Frequency Modulation, signal,
spectrum and phasor.
Cuthbert Nyack
In frequency modulation the amplitude is kept constant and the
frequency is modulated by the amplitude of the modulating
signal. The modulation index for fm is
d = maximum frequency
deviation/modulating frequency.
FM signal can be represented
as:-
v = ac sin(wct
+ m sin wmt )
The frequency spectrum can be found by rewriting the above
expression as a sum of components of constant frequency using
the properties of the Bessel Functions. This gives:-
v = ac{Jo
(d)
sin(wct)
+ J1
(d)[sin(wc
+ wm)t - sin(
wc
- wm)t]
+ J2
(d)[sin(wc
+ 2wm)t + sin(
wc
- 2wm)t]
+ J3
(d)[sin(wc
+ 3wm)t - sin(
wc
- 3wm)t]
+ ...
This expression implies that the FM spectrum consists of a component at
wc and an infinite number
of lines at wc
± nwm
and that the amplitude of the components are given by the bessel
functions.
The Applet below shows several features of FM signals which can be
seen by changing the parameter Fn which is set by Scrollbar 0.
Fn = 0 Shows the FM spectrum and signal for varying
carrier frequency, modulating frequency and modulating index.
Fn = 1 Shows the FM spectrum and signal with the sum of
the first and second pair of sidebands.
Fn = 2 Shows the Carrier plus the first pair of sidebands.
The result is a signal with varying phase and amplitude.
Fn = 3 Shows the Carrier plus the second pair of sidebands.
The result is a signal with varying amplitude but no phase
variation. This partly compensates for the amplitude variation
introduced by the first pair of sidebands.
Fn = 4 Shows how the sidebands add up to produce a constant
amplitude signal with varying phase and frequency.
Fn = 5 Shows a simplified phasor representation of an FM
signal. The diagram gets confusing if all individual vectors
are shown. Instead the resultant of pairs of sidebands are shown.
(For reference the vectors for even n behave similarly to the
phasors in an AM signal and for odd n they behave similar to a DSB
signal with the carrier shifted by 90°.)
The resultant for the first/second/third/fourth/fifth pair of
sidebands are shown in
pink/yellow/green/cyan/light magenta.
The overall resultant is shown by the magenta line and as
time is changed by Pt(15), the overall resultant moves along the
circumference of the Dark Orange circle. This corresponds to a
signal with constant amplitude and varying phase and frequency.
The amplitude of the phase variation is changed by
d.
Fn = 6 shows the signal produced when a carrier is modulated
by 2 frequencies
wm and
wm2 with modulating indices
d and d2
respectively.
Fn = 7 Shows the frequency spectrum of an FM signal with 2 modulating
frequencies. Unlike AM, FM is a 'nonlinear' form of modulation. The
spectrum with 2 modulating frequencies contains components
produced by each modulating frequency
(wc ± n wm
and wc ± m wm2)
PLUS mixing or beating
components at
(wc ± n wm
± m wm2).
When activated, an image of the applet is shown below.
Here the applet is used to show the spectrum of an FM signal with
2 modulating frequencies.
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COPYRIGHT © 1996,2010 Cuthbert Nyack.