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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 $ k$ 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 $ t$ 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 $ k$th harmonic in the Fourier-series expansion of the periodic signal $ x_m(t)$. 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, $ J_0(0)=1$ and $ J_k(0)=0$ for all $ k>0$. 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.
Figure 4.15: Bessel functions of the first kind for a range of orders $ k$ and argument $ \beta $.
\includegraphics[width=\twidth]{eps/bessel}
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