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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.

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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

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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

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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

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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.

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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).

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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)

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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).

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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

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• 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

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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

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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

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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.

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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

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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).

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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

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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.

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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

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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 .

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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

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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.

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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

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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

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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

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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

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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

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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

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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)

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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,

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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

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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.

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Figure 3.27 Schematic diagram of the proposed research work