Frequency Modulation Review

Frequency Modulation (FM) results in an identical signal to phase modulation,  however the terminology is different.   For an ideal oscillator

s(t) = A sin(w_ct)

.. (1.5)     

the phase at any time is given by w_ct,    note that it changes steadily with time with slope w_c. This leads to the definition of the instantaneous frequency as the rate of change of the phase of the signal with time.   So for the phase modulated signal

s(t) = A sin(w_ct + q_pk sin(w_mt))

.. (1.6)     

the phase at any time is w_ct + q_pksin(w_mt), and differentiating (1.6)  leads to 

instantaneous frequency = w_c + q_pk w_m cos(w_mt) 

Thus the carrier frequency w_c   is sinusoidally modulated with peak frequency deviation w_pk  given by

w_pk = q_pk w_m 

define     b = q_pk  = w_pk / w_m  =  f_pk / f_m

.. (1.7)     

where b is the Modulation Index.  Notice that the modulation index is the ratio of the peak frequency deviation to the modulating frequency,   and that this can be calculated using both frequencies as radian frequencies,  or both in Hz.   The Modulation Index also represents the peak phase deviation in radians. 

The frequency modulated signal becomes

s(t) = A sin(w_ct + b cos(w_mt))

.. (1.8)     

Thus we can repeat the spectra shown on the previous page for PM,  but in this case simply change the labels from phase deviation to modulation index.   So for a 0dBm carrier at 100MHz,  being frequency modulated by a 100kHz sine wave with various values of the modulation index,  the spectra are shown in the following table:

As it is equivalent to Phase Modulation,  Frequency Modulation is also non-linear modulation.