The SIJ Transactions on Computer Networks & Communication Engineering (CNCE), Vol. 1, No. 4, September-October 2013
Application of Bessel Function of the
First Kind in Frequency Modulated
Transmission
Vincent M. Bulinda*, J.A. Okelo**, J.K. Sige*** & J. Okwoyo****
*Student, Department of Pure and Applied Mathematics, Jomo Kenyatta University of Agriculture and Technology, KENYA.
E-Mail: majorblinda@yahoo.com
**Supervisor, Department of Pure and Applied Mathematics, Jomo Kenyatta University of Agriculture and Technology, KENYA.
E-Mail: masenoo@gmail.com
***Supervisor, Department of Pure and Applied Mathematics, Jomo Kenyatta University of Agriculture and Technology, KENYA.
E-Mail: jsigey2002@jkuat.ac.ke
****Supervisor, School of Mathematic, University of Nairobi, KENYA. E-Mail: jmkwoyo@uonbi.ac.ke
Abstract—The oscillatory nature of the Bessel Functions of the first kind is used to determine the spectrum of
a frequency modulated signal in broadcasting music and speech. A novel method proposed in the synthesis of
FM to develop the spectral characteristics of the FM equation is discussed. The frequency modulated
transmission is mathematically represented by a harmonic distribution of a sine wave carrier modulated by a
sine wave signal which will be represented graphically using Matlab software in a user specified range in order
to confirm oscillatory nature of these functions. The graph obtained will be compared with the one in the final
expression which describes the motion of sound when acted on by a sinusoidal forcing function. This type of
analysis is highly crucial and significant for FM transmitters commonly used by business band for mobile
communication and FM radio services for voice transmission.
Keywords—Amplitude; Bessel Functions of the First Kind; Frequency Modulated Transmission; Gamma
Function; Matlab Plots; Sinusoidal; Voice Transmission.
Abbreviations—Frequency Modulation (FM); Jomo Kenyatta University of Agriculture and Technology
(JKUAT).
I.
W
INTRODUCTION
AVES are many of the phenomena encountered in
daily life. Music is transmitted from radio stations
in form of electromagnetic waves which is
decoded and made to form the membrane in the speaker to
vibrate. This vibration causes pressure in a wave in the air
which represents a simple harmonic motion. According to
Chowning (1986), the nature of sound and human hearing are
connected to mathematical investigation whereby human
hears are able to detect sound and how mathematics gives a
powerful way to understand sound and then create and
manipulate it. 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. These applications have solutions that are based on
the Bessel functions, Jennifer (2008). One such example is
that of a uniform density chain fixed at one end undergoing
small oscillations; therefore these functions are associated
with a wide range of problems in important areas of
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mathematical physics. A single frequency travelling wave
will take the form of a sine wave, Sebastian (2011). There are
two major features of the sinusoidal vibration; namely
frequency and amplitude. This is a general wave relationship
which applies to transverse waves whose examples are sound
and light waves, Erwin (2006), other electromagnetic waves,
and waves in mechanical media. Both music and mathematics
are expressive languages whose relations are revealed
through pattern and serendipity, Frederick (1947). Bessel
functions were first used to describe body motions, with the
Bessel functions appearing in the series expansion on
planetary perturbation, Niedziela (2008). These mathematical
functions were derived around 1817 by the German
astronomer Friedrich Wilhelm Bessel during an investigation
of solutions of one of Kepler’s equations of planetary motion.
Chowning (1986) discovered that by bringing the modulation
rate down into the range of human hearing, and by using a
computer to strictly control the relationship between the
carrier frequency, the modulating frequency, and the amount
of modulation, musical timbres would result. The modulation
© 2013 | Published by The Standard International Journals (The SIJ)
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The SIJ Transactions on Computer Networks & Communication Engineering (CNCE), Vol. 1, No. 4, September-October 2013
of those frequencies was far above the range of human
hearing. Frequency modulation synthesis also turned out to
be a much less computationally intense process that simple
additive synthesis, where sine tones are stacked up to create
artificial musical tones. The problem under study is the
analysis of Bessel functions of the first kind to determine the
amplitudes of frequency modulation component.
The problem involves studying the motion of sound
which is a wave subjected to a sinusoidal forcing function.
The intention is to obtain an expression that describes the
motion of the mass in one dimension when acted on by a
sinusoidal forcing function. Bessel function of the first kind
would be used to obtain the equation of motion in series
form. Bessel functions of the first kind are used to determine
the amplitudes of an oscillating wave. In this work, this will
be achieved by;
 Derivation of the general equation of a simple
harmonic motion that is periodic will be done by
using the knowledge obtained from differential
equations as a whole to simplify the equations
resulting to the general equation of the wave by
using Newton’s second law of motion and Hooks’
law.
 Plotting graphs to show the nature and behaviour of
sound waves. The graphs drawn will show that the
function 𝐽𝑛 𝑥 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
Jihao (2009). He 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.
II.
MATHEMATICAL FORMULATION
In this work, it is assumed 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
According to Riley et al., (1998) a wave that has its profile
sinusoidal is said to be harmonic if
(1)
𝜑 𝑥, 𝑡 = 𝐴𝑠𝑖𝑛(𝑘𝑥 ± 𝑣𝑡)
Equation (1) is the equation of a harmonic wave, Jain
(1984), resulting to the basic equation for FM which is given
by
(2)
𝑥 𝑡 = 𝐴𝑠𝑖𝑛 𝜔𝑐 𝑡 + 𝑙𝑠𝑖𝑛 𝜔𝑚 𝑡
We consider a particle at distance 𝑥 from a particle from
its right, Straus (1992), Kythe et al., (1997) and Smith &
Roland (2002). Let the wave travel from left to right with
velocity 𝑣. Then the displacement at point 𝐴 is given by
(3)
𝑦 = 𝑎𝑠𝑖𝑛⁡
(𝜔𝑡 − 𝛼𝑥)
2𝜋
2𝜋𝑣
where 𝜔 = =
and 𝛼 = 2π. Substituting these in
𝑇
𝜆
equation (3) and then differentiating twice with respect to
both x and t and substituting, we find the equation of wave
motion, equation (4), which is applicable to all types of
transverse waves, Kreyszig (1993).
𝜕2 𝑦
𝜕2 𝑦
2
(4)
=
𝑣
𝜕𝑡 2
𝜕𝑥 2
Bessel differential equations given by;
𝑑2 𝑦
𝑑𝑦
(5)
𝑥2 2 + 𝑥
+ 𝑥 2 − 𝑛2 𝑦 = 0
𝑑𝑥
𝑑𝑥
Where n is the order of differential equation and it is a
given number, real or complex. The point 𝑥 = 0 is a regular
singularity, and is the Bessel functions which is a solution of
equation (5) which has a solution of the form
∞
𝑎𝑘 𝑥 𝑚 +𝑘
𝑦=
(6)
𝐾=0
Using power series solutions and substituting in equation
(5), we get the solution
𝑥𝑛
𝑥2
𝐽𝑛 (𝑥) = 𝑛
1−
2 𝛤(𝑛 + 1)
2(2𝑛 + 2)
(7)
𝑥4
+
−⋯
2.4 2𝑛 + 2 (2𝑛 + 4)
The Bessel functions 𝐽𝑛 𝑥 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
(2002), where n-is a positive integer and not zero. It can be
written as an infinite polynomial with terms derived from the
gamma function, Γ, Watson (1995),
𝑥 𝑛+2𝑘
∞
−1 𝑘
2
(8)
𝐽𝑛 𝑥 =
𝑘! 𝛤 𝑛 + 𝑘 + 1
𝐾=0
Here 𝐽𝑛 𝑚𝑓 is the Bessel function of the first kind,
argument 𝑚𝑓 and order n.
The relationship between the signs of n is given by
(9)
𝐽−𝑛 𝑚𝑓 = (−1)𝑛 𝐽𝑛 𝑚𝑓
IV.
RESULTS AND DISCUSSIONS
This modulated signal is consisting of three or more
frequency components added together to give the appearance
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The SIJ Transactions on Computer Networks & Communication Engineering (CNCE), Vol. 1, No. 4, September-October 2013
of a sine wave. We can easily see that the frequency is
varying with time when displayed in the time domain. From
the two graphs of Bessel Functions above, the values of the
term 𝐽𝑛 𝑚𝑓 which gives amplitude of n-th side band with
modulation index mf are determined. Using series solution as
mentioned in equation (7), the values of the 𝐽𝑛 𝑚𝑓 terms are
calculated. Mathematically, the results of the numerical
computation of the values of 𝐽0 𝑚𝑓 , 𝐽1 𝑚𝑓 , 𝐽2 𝑚𝑓 ,
𝐽3 𝑚𝑓 , 𝐽4 𝑚𝑓 and so forth are plotted on graphs as shown
in figure 1 and figure 2.
Figure 2: Plots of Bessel Functions of the First Kind at 0≪ mf≪60
V.
Figure 1: Plots of Bessel Functions of the First Kind at 0≪ mf≪20
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 𝐽0 𝑚𝑓 and 𝐽1 𝑚𝑓 while the
amplitude of the higher-order (n > 1) sideband pairs is very
small, Saxena et al., (2009). 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 & Stegun (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.
The application of Bessel function in analyzing side band
frequency is discussed in analytical manner. In all cases,
including the rest frequency, 𝐽0 𝑚𝑓 , the amplitude of the
Bessel function goes to zero for numerous values of mf,
meaning that the rest-frequency component of the FM wave
can disappear. As mf increases, the bandwidth increases too
and individual spectral lines do not increase in amplitude
monotonically, Saxena et al., (2009). Their amplitudes are
determined by 𝐽𝑛 𝑚𝑓 , plots that appear in figure 1 and 2.
From our analysis, we have therefore asserted that sine
waves describe many oscillating phenomena. When the wave
is damped, each successive peak decreases as time goes on.
A true sine wave starting at t = 0 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.
In summary, from the results we can see that sound is
caused by variations in air pressure and is perceived by the
human ear's ability to detect those variations. These
vibrations in the air's pressure propagate through the air as a
longitudinal wave moving at a given speed. For instance, if
one is standing in one place, he may notice the air pressure
changing periodically as shown by the two figures.
REFERENCES
[1]
[2]
[3]
[4]
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CONCLUSION
E.T. Frederick (1947), “Radio Engineering”, McGraw -Hill,
New York.
M. Abramowitz & I.A. Stegun (1965), “Handbook of
Mathematical Functions”, Dover, New York, Pp. 65–85.
M.K. Jain (1984), “Numerical Solutions of Differential
Equations”, Ed. 2, Wiley, Eastern Limited, New Delhi, Pp. 28.
J. Chowning (1986), “FM Theory & Applications”, Yamaha
Music Foundation.
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The SIJ Transactions on Computer Networks & Communication Engineering (CNCE), Vol. 1, No. 4, September-October 2013
[5]
[6]
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G.N. Watson (1995), “A Treatise on the Theory of Bessel
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P. Kythe, Pretap & M. Shaferkotter (1997), “Partial Differential
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Modeling of Physical Systems”, Oxford University Press, New
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Ohio State University, Columbus, Ohio.
J. Niedziela (2008), “Bessel Functions and their Applications”,
Univ. of Tennessee –Knoxville.
Y. Jihao (2009), “On the use of a Linear Data Model for
Parameter Estimation of Sinusoidal FM Signals”, Univ. of
Electron. Sci. & Technol. of China, Chengdu, China.
D. Saxena, M.M. Kumar & S. Loonker (2009), “Determination
and Analysis of Sidebands in FM Signals
using
Bessel
Function”, Lachoo Memorial College of Science and
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World”, International Centere for Theoretical Physics, Italy.
Vincent Major Bulinda. Mr. Bulinda holds
a Bachelor of Science degree in Mathematics
and Physics from University of Eastern
Africa, Baraton Kenya. He is undertaking his
final project for the requirement of Master
Science degree in Applied Mathematics from
Jomo Kenyatta university of Agriculture and
Technology, Kenya.
He is currently an associate faculty at the
Mount Kenya University and Kisii University respectively (January
2013 – Present) responsible for carrying out teaching and research
duties.
Dr. Okelo Jeconia Abonyo. He holds a PhD in Applied
Mathematics from Jomo Kenyatta University of Agriculture and
Technology as well as a Master of science degree in Mathematics
and first class honors in Bachelor of Education, Science; specialized
in Mathematics with option in Physics, both from Kenyatta
University. He has a dependable background in Applied
Mathematics in particular fluid dynamics, analyzing the interaction
between velocity field, electric field and magnetic field. Has a hand
on experience in implementation of curriculum at secondary and
university level. He has demonstrated sound leadership skills and
have ability to work on new initiatives as well as facilitating teams
to achieve set objectives. He has a good analytical, design and
problem solving skills.
He is working as a Deputy Director in School of Open learning and
Distance e Learning SODeL Examination, Admission & Records
(JKUAT), Senior lecturer Department of Pure and Applied
Mathematics and Assistant Supervisor at Jomo Kenyatta University
of Agriculture and Technology. Work involves teaching research
methods and assisting in supervision of undergraduate and
postgraduate students in the area of applied mathematics. He has
published 10 papers on heat transfer in respected journals.
Dr. Okwoyo James Mariita. James holds a
Bachelor of Education degree in Mathematics
and Physics from Moi University, Kenya,
Master Science degree in Applied
Mathematics from the University of Nairobi
and PhD in applied mathematics from Jomo
Kenyatta University of Agriculture and
Technology, Kenya.
He is currently a lecturer at the University of
Nairobi (November 2011 – Present) responsible for carrying out
teaching and research duties. He plays a key role in the
implementation of University research projects and involved in its
publication. He was an assistant lecturer at the University of Nairobi
(January 2009 – November 2011). He has published 7 papers on
heat transfer in respected journals.
Dr. Johana Kibet Sigey. Sigey holds a
Bachelor of Science degree in mathematics
and computer science first class honors from
Jomo Kenyatta University of Agriculture and
Technology, Kenya, Master of Science
degree in Applied Mathematics from
Kenyatta University and a PhD in applied
mathematics from Jomo Kenyatta University
of Agriculture and Technology, Kenya.
He is currently the acting director, JKUAT, Kisii CBD where he is
also the deputy director. He has been the substantive chairman department of Pure and Applied Mathematics –, JKUAT (January
2007 to July- 2012). He holds the rank of senior lecturer, in Applied
Mathematics Pure and Applied Mathematics department, JKUAT
since November 2009 to date. He has published 9 papers on heat
transfer in respected journals.
ISSN: 2321 – 2403
© 2013 | Published by The Standard International Journals (The SIJ)
87