# Phase modulation

(Redirected from Phase modulation index)

Phase modulation (PM) is a modulation pattern that encodes information as variations in the instantaneous phase of a carrier wave.

Unlike frequency modulation (FM), phase modulation is not widely used for transmitting radio waves. This is because PM requires 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 for signal and waveform generation in digital synthesizers, such as the Casio CZ synthesizers, and to implement FM synthesis in such digital synthesizers as the Yamaha DX7.

## Theory

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

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

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.