The following trigonometric identity facilitates quantitative understanding of the spectrum produced by the basic FM equation of Equation 1:

sin(θ+asinβ)=
J
0
(a)sinθ+
∑
k=1
∞
J
k
(a)[
sin(θ+kβ)+
(−1)
k
sin(θ−kβ)
]
sin(θ+asinβ)=
J
0
(a)sinθ+
∑
k=1
∞
J
k
(a)[
sin(θ+kβ)+
(−1)
k
sin(θ−kβ)
]

(2)
The term
J
k
(a)
J
k
(a)
defines a Bessel function of the first kind of order
k
k
evaluated at the value
a
a
.

Note how the left-hand side of the identity possesses exactly the same form as the basic FM equation of Equation 1. Therefore, the right-hand
side of the identity explains where the spectral components called sidebands are located, and indicates the amplitude of each
spectral component. The Figure 3 screencast video continues the discussion by explaining the significance of each part of
Equation 2, especially the location of the sideband spectral components.

As discussed Figure 3 video, the basic FM equation produces an infinite number of sideband components; this is also evident by
noting that the summation of Equation 2 runs from k=1 to infinity. However, the amplitude of each sideband is controlled by the Bessel function,
and non-zero amplitudes tend to cluster around the central carrier frequency.
The Figure 4 screencast video continues the discussion by examining the behavior of the Bessel function
J
k
(a)
J
k
(a)
when its two parameters are varied, and shows how these parameters link to the modulation index and sideband number.

Now that you have developed a better quantitative understanding of the spectrum produced by the basic FM equation, the Figure 5
screencast video revisits the earlier audio demonstration of the FM equation to relate the spectrum to its quantitative explanation.

Comments:"A multimedia educational resource for signal processing students and faculty."