
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

 |
(4.6) |
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
 |
(4.7) |
The last expression can be interpreted as the Fourier superposition of the
sinusoidal harmonics of
![$ \exp[j\beta\sin(\omega_m t)]$](http://www.dsprelated.com/josimages_new/mdft/img528.png)
,
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
![$ \exp[(\beta/2) z]$](http://www.dsprelated.com/josimages_new/mdft/img529.png)
times that for
![$ \exp[-(\beta/2)/z]$](http://www.dsprelated.com/josimages_new/mdft/img530.png)
(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.
Figure 4.15:
Bessel functions of the first kind
for a range of orders
and argument
.
![\includegraphics[width=\twidth]{eps/bessel}](http://www.dsprelated.com/josimages_new/mdft/img537.png) |
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