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i

Modeling Propagation Characteristics for Arm

Motion in Wireless Body Area Networks

By

Miss Aamrah Ikram

Registration Number: CIIT/FA11-REE-065/ISB

MS Thesis

In

Electrical Engineering

COMSATS Institute of Information Technology

Islamabad – Pakistan

ii

Modeling Propagation Characteristics for Arm-

Motion in Wireless Body Area Networks

A Thesis presented to

COMSATS Institute of Information Technology

In partial fulfillment

of the requirement for the degree of

MS (Electrical Engineering)

By

Miss Aamrah Ikram

CIIT/FA11-REE-065/ISB

Fall, 2012

iii

Modeling Propagation Characteristics for Arm-

Motion in Wireless Body Area Networks

A Graduate Thesis submitted to Department of Electrical Engineering as

partial fulfillment of the requirement for the award of M. S. Degree

(Electrical Engineering).

Name

Registration Number

Aamrah Ikram

CIIT/FA11-REE-065/ISB

Supervisor:

Dr. Mahmood Ashraf Khan,

Director,

Center for Advanced Studies in Telecommunications (CAST),

COMSATS Institute of Information Technology (CIIT),

Islamabad, Pakistan

December, 2012

Co-supervisor:

Dr. Nadeem Javaid,

Assistant Professor,

Center for Advanced Studies in Telecommunications (CAST),

COMSATS Institute of Information Technology (CIIT),

Islamabad, Pakistan

December, 2012

iv

Final Approval

This thesis entitled

Modeling Propagation Characteristics for Arm-

Motion in Wireless Body Area Networks

By

Miss Aamrah Ikram

CIIT/FA11-REE-065/ISB

has been approved

for the COMSATS Institute of Information Technology, Islamabad

External Examiner: __________________________________

(To be decided)

Supervisor: ________________________

Dr. Mahmood Ashraf Khan/Director,

Center for Advanced Studies in Telecommunications (CAST),

CIIT, Islamabad, Pakistan.

Co-Supervisor: ________________________

Dr. Nadeem Javaid /Assistant professor,

Center for Advanced Studies in Telecommunications (CAST),

CIIT, Islamabad, Pakistan.

Head of Department:________________________

Dr. Raja Ali Riaz / Associate professor,

Department of Electrical Engineering,

CIIT, Islamabad, Pakistan.

v

Declaration

I Miss Aamrah Ikram CIIT/FA11-REE-065/ISB herebyxdeclare that I

havexproduced the workxpresented in this thesis, duringxthe

scheduledxperiod of study. I also declare that I havexnot taken anyxmaterial

from anyxsource exceptxreferred toxwherever due that amountxof

plagiarism isxwithin acceptablexrange. If a violationxof Higher Education

Commission rulesxon research hasxoccurred in thisxthesis, I shall be

liablexto punishablexaction under the plagiarismxrules of the Higher

Education Commission, Pakistan.

Date: ________________

________________

Miss Aamrah Ikram

CIIT/FA11-REE-065/ISB

vi

Certificate

It is certified that Miss Aamrah Ikram, CIIT/FA11-REE-065/ISB has carried

out all the work related to this thesis under my supervision at the

Department of Electrical Engineering, COMSATS Institute of Information

Technology, Islamabad and the work fulfills the requirements for the

award of MS degree.

Date: _________________

Supervisor: ________________________

Dr. Mahmood Ashraf Khan/Director,

Center for Advanced Studies in Telecommunications (CAST),

CIIT, Islamabad.

Co-Supervisor:____________________

Dr. Nadeem Javaid /Assistant professor,

Center for Advanced Studies in Telecommunications (CAST),

CIIT, Islamabad

____________________________

Head of Department:

Dr. Raja Ali Riaz/Associate Professor,

Department of Electrical Engineering,

CIIT, Islamabad.

vii

DEDICATION

Dedicated to my parents.

viii

ACKNOWLEDGMENT

I am heartily grateful to my supervisor, Dr. Mahmood Ashraf Khan, and co-

supervisor Dr. Nadeem Javaid whose patient encouragement, guidance

and insightful criticism from the beginning to the final level enabled me

have a deep understanding of the thesis.

Lastly, I offer my profound regard and blessing to everyone who supported

me in any respect during the completion of my thesis especially my friends

in every way offered much assistance before, during and at completion

stage of this thesis work.

Miss Aamrah Ikram

CIIT/FA11-REE-065/ISB

ix

List of Abbreviations

WBANs

Wireless Body Area Networks

WBASNs

Wireless Body Area Sensor Networks

DGF

Dyadic Green’s Function

MF

Mathieu Functions

AMF

Angular Mathieu Functions

RMF

Radial Mathieu Functions

PDA

Personal Digital Assistant

BAN

Body Area Networks

UWB

Ultra-Wide Band

EM

Electromagnetic

IBC

Impedance Boundary Conditions

FT

Fourier Transform

x

List of Publications

[1] Ain.Q, Ikram. A, Javaid. N, Qasim. U, Khan. Z. A, “Modeling Propagation Characteristics

for Arm- Motion in Wireless Body Area Sensor Networks”, published in 7th International

Conference on Broadband and Wireless Computing, Communication and Applications

(BWCCA-2012), Victoria, Canada, 2012.

[2] Ikram. A, Javaid.N “Modeling Wave Propagations for Bio-sensors with Arm-Motion",

submitted in 4th IEEE International Conference on Ambient Systems, Networks and

Technologies (ANT-13) June 25-28, 2013, Halifax, Nova Scotia, Canada.

xv

Abstract

To monitor health information using wireless sensors on body is a promising new application. In

health care technology Wireless Body Area Networks (WBANs) are expected to be a

breakthrough technology. Human body acts as a transmission channel in wearable wireless

devices, so electromagnetic propagation modeling is well thought-out for transmission channel in

Wireless Body Area Sensor Network (WBASN). In this research, I have presented the wave

propagation in WBASN which is modeled as point source (Antenna), close to the arm of the

human body. In this thesis, I have deduced the wave propagation model in WBASNs;

transmission sensor is close to the arm of the human body. To model the arm motion of human

body, we used Dyadic Green's Function (DGF) and Mathieu Functions (MF). DGF is specifically

used to propose a channel model for arm motion of human body model. Using terms of vector

wave function and scattering superposition this function can be expanded. MF is developed for

Angular Mathieu Functions (AMF), Radial Mathieu Functions (RMF), sine and cosine-sphere

terms. This thesis, describes the analytic derivation of electric field distribution derived by both

DGF and MF, in spherical pattern and the simulation results of those derivations.

Table of Contents

1 Introduction 1

2 Related Work 4

3 Motivation 5

4 Functions for Boundary Value Problems 7

4.1 Dyadic Green’s Functions: . . . . . . . . . . . . . . . . . . . . . . . 7

4.2 MathieuFunctions ........................... 8

4.3 HelmholtzEquation........................... 9

4.4 Inhomogeneous Helmholtz Equation: . . . . . . . . . . . . . . . . . 9

4.5 HankelFunctions:............................ 9

4.6 Wronskian Relationship: . . . . . . . . . . . . . . . . . . . . . . . . 10

4.7 FourierTransform............................ 10

4.8 Spherical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . 11

5 Mathematical Modeling for Arm Motion using DGF. 13

5.0.1 Electric Field Propagation Characteristics . . . . . . . . . . 14

5.0.2 Spherical Wave Vector Eigen Function . . . . . . . . . . . . 14

5.0.3 Scattering Superposition . . . . . . . . . . . . . . . . . . . . 16

5.0.4 Superposition of Direct Wave . . . . . . . . . . . . . . . . . 18

5.0.5 Superposition of Scattered Wave . . . . . . . . . . . . . . . . 18

5.1 Transmitter and Receiver Located Outside the Body . . . . . . . . 21

6 Propagation Model Geometry for Arm-Motion using MF 22

6.1 Conversion of three-D problems to two-D problems . . . . . . . . . 22

6.1.1 Vector of Scattered Wave . . . . . . . . . . . . . . . . . . . . 24

6.1.2 Vector of Incident Wave . . . . . . . . . . . . . . . . . . . . 25

6.1.3 Resultant Vector Field . . . . . . . . . . . . . . . . . . . . . 25

6.2 Impedance Boundary Conditions for Human Body . . . . . . . . . . 27

6.3 Field Distribution (On-Body Sensor) . . . . . . . . . . . . . . . . . 28

i

6.4 Pathloss................................. 30

7 Simulations and Discussions 32

7.1 Simulations using DGF . . . . . . . . . . . . . . . . . . . . . . . . . 32

7.2 Simulations using MF . . . . . . . . . . . . . . . . . . . . . . . . . 35

8 Conclusion 40

References 40

ii

List of Figures

5.1 Human body model showing arm motion in 3D. . . . . . . . . . . . 13

5.2 Scattering Superposition . . . . . . . . . . . . . . . . . . . . . . . . 17

6.1 2-D model of wave propagation for WBANs . . . . . . . . . . . . . 23

7.1 Magnitude of scattered ﬁeld component Eϕversus angle ϕ,with dif-

ferent values of dand the angle is θ=π

6............... 33

7.2 Magnitude of scattered ﬁeld component Eϕversus angle ϕ,with dif-

ferent values of dand the angle is θ=π

3............... 34

7.3 Magnitude of scattered ﬁeld component Eϕversus angle ϕ,with dif-

ferent values of dand the angle is θ=π............... 35

7.4 Magnitude of electric ﬁeld Erat diﬀerent values of d, and θ=π

2. . . 36

7.5 Magnitude of electric ﬁeld Erat diﬀerent values of d, and θ=π

4. . . 37

7.6 Magnitude of electric ﬁeld Erat diﬀerent values of d, and θ=π

6. . . 37

7.7 Magnitude of electric ﬁeld Erat diﬀerent values of d, and θ=π

10 . . 38

7.8 Magnitude of electric ﬁeld Erat diﬀerent values of d, and θ=π

3. . . 38

7.9 Magnitude of electric ﬁeld Erat diﬀerent values of d, and θ=π. . . 39

iii

List of Tables

4.1 Functions and their uses in thesis. . . . . . . . . . . . . . . . . . . . 12

6.1 Human body Impedance parameters at diﬀerent frequencies . . . . 28

iv

Chapter 1

Introduction

Hospitals throughout the world are facing a unique problem, as the aged popu-

lation is increased, health-care population is decreased. Telecommunication com-

munity is not doing much work in the ﬁeld of medicine however, there is a need of

remote patient monitoring technology. To fulﬁll this task, it is required to build

communication network between an external interface and portable sensor devices

worn on and implemented within the body of the user which can be done by Body

Area Networks.

BAN is not only useful for remote patient monitoring, but can also establishes

within the hospitals; like in operation theaters and intensive care units. It would

enhance patient comfort as well as provide ease to doctors and nurses to perform

their work eﬃciently. BAN is used for connecting body to wireless devices and

ﬁnds applications in various areas such as entertainment, defense forces and sports.

Wireless Body Area Networks (WBANs) is providing promising applications in

bio-medical sensor systems. The large range and prospective of these applications

makes it an exhilarating new research track. There is a requirement of home

health care monitoring for elder or aged population. Particularly, advancement is

required in physiological sensors, wireless communications and channel modeling.

These sensors measure the parameters, which are under observation and sends the

data to the nearest Personal Digital Assistant (PDA). The sensor and PDA both

are worn directly on the body. Automatic emergency calls are built, when their is

rapid change in brain, heart, diabetes and high blood pressure etc for diagnosis.

Importantly, range of sensors and its’s eﬀect on ﬁeld with the movement of human

1

body should be under consider.

The basic step in building any wireless device is to study the transmission chan-

nel and to model it accurately. Channel modeling is a technique that has been

initiated by a group of researchers throughout the world [1]. They have studied

path loss and performed measurement campaigns for wireless node on the body

[2-8]. Some researchers have taken into account, the implanted devices which are

the area of BAN called as intra-body communication [9].

Many researchers paid attention on developing the propagation model around

cylinder. They have developed models by taking the human body as lossy cylin-

der. For the short range low data rate communication in BAN, measurement

groups have considered UWB (Ultra-Wide Band) as the appropriate air interface.

The models developed by measurement campaigns are only path loss models and

do not provide any description of propagation channel.

For this we need simple and generic body area propagation model to develop

low power radio systems for inspection of human body. To model the transmis-

sion channel correctly is the main step to build up the wireless devices. Channel

modeling also assists to understand the wave propagation in and around human

body. It is important to study the propagation mechanism of radio waves on

and inside the body in order to develop an accurate BAN channel model. This

study will show the underlying propagation characteristics. It would help in the

development of BAN transceivers which are much suited to the body environment.

In this thesis, wave propagation model for Upper half of human body (spherical

symmetry) is developed by using two ways; DGF and MF. These propagation

models can tackle as electromagnetic boundary-value problem and solved by es-

tablished methods but with complicated process. The course of action of these

two functions are:

1. DGF:

For a given position of the transmitter on or inside the body it is required to ﬁnd

out the electromagnetic ﬁeld on or inside the body for a BAN channel model. This

is quite a critical problem that requires a large amount of computational power.

Therefore, it is necessary to derive an analytical expression which will perform this

objective. In short this determines which propagation mechanism takes place, that

2

is reﬂection, diﬀraction and transmission [10]. An appropriate method of doing

this task is by using Dyadic Green’s function. The solution of canonical problems,

such as cylinder, multi layer and sphere have been solved in electromagnetic (EM)

theory, using Dyadic Green’s Functions [11-13].

2. MF:

Mathieu functions allow an accurate and time saving computation. The expan-

sion coeﬃcients are derived and can be easily implemented in computer code.

Compared to the calculation with series of Bessel function products. The derived

solution of 2-D problem is expanded in term of eigenfunctions in the spherical

coordinates by MF, using separation of variables. Solving boundary conditions on

sphere’s surface, expansion coeﬃcients can be calculated. Mainly, focus is on the

ﬁeld outside the body as surface impedance of human body is easy to be tested.

Also human body is not an ideal for the radio frequency transmission. This will

be very helpful for the Channel modeling of WBANs.

3

Chapter 2

Related Work

Wen Xun Zhang [10], ﬁnd out the ﬁeld distribution in cylindrical coordinates.

He formulated the path gain of electromagnetic ﬁeld around the human body at

15-MHz and 2.40-GHz frequencies.

Attaphongse Taparugssanagorn, Carlos Pomalaza-Rez, Ari Isola, Raﬀaello Tesi,

Matti Hmlinen [11], they worked on Wideband (UWB) for the channel modeling

of WBANs. Contribution is, they toke measurements in the frequency.

Arthur Astrin, Takahiro Aoyagi [14], worked on WBAN for the devices which are

implanted inside the body. They compared the path loss of BAN devices by taking

into account obstacles near the human body an diﬀerent postures of human body

in daily life.

L. An, M. J. Bentum [15], have investigated propagation around the human body

and proposed the channel modeling for various scenarios. That channel model-

ing is done in the way to understand wave propagation in and around human body.

Asthag Gupta, Thushara D [16], they developed analytical channel model for hu-

man body in cylindrical form. They have used DGF for the channel modeling for

BAN communication.

4

Chapter 3

Motivation

Recently, WBASNs shows potential due to increasing application in medical health

care. New applications in WBANs are facilitating the patient to being monitored

and stay connected to its health care system. Characterization of the electro-

magnetic wave propagation is an important step for the development of WBANs.

Although, complexity of human tissue structure and geometry makes it diﬃcult

to derive wave propagation model.

In WBASNs, each sensor in the body sends it’s data to antenna, both sensors

and antenna are worn directly on the body. Examples include sensors which can

measure brain activity, blood pressure, body movement and automatic emergency

calls. We require straightforward and basic body area propagation models to de-

velop eﬃcient and low power radio systems near the human body. To achieve bet-

ter performance and reliability, wave propagation needs to be modeled correctly.

Few studies have focused on analytic model of propagation around a cylinder (as

human body resembles a cylinder) using diﬀerent functions.

The disadvantage is that, the earlier studies have not concentrated on wave prop-

agation, and only focused on propagating around a circular cylinder based on

analytic model [17-19]. Most of the researchers have focused on the analytical

wave propagation model for cylinder using diﬀerent diﬀerential functions. All of

them, discussed and developed the wave propagation model for cylindrical distri-

bution as geometry of human body resembles to cylinder.

BAN channel model requires extensive study for manipulation of the body on the

radio propagation as antennas are placed on or inside the human body. As I have

described before that a simple and standard wave propagation model is needed.

The propagation characteristics of cylindrical shaped human body have been de-

5

rived using Dyadic Green’s functions [20]. The model includes the cases of trans-

mitter and receiver presents either inside or outside of the body and also provides

simulation plots of Electric ﬁeld with diﬀerent values of angle (θ).

All the above proposals describe the propagation characteristics of cylindrically

shaped human model [21]. I have developed a simple but generic approach to body

area propagation derived from Dyadic Green’s Function (DGF) and MF. These

approaches are for arm motion of human body. When the human arm is moved in

r, θ, ϕ direction, propagation characteristics of spherical shaped have been derived

using DGF and MF.

6

Chapter 4

Functions for Boundary Value

Problems

4.1 Dyadic Green’s Functions:

Green’s function is a kind of function which solves inhomogeneous diﬀerential

equations having speciﬁc initial conditions or boundary-value problem. Dyadic

Green’s function is basically depends on the spherical vector eigen functions. These

eigen functions are Lnhk(χ), Mnhk (χ) and Nnhk (χ).

Lnhk(χ) = ∇[Ψnhk (χ)] (4.1)

Mnhk(χ) = ∇ × [Ψnhk (χ)] (4.2)

Nnhk(χ) = 1

k∇×∇[Ψnhk(χ)] (4.3)

where kis the wave number of medium, nis an integer, his a real number and x

is a point in space.

7

4.2 Mathieu Functions

Mathieu Functions are kind of special functions use to solve the problems in applied

mathematics.

It can solve the problems related to:

1. Elliptical cylinder coordinates.

2. Electrical dipoles.

3. Wave motion in periodic medium.

4. Spherical Coordinates.

The main equation for Mathieu diﬀerential equation is given as:

d2y

dx2+ [a−2qcos(2x)]y= 0 (4.4)

ais the constant.

Mathieu function is further divided into two parts:

1. Angular Mathieu function.

It is denoted by Sn(x) and have even and odd components Meand Mo.

2. Radial Mathieu function.

It is denoted by Rn(x) and have even and odd components Ceand So.

Even and odd Mathieu functions:

1. Even Mathieu Function

Me=

∞

l=0

Be

l(h, m)cosnθ (4.5)

mand nare both even or both odd.

2. Odd Mathieu function

Mo=

∞

l=0

Bo

l(h, m)cosnθ (4.6)

In above equations (Be

l) and (Bo

l) are the expansion coeﬃcients for even and odd

functions respectively.

8

4.3 Helmholtz Equation

This equation is used solve the inhomogeneous and homogeneous partial diﬀeren-

tial equations.

1. Homogeneous Helmholtz equation:

(∇2+k2)U(y, t) = 0 (4.7)

∇2= laplacian operator, k= Wave Number and U(y, t)= Vector function.

4.4 Inhomogeneous Helmholtz Equation:

(∇2+k2)U(y, t) = −δ(4.8)

i. 2-D Inhomogeneous Helmholtz equation:

(∇2+k2)U(y, t) = −δ(x−x0)δ(y−y0)δ(z−z0) (4.9)

ii. 3-D Inhomogeneous Helmholtz equation:

(∇2+k2)U(y, t) = −e−jkzz0δ(x−x0)δ(y−y0) (4.10)

4.5 Hankel Functions:

Hankel functions are solution of bessel’s equations which are linearly Independent.

Mathematical representation of Hankel Function is:

Hl=Jl(Z) + ıYl(z) (4.11)

Jl(z)=First kind of bessel function and Yl(z)=Second kind of bessel function .

9

Orders of 2nd kind of spherical Hankel Function:

→Zero order of Hankel Function of 2nd kind:

h(2)

o=ıe−ız

z(4.12)

→First order of Hankel Function of 2nd kind:

h(2)

1=−e−ız(z−ı)

z2(4.13)

→Second order of Hankel Function of 2nd kind:

h(2)

1=−ıe−ız(z2−3ıZ −3)

z3(4.14)

→Third order of Hankel Function of 2nd kind:

h(2)

1=e−ız(z3−36ıZ2−15z+ 15ı)

z4(4.15)

4.6 Wronskian Relationship:

Wroskian is determinant, used to study diﬀerential equations and to show the set

of solution is linearly independent. let xand yare two functions, wronskian of

these two functions are given as:

W(x, y) = x´y−´xy

or

W(f1, f2, f3) =

f1(x)f2(x)f3(x)

´

f1(x)´

f2(x)´

f2(x)

f′′

1(x)f′′

2(x)f′′

2(x)

(4.16)

4.7 Fourier Transform

The Fourier transform is a standardization of the complex Fourier series in the

limit as n→ ∞ . It replace the discreet Blto continuous F(z)dz, in this way sum

is changed to integral form.

10

Generalized equation is :

F(z) = ∞

−∞

F(k)e2πıkz dz (4.17)

and Inverse Fourier Transform is given as:

F(k) = ∞

−∞

F(z)e−2πıkz dk (4.18)

4.8 Spherical Coordinates

It is a three coordinate system having three-dimensional space. A single point is

deﬁned by three diﬀerent numbers:

1. The radial angle(also called radius and radial coordinate) denoted by r,

2. The polar angle represented by θ,

3. The azimuthal angle as ϕ.

Mathematically they are deﬁned as:

x=rsin θcos ϕ(4.19)

y=rsin θsin ϕ(4.20)

z=rcos θ(4.21)

11

centering

Functions Used to Solve Equations

Dyadic Green’s Functions Solves inhomoge-

neous boundary value

problems with the help of vector eigen functions. G(x, x0) = δ(x, x0)

Mathieu Functions Gives near and far electric ﬁeld magnitude. d2y

dx2+ [a−2qcos(2x)]y= 0

Helmholts Equation To ﬁnd out magnetic potential vector in 3-D

model.

(∇2+k2)U(x, t) = 0

Hankle Function Combining the scattered and Incident waves. H(1)

n=Jn(Z) + ıYn(z)

Wronskian relationship Simpliﬁes the equation for even and odd Mathieu

Functions.

W(x, y) = x´y−´xy

Fourier Transform Converts three dimension problem to two dimen-

sion

F(z) = ∞

−∞ F(k)e2πıkz dz

Inverse Fourier Transform Converts two dimension problem to two dimension

problem

F(k) = ∞

−∞ F(z)e−2πıkz dk

Spherical Coordinates Use for the representation of polar coordinates to

spherical coordinates

x=rsin θcos ϕ,y=rsin θsin ϕ,z=rcos θ

Table 4.1: Functions and their uses in thesis.

12

Chapter 5

Mathematical Modeling for Arm

Motion using DGF.

In this thesis, spherical symmetry is used to represent in and around the arm of

the human body. A point on body is a sensor, denoted by x which represents

(r,Θ,ϕ) coordinates in the spherical coordinate system and x0is the location of

transmitting antenna. (r,Θ,ϕ) are unit vectors along radial, angle of elevation

from z-axis and azimuthal angle from x-axis as shown in ﬁgure 3.1.

Figure 5.1: Human body model showing arm motion in 3D.

13

5.0.1 Electric Field Propagation Characteristics

Let E(x) be electric ﬁeld at point x due to current source J(x0). The general

formula for Electric ﬁeld can be written as:

E(x) = iωµp V

G(x, xo)J(x, x0)dv (5.1)

where V is volume of source,J(X0) is the current source, G(x, x0) is the Dyadic

Green’s function ′ω′is the radian frequency of transmission and ′µ′

pis magnetic

permeability of the medium. A Dyadic Green’s function is a type of function used

to solve inhomogeneous diﬀerential equations subject to speciﬁc initial conditions

or boundary condition.

5.0.2 Spherical Wave Vector Eigen Function

As we are considering arm motion of human body, therefore, spherical symmetry

is used by taking shoulder as center. For this, spherical eigen functions are used

to write the Dyadic Green’s function.

Dyadic Green’s function is basically depends on the spherical vector eigen func-

tions [14]. These eigen functions are Lnhk (χ), Mnhk(χ) and Nnhk (χ), where kis

the wave number of medium, nis an integer, his a real number and x is a point

in space. These all are the solutions to the Helmholtz equation having three com-

ponents in r, Θ and ϕ. These vector eigen functions are given by [22]:

Lnhk(χ) = ∇[Ψnhk (χ)] (5.2)

Mnhk(χ) = ∇ × [Ψnhk (χ)] (5.3)

Nnhk(χ) = 1

k∇×∇[Ψnhk(χ)] (5.4)

14

In above eigen functions, Laplacian operator in the spherical coordinate system is

∇. It’s mathematical expression is given as:

∇=∂

∂r +∂

r∂θ +∂

rsin θ∂ϕ (5.5)

where x represents the point in space having components r,Θ and ϕ. Solution of

Helmoltz equation is Ψnhk(x) which is the scalar eigen function [22].

[Ψnhk(χ)] = Zn(ηr)Ph

n(cos θ)cos

sin hϕ (5.6)

where Znis a general spherical function of order n. For sphere we use Hankle

function of ﬁrst and second order which are deﬁned as:

[Zn(ηr)] = (−1)n(ηr)( d

drη2r)n(sin(ηr)

ηr )n(5.7)

ηis the propagation constant in direction of ϕ,whereas k2=η2+h2. The laplace

operator is applied and ﬁnd the eigen values Lnhk,Mnhk and Nnhk by using eigen

function. The vector eigen function in (3.2), (3.3) and (3.4) becomes:

Lnhk(χ) = ∂Zn(ηr)

∂r Ph

n(cos θ)cos

sin hϕ +zn(ηr)

r

∂

∂θ Ph

n(cos θ)cos

sin hϕ +hZn(ηr)

rsin θPh

n(cos θ)sin

coshϕ

(5.8)

Mnhk(χ) = ∓hZn(ηr)

sin θPh

n(cos θ)sin

coshϕ −Zn(ηr)

∂

∂θ Ph

n(cos θ)cos

sin hϕ

(5.9)

Nnhk(χ) = nZn(ηr)

kr Ph

n(cos θ)cos

sin hϕ +1

kr

∂rZn(ηr)

∂r Ph

n(cos θ)cos

sin hϕ ∓h

sin θPh

n(cos θ)sin

coshϕ

(5.10)

15

These three vector eigen function are perpendicular among themselves as well as

with respect to each other [11]. In the form of matrices, vector Eigen functions

can be written in this form,

Lnhk(χ) =

∂Zn)(ηr)

∂r Ph

n(cos θ)cos

sin hϕ

Zn(ηr)

rPh

n(cos θ)cos

sin hϕ

hZn(ηr)

sin θPh

n(cos θ)sin

coshϕ

(5.11)

Mnhk(χ) =

0

∓hZn(ηr)

rPh

n(cos θ)cos

sin hϕ

−Zn(ηr)∂P h

n(cos θ)sin

coshϕ

∂θ

(5.12)

Nnhk(χ) =

hZn(ηr)

kr Ph

n(cos θ)cos

sin hϕ

∂n(ηr)

kr∂r

Ph

n(cos θ)cos

sin hϕ

∂θ

∓h

sin θPh

n(cos θ)sin

coshϕ

(5.13)

5.0.3 Scattering Superposition

In scattering problems, it is desirable to determine an unknown scattered ﬁeld that

is due to a known incident ﬁeld. Using the principle of scattering superposition we

can write Dyadic Green’s equation as superposition of direct wave and scattering

wave. In Figure 3.2, concept of scattering superposition is shown in which there is

a sensor located inside the arm of body considered as sphere. The sensor transmits

the wave to antenna which is divided in two parts as Direct wave and Scattered

wave. The Direct wave is considered as wave directly transmits from sensor to

transmitter and scattered wave is composed of reﬂection and transmission waves.

Therefore, general equation of scattering superposition is illustrated as:

G(x, x0) = Gd(x, x0) + Gs(x, x0) (5.14)

16

Figure 5.2: Scattering Superposition

17

Dyadic Green’s equation is divided in to two parts as direct wave [Gd(x, x0)] and

scattered wave [Gs(x, x0)]. The direct wave corresponds to direct from source to

measuring point and scattered is the reﬂection and transmission waves due to

presence of dielectric interface.

5.0.4 Superposition of Direct Wave

The direct component of DGF is given as [11]:

Gd(x, x0) = rr

k2(δ(x−x0) + ȷ

8π∞

−∞

dh

∞

n=−∞

1

n2x×

M(1)

nhk(X)M∗

nhk(X0) + N(1)

nhk(X)N∗

nhk(X0)

Mnhk(X)M(1∗)

nhk (X0) + Nnhk(X)N(1∗)

nhk (X0)

(5.15)

In the above equation of DGF, r > r0is for ﬁrst case and r < r0is second case.The

∗denotes the conjugation and is for the Dyadic product. Here we introduces

superscript (1) for outgoings wave and other for standing waves. If the vector

eigen function has the superscript (1) then, H(1)

nis chosen for Znand Jnshould

be used otherwise.

5.0.5 Superposition of Scattered Wave

Here we discuss four diﬀerent scenarios for the scattering components of DGF

along with boundary conditions Gs(x, x0).

(i). Both receiver and transmitter are inside the body.

(ii). The receiver is located outside and transmitter is located inside the body.

(iii). The receiver is located inside and transmitter is outside the body.

(iv). Both transmitter and receiver are located outside the body.

Receiver and transmitter are in the order: 1 denotes the medium inside human

body and 2 is for free space medium.

1) Transmitter and Receiver located inside the body. In this case, Receiver

and Transmitter both located inside the body so we can write Dyadic Green’s

equation as,

18

G(11)

s(x, x0) = ȷ

8π∞

−∞

dh

∞

−∞

1

η2x

×[Mnhk1Nnhk1]R12×

Nnhk1(X0)T

Mnhk1(X0)T

(5.16)

where R12 contains reﬂection coeﬃcients. R12 is calculated in literature using

boundary conditions, its matrix is given by [16]:

R12 = [Jn(η1d)Hn(η2d)−Hn(η2d)Jn(η1d)]−1

×[Hn(η2d)Hn(η1d)−Hn(η1d)Jn(η2d)]−1(5.17)

In the above equation of reﬂection coeﬃcient ’d’ represents radius of spherical

body model, η2

1=k2

1−h2, η2

2=k2

2−h2, k2

1=ω2µ1ϵ1, k2

2=ω2µ2ϵ2. The 2x2

matrices for jn(ηd) and Hn(ηd) are expressed as:

Bn(ηpd) = 1

η2

pd×ȷωϵpηpdBn(ηpd)−nhBn(ηp

−nhBn(ηp−ȷωµpηpdBn(ηpd)(5.18)

Bnis either H(1orJn)

n,B(.) is the derivative of Bw.r.t the whole argument, and

p=1,2

2) Transmitter located inside and Receiver located outside the body

In this case DGF can be written as :

G(21)

s(x, x0) = ȷ

8Π ∞

−∞

dh

∞

n=−∞

1

η2

×[Nnhk.Mnhk ]T12 N∗

nhk1(x0)T

M∗

nhk1(x0)T(5.19)

In the above equation T12 is a transmission coeﬃcient Matrix and given as:

19

T12 = 2ω

πη2

1d[Jn(η1d)Hn(η2d)−Hn(η2d)Jn(η1d)]−1

×ε10

0ε(5.20)

3) Both Transmitter and Receiver located outside the body

Gs(x, x0) = ȷ

8Π ∞

−∞

dh

∞

n=−∞

1

n2

×[MnhkNnhk ]R21

Nnhk(X0)TMnhk(X0)T

(5.21)

Similarly as R12,R21 is the reﬂection coeﬃcient matrix and it is given as:

R21 = [Jn(η1d)Hn(η2d)−Hn(η2d)Jn(η1d)]−1

×[Jn(η2d)Jn(η1d)−Jn(η1d)Jn(η2d)] (5.22)

4) Transmitter located outside and Receiver is inside the body In this

case we can write DGF as:

Gs(x, x0) = ȷ

8Π ∞

−∞

dh

∞

n=−∞

1

n2

×[MnhkNnhk ]T21

N∗

nhk1(x0)T

M∗

nhk1(x0)T

(5.23)

where T12 is the transmission coeﬃcient matrix, given as:

T21 = 2ω

Πηd [Jn(η1d)Hn(η2d)−Hn(η2d)Jn(η2

2d)]−1

×ε20

0−µ2(5.24)

20

5.1 Transmitter and Receiver Located Outside

the Body

In this section I present the equation which is required for simulation. With the

help of simulation it will be easy to study the propagation characteristics of arm

motion making spherical pattern.

Gs(x, x0) = ȷ

8Π ∞

−∞

dh

∞

n=−∞

1

n2Gnh(x, x0)dh

(5.25)

Gnh(x, x0) is stated as:

Gnh(x, x0) = Nnhk (X)1Mnhk (X)1×R21

Nnhk(X0)TMnhk (X0)T(5.26)

21

Chapter 6

Propagation Model Geometry for

Arm-Motion using MF

The geometry of the WBAN channel model for arm motion is spherical [23]. To

understand the geometry of model, cartesian coordinates (x, y, z) are translated

to spherical polar coordinates (r, θ, ϕ) using the following general equations:

x=rsin θcos ϕ(6.1)

y=rsin θsin ϕ(6.2)

z=rcos θ(6.3)

where as θϵ[0, π], ϕϵ[0,2π] and rϵ[0.∞) and ris sphere’s radius. ϕis describing the

diﬀerent angles at which arm is moving. Let point source is located at (r0, θ0, ϕ0)

and r=r1is deﬁning the interface between the body and free space.

6.1 Conversion of three-D problems to two-D

problems

First, it is needed to covert the three-D problem to two-D problem. Resultant

ﬁeld will be the combination of ﬁrst and fourth kind of MF. We know MF involves

complex computations, to avoid this complexity problem is transformed to two-D.

22

Figure 6.1 is showing the 2-D transform of spherical polar coordinates. Sensor is

place on (r1,θ1,ϕ1) coordinates, whereas (r,θ,ϕ) are representing the coordinates

at origin.

Figure 6.1: 2-D model of wave propagation for WBANs

Transformation of sphere into two-D presents circular geometry. Magnetic po-

tential vectors have to be calculated before measuring magnetic ﬁeld. Magnetic

potential vector ⃗

His the solution to inhomogeneous Helmholtz equation [22]. It’s

general formula is stated as:

(∇2+k2)⃗

H(x, y, z) = −δ(x−x0)δ(y−yo)δ(z−z0).(6.4)

Mathematically three-D geometry can be converted to two-D by using fourier

transform with respect to z, on both sides of equation (4) [24]. Therefore, equa-

tion 4.4 becomes:

(∇2

t+k2)⃗

H(x, y, z) = −ˆre−ikzroδ(x−x0)δ(y−yo) (6.5)

23

This equation is combination of two parts; inhomogeneous Helmholtz equation

and homogeneous Helmholtz equation.

where, ∇2

t=∂2

∂x2+∂2

∂y2and kt2=k2−k2

r. By expanding both scattered wave

and incident wave using MF ( i. e eigen functions for spherical coordinates). Us-

ing IBC (Impedance boundary conditions) [25], expansion coeﬃcient of scattered

wave can be measured on the surface of sphere.

6.1.1 Vector of Scattered Wave

Here, we are only considering the homogeneous equation for scattered wave. Po-

tential scattered wave ⃗

Hsc in spherical coordinates can be written as:

2

d2(cosh2r−cos2θ)(∂2

∂r2+∂2

∂θ2) + k2

r⃗

Hsc(r, θ, kr) = 0 (6.6)

In above equations (Be

n) and (Bo

n) are the expansion coeﬃcients for even and odd

functions respectively. General solution of this equation is the combination of

RMF and AMF, the notations are same as in [26]:

⃗

Hsc(r, θ, kr) = ˆz

∞

n=0

Rn(r)Sn(θ)e−ikrr0(6.7)

where Rn(r) is RMF and Sn(θ) is AMF. Potential vector of scattered wave outside

the sphere (r≥r1) can be expressed using 4th kind of MF. Fourth kind of MF is

used to express the far ﬁeld and combination of even and odd MF as:

⃗

Hsc(r, θ, kr) = ˆz

∞

n=0 BenM(4)

en(r, q0)cen(θ, q0)

+BonM(4)

sn(r, q0)sen(θ, q0)e−ikrr0

(6.8)

where, q0= (kr0

d

2)2is parameter of MF, krois representing wave number. Ben and

Bon are expansion coeﬃcient of even and odd MF respectively.

Similarly, AMF are cenand sen,′n′is an integer describing the order. Notation

′ce′and ′se′comes from sine-sphere and cosine-sphere. These functions are peri-

odic from πto 2π.

24

6.1.2 Vector of Incident Wave

The potential vector of the incident wave can be derived from equation (4.5) using

the 2nd kind of the Hankel function of 0(th)order [24]. It’s mathematical formula

is stated as:

⃗

Hinc(k0, r, kr) = ˆre−ikrr0

4ih(2)

0(kuOr) (6.9)

In this equation h(2)

0is the 2nd kind of Hankel Function with 0(th)order. The

source point (x0, y0) and the ﬁeld point (x, y) are separated with the distance ′r′

and mathematically formulated as:

r=(x−x0)2+ (y−y0)2(6.10)

If ﬁeld points close to the body are (r1≤r≤r0), by applying addition theorem

of Hankel Function on MF [27], equation (4.9) becomes:

⃗

Hinc(r, θ, kr) = ˆre−ikrr0

2i×∞

n=0

Mc(4)

n(r0, q0)Mc(1)

n(r.q0)

×cen(θ0, q0)cen(θ, q0) +

∞

n=1

Ms(4)

n(r0, q0)Ms(1)

n(r, q0)

×sen(θ0, q0)sen(θ, q0)

(6.11)

For the ﬁeld points far away from the surface of body, we use boundary conditions

(r0≤r≤ ∞).

⃗

Hinc(r, θ, kr) = ˆre−ikrr0

2i×∞

n=0

Mc(1)

n(r0, q0)Mc(4)

n(r.q0)

×cen(θ0, q0)cen(θ, q0) +

∞

n=1

Ms(1)

n(r0, q0)Ms(4)

n(r, q0)

×sen(θ0, q0)sen(θ, q0)

(6.12)

6.1.3 Resultant Vector Field

Summation of potential vectors of incident and scattered waves gave resultant

potential vectors.

25

In the form of MF, this vector can be expressed by the following equation:

⃗

H= ˆr˜

Hr(6.13)

ˆr˜

Hr= ˜re−ikrr0

4i

+∞

n=0

Mn(r, θ) (6.14)

For the ﬁeld close to the body r1≤r≤r0, generalize term for Mn(r, θ) becomes:

Mn(r, θ) = BenM(4)

cn (r, q0)cen(θ, q0)

+BenM(4)

sn (r, q0)senθ, q0

+2M(4)

cn (r0, q0)×M(1)

cn (r, q0)cen(θ0, q0)cen(θ, q0)

+M(4)

sn (u0, q0)M(1)

sn (r, q0)×sen(θ0, q0)sen(θ, q0)

(6.15)

For the ﬁeld far away from the body r1≤r≤ ∞, can be written as:

Mn(r, θ) = BenM(4)

cn (r, q0)cen(θ, q0)

+BenM(4)

sn (r, q0)senθ, q0

+2M(1)

cn (r0, q0)×M(4)

cn (r, q0)cen(θ0, q0)cen(θ, q0)

+M(1)

sn (u0, q0)M(4)

sn (r, q0)×sen(θ0, q0)sen(θ, q0)

(6.16)

In spectral domain the total radiated ﬁeld has ˆrcomponent, mathematically it

can be written as:

˜

Er=k2

r

ıωϵ e−ıkrr0⃗

H

˜

Er=−k2

r

4ωϵ e−ıkrr0

+∞

n=o

Mn(r, θ)

(6.17)

As spectral domain has only ˆrcomponent, Using equation of magnetic potential

26

vector transversal component can be deduced; which are as follow:

˜

hr=µ

4πr e−ıkrr0cos θ∂˜

hr

∂θ (6.18)

˜

hθ=−µ

4πr e−ıkrr0sin θ∂˜

hr

∂r (6.19)

6.2 Impedance Boundary Conditions for Human

Body

Human body impedance is deﬁned as:

Z=µb

εb

(6.20)

The generalized form of Impedance Boundary Condition is:

ˆ

l×(ˆ

l×˜

E) = −Z(ˆ

l×˜

h) (6.21)

Spectral domain can be deﬁned by total magnetic ﬁeld deﬁned by ˜

Eand ˜

hand

given as:

ˆ

l×(ˆ

l×˜

E)|(r=r1)=−ˆr˜

Er|(r=r1)(6.22)

Z(ˆ

l×˜

h)|(r=r1)= ˆrZ ˜

hθ|(r=r1)(6.23)

so equation (21) becomes:

−ˆr˜

Er|(r=r1)= ˆrZ ˜

hθ|(r=r1)(6.24)

Substituting values of ˜

Erand ˜

hθin equation (4.24), we get the following equation:

k2

r

ıωε ˜

Hr|(r=r1)=−√2Ze−ıkrr0

4πr sin θ∂˜

Hr

∂r |(r=r1)(6.25)

potential vector ˜

Hr|(r=r1)can be expressed as (4.14) and expanded up to ﬁnite

term using (15).

27

Frequency (Hz) Electric Permittivity Conductivity (S.m−1) Wavelength (m)

1MHz 1 0 299.79

915MHz 36.6642 0.6231 0.32764

2GHz 33.089 1.1708 0.0226

2.40MHz 35.1940 1.1367 0.0226

Table 6.1: Human body Impedance parameters at diﬀerent frequencies

suppose

B(θ) = −ıωε0√2Z

k2

rπr sin θ(6.26)

Even and odd terms are expressed as:

Ben =−2M(4)

cn (r0, q0)cen(θ0, q0) (6.27)

and

Bon =−2M(4)

sn (r0, q0)sen(θ0, q0) (6.28)

These are the basic formulas for even and odd expansion coeﬃcients and expanded

terms are deﬁned as:

Ben =k2

r(4πr) sin θM(1)

cn (r1, q0) + ık0˘

Z´

M1

cn(r1, q0)

k2

r(4πr) sin θM(4)

cn (r1, q0) + ık0˘

Z´

M4

cn(r4, q0)(6.29)

and

Bon =k2

r(4πr) sin θM(1)

sn (r1, q0) + ık0˘

Z´

M1

sn(r1, q0)

k2

r(4πr) sin θM(4)

sn (r1, q0) + ık0˘

Z´

M4

sn

˘(r4, q0)(6.30)

In above equations ˘

Zis the normalized surface impedance it’s formula is ˘

Z=Z

Z0.

Whereas Z0is the wave impedance in the air.

Diﬀerent parameters, which eﬀects the human body impedance at diﬀerent fre-

quencies are given in table 6.1.

6.3 Field Distribution (On-Body Sensor)

The ﬁeld distribution around the sphere can be solved in the spatial domain using

inverse fourier transform [28].

Er=1

2π∞

−∞

˜

Ereıkrrdkr(6.31)

28

For WBANs applications, on-body sensors are near the body surface of human.

Therefore, radial components of sensor and receiver (ﬁeld point) considered same

as the body surface i.e, (r0≈r≈r1).

Mn(r, θ) = Me

n(r1, θ) + Mo

n(r1, θ) (6.32)

Me

n(r1, θ) and Mo

n(r1, θ) are deﬁned as:

Me

n(r1, θ) = M(4)

cn (r1, q0)cen(θ0, q0)

×[Ben + 2M(1)

cn (r1, q0)cen(θ0, q0)] (6.33)

and

Mo

n(r1, θ) = M(4)

sn (r1, q0)sen(θ0, q0)

×[Bon + 2M(1)

sn (r1, q0)sen(θ0, q0)] (6.34)

Substituting the values of (Ben) and (Bon) in equation (4.33) and (4.34), we get

the following equations:

Me

n(r, θ) = M(4)

sn (r1, q0)cen(θ, q0)

×k2

r(4πr) sin θM(1)

cn (r1, q0) + ık0˘

ZM′(1)

cn

k2

r(4πr) sin θM(4)

cn (r1, q0) + ık0˘

ZM′(4)

cn

+2M(1)

cn (r1, q0)cen(θ0, q0)

(6.35)

By applying Wronskian relationship on equation (4.35) [29], we get the simpliﬁed

equation for even and odd MF:

Me

n(r, θ) = K0(4πr)M(4)

cn (r1, q0)cen(θ, q0)cen(θ0, q0)

k2

r(4πr) sin θM(4)

cn (r1, q0) + ık0˘

ZM′(4)

cn

(6.36)

Similarly,

Mo

n(r, θ) = K0(4πr)M(4)

sn (r1, q0)sen(θ, q0)sen(θ0, q0)

k2

r(4πr) sin θM(4)

sn (r1, q0) + ık0˘

ZM′(4)

sn

(6.37)

The ﬁnal equation for electric ﬁeld distribution by substituting equation (4.36),

(4.37) and (4.32)in equation (4.31):

29

Er(r, θ, ϕ) = 2×Zs

ωϵ ∞

0

eıkrr

∞

n=0

cen(θ, q0)cen(θ0, q0)

kr24πsinθ +ıkrM(4)

cn ˇ

Z

M4

cn(r1,q0)

+

∞

0

eıkrr

∞

n=0

sen(θ, q0)sen(θ0, q0)

kr24πsinθ +ıkrM(4)

sn ˇ

Z

M4

sn(r1,q0)dkr

(6.38)

6.4 Path loss

WBANs are greatly inﬂuenced by the amount of path loss that occurs due to

diﬀerent impairments. Devices for WBANs are generally placed inside or on the

body surface, so, losses between these devices would aﬀect the communication and

can degrade the performance monitoring in. In the following sections, we study in

detail about the WBANs communication and path loss that occurs in it and how

it aﬀects the performance.

Reduction in intensity of electromagnetic wave introduces path loss. Path loss

is mainly caused by free space impairments of propagating signal like refraction,

attenuation, absorption and reﬂection etc. It also depends on the distance be-

tween transmitter and receiver antennas, the height and location of the antennas,

propagation medium such as moist or dry air etc, and environment around the

antennas like rural and urban etc. Path loss for WBANs is diﬀerent from tradi-

tional wireless communication because it depends on both distance and frequency.

Frequency is catered because body tissues are greatly aﬀected by the frequency

on which sensor device is working.

Path loss in WBANs is very important parameter. WBANs work eﬀectively when

the path loss between the transmitter and receiver is less. Path loss in WBAN

occurs due to many factors such as reﬂection, diﬀraction and refraction etc, from

the body parts. So, data may face distortion due to path loss which causes dif-

ﬁculty for medical team located at far distance to correctly retrieve data. Path

loss will decrease the eﬃciency of monitoring diﬀerent vital signs in human body

at patient’s level as well as at medical team’s level. The main focus of this section

is to minimize the path loss that occurs at diﬀerent stages in WBANs. Path loss

depends on the frequency of the transmitter and distance of the receiver. This

increases the eﬃciency of monitoring in BAN which is our main goal. Path loss

dependence on distance as well as frequency is given in following equations:

P L =P L0+ 10nlog10

d

d0

+σs(6.39)

30

Whereas Lis the path loss, dis the distance between transmitting antenna and

receiving antenna, d0is the notation for reference distance, nis representing the

path loss exponent and σsis standard deviation. The above equation of path loss

can also expressed as:

P L = 20log10 (4πdf

c) (6.40)

31

Chapter 7

Simulations and Discussions

7.1 Simulations using DGF

As I have deﬁned earlier, arm motion at diﬀerent angles are presenting spherical

pattern. Therefore, I simulate the radio propagation environment having radius

d= 15cm, magmatic permeability for human body (assume that permeability of

human body is approximately equal to air) µ2= 1.256 ×10−6, similarly electric

permittivity ε2= 2.563 ×10−10. The dielectric constant is mean value of all tis-

sues of human body.

I consider the surrounding homogeneous medium to be air with magmatic perme-

ability µ1= 1.256×10−6 and electric permittivity ε1= 8.8542×10−12. Frequency

up to GHz is used for BAN communication, which is for ISM band. The Transmis-

sion frequency for simulation is 1GHz. We assumed that the transmitter is acting

as point source at x0= (16cm, π

2,0). The radial distance of receiver is r0= 18cm

from the central spherical axis of shoulder. For the simulation, we assumed that

receiver move along the azimuthal angle for varying values of ϕ0and diﬀerent

heights from the center of shoulder.

For simulation, we consider equation (3.26) in which Gnh(x, x0) is used in matrix

form of eigen functions. This equation has an integration which is not possible

so, approximated to summation. Thus, I approximate equation (3.26) in to this

form:

32

Gs(x, x0) = ȷ

8Π

L

l=−L

Q

n=−Q

1

n2Gnh(x, x0)dh

(7.1)

where L and Q are the truncation limits and ∆Hare the step size of integration.

N and ∆Hare so small that could be ignored and has no eﬀect on calculations. I

only presents electric propagation of multipath reﬂection and transmission waves

of scattering DGF.This is more signiﬁcant to represent the attribute of arm mo-

tion as compared to the direct DGF.Figure 2, 3 and 4 shows the scattering DGF

(simulation) of electric ﬁeld with the change in θ.

Figure 7.1: Magnitude of scattered ﬁeld component Eϕversus angle ϕ,with diﬀerent

values of dand the angle is θ=π

6

Using equation (8.1), there are three components in r,θand ϕdirection. Every

Component of electric ﬁeld is plotted as a function of azimuthal angle ϕ. The val-

ues of ϕis (0 to 2π), whereas at z coordinate diﬀerent values of receiver has been

plotted. The electric ﬁeld is plotted, which is vector addition of three components.

These all parameters are shown in the simulation graph.

33

By taking the value of θ=π

6, ﬁgure 8.2 shows that magnitude of electric ﬁeld

(Eϕ) is decreasing as the distance of receiving antenna is increasing from the sen-

sor (transmitting antenna). The plot shows electric ﬁeld component at diﬀerent

values of ϕ, varying from 0 to 2π. In this case, Eϕis decreasing from (4080 to

4065)dB by replacing the receiving antenna from 0 cm to 10 cm.

Figure 7.2: Magnitude of scattered ﬁeld component Eϕversus angle ϕ,with diﬀerent

values of dand the angle is θ=π

3

In Figure 8.3, having value of θ=π

3, magnitude of electric ﬁeld (Eϕ) again de-

creases as the antenna moves away from sensor. For the values of ϕfrom 0 to

2π,Eϕhas diﬀerent values from (4060 from 4068)dB. By changing position of

receiving antenna from 0 cm to 10 cm.

The values of distance and ϕare same, as described in the above graphs by only

replacing the parameter θ=π. Similarly in ﬁgure 4 values of Eϕchange from (4082

to 4074)dB by moving the position of receiver away from transmitting antenna,

which in return decreases the electric ﬁeld intensity.

34

Figure 7.3: Magnitude of scattered ﬁeld component Eϕversus angle ϕ,with diﬀerent

values of dand the angle is θ=π

7.2 Simulations using MF

For simulation I take electric permittivity of air, ε0=107

4πc2where cis speed of

light and relative permittivity εr= 36.6647. The conductivity of human body is

approximated to σ= 0.6321 at frequency of 915MHz.

By taking the value of θ=π/6 for d= 20cm, which is the radius of propagation

environment. We set the transmission frequency at 915Mhz and radial distance

from the central spherical axis is r= 0.15cm. We also assumed that with the

movement of arm, ϕ0varies at diﬀerent angles.

Using equation 4.38, we have three components r, θ, ϕ direction, each component

of electric ﬁeld is plotted as a function of azimuthal angle ϕ. At z-coordinates

values of receiver is plotted at diﬀerent distances by changing the values of ϕfrom

0toπ. Electric ﬁeld is plotted as a function of all three components, all these pa-

rameters are shown in the simulation graphs.

I have done simulation in six scenarios by varying θat diﬀerent angles. Taking

θ=π

2, ﬁgure 5.4 shows that magnitude of electric ﬁeld is decreasing as receiving

35

antenna is moving away from the source. The simulation graph is also presenting

the change in electric ﬁeld by varying ϕfrom 0 to π. from simulation, we can also

understand change in magnitude with motion of arm at diﬀeren angles of Φ in

radians.

0

5

10

1

1.2

1.4

1.6

1.8

2x 104

0

5

10

15

20

phi(radians)

magnnitude(dB)

phi vs magnitude vs distance

distance(cm)

Figure 7.4: Magnitude of electric ﬁeld Erat diﬀerent values of d, and θ=π

2.

In second scenario we repositioned theta at θ=π

4and observed that magnitude of

electric ﬁeld is diminishing as replacing the receiving antenna away from source.

We can also study the ﬁeld distribution at diﬀerent angular positions of ϕ, as

shown in ﬁgure 8.5.

In third scenario we have replaced θ=π

6and monitor that magnitude of electric

ﬁeld intensity is lowing down as receiving antenna is moving far from the sensor,

which is implanted on the center of shoulder.The ﬁeld distribution at diﬀerent

azimuthal positions, are depicted in 8.6.

We take θ=π

10 and study that magnitude of electric ﬁeld is exponentially decreas-

ing by increasing distance of the receiver away from the source. In spherical ﬁeld

distribution, how magnitude of electric ﬁeld is distributed by changing the angle

ϕfrom 0 t0 πb8.7.

36

0

5

10

0

2

4

6

8

10

x 104

0

5

10

15

20

phi(radians)

magnnitude(dB)

phi vs magnitude vs distance

distance(cm)

Figure 7.5: Magnitude of electric ﬁeld Erat diﬀerent values of d, and θ=π

4.

0

5

10

0

2

4

6

8

10

x 104

0

5

10

15

20

phi(radians)

phi vs magnitude vs distance

magnnitude(dB)

distance(cm)

Figure 7.6: Magnitude of electric ﬁeld Erat diﬀerent values of d, and θ=π

6.

By putting θ=π

3, again we have observed the same pattern of decreasing magni-

tude of ﬁeld, when receiver is going away from the transmitter. As shown in 3-D

plot of 8.8.

In last Scenario values of distance and ϕare same, as mentioned in above graphs.

37

0

5

10

1

2

3

4

5

6

7

8

x 104

0

5

10

15

20

phi(radians)

phi vs magnitude vs distance

magnnitude(dB)

distance(cm)

Figure 7.7: Magnitude of electric ﬁeld Erat diﬀerent values of d, and θ=π

10 .

0

5

10

0

2

4

6

8

10

x 104

0

5

10

15

20

phi(radians)

phi vs magnitude vs distance

magnnitude(dB)

distance(cm)

Figure 7.8: Magnitude of electric ﬁeld Erat diﬀerent values of d, and θ=π

3.

By replacing the value of θ=π, in return we can see in ﬁgure 8.9 as distance

of receiver is increasing from the transmitting antenna intensity of electric ﬁeld is

decreasing.

By studying and comparing all above six 3-D plots we I have prove that magni-

38

0

5

10

1

2

3

4

5

6

7

x 104

0

5

10

15

20

phi(radians)

phi vs magnitude vs distance

magnnitude(dB)

distance(cm)

Figure 7.9: Magnitude of electric ﬁeld Erat diﬀerent values of d, and θ=π.

tude of electric ﬁeld and distance of receiving antenna has inversely proportional

relationship.

39

Chapter 8

Conclusion

I have proposed a standard approach to derive an analytical channel modeling

and propagation characteristics of arm motion as spherical model. To predict the

electric ﬁeld around body, I have formulated two procedures based on DGF and

MF.

First, I derive Eigen functions of spherical model then calculated the scattering

superposition to come across reﬂection and transmission waves of antenna. The

model includes four cases where transmitter or receiver is located inside or outside

of the body. This model is presented to understand complex problem of wave

propagation in and around arm of human body. Simulation shows that Electric

ﬁeld decreases when receiver moves away from the shoulder with change of angle

θ.

Secondly, I develop mathematical formulation in terms of MF. This function in-

volve complicated terms of AMF, RMF, cosine and sine-sphere. Using this func-

tion I derive only one case, when transmitter and receiver both are located outside

human body. Simulation results show, how magnitude of electric ﬁeld varies with

the change of arm motion. Similarly, graphical results describe if distance between

receiver and sensor increases, electric intensity decreases.

40

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