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. 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°.

Theory

Suppose that the signal to be sent (called the modulating or message signal) is $m(t)$.

The carrier onto which the signal is to be modulated is

$c(t)\; =\; A\_csinleft(omega\_mathrm\{c\}t\; +\; phi\_mathrm\{c\}right).$

Then the modulated signal is

$y(t)\; =\; A\_csinleft(omega\_mathrm\{c\}t\; +\; m(t)\; +\; phi\_mathrm\{c\}right),$

This shows how $m(t)$ modulates the phase. Clearly, it can also be viewed as a change of the frequency of the carrier signal. PM 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 spectral behaviour of phase modulation is difficult to derive, but the mathematics 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

$2left(h\; +\; 1right)f\_mathrm\{M\}$,

where $f\_mathrm\{M\}\; =\; omega\_mathrm\{m\}/2pi$ 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.

See also

• Automatic frequency control
• Modulation for a list of other modulation techniques
• Phase-shift keying (PSK), the digital version of PM
• Electro-optic modulator for Pockel's Effect phase modulation for applying sidebands to a monochromatic wave