International Journal of Electronics and Computer Science Engineering 454 Available Online at www.ijecse.org ISSN: 2277-1956 Determinationn and Analysis of Sidebands in FM Signals using Bessel Functionn 1 Dhiraj Saxena, 2 Mridul Kumar Mathur, 3 Seema Loonker 1 Department of Physics & Electronics 2,3 Department of Computer Science 1,2,3 Lachoo Memorial College of Science and Technology, Jodhpur (Raj.), INDIA Email: 1 dhirajm_in@yahoo.com, 2 Mathur_mridul@yahoo.com, 3 seemasurana@gmail.com Abstract- In Frequency Modulation the components of the modulated wave are much more complex in comparison to other analog modulation techniques. Here a single frequency modulating signal produces an infinite number of pairs of sidebands frequencies. However the sideband frequencies are negligibly small in amplitude but they increase the bandwidth of the FM signal. An exact analysis of these sidebands is essential so as to find the exact bandwidth in order to overcome the problems of overlapping of adjacent signals and cross talk. Here we have analyzed the FM signals using the Bessel functions in order to determine the amplitudes of the available sidebands and thereby the bandwidth. We find that larger the value of modulation index, more sets of sideband frequencies is produced. Keywords: Frequency modulation, Bessel Function, Sidebands I. INTRODUCTION Bessel functions arises from the solution of a differential equation frequently used in various applications of physics, communication and signal processing [1,2]. Bessel's equation originates when finding separable solutions to Laplace's equation and the Helmholtz equation in cylindrical or spherical coordinates. Bessel functions are therefore especially important for many problems of wave propagation and static potentials. Bessel functions have extensive application especially in the case of handling cylindrical and spherical coordinate systems based problems. Bessel functions have extensive applications in studying electromagnetic waves in a cylindrical waveguide, heat conduction in a cylindrical object, modes of vibration of a thin circular artificial membrane, diffusion problems on a lattice, solutions to the radial Schrödinger equation for a free particle, solving for patterns of acoustical radiation etc.[3] Bessel functions have also been found useful in the applications regarding signal processing such as FM synthesis, Kaiser window, or Bessel filter etc. This paper presents application of Bessel function in analyzing all side bands in the process of frequency modulation for distortion less transeption. II. BESSEL FUNCTION In mathematics, Bessel functions, first defined by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are canonical solutions y(x) of Bessel's differential equation 1.1: -------------------------(1.1) Where n is a non-negative real number. The solutions of this equation are called Bessel Functions of order n. most of the application n is taken as non-negative integers, i.e., n=0,1,2,3 or half-integer. Bessel functions are also known as cylinder functions or cylindrical harmonics because they are found in the solution to Laplace's equation in cylindrical coordinates. Since Bessel's differential equation is a second order ordinary differential equation, two sets of functions, the Bessel function of the first kind and the Bessel function of the second kind (also known as the Weber Function), are needed to form the general solution 1.2: ----------------------------(1.2)
Determination and Analysis of Sidebands in FM Signals using Bessel Function 455 However, Yn(x) is divergent at x=0. The associated coefficient c2 is forced to be zero to obtain a physically meaningful result when there is no source or sink at x=0. [4, 5, 6] The Bessel function of the first kind of order n can be expressed as a series of gamma functions. ---------------------------(1.3) The generating function of the Bessel Function of the first kind is expressed as -------------------------(1.4) Various functions and their special cases can be expressed in terms of Bessel functions as mentioned below in equation 1.5 and 1.6. -------------------------(1.5) & (1.6) III. FREQUENCY MODULATION The key process of analog transmission is modulation, which requires manipulation of one or more of the parameters of the carrier that characterizes the analog signal. If the frequency of the carrier signal is varied in accordance with the modulating signal then the process is refereed as frequency modulation. The classic definition of FM is that the instantaneous output frequency of a transmitter is varied in accordance with the modulating signal Figure 1- FM signal with the modulating signal in frequency modulation As shown in the figure, the carrier s instantaneous frequency deviation from its un modulated value varies in proportion to the instantaneous amplitude of the modulating signal in the process of frequency modulation. In an FM signal, frequency deviation, δ, is the maximum frequency deviation of the carrier frequency caused by the amplitude of the modulating signal. The modulation index for an FM signal can be defined as the ratio of the maximum frequency deviation to the modulating signal s frequency
IJECSE,Volume1,Number 2 Dhiraj Saxena et al. δ m = --------------------------------(1.7) f f m mf is proportional to the amplitude of the modulating signal and inversely proportional to the frequency of the modulating signal. Also, it is independent from the modulation frequency. A-MATHEMATICAL ANALYSIS OF FM Equation for a carrier wave can be written as a sine wave : e c (t) = E c sin(ωt + φ) ---------------------(1.8) Modulating signal can also be expressed as sine wave. e M =E M sinω M t --------------------(1.9) Frequency modulation is realized by varying ω in accordance with the modulating signal. Thus an equation for the instantaneous voltage can be written for the signal frequency of an FM wave as a function of time: efm(t) = EC sin(ωct + mf sinωmt) efm(t) = EC [ sin(ωct) cos( mf sinωmt) + cos(ωct) sin ( mf sinωmt)] -------(1.10) Where EC is the rest-frequency peak amplitude, ωc and ωm represent the rest and modulating frequencies, and mf is the index of modulation. Equation 1.10 represents a single low-frequency sine wave, fm, frequency modulating another high-frequency sine wave, fc. The argument of the sine wave is itself a sine wave in this equation. This modulated wave has actually the vector sum of three sine waves. This modulated signal is consisting of three or more frequency components vectorially added together to give the appearance of a sine wave that s frequency is varying with time when displayed in the time domain B.APPLICATION OF BESSEL FUNCTIONS IN ANALYZING SIDE BANDS OF FM Equation 1.10 cannot be solved with algebra or trigonometric identities. The only way out is to use Besselfunction identities to yield solutions to equation 7 and to determine the frequency components of an FM wave. Equation 1.10 can be expressed in terms of Bessel function using equation 1.5 and 1.6 in the following manner: sin +2 cos2 + = -- -----------------(1.11) cos 2 sin 2 1 Equation (1.11) can be further solved and can be written as --------------(1.12) It itemizes the various signal components in an FM wave and their amplitudes. This equation indicates that there are an infinite number of sideband pairs for an FM wave. Each sideband pair is symmetrically located about the transmitter s rest frequency, fc, and separated from the rest frequency by integral multiples of the modulating frequency, n fm, where n = 1, 2, 3,.... The magnitude of the rest frequency and sideband pairs is dependent upon the index of modulation, mf, and given by the Bessel function coefficients, Jn(mf), where the subscript n of Jn is the order of the sideband pair. For example J0(1.0) represents the rest-frequency amplitude of an FM wave with an index of modulation equal to 1.0. Similarly J1(2.5) is the amplitude of the first pair of sidebands for an FM wave with mf = 2.5.
Determination and Analysis of Sidebands in FM Signals using Bessel Function 457 To determine sideband frequencies of FM signal, it is required to determine value of the term Jn(mf) which gives amplitude of nth side band with modulation index mf.. The values of the Jn(mf) terms can be calculated from series solution as mentioned in equation 4.. For the sake of simplification, the results of the numerical computation of the values of J0(mf), J1(mf), J2(mf), and so forth are usually plotted on a graph as shown in figure 2. Figure (2): plotting of amplitude of side bands as a function of modulation index using Bessel Functions 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) i.e. the rest frequency and the first sideband pair), while the amplitude of the higherorder (n > 1) sideband pairs is very small. As mf increases, the amplitude of the rest frequency decreases and the amplitude of the higher-order sidebands increases, which would seem to indicate an increasing signal bandwidth? It can be further 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 act like an exponentially decaying sine wave. Therefore, the amplitudes of the higher-order sideband pairs eventually approach zero. Figure (3): Bessel Function of First Kind J n (m)
IJECSE,Volume1,Number 2 Dhiraj Saxena et al. 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. These values as plotted on the graph can also be consolidated in a table for integer or fractional values of mf as shown below: Amplitude values with minus signs in this table represent phase shifts of 180 degrees and that amplitude values less than 0.01 have been left out as they represent component frequencies with insignificant power content. Power of these side band can also be analyzed by calculating amplitude of side bands using Bessel function. Significance of this side band analyzed can be understood using an example. Consider an FM signal resulting from a modulating signal of 10 khz, an index of modulation of 0.25, and rest frequency of 500 khz. Now from Bessel function analysis and using graph, it can be seen that for this case, there is only one pair of sidebands with appreciable power. This type of FM signal is also referred as narrow-band FM (NBFM), where mf 0.5. 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. IV. CONCLUSION This paper presents the application of Bessel functions in analyzing side bands as generated in the process of frequency modulation. This type of analysis is extremely useful for efficient FM transmission as employed in mobile and other commercial communication services. This paper first introduced the Bessel function and some of its special forms as used in wide number of application. The application of Bessel function in analyzing side band frequency is discussed in analytical manner. It explains how Bessel functions determine amplitude and power of significant side bands as a function of modulation index in the process of FM transmission. REFERENCES [1] F. E. Relton, Applied Bessel Functions (Blackie and Son Limited, 1946). [2] H. J. Arfken, G. B., Weber, Mathematical Methods for Physicists (Elsevier Academic Press, 2005). [3] Arfken, George B. and Hans J. Weber, Mathematical Methods for Physicists, 6th edition (Harcourt: San Diego, 2005). ISBN 0-12-059876-0. [4] Bayin, S.S. Mathematical Methods in Science and Engineering, Wiley, 2006 [5] Bayin, S.S., Essentials of Mathematical Methods in Science and Engineering, Wiley, 2008 [6] Bowman, Frank Introduction to Bessel Functions (Dover: New York, 1958). ISBN 0-486-60462-4