Phase modulation

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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 counterpart, frequency modulation (FM), PM is not very widely used (except perhaps for in the inappropiately named FM-synthesis for musical instruments, introduced by Yamaha around 1982.) 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

Image:Phase Modulation.png Suppose that the signal to be sent, the modulating signal with frequency <math>\omega_\mathrm{m}</math> and phase <math>\phi_\mathrm{m}</math>, is

<math>m(t) = M\sin\left(\omega_\mathrm{m}t + \phi_\mathrm{m}\right)</math>,

and the carrier onto which the signal is to be modulated is

<math>c(t) = C\sin\left(\omega_\mathrm{c}t + \phi_\mathrm{c}\right) </math>.

Then the modulated signal,

<math>y(t) = C\sin\left(\omega_\mathrm{c}t + m(t) + \phi_\mathrm{c}\right)</math>,

which shows how <math>m(t)</math> 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:

• 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

<math>2\left(h + 1\right)f_\mathrm{M}</math>Hz,

where <math>f_\mathrm{M} = \omega_\mathrm{m}/2\pi</math> and <math>h</math> 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:

<math> h = \Delta \theta</math>,

where <math>\Delta \theta</math> is the peak phase deviation. Compare to the modulation index for frequency modulation.

See also

• Modulation for a list of other modulation techniques
• Phase-shift keying (PSK), the digital version of PMde:Phasenmodulation

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