#### 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*
[84, 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.

**Next Section:**

FM Spectra

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Example AM Spectra