# Phase modulation

Phase modulation (PM) is a form of modulation that represents information as variations in the instantaneous phase of a carrier wave. Modification in phase according to low frequency will give phase modulation.

Unlike its more popular counterpart, frequency modulation (FM), PM is not 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 Casio CZ synthesizers, or to implement FM Synthesis in digital synthesizers such as the Yamaha DX7. (see FM Synthesis)

## Theory

An example of phase modulation. The top diagram shows the modulating signal superimposed on the carrier wave. The bottom diagram shows the resulting phase-modulated signal.

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 $m(t)$ and the carrier onto which the signal is to be modulated is

$c(t) = A_c\sin\left(\omega_\mathrm{c}t + \phi_\mathrm{c}\right).$

Annotated:

carrier(time) = (carrier amplitude)*sin(carrier frequency*time + phase shift)

This makes the modulated signal

$y(t) = A_c\sin\left(\omega_\mathrm{c}t + m(t) + \phi_\mathrm{c}\right).$

This shows how $m(t)$ 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:

$2\left(h + 1\right)f_\mathrm{M}$,
where $f_\mathrm{M} = \omega_\mathrm{m}/2\pi$ and $h$ is the modulation index defined below. This is also known as Carson's Rule for PM.

## Modulation index

As with other modulation indices, this quantity indicates by how much the modulated variable varies around its unmodulated level. It relates to the variations in the phase of the carrier signal:

$h\, = \Delta \theta\,$,

where $\Delta \theta$ is the peak phase deviation. Compare to the modulation index for frequency modulation.