A New Narrowband Phase Modulation Mathematical Identity

Author(s)
Shihadeh M. Saadeh

ABSTRACT

A new mathematical identity is suggested to describe narrow band phase modulation and other similar physical problems instead of using the Bessel function. Bessel functions are extensively used in mathematical physics [1,2], electromagnetic wave propagation and scattering [3,4], and communication system theory [3,5,6]. Such phenomena must often be approximated by appropriate formulas since there is no closed form solution or expression, which usually leads to complex mathematical solutions [5,7]. Comparisons are made between the exact solution numerically calculated and graphed with the new mathematical identities’ prediction of phase modulation behavior. The proposed mathematical identity matches the results very well, leading to simpler analysis of such physical behavior.

A new mathematical identity is suggested to describe narrow band phase modulation and other similar physical problems instead of using the Bessel function. Bessel functions are extensively used in mathematical physics [1,2], electromagnetic wave propagation and scattering [3,4], and communication system theory [3,5,6]. Such phenomena must often be approximated by appropriate formulas since there is no closed form solution or expression, which usually leads to complex mathematical solutions [5,7]. Comparisons are made between the exact solution numerically calculated and graphed with the new mathematical identities’ prediction of phase modulation behavior. The proposed mathematical identity matches the results very well, leading to simpler analysis of such physical behavior.

Cite this paper

S. Saadeh, "A New Narrowband Phase Modulation Mathematical Identity,"*Journal of Modern Physics*, Vol. 3 No. 6, 2012, pp. 511-515. doi: 10.4236/jmp.2012.36069.

S. Saadeh, "A New Narrowband Phase Modulation Mathematical Identity,"

References

[1] A. V. Alekseev and N. V. Sushilov, “Analytic Solutions of Bloch and Maxwell-Bloch Equations in the Case of Arbitrary Field Amplitude and Phase Modulation” Physical Review A, Vol. 46, No. 1, 1992, pp. 351-355. doi:10.1103/PhysRevA.46.351 PMid:9907870

[2] N. Nayak and G. S. Agarwal, “Absorption and Fluorescence in Frequency-Modulated Fields under Conditions of Strong Modulation and Saturation” Physical Review A, Vol. 31, No. 5, 1985, pp. 3175-3182. doi:10.1103/PhysRevA.31.3175 PMid:9895871

[3] A. Hund, “Frequency Modulation,” McGraw-Hill, New York, 1942.

[4] J. D. Jackson, “Classical Electrodynamics,” 3rd Edition, Wiley, New York, 1998.

[5] J. G. Proakis and M. Salehi, “Communication Systems Engineering,” Prentice Hall, Upper Saddle River, 2001.

[6] N. M. Blachman, “Noise and Its Effect on Communication,” 2nd Edition, Krieger Publishing Co., Malabar, 1982.

[7] G. N. Watson, “A Treatise on the Theory of Bessel Functions,” 2nd Edition, Cambridge University Press, Cambridge, 1995.

[8] S. Saadeh, J. Shultz and G. Salamo, “Experimental Observation of Chirped Continuous Pulse-Train Soliton Solutions to the Maxwell-Bloch Equations,” Optics Express, Vol. 8, No. 2, 2001, pp. 153-158. doi:10.1364/OE.8.000153

[9] L. A. Pipes, “Applied Mathematics for Engineers and Physicists,” 2nd Edition, McGraw-Hill, New York, 1958.

[10] M. Abramowitz and I. A. Stegun, “Handbook of Mathematical Functions,” National Bureau of Standards, Washington DC, 1964.

[11] N. M. Blachman and S. H. Mousavinezhad, “Trigonometric Approximation for Bessel Functions,” IEEE Tran- sactions on Aerospace and Electronic Systems, Vol. 22, No. 1, 1986, pp. 2-7. doi:10.1109/TAES.1986.310686

[12] A. Yarvis, “Introduction to Optical Electronics,” 2nd Edition, Holt McDougal, Geneva, 1977.

[1] A. V. Alekseev and N. V. Sushilov, “Analytic Solutions of Bloch and Maxwell-Bloch Equations in the Case of Arbitrary Field Amplitude and Phase Modulation” Physical Review A, Vol. 46, No. 1, 1992, pp. 351-355. doi:10.1103/PhysRevA.46.351 PMid:9907870

[2] N. Nayak and G. S. Agarwal, “Absorption and Fluorescence in Frequency-Modulated Fields under Conditions of Strong Modulation and Saturation” Physical Review A, Vol. 31, No. 5, 1985, pp. 3175-3182. doi:10.1103/PhysRevA.31.3175 PMid:9895871

[3] A. Hund, “Frequency Modulation,” McGraw-Hill, New York, 1942.

[4] J. D. Jackson, “Classical Electrodynamics,” 3rd Edition, Wiley, New York, 1998.

[5] J. G. Proakis and M. Salehi, “Communication Systems Engineering,” Prentice Hall, Upper Saddle River, 2001.

[6] N. M. Blachman, “Noise and Its Effect on Communication,” 2nd Edition, Krieger Publishing Co., Malabar, 1982.

[7] G. N. Watson, “A Treatise on the Theory of Bessel Functions,” 2nd Edition, Cambridge University Press, Cambridge, 1995.

[8] S. Saadeh, J. Shultz and G. Salamo, “Experimental Observation of Chirped Continuous Pulse-Train Soliton Solutions to the Maxwell-Bloch Equations,” Optics Express, Vol. 8, No. 2, 2001, pp. 153-158. doi:10.1364/OE.8.000153

[9] L. A. Pipes, “Applied Mathematics for Engineers and Physicists,” 2nd Edition, McGraw-Hill, New York, 1958.

[10] M. Abramowitz and I. A. Stegun, “Handbook of Mathematical Functions,” National Bureau of Standards, Washington DC, 1964.

[11] N. M. Blachman and S. H. Mousavinezhad, “Trigonometric Approximation for Bessel Functions,” IEEE Tran- sactions on Aerospace and Electronic Systems, Vol. 22, No. 1, 1986, pp. 2-7. doi:10.1109/TAES.1986.310686

[12] A. Yarvis, “Introduction to Optical Electronics,” 2nd Edition, Holt McDougal, Geneva, 1977.