Phase Modulation Review

As shown in the previous section,  the general signal with phase noise can be represented by

s(t) = A sin(w_ct + f(t))

f(t) is a noise-like signal that phase modulates the carrier.   Before analyzing the case of modulation by noise,  we will first review sinusoidal phase modulation.

A signal consisting of a carrier of frequency w_c phase modulated by a sinusoidal signal of frequency w_m is given by

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

.. (1.4)     

where q_pk is the peak phase deviation (in radians)

The time-domain waveform in (1.4) is quite simple,  the signal has a phase offset that varies according to the modulation signal.   Large-level phase modulation is easily observed on an oscilloscope as jitter in the zero crossings.   

The spectrum of the waveform in (1.4) is anything but simple,  consisting of a sidebands around the carrier w_c  offset by  ±w_m and (theoretically) an infinite number of harmonics of w_m .  The result is an infinite series of sidebands with amplitudes given by Bessel functions.   The following table shows the spectrum of an 0dBm, 100MHz carrier phase modulated by a 100kHz sinusoid,   with varying phase deviations:

Phase Modulation is a non-linear modulation.  This means that if we want to know what happens if the signal is phase modulated by a signal that is a sum of other signals,   we cannot work it out by adding up the modulation responses to each signal individually.  This is fairly clear from the spectra above,  increasing the modulation index does not just increase the levels of the spectral lines in a linear manner,  a whole lot of new spectral lines become important.

The non-linearity  is potentially a problem for when we analyze noise modulation,  as one of the main techniques for treating noise is to consider it as a sum of sine waves,  and to consider the response to each sine wave separately.      Luckily,  for low levels of phase modulation,  the modulation behaves in a linear manner,  allowing phase noise to be treated using conventional linear techniques.