electrode over the chest is the positive electrode, while the
limb electrodes are all averaged together to form a general ground electrode.
Figure 3.7 Chest Leads
The precordial (chest) leads start with V1, placed beneath the 4th rib
to the right of the sternum. Lead V2 is opposite to V1 at the left side of the
sternum. V3 is halfway to lead V4, which is placed below rib 5 directly down
from the middle of the clavicle. Lead V5 is straight around the chest from V4,
in line with the front of the armpit. V6 is directly around from V5, straight
down from the middle of the armpit (Leslie Cromwell et al 1997; Khandpur,
2003).
33
3.1.5 Normal Sinus Rhythm
Each P wave is followed by a QRS complex. P wave rate is 60-100
beats per minute (bpm). If rate is less than 60bpm it is called sinus
bradycardia. If rate is greater than 100 it is called sinus tachycardia
Figure 3.8 presents the normal ECG signal.
Figure 3.8 Normal ECG Signal
3.1.6 Standard conventions for reading an ECG
The rate of paper (i.e. of recording of the ECG) is 25 mm/s, which
results in:
1 mm = 0.04 sec (or each individual block)
5 mm = 0.2 sec (or between 2 dark vertical lines)
34
The voltage recorded from the leads is also standardized on the
paper where 1 mm = 0.1 mV (or between each individual block vertically).
Figure 3.9 shows the recording chart. The standards are:
5 mm = 0.5 mV (or between 2 dark horizontal lines)
10 mm = 1.0 mV (this is how it is usually marked on the
ECG)
Figure 3.9 ECG Recording Chart
3.1.7 Waves and Intervals of ECG
3.1.7.1 P Wave
During normal atrial depolarization, the main electrical vector is
directed from the SA node towards the AV node, and spreads from the right
atrium to the left atrium. This turns into the P wave on the ECG, which is
upright in II, III and a VF (since the general electrical activity is going toward
the positive electrode in those leads) and inverted in a VR (since it is going
away from the positive electrode for that lead). A P wave must be upright in
leads II and a VF and inverted in lead aVR to designate a cardiac rhythm as
Sinus Rhythm (Brendan Phibbs 2005).
35
The P wave height and width depends not only on the size of the RA
and LA but also the site of origin of atrial impulse. A normal SA nodal origin
of P wave produce the normal shaped P Waves. Ectopic P waves can have a
wide variation of morphology (fully inverted, partially inverted, slurred,
biphasic, notched, rounded, deformed, etc. The morphology is dictated by the
direction of P wave vector and thus it is quite variable in different leads.
Further it is also determined by the inter atrial and intra atrial conduction
(Ariyarajah et al 2005). A P wave can also be of very low amplitude and it
may be entirely isoelectric, which could actually mean the P waves are as
good as absent. This can happen in all leads or in few leads. Atria get
electrically activated but fail to inscribe a P wave. This is termed as isoelectric
P waves.
3.1.7.1.1 The Importance of Isoelectric P Waves
It can happen, both in sinus rhythm and in ectopic atrial rhythm.
Absent P waves should be differentiated form isoelectric P waves. It is
typically described in focal atrial rhythm arising from the right side of the
inter atrial septum near the perinodal tissue. The atrial tachycardia arising
from this site has isoelectric P waves in most of the leads especially in lead V.
The relationship between P waves and QRS complexes helps
distinguish various cardiac arrhythmias.
• The shape and duration of the P waves may indicate atrial
enlargement. The PR interval is measured from the beginning
of the P wave to the beginning of the QRS complex. It is 120
to 200 millisecond long. On an ECG tracing, this corresponds
to 3 to 5 small boxes.
• A PR interval of over 200 millisecond may indicate a first
degree heart block
36
• A short PR interval may indicate a pre-excitation syndrome
via an accessory pathway that leads to early activation of the
ventricles, such as seen in Wolff-Parkinson White syndrome.
• PR segment depression may indicate atrial injury or
pericarditis.
• Variable morphologies of P waves in a single ECG lead are
suggestive of an ectopic pacemaker rhythm such as wandering
pacemaker or multifocal atrial tachycardia. Table 3.1 presents
the abnormalities due to variations in P waves.
Table 3.1 Causes of abnormalities and their characteristic features
S No Characteristic Feature of P wave Causes 1 P wave inversion (other than aVR) a) Ectopic atrial focus
b) AV nodal rhythm 2 High amplitude P wave Atrial Hypertrophy (or)
Atrial Dilation a) Mitral valve disease b) Hypertension c) Cor Pulmonale d) Congenital Heart Disease
3 Wide P wave (over 0.11s) Left Atrial Enlargement 4 Biphasic P wave
(2nd half negative in III or V1) Left Atrial Enlargement
5 M shaped or notched P wave Findings: a) Over 0.04s between peaks b) Taller in I than II
a) M-Mitral: Left Atrial Enlargement
6 Peaked P-wave Findings: 1)Tall and pointed 2)Taller in Lead III than in I
P-Pulmonale: Right Atrial Enlargement
7 P-wave absent a) Sinoatrial node block b) AV nodal rhythm
8 Inverted P-wave Dextrocardia
37
3.1.7.2 QRSComplex
The QRS complex is a structure on the ECG that corresponds to the
depolarization of the ventricles. Because the ventricles contain more muscle
mass than the atria, the QRS complex is larger than the P wave. In addition,
since the bundle of his (Purkinje system) also coordinates the depolarization
of the ventricles, the QRS complex tends to look "spiked" rather than rounded
due to the increase in conduction velocity. A normal QRS complex is 0.06 to
0.10 sec (60 to 100 ms) in duration. The duration, amplitude, and morphology
of the QRS complex is useful in diagnosing cardiac arrhythmias, conduction
abnormalities, ventricular hypertrophy, myocardial infarction and other
disease states. Q waves can be normal (physiological) or pathological. Normal
Q waves represent depolarization of the inter-ventricular septum. For this
reason, they are referred to as septal Q waves and can be appreciated in the
lateral leads I, aVL, V5 and V6 (Khandpur 2003).
3.1.8 Diseases Related with Abnormal QRS Complex
3.1.8.1 Tachycardia
Tachycardia typically refers to a heart rate that exceeds the normal
range for a resting heart rate. Ventricular tachycardia (VT or V-tach) is a
potentially life-threatening cardiac arrhythmia that originates in the ventricles.
It is usually an irregular, wide QRS complex with a rate between 120 and 250
beats per minute (Holly L., 2009) Ventricular tachycardia has the potential of
degrading the more serious ventricular fibrillation. Ventricular tachycardia is
a common and often lethal, complication of a myocardial infarction
(heart attack).
3.1.8.2 Ventricular Fibrillation
Ventricular fibrillation occurs in the ventricles (lower chambers) of
the heart; it is always a medical emergency. If left untreated, ventricular
38
fibrillation (VF or V-fib) can lead to death within minutes. When a heart goes
into V-fib, effective pumping of the blood stops. V-fib is considered as a form
of cardiac arrest, and an individual suffering from it will not survive unless
cardiopulmonary resuscitation (CPR) and defibrillation are provided
immediately.
3.1.8.3 Bradycardia
A slow rhythm, (less than 60 beats/min), is labeled Bradycardia.
This may be caused by a slowed signal from the sinus node (termed sinus
Bradycardia), a pause in the normal activity of the sinus node (termed sinus
arrest), or by blocking of the electrical impulse on its way from the atria to the
ventricles (termed AV block or heart block).Bradycardia may also be present
in the normally functioning heart of athletes or other well-conditioned
persons.
3.1.8.4 Bundle Branch Block
A bundle branch block refers to a defect of the heart's electrical
conduction system. When a bundle branch becomes injured due to underlying
heart disease, myocardial infarction, or cardiac surgery, it ceases to conduct
electrical impulses appropriately. This results in altered pathways for
ventricular depolarization. Since the electrical impulse can no longer use the
preferred pathway across the bundle branch, it may move instead through
muscle fibers in a way that both slows the electrical movement and changes
the directional propagation of the impulses. As a result, there is a loss of
ventricular synchrony, prolonged ventricular depolarization and
corresponding drop in cardiac output (George A. Perera et al 1941).
3.1.8.5 ST Segment
The ST segment connects the QRS complex and the T wave and has
duration of 0.08 to 0.12 sec (80 to 120 ms). It starts at the J point (junction
39
between the QRS complex and ST segment) and ends at the beginning of the
T wave. However, since it is usually difficult to determine exactly where the
ST segment ends and the T wave begins, the relationship between the ST
segment and T wave should be examined together. The typical ST segment
duration is usually around 0.08 sec (80 ms). It should be essentially level with
the PR and TP segment.
3.1.9 Types of ST-Segment
ST-segment elevation was classified into three types according to
the morphology of the ST elevation after the J point on any pericardial
derivation: figure 3.10 shows the three different types of ST-Segment
elevation in ECG signal. Concave type where ST-T segment rises with
downward convexity, straight type where ST-T segment rises obliquely like
an inclined plane and convex type where ST-T segment rises with an upward
convexity (Gettes and Cascio 1991). Figure 3.11 presents various disorders
related to the morphological changes in ST segment.
Figure 3.10 Three different types of ST-Segment elevation in ECG
signal
A=concave-type;B=straight type; C,D,E=convex-type.
40
Figure 3.11 different types of disorders in ST segment
3.1.10 QT Interval
The QT interval is measured from the beginning of the QRS
complex to the end of the T wave. A normal QT interval is usually about 0.40
seconds. The QT interval as well as the corrected QT interval is important in
the diagnosis of long QT syndrome and short QT syndrome. The QT interval
varies based on the heart rate, and various correction factors have been
developed to correct the QT interval for the heart rate.
3.1.11 U Wave
The U wave is not always seen. It is typically small and follows the
T wave. U waves are thought to represent repolarization of the papillary
muscles or Purkinje fibbers. Prominent U waves are most often seen in
hypokalaemia, but may be present in hypercalcemia, thyrotoxicosis, or
41
exposure to digitalis, epinephrine, and Class 1A and 3 antiarrhythmic, as well
as in congenital long QT syndrome and in the setting of intracranial
haemorrhage. An inverted U wave may represent myocardial ischemia or left
ventricular volume overload.
3.1.12 T Wave Abnormalities
T wave inversion in lead III is a normal variant. New T-wave
inversion (compared with prior ECGs) is always abnormal. Pathological T
wave inversion is usually symmetrical and deep (>3mm).Inverted T-waves in
the right precordial leads (V1-3) are a normal finding in children, representing
the dominance of right ventricular forces.
3.1.12.1 Peaked T Waves
Tall, narrow, symmetrically peaked T-waves are characteristically
seen in hyperkalaemia.
3.1.12.2 Hyper Acute T Waves
Broad, asymmetrically peaked or ‘hyperacute’ T-waves are seen in
the early stages of ST-elevation MI (STEMI) and often precede the
appearance of ST elevation and Q waves. They are also seen with Prinzmetal
angina.
3.1.12.3 Inverted T Waves
Inverted T waves are seen in the following conditions
Normal finding in children
Persistent juvenile T wave pattern
Myocardial ischemia and infarction
42
Bundle branch block
Ventricular hypertrophy (‘strain’ patterns)
Pulmonary embolism
Hypertrophic cardiomyopathy
Raised intracranial pressure
3.1.12.4 Biphasic T Waves
There are two main causes of biphasic T waves: Myocardial
ischemia and Hypokalemia. In Ischemic condition T wave will go up and then
downwards. In hypokalemic conditions T waves go down and then upwards.
3.1.12.5 Flattened T Waves
Flattened T waves are a non-specific finding, but may represent
ischemia (if dynamic) or electrolyte abnormality e.g. hypokalemia.
3.1.13 The Limitations of the ECG
The ECG reveals the heart rate and rhythm only during the
time that the ECG is taken. If intermittent cardiac rhythm
abnormalities are present, the ECG is likely to miss them.
Ambulatory monitoring is needed to record transient
arrhythmias.
The ECG can often be normal or nearly normal in patients
with undiagnosed coronary artery disease or other forms of
heart disease (false negative results.)
Many “abnormalities” that appear on the ECG turn out to have
no medical significance after a thorough evaluation is done
(false positive results).
43
3.2 PLETHYSMOGRAPHY
Plethysmograph is a combination of the ancient Greek words
‘plethysmos’, meaning increase, and ‘grapho’ is the word for write, and is an
instrument mainly used to determine and register the variations in blood
volume or blood flow in the body which occur with each heartbeat. PPG was
one of the earliest methods devised for measuring blood flow in the
extremities, having first been employed for this purpose around the turn of the
century. By 1938, Hertzman found a relationship between the intensity of
backscattered light and blood volume in the skin. Indeed much of our basic
knowledge of vascular physiology and Pathophysiology has been derived
from PPG studies.
3.2.1 Photoplethysmography
Photoplethysmography (PPG) is an optical technique which
typically operates using infrared light, allowing the transcutaneous
registration of venous and/or arterial blood volume changes in the skin
vessels. The complex interaction between the heart and connective
vasculature are the components of the mechanism that generates the PPG
signal.
Figure 3.12 PPG signal and its component waves
44
The first part of the PPG waveform (systolic component) is formed
as a result of pressure transmission along a direct path from the aortic root to
the finger. The second part (diastolic component) is formed by pressure
transmitted from the ventricle along the aorta to the lower body where it is
reflected back along the aorta to the finger. The upper limb provides a
common channel for both the directly transmitted pressure wave and the
reflected wave and, therefore, has little influence on the contour of the PPG
signal as is shown in the above Figure 3.12.
3.2.2 Working Principle
The fundamental of this technology is the detection of the dynamic
cardiovascular pulse-wave, generated by the heart, as it travels throughout the
body. The cardiovascular pulse wave is propagated by the elastic nature of the
peripheral arteries, as they are excited by the contractions of the heart. The
heart instigates a pulse pressure wave that travels throughout the arteries into
deeper vasculature. Generally, the illuminating PPG wavelength is chosen to
provide weak absorption in tissue, yet stronger absorption by blood, to
provide a high degree of optical contrast. Infrared radiation is often employed
and provides a convenient illumination source. It provides a signal
proportional to changes in skin blood volume.
The wavelength of the light used is crucial when determining the
parameters of interest. Wave lengths between 650-950 nm are commonly
used because they combine good penetration with good contrast between the
dark vessels (veins and arteries) and the light tissue. In this wavelength range
hemoglobin in the blood absorbs much more strongly than the remaining
tissue. Light is reflected after it reaches the skin and part of the light
penetrates into deeper layers where it may be either scattered or absorbed.
Absorption is predominant in the epidermis and upper dermis, whereas
scattering is predominant in deeper layers.
45
While absorption is due to specific chromophores such as water,
hemoglobin and melanin, scattering is caused by the different refractive
indices of tissue components such as cell organelles and membranes. In the
dermis, collagen fibers are believed to be a major source of light scattering. In
the epidermis, the major absorbing entity in this spectral region is melanin.
For example: the wavelength of 400-600 nm is absorbed in the dermis by
blood chromophores: hemoglobin, oxyhemoglobin, bilirubin and carotene. A
weak absorption by blood occurs at wavelengths of 700-1300 nm with a low
scattering in the dermis (Xun Shen and Roeland Van Wijk 2005).
3.2.3 Optical Characteristics of Biological Tissue
A decreased absorption of the skin in the visible spectral region is
caused by a considerably lower amount of biologically important
chromophores in comparison with ultraviolet (UV) radiation (melanin, DNA,
urocanic acid and aromatic amino acids).
Figure 3.13 Optical characteristics of biological tissue in visible and
infrared range
46
Figure 3.13 shows the optical characteristics of biological tissue in
the visible and infrared range. The signal produced by PPG also depends on
the location and the properties of the subject's skin at that site, including skin
structure, blood oxygen saturation, blood flow rate and temperature (Lev
Vladimirovich et al 2007).
Furthermore, the amount of reflected light varies with the number of
red blood cells in the cutaneous microcirculation. Slight dilatation and
contraction of arterioles and capillaries during each cardiac cycle attenuate
light reflection. PPG requires a light source and a detector, and their relative
positions may vary. Differing PPG sensors have been designed with different
aims: reflection mode (the light source and the detector are placed side by
side with mean volume of interaction between infrared photons and
measuring up to 4 mm in tissue depth) allows placing on virtually any tissue
site or transmission mode (light source and the detector opposite each other
on the skin surface, illuminating a large tissue volume for strong signals)
typically for application on the peripheral digits, or with fiber optic lines for
use in highly magnetic environments such as MRI. In quantitative PPG the
optical illumination in the measuring area is automatically adjusted for each
different type of skin until a predetermined level of reflected light is reached.
With this technology, PPG measurements are independent of skin color,
thickness and individual blood volume.
3.2.4 The Steady and Pulsatile Components of PPG Signal
The PPG signal consists of AC and DC components. The AC
component (for arterial pulse detection) is synchronous with the heart rate and
depends on the pulsatile blood volume changes. Figure 3.14 shows the
Pulsatile components in PPG signal. It has been suggested that there is AC
orientation during each cardiac cycle .
47
Figure 3.14 Pulsatile components in PPG signal
Conversely, the DC component (commonly used for venous
evaluation) of the signal varies slowly and reflects variations in the total blood
volume of the examined tissue .There is variation in the wave due to aging of
the artery which leads to stiffening. Figure 3.15 presents the variation in pulse
wave due to ageing.
Figure 3.15 Variation in pulsatile wave in artery
48
3.3 WAVELET TRANSFORM
3.3.1 Fourier analysis
The most well-known of the transforms is Fourier analysis, which
breaks down a signal into constituent sinusoids of different frequencies. For
many signals, Fourier analysis is extremely useful because the signal’s
frequency content is of great importance. Fourier analysis has a serious
drawback, i.e., in transforming to the frequency domain, time information is
lost. If the signal properties do not change much over time i.e., if it is what is
called a stationary signal—this drawback isn’t very important. But if it is a
non- stationary signal, Fourier analysis is not suitable in detecting them.
dt.e*)t(x)f(X ftj2
(3.1)
df.e*)f(X)t(x ftj2
(3.2)
Figure 3.16 Fourier analysis
Figure 3.16 shows Fourier analysis by which a signal is broken
down into its constituent sinusoids of different frequencies and is thus
49
transformed into frequency domain (Duhamel and Vetterli 1990, Frigo and
Johnson, 1998).
3.3.2 Short-Time Fourier Analysis
This is a technique called windowing the signal in which the Fourier
transform is used to analyze only a small section of the signal at a time. This
maps a signal into a two-dimensional function of time and frequency. It
provides some information about both when and at what frequencies a signal
event occurs. Its drawback is that once we choose a particular size for the time
window, that window is the same for all frequencies.
Figure 3.17 Short time Fourier analysis
Figure 3.17 shows the translation of a short section of a signal from
time domain to frequency domain using STFT.
t
ft2j*)w(X dt.e*)]'tt(w*)t(x[)f,t(STFT
(3.3)
Here, x(t) is the signal itself, w(t) is the window function, and * is
the complex conjugate.
50
3.3.3 Wavelet Transform
It is possible to analyze any signal by using an alternative approach
called the multi resolution analysis (MRA) (Burrus et al 1999). MRA, as
implied by its name, analyzes the signal at different frequencies with different
resolutions. Every spectral component is not resolved equally as was the case
in the STFT. MRA is designed to give good time resolution and poor
frequency resolution at high frequencies and good frequency resolution and
poor time resolution at low frequencies. The Wavelet analysis does this by
using a windowing technique with variable-sized regions.
Figure 3.18 Wavelet transform
Figure 3.18 shows wavelet transform by which a signal can be
analyzed at different frequencies with different resolutions.
3.3.3.1 Continuous Wavelet Transform
The continuous wavelet transform (CWT) is defined as the sum over
all time of the signal multiplied by scaled, shifted versions of the wavelet
function ψ. The result of the CWT is many wavelet coefficients, which are a
function of scale and position. Multiplying each coefficient by the
appropriately scaled and shifted wavelet yields the constituent wavelets of the
original signal. The CWT can operate at every scale, from that of the original
51
signal up to some maximum scale. The CWT is also continuous in terms of
shifting: during computation, the analyzing wavelet is shifted smoothly over
the full domain of the analyzed function.
dt
st)t(x
|s|1)s,()s,(CWT *
xx
(3.4)
The transformed signal is a function of two variables, and s, the
translation and scale parameters, respectively (t) is the transforming
function, and it is called the mother wavelet.
3.3.3.1.1 Scaling Function (T)
A signal or function x (t) can often be better analyzed as a linear
combination of expansion functions
)()( ttx k
kk
(3.5)
where k is an integer index of the finite or infinite sum, k are real-valued
expansion coefficients, and k(t) are real valued expansion functions.
Consider the set of expansion functions composed of integer translations and
binary scaling of the real, square integrable function (t); i.e., the set {j,k (t)}
where
)kt2(2)}t({ j2/jk,j (3.6)
For all j, Z and (t) L2(R). Here, k determines the position of
j,k (t) along the time axis, j determinesj,k (t) ’s width- how broad or narrow it
52
is along the t-axis and 2/j2 controls its height or amplitude. The shape of
j,k (t) changes with j, (t) and is called as scaling function.
3.3.3.1.2 MRA Requirements
1. The scaling function is orthogonal to its integer translated.
2. The subspaces spanned by the scaling function at low scales
are nested within those spanned at higher scales.
V.....VVVV....V 2101
3. The only function that is common to all Vjis x(t)=0
}0{V
4. Any function can be represented with arbitrary precision.
3.3.3.1.3 Wavelet Function (T)
The term wavelet means a small wave. The smallness refers to the
condition that this (window) function is of finite length (compactly
supported). The wave refers to the condition that this function is oscillatory.
The term mother implies that the functions with different region of support
that are used in the transformation process are derived from one main
function, or the mother wavelet. In other words, the mother wavelet is a
prototype for generating the other window functions.
A wavelet function (t), together with its integer translated and
binary scaling, spans the difference between any two adjacent scaling
subspaces Vj and Vj+1.
53
We can define the set of wavelets as
)kt2(2)t( j2/jk,j (3.7)
The two most important properties of wavelet function are-
1. The function integrates to zero.
0dt)t(
2. It is square integrable or, equivalently, has finite energy;
dt|)t(| 2
We can now express the space of all measurable, square- integrable
functions as
....WWV)R(L 1002
Or,
....WWV)R(L 2112
Or even ....WWWWW....)R(L 210122
54
Figure 3.19 Scaling and Wavelet function spaces
Figure 3.19 shows the relationship between scaling and wavelet
function spaces.
3.3.3.1.4 Dilation
Scaling a wavelet simply means stretching (or compressing) it. It is
denoted by the scale factor, often denoted by the letter a. The smaller the scale
factor, the more “compressed” the wavelet. The higher scales correspond to
the most stretched wavelets. The more stretched the wavelet, the longer the
portion of the signal with which it is being compared, and thus the coarser the
signal features being measured by the wavelet coefficients.
V2 = 10011 WWVWV
W0 W1
V1=V0 W0
V0
55
Figure 3.20 Dilation
Figure 3.20 shows the relationship between scale factor and wavelet
signal features.
Thus, there is a correspondence between wavelet scales and
frequency as revealed by wavelet analysis:
• Low scale a Compressed wavelet Rapidly changing
details High frequency.
• High scale a Stretched wavelet Slowly changing,,
coarse features Low frequency.
Figure 3.21 Different scales in dilation
56
Figure 3.21 shows the effect of different values of scale factor on
wavelet. Smaller value of scale factor compresses the wavelet while higher
value of scale factor stretches the wavelet.
3.3.3.1.5 Translation
Shifting a wavelet simply means delaying (or hastening) its onset.
Mathematically, delaying a wavelet function by k is represented by
Figure 3.22 Translation
Figure 3.22 Shows the delaying of wavelet function by time k.
3.3.3.2 Discrete Wavelet Transform
A main goal of wavelet research is to create a set of expansion
functions and transforms that give informative, efficient, and useful
description of a function or signal. In applications working on discrete
signals, one never has to directly deal with expansion functions. Discrete
wavelet transform (DWT) is obtained simply by passing a discrete signal
through a filter bank. Wavelet theory can be understood and developed only
by using such digital filters. This is the meeting point between wavelets and
sub band coding and the origin of two different nomenclatures for the same
concepts. In fact, wavelet transform and sub band coding are so closely
connected that both terms are often used interchangeably. Filter banks are
57
structures that allow a signal to be decomposed into sub signals through
digital filters, typically at a lower sampling rate. Figure 2.1shows a two-band
filter bank.
Figure 3.23 One-level two band perfect reconstruction filter bank
Figure 3.23 shows the analysis and synthesis blocks of a filter bank.
The sampling rates are the same at both banks. Downsampling is performed at
the analysis end and upsampling at the synthesis end.
It is formed by the analysis filters (Hi(z), i = 0, 1) and the synthesis
filters (Gi(z), for i = 0, 1).Filters H0(z) and G0(z) are low-pass filters. In an
M-band filter bank, Hi(z) and Gi(z) for 0 < i < M − 1 are band-pass filters, and
HM−1(z) and GM−1(z) are high-pass filters. For a two-band filter bank, M = 2
and H1 (z) and G1(z) are high-pass filters. If the input signal can be recovered
without errors from the sub signals, the filter bank is said to be a perfect
reconstruction (PR) or a reversible filter bank. To enable PR, the analysis and
synthesis filters have to satisfy a set of bilinear constraints.
Every finite impulse response (FIR) filter bank with an additional
linear constraint on the low-pass filter is associated with a wavelet basis. The
low-pass synthesis filter G0 (z) is associated with the scaling function, and the
remaining band-pass synthesis filters (G1 (z) in the 2-band case) are each
58
associated with the wavelet functions. Analysis low-pass filter H0 (z) is
associated with the so-called dual scaling function and analysis band-pass
filters with the dual wavelet functions.
The notion of channel refers to each of the filter bank branches. A
channel is the branch of the 1-D scaling coefficients (or approximation signal)
and also each branch of the wavelet coefficients (or detail signals). The
concept of band involves the concept of frequency representation, but it is
commonly used in image processing to refer to each set of samples which are
the output of the same 2-D filter. In 1-D linear processing both concepts are
interchangeable.
The coefficients of the discrete wavelet expansion of a signal may
be computed using a tree structure where the filter bank is applied recursively
along the low-pass filter channel. Every recurrence output is a different
resolution level, which is a coarser scale representation of the original signal.
In summary, a DWT is a dyadic tree-structured transform with a multi-
resolution structure.
An alternative approach to the filter bank structure for computing
DWT is the lifting scheme (LS). Lifting is more flexible and may be applied
to more general problems.
3.3.3.3 Biorthogonal Wavelets:
In many filtering applications we need filters with symmetrical
coefficients to achieve linear phase. None of the orthogonal wavelet systems
except Haar are having symmetrical coefficients. But Haar is too adequate for
many practical applications. Biorthogonal wavelet systems can be designed to
have this property. This is our motivation for designing such wavelet system.
But the price is that non-zero coefficients in analysis filters and synthesis
59
filters are not same. In orthogonal wavelet system, Ø(t) is orthogonal to Ψ(t)
and its translates. In biorthogonal system our requirement is that Ø(t) be
orthogonal to Ψ’(t) and its translates.
3.3.3.4 Vanishing Moments
Vanishing moments is a core concept of wavelet theory. In fact, the
number of vanishing moments was a more important factor than spectral
considerations for the choice of the wavelet transforms for the JPEG2000
standard.
Vanishing the nth moment means that given a polynomial input up
to degree n, the filter output is zero. A wavelet has N vanishing moments if
Nnfordttt
n
0,0)( (3.8)
The same definition applies for the dual wavelet to have N’
vanishing moments. N and N’ are also the multiplicity of the origin as a root
of the Fourier transform of the synthesis and analysis high-pass filter,
respectively. Also, it is the multiplicity of the regularity factor (1+z−1) in
H1(z) and G1(z) (which are the z-transform of the filters h1[n] and g1[n]).
3.3.3.5 Discrete wavelet transform in signal processing
The DWT of a signal x is calculated by passing it through a series of
filters. First the samples are passed through a low pass filter with impulse
response g resulting in a convolution of the two:
ky n (x *g)[n]) x[k] g [n k]
(3.9)
60
The signal is also decomposed simultaneously using a high-pass
filter h. The outputs are detail coefficients (from the high-pass filter) and
approximation coefficients (from the low-pass). It is important that the two
filters are related to each other and they are known as a quadrature mirror
filter. However, since half the frequencies of the signal have now been
removed, half the samples can be discarded according to Nyquist’s rule. The
filter outputs are then sub sampled by 2. This decomposition is repeated to
further increase the frequency resolution and the approximation coefficients
are decomposed with high and low pass filters and then down-sampled. This
is represented as a binary tree with nodes representing a sub-space with a
different time-frequency localisation. The tree is known as a filter bank.
Figure 3.24 Wavelet transform decomposition diagram
Figure 3.24 shows the different stages of decomposition of wavelet
transform.
The most commonly used set of discrete wavelet transforms was
formulated by the Belgian mathematician Ingrid Daubechies in 1988. This
formulation is based on the use of recurrence relations to generate
progressively finer discrete samplings of an implicit mother wavelet function;
each resolution is twice that of the previous scale. In her seminal paper,
61
Daubechies derives a family of wavelets, the first of which is the Haar
wavelet. Interest in this field has exploded since then, and many variations of
Daubechies original wavelets were developed (Dokur 2006; Vajarian and
Splinter 2006).
Figure 3.25 Haar wavelet
Haar wavelet is shown in Figure 3.25. Studies have reported that
Daubechies wavelet is most suitable for ECG signal processing and for the
proposed work Daubechies wavelet is used. Daubechies wavelet families are
similar in shape to QRS complex and their energy spectrum is concentrated
around low frequencies.Figure 3.26 presents the wavelet and scaling function
of Daubechies wavelet.
Figure 3.26 Daubechies wavelet
62
3.4 OVERVIEW OF THE PROPOSED WORK
The aim of the proposed work is to
1. Monitor ECG and PPG signals simultaneously and
continuously to extract essential cardiac parameters like blood
pressure, heart rate and pulse rate.
2. Predict the cardiac risk of a person by calculating
augmentation Index using PPG signal thus assessing arterial
stiffness.
3. Analyse ECG signal by segmenting ECG into P wave, QRS
complex, ST segment and T wave is also proposed.
Figure 3.27 presents the schematic diagram of the proposed work.
ECG and PPG of patients are acquired simultaneously. R peaks of the ECG
and pulse peaks of the PPG are detected and the time difference between the
peaks is pulse transit time and it is calculated. PTT is inversely proportional to
blood pressure and is calculated. Using the R peak interval, heart rate is
estimated. Similarly using pulse peaks, pulse rate is estimated. From the
acquired PPG the points of interest (Ps, Pd and Pi) are measured and using
this augmentation index is calculated. Cardiac risk assessment is done based
on the obtained augmentation index. From the acquired ECG signal, segments
of ECG such as P wave, QRS complex, ST segment and T wave are extracted
and their characteristic features are obtained and using these ECG signal
analysis is done.
63
Figure 3.27 Schematic diagram of the proposed research work