Phase modulation
Phase modulation (PM) is a form of modulation that represents information as variations in the instantaneous phase of a carrier wave.
Unlike its more popular counterpart, frequency modulation (FM), PM is not very widely used for radio transmissions. This is because it tends to require more complex receiving hardware and there can be ambiguity problems in determining whether, for example, the signal has changed phase by +180° or -180°. PM is used, however, in digital music synthesizers such as the Yamaha DX7, even though these instruments are usually referred to as "FM" synthesizers (both modulation types sound very similar, but PM is usually easier to implement in this area).
PM changes the phase angle of the complex envelope in direct proportion to the message signal.
Suppose that the signal to be sent (called the modulating or message signal) is 
and the carrier onto which the signal is to be modulated is
Annotated:
- carrier(time) = (carrier amplitude)*sin(carrier frequency*time + phase shift)
This makes the modulated signal
This shows how modulates the phase - the greater m(t) is at a point in time, the
greater the phase shift of the modulated signal at that point. It can
also be viewed as a change of the frequency of the carrier signal, and
phase modulation can thus be considered a special case of FM in which
the carrier frequency modulation is given by the time derivative of the phase modulation.
The mathematics of the spectral behavior reveals that there are two regions of particular interest:
- For small amplitude signals, PM is similar to amplitude modulation (AM) and exhibits its unfortunate doubling of baseband bandwidth and poor efficiency.
- For a single large sinusoidal signal, PM is similar to FM, and its bandwidth is approximately
-
-
,
-
-
where
and 
is the modulation index defined below. This is also known as Carson's Rule for PM.
