Narrowband Phase and Frequency Modulation Theory

When the modulation index b << 1 (FM),  or equivalently the peak phase deviation  q_pk <<  1 (PM),  then there are some approximations that can be made which simplify the expressions for the signal waveforms.   This is called Narrowband modulation.

Starting with (1.4) for sinusoidal phase modulation

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

if  q_pk <<  1  then we can approximate (1.4) by

.. (1.9)     

Where we used the fact that sin(x) ~ x   and cos(x) ~ 1    for     x << 1.  

So low level phase (narrow-band frequency) modulation produces a pair of sidebands with amplitude of 0.5 q_pk relative to the carrier.   This relationship is linear.   It is easy to repeat the analysis in (1.9) for modulation consisting of a two sinusoidal signals,   the result is a pair of sidebands for each sinusoidal modulation.. 

This is the relationship that enables us to easily investigate the spectrum of phase noise.   As long as total phase modulation results on a phase deviation much less than one radian (about 60 degrees) then we can associate the spectral components of the modulating signal with those in the modulation.   

Narrow-Band Frequency Modulation:  as the modulation index  b  was shown to be equal to  q_pk,  then for frequency modulation with modulation index b << 1,   the spectrum consists of a pair of sidebands,  offset from the carrier by the modulating frequency,  of amplitude 0.5 b relative to the carrier.