# Phase modulation

Phase modulation (PM) is a form of modulation which represents information as variations in the instantaneous phase of a carrier wave.

Unlike its more popular couterpart, frequency modulation (FM), PM is not very widely used. This is because it tends to require more complex receiving hardware and there can be ambiguity problems with determining whether, for example, the signal has 0° phase or 180° phase.

## Theory

Suppose that the signal to be sent, the modulating signal with frequency [itex]\omega_m[itex] and phase [itex]\phi_m[itex], is

[itex]m(t) = M\sin\left(\omega_m + \phi_m\right)[itex],

and the single that will be broadcast, the carrier signal, is

[itex]c(t) = C\sin\left(\omega_c + \phi_c\right) [itex].

Then the modulated signal,

[itex]y(t) = C\sin\left(\omega_c + M + \phi_c\right)[itex],

which shows how [itex]M[itex] modulates the phase. Clearly, it could also be viewed as a change to the frequency of the signal and PM can be considered a special case of FM where the carrier frequency modulation is the time derivative of the modulating signal.

The spectral behaviour of PM is difficult to derive, but the mathematics reveals that there are two regions of particular interest:

[itex]2\left(h + 1\right)f_M[itex]Hz,
where [itex]f_M = \omega_m/2\pi[itex] and [itex]h[itex] is the modulation index defined below. This is also known as Carson's Rule for PM.

## Modulation Index

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

[itex] h = \Delta \theta[itex],

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

• Art and Cultures
• Countries of the World (http://www.academickids.com/encyclopedia/index.php/Countries)
• Space and Astronomy