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.
Highlights
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]
Frequency or phase modulation is an efficient form of communication, where information is transmitted over a carrier signal by changing the instantaneous frequency [4]
Summary
Frequency or phase modulation is an efficient form of communication, where information is transmitted over a carrier signal by changing the instantaneous frequency [4]. While trying to better understanding the evolution of an electric field envelope as the previously studied laser beam propagated through the two level sample, and the effects of phase modulation by using custom in house developed software for a numerical based computer simulation study, it was observed that at a certain beer’s absorption length inside the sample, the electric field envelope looks like an analytical function. During fitting trials of the observed electric field in several trigonometric form functions, an expression for the cos sin mt function in two terms was developed, and the idea of a alterative to the complex Bessel Functions approach was conceived [9,10,11] This new expression was compared to the exact cos sin mt , the agreement and fit are excellent for δ < 1 i.e. for small index of modulation, which is the case in optical frequency modulation and in communications
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