Bessel Functions

The Bessel functions of the first kind may be defined as the coefficients in the two-sided Laurent expansion of the so-called generating function [81, p. 14],^4.10

where is the integer order of the Bessel function, and is its argument (which can be complex, but we will only consider real ). Setting , where will interpreted as the FM modulation frequency and as time in seconds, we obtain

The last expression can be interpreted as the Fourier superposition of the sinusoidal harmonics of , i.e., an inverse Fourier series sum. In other words, is the amplitude of the th harmonic in the Fourier-series expansion of the periodic signal .

Note that is real when is real. This can be seen by viewing Eq.(4.6) as the product of the series expansion for times that for (see footnote pertaining to Eq.(4.6)).

Figure 4.15 illustrates the first eleven Bessel functions of the first kind for arguments up to . It can be seen in the figure that when the FM index is zero, and for all . Since is the amplitude of the carrier frequency, there are no side bands when . As the FM index increases, the sidebands begin to grow while the carrier term diminishes. This is how FM synthesis produces an expanded, brighter bandwidth as the FM index is increased.

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.