Phase modulation

Passband modulation
Analog modulation
AM · SSB · QAM · FM · PM · SM
Digital modulation
FSK · MFSK · ASK · OOK · PSK · QAM MSK · CPM · PPM · TCM · SC-FDE
Spread spectrum
CSS · DSSS · FHSS · THSS
See also: Demodulation, modem, line coding, PAM, PWM, PCM

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 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 Yamaha DX7, even though these instruments are usually referred to as "FM" synthesizers (both modulation types sound very similar, but PM is usually easier to implement in this area).

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 behaviour 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

$2\left(h + 1\right)f_\mathrm{M}$,

where f_M = ω_m / 2π 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 Δθ is the peak phase deviation. Compare to the modulation index for frequency modulation.

See also

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