Bessel Functions

The Bessel functions of the first kind may be defined as the coefficients $J_k(\beta)$ in the two-sided Laurent expansion of the so-called generating function [84, p. 14],^4.10

$\displaystyle e^{\frac{1}{2}\beta\left(z-\frac{1}{z}\right)} = \sum_{k=-\infty}^\infty J_k(\beta) z^k \protect$

(4.6)

where

is the integer order of the Bessel function, and $\beta$ is its argument (which can be complex, but we will only consider real $\beta$ ). Setting $z=e^{j\omega_mt}$ , where $\omega_m$ will interpreted as the FM modulation frequency and

as time in seconds, we obtain

$\displaystyle x_m(t)\isdef e^{j\beta\sin(\omega_m t)} = \sum_{k=-\infty}^\infty J_k(\beta) e^{jk\omega_m t}. \protect$

(4.7)

The last expression can be interpreted as the Fourier superposition of the sinusoidal harmonics of $\exp[j\beta\sin(\omega_m t)]$ , i.e., an inverse Fourier series sum. In other words, $J_k(\beta)$ is the amplitude of the

th harmonic in the Fourier-series expansion of the periodic signal

Note that $J_k(\beta)$ is real when $\beta$ is real. This can be seen by viewing Eq. $\,$ (4.6) as the product of the series expansion for $\exp[(\beta/2) z]$ times that for $\exp[-(\beta/2)/z]$ (see footnote pertaining to Eq. $\,$ (4.6)).

Figure 4.15 illustrates the first eleven Bessel functions of the first kind for arguments up to $\beta=30$ . It can be seen in the figure that when the FM index $\beta$ is zero, and for all . Since $J_0(\beta)$ is the amplitude of the carrier frequency, there are no side bands when $\beta=0$ . 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.