The SIJ Transactions on Computer Networks & Communication Engineering (CNCE), Vol. 3, No. 6, September 2015
Long Distance Data Transmission using
Bessel Functions of the First Kind
Vincent M. Bulinda*
*Lecturer, School of Pure and Applied Sciences, Department Mathematics and Actuarial Science, Kisii University, KENYA.
E-Mail: majorblinda{at}yahoo{dot}com
Abstract—The Bessel functions of the first kind behave like a wave that can allow light energy to be modelled,
which can be a cylindrical dielectric waveguide made of silica glass, in transmission of data and voice over a
long distance. This is applicable in communication for they transmit sound or light waves carrying information
over a long distance. Amplitude and frequency domains are representations of periodic signals that show
energy that is spread across the spectrum. The spectrum is continuous and has different appearance from the
source of information. Bessel functions analysis method will determine both the amplitude spectrum and the
phase spectrum of any signal. This technique is aimed at explaining variations in amplitude of sound/light in a
medium before amplification in digital signal processing of nonlinear impairments. This will be represented
graphically using Matlab plots.
Keywords—Amplitude; Bessel Functions of the First Kind; Gamma Function; Long Distance Data
Transmission; Matlab Plots.
Abbreviations—Gigabit per second (Gbit/s); Terabits per second (Tb/s); Wavelength-Division Multiplexing
(WDM).
I.
L
INTRODUCTION
IGHT energy can be lounged into a fiber optic which
consists of a central core in which light is guided,
embedded in an outer cladding of slightly lower
refractive index. Light rays incident on the core-cladding
boundary angles greater than the critical angle undergo total
internal reflection and are guided through the core without
refraction.
According to Gallager et al., [13], the reason why the
distinction between “data” and “information” is not made is
that people implicitly identify the linguistic notion of
information with the technical notion of in Theory of
Communication. The technical notion of information is
syntactic and devoid of any notion of meaning and, hence,
context independent. In order to attach “meaning” to data,
context must be brought into the picture and then any notion
of universality will have to be given up. A theory of
information which is context dependent and attaches
meaning, which does not exist today, needs to be developed.
We would then have the hope of attaching “interpretations”
to data and thereby make the data relevant for the purpose of
taking action to reach desirable goals.
Optical communication systems can support Tb/s
capacities over many thousands kilometers, which makes
them the ideal communication technology [Ladányi, 14]. The
most used broadband technologies nowadays are WDM
systems with 64 or higher number of channels and the
ISSN: 2321-2403
40Gbit/s per channel rates. By use of such capacity in these
systems we consider the effects that may occur by
propagating of pulses in these systems. Increasing the number
of channels leads to various nonlinear phenomena that focus
on the propagation of pulses of the width of several tens to
hundreds of picoseconds, for the investigation of dispersion
effects.
Fibre communication systems are fulfilling the increased
demand on communication links, especially with the
proliferation of the internet [Chynoweth, 1]. These light
waves proven to provide faster and cheaper internet
connectivity than the traditional satellites; such as streaming
videos live or downloading high-definition videos in a very
short time. As a result of recent technological advances in
fabrication, light can be guided through fibre optic with very
minimal loss and these optical fibers are replacing copper
coaxial cables as the preferred transmission medium for
electromagnetic waves. The low attenuation and superior
signal quality of fibre optic communication systems have
similar characteristics of the Bessel functions of the first
kind; whose signals are transmitted over much longer
distances than metallic-based systems without signal
regeneration [Ishigure et al., 2].
Frequency modulation synthesis is an elegant technique
for creating complex sounds that was used in the first
commercially successful music synthesizers produced in the
1980s. One such example is that of a uniform density chain
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The SIJ Transactions on Computer Networks & Communication Engineering (CNCE), Vol. 3, No. 6, September 2015
fixed at one end undergoing small oscillations; therefore
these functions are associated with a wide range of problems
in important areas of mathematical physics. A single
frequency travelling wave will take the form of a sine wave
[Sebastian, 12]. The frequency and amplitude relationship
applies to transverse waves whose examples are sound and
light waves [Erwin, 8], other electromagnetic waves, and
waves in mechanical media. Bessel functions were first used
to describe body motions, with the Bessel functions
appearing in the series expansion on planetary perturbation
[Niedziela, 9]. They express and describe the motion of
sound in series form when acted on by a sinusoidal forcing
function in determining the amplitudes of an oscillating
wave. The graphs drawn will show that the function
is
oscillating with a decreasing amplitude and varying period.
The roots of these functions are not completely regularly
spaced and the amplitude of the wave decreases with the
increase in modulating frequency which looks similar to a
sine function.
Similar research on Bessel functions has been done by
Vincent M. Bulinda et al., [3] in analysing Bessel functions
of the First Kind in Frequency Modulated Transmission.
They used a distance of 20 and 60 to analyse the flow of the
sound waves.
Jihao [10] presented a linear data model for multicomponent sinusoidal frequency-modulated (FM) signals,
based on the fact that a sinusoidal FM signal can be
decomposed into a set of harmonic frequencies, with
harmonic amplitudes given by Bessel functions of the first
kind and via several properties derived from the present data
model, a novel method was proposed to estimate the
parameters of the multi-component sinusoidal FM signals.
The feasibility of the proposed method was demonstrated by
simulation results.
Kundaeli [4] analysed an optical fiber communication
system using laser rate equations where the effect of
electrical pulse shaping on dispersion include pulse distortion
was investigated using computer simulation techniques. The
results showed that the detrimental effects of the dispersion
can be greatly reduced without incurring very high costs.
More research has been done on dispersion-dominated
nonlinear fiber-optic channels, Verma et al., [5] used a
technique aimed at simplification of the following digital
signal processing of nonlinear impairments by using a model
describing pulse propagation in the dispersion-dominated
nonlinear fiber channel. In the limit of very strong initial predispersion the nonlinear propagation equations for each
Fourier mode become local and decoupled.
There are several schemes proposed to solve differential
equations, utilizing photonic temporal integrator based on
fiber gratings [Slavík et al., 2008] or silicon micro-ring
resonator [Lu et al., 16]. For the integrator-based scheme, due
to the intrinsic limited time window of an optical integrator, it
has a limitation in the processing bandwidth (speed);
meanwhile the currently proposed micro-ring based
differential-equation solver lacks of tunability and flexibility,
which can only be used to solve one specific differential
ISSN: 2321-2403
equation with an invariable constant coefficient. In this work,
we use the Bessel functions of the first kind to illustrate the
first-order linear ordinary differential equations and
experimentally demonstrate it. Based on an all-optical
intensity waves, an excellent agreement with theoretical
results can be obtained experimentally.
II.
MATHEMATICAL FORMULATION
We still use the assumptions made by Vincent M. Bulinda et
al., [3] that;
 The frequencies in the spectrum are in the harmonic
series,
 Both odd and even numbered harmonics are at some
times present,
 Harmonics increase in significance with intensity,
 The rise-time of the amplitude is rapid.
III.
SOLUTIONS OF THE EQUATIONS
The Bessel differential equations given by;
(1)
Where n is the order of differential equation and it is a
given number, real or complex [Basmadjian, 7]. The point
is a regular singularity. The Bessel functions which is a
solution of equation (1) which has a solution of the form
(2)
Using power series solutions and substituting in equation
(1), we get the solution
(3)
The Bessel functions
has power series that is
convergent, with better convergence than the familiar series
for the exponential or trigonometric functions which can also
be expressed as the sum for integral values of n [Basmadjian,
7], where n-is a positive integer and not zero. It can be
written as an infinite polynomial with terms derived from the
gamma function, [Watson, 6],
(4)
Here
is the Bessel function of the first kind,
argument
and order n.
The relationship between the signs of n is given by
(5)
IV.
PLOTS OF BESSEL FUNCTIONS,
Using equation (4) and the relations in the signs in equation
(5), we use Matlab plots to plot the Bessel function graphs
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The SIJ Transactions on Computer Networks & Communication Engineering (CNCE), Vol. 3, No. 6, September 2015
which shows the nature and behaviour of sound waves in
Figures 1-4. Modulated signal consists of five frequency
components added together to give a sine wave, varying
between the distances 0-70, 0-120, 0-250 and 0-300. In this
case, the frequency is varying with time when displayed in
the distance domain.
Figure 4: Plots of Bessel Functions of the First Kind at 0≪mf≪300
V.
Figure 1: Plots of Bessel Functions of the First Kind at 0≪mf≪70
Figure 2: Plots of Bessel Functions of the First Kind at 0≪mf≪120
Figure 3: Plots of Bessel Functions of the First Kind at 0≪mf≪210
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RESULTS AND DISCUSSIONS
From the four graphs of Bessel Functions above, the values
of the term
which gives amplitude of n-th side band
with modulation index mf are determined using series
solution as mentioned in equation (2) and the values of the
terms are calculated. Mathematically, the results of
the numerical computation of the values of
,
,
,
,
and
are plotted. It can be
observed from the graph that for small values of mf, the only
Bessel functions with any significant amplitude are
and
, while the amplitude of the higher-order (n > 1)
sideband pairs is very small [Saxena et al., 11]. As mf
increases, the amplitude of the rest frequency decreases and
the amplitude of the higher-order sidebands increase, thus an
increasing signal bandwidth. Therefore, the amplitudes of the
higher-order sideband pairs eventually approach zero.
From Matlab plots used in plotting amplitude of side
bands as a function of modulation index of Bessel Functions
of the first kind, it can be observed from the graph that for
small values of mf, the only Bessel functions with any
significant amplitude are
and
while the
amplitude of the higher-order (
) sideband pairs is very
small [Saxena et al., 11]. As mf increases, the amplitude of
the rest frequency decreases and the amplitude of the higherorder sidebands increase, thus an increasing signal
bandwidth. It is therefore observed that as mf keeps
increasing, the sideband pairs are essentially zero amplitude
until about mf = n, at which point they increase in amplitude
to a maximum and then decrease again. In all cases, as mf
keeps increasing, each Bessel function appears to behave like
an exponentially decaying sine wave. Therefore, the
amplitudes of the higher-order sideband pairs eventually
approach zero, Abramowitz (1965).
For particular values of mf the amplitude of particular
side frequency pairs becomes zero. The amplitude of the
carrier and each pair of sidebands is given by Bessel
functions of the first kind and the amplitude of higher order
side frequencies decreases rapidly with increase in time. This
presents application of Bessel functions of the first kind in
analysis of side bands in the process of frequency
modulation.
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The SIJ Transactions on Computer Networks & Communication Engineering (CNCE), Vol. 3, No. 6, September 2015
VI.
CONCLUSION
From the analysis of determining the characteristics of
damping of oscillations for a system with an amplitudedependent resistance, depends on the nature of the dissipative
forces and method of determining the characteristics of
damping, which is not only of theoretical but also of practical
importance.
A rather simple practical method is proposed for
determining the reliable value of the damping characteristic
from the envelope of damped oscillations, and methods of
using the existing expressions for the logarithmic decrement
when determining the damping characteristics of an
oscillatory system in the case of large damping as shown.
Their amplitudes are determined by
, plots that appear
in figure 1 to 4.
A true sine wave starting at
begins at the origin
(amplitude is zero). A cosine wave begins at its maximum
value due to its phase difference from the sine wave. In
practice a given waveform may be of intermediate phase,
having both sine and cosine components. The term "damped
sine wave" describes all such damped waveforms, whatever
their initial phase value. Simulation results prove the
correctness of the proposed theory.
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Vincent Major Bulinda. Bulinda is currently
a PhD student in Applied Mathematics at the
Jomo Kenyatta University of Agriculture and
Technology He holds an Msc in Applied
Mathematics from Jomo Kenyatta University
of Agriculture and Technology, BSc in
Mathematics and Physics from University of
Eastern Africa, Baraton. He has a dependable
background in Applied Mathematics in
particular calculus and its applications in communication and fluid
mechanics.
He has an experience in implementation of curriculum at university.
He is working as an assistant lecturer, department of Mathematics at
Kisii University, Kenya. The work involves teaching mathematics
units and assisting in supervision of undergraduate and postgraduate
students in the area of applied mathematics.
[6]
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