Frequency
Modulation and Bessel Functions
Cuthbert Nyack
The spectrum of a frequency modulated signal must be determined using
Bessel Functions. Applet below shows how this can be done. The first
4 Bessel functions are plotted in the applet. To find the amplitude of
the carrier frequency in the Fourier Spectrum
of the modulated signal (carrier amplitude without modulation is
assumed to be 1) the horizantal and vertical red lines must
be used. Red curve shows Jo(m) where m is the modulation index. The
vertical red line is drawn at the horizontal coordinate equal to m, a
horizontal line passing through the intersection of Jo(m) and the vertical
line is now drawn. The intersection of the horizontal line with the
vertical axis gives the amplitude at the carrier frequency. Note that
this becomes zero at m ~ 2.4. . Orange curve shows J1(m) and horizontal
and vertical orange lines can be used to find the amplitude of the first
sideband. Other lines in the spectrum can be found similarly using the
green(J2(m)) and purple(J3(m)) curves. If the amplitude of the signal
without modulation is V, then all the numbers obtained from the above
analysis must be multiplied by V.
Vertical axis in above applet is from -1 to +1(with vgain=1),
Horizontal axis is 0 to 8 when hgain=1.
As m is varied, the amplitude of the carrier and first sideband is
indicated numerically.
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COPYRIGHT © 1996 Cuthbert Nyack.