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11
Frequency modulation introduces
control of the phase argument,
Φ, in the basic carrier equation:
A sin (ω
c
t + Φ ).
FM is implemented by varying
Φ in direct proportion to the
integral of the modulating
signal. Thus, for a modulating
signal m(t), the FM signal can be
written:
A sin (ω
c
t + k ∫ m(x) dx )
where k sets the peak frequency
deviation. For the special case of
a modulating tone cos (ω
m
t), the
phase argument becomes:
k/ω
m
sin (ω
m
t ),
where k is the peak frequency
deviation and k/ω
m
is the FM
modulation index.
The FM equation is entered
directly into the AWG’s equation
editor (Figure 10). The modula-
tion tone is 1000 Hz, so the
unique or arbitrary portion of
the signal repeats every 1 ms.
Choosing a common FM IF
carrier frequency of 10.7 MHz,
note that the carrier frequency is
a multiple of the modulating
frequency. This means that the
carrier signal will be phase
continuous when the 1 ms
record repeats.
Figure 11 shows a spectrum
analyzer plot of the modulated
signal. The peak deviation of
5.52 kHz was selected because a
modulation index of 5.52 causes
the carrier component in the
modulated signal to vanish. This
is confirmed by noting that the
0th order Bessel function for a
modulation index of 5.52,
J
0
(5.52), is zero.
Frequency Modulation
4
Frequency (kHz
-80
-70
-60
-50
-40
-30
-20
-10
0
10680 10685 10690 10695 10700 10705 10710 10715 10720
Figure 11. Spectrum analyzer plot of the FM
signal. The carrier component vanishes for a
modulation index of 5.52. The carrier would also
vanish for indices of 2.40, 8.65, and 11.79. This
is a simple way to verify that the peak deviation of
an FM signal has been set properly.
Figure 10. AWG equation for FM single-tone
modulation. The peak deviation is 5.52 kHz with a
modulating tone of 1000 Hz. The carrier
frequency is 10.7 MHz. A 1 ms period is used
with a 32,768 point record length; this sets the
AWG sampling rate to 32.768 MHz.
Magnitude (dBm)