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In angle modulation, the amplitude of the signal is held constant and the phase is being varied with the message. An angle modulated signal is of the form:
![$\displaystyle x_c(t)=A_c\cos(\omega_c t+\phi(t)).$](img4.png) |
(1) |
The instantaneous phase of
is given by
![$\displaystyle \theta_i(t)=\omega_c t+\phi(t),$](img6.png) |
(2) |
and the instantaneous frequency is given by
![$\displaystyle \omega_i(t)=\frac{d\theta_i(t)}{dt}=\omega_c+\frac{d\phi(t)}{dt}.$](img7.png) |
(3) |
Using this approach, if the message is proportional to
, which is the phase deviation, then we have phase modulation. If the message is proportional to
, which is the frequency deviation, then we have frequency modulation.
In order to have phase modulation,
![$\displaystyle \phi(t)=k_p m(t),$](img10.png) |
(4) |
where
is known as the deviation constant. For frequency modulation,
![$\displaystyle \frac{d\phi(t)}{dt}=k_f m(t),$](img12.png) |
(5) |
where
is known as the frequency deviation constant. Consequently, an FM modulated signal is of the form
![$\displaystyle x_c(t)=A_c\cos(\omega_c t + k_f\int_t m(\alpha)d\alpha).$](img14.png) |
(6) |
An FM signal is shown in Figure 1
Figure 1:
Frequency modulation
![\begin{figure}
\epsfig{width=5in,file=FM_signal.eps}
\end{figure}](img15.png) |
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Copyright © 2003, Aly El-Osery
Last Modified 2003-11-02