PHASE MODULATION

A technique used in telecommunications transmission systems whereby the phase of a periodic carrier signal is changed in accordance with the characteristics of an information signal, called the modulating signal. Phase modulation (PM) is a form of angle modulation. For systems in which the modulating signal is digital, the term “phase-shift keying” (PSK) is usually employed. See also Angle modulation.
In typical applications of phase modulation or phase-shift keying, the carrier signal is a pure sine wave of constant amplitude, represented mathematically as Eq. (1),
1.
where the constant A is its amplitude, θ(t) = ωt is its phase, which increases linearly with time, and ω = 2πf and f are constants that represent the carrier signal's radian and linear frequency, respectively.
Phase modulation varies the phase of the carrier signal in direct relation to the modulating signal m(t), resulting in
2.
Eq. (2), where k is a constant of proportionality. The resulting transmitted signal s(t) is therefore given by Eq. (3).
3.
At the receiver, m(t) is reconstructed by measuring the variations in the phase of the received modulated carrier.
Phase modulation is intimately related to frequency modulation (FM) in that changing the phase of c(t) in accordance with m(t) is equivalent to changing the instantaneous frequency of c(t) in accordance with the time derivative of m(t). See also Frequency modulation.
Among the advantages of phase modulation are superior noise and interference rejection, enhanced immunity to signal fading, and reduced susceptibility to nonlinearities in the transmission and receiving systems. See also Distortion (electronic circuits); Electrical interference; Electrical noise.
When the modulating signal m(t) is digital, so that its amplitude assumes a discrete set of values, the phase of the carrier signal is “shifted” by m(t) at the points in time where m(t) changes its amplitude. The amount of the shift in phase is usually determined by the number of different possible amplitudes of m(t). In binary phase-shift keying (BPSK), where m(t) assumes only two amplitudes, the phase of the carrier differs by 180°. An example of a higher-order system is quadrature phase-shift keying (QPSK), in which four amplitudes of m(t) are represented by four different phases of the carrier signal, usually at 90° intervals. See also Modulation.

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

Annotated:

carrier(time) = (carrier amplitude)*sin(carrier frequency*time + phase shift)

This makes the modulated signal

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 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

,

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:

,

where Δθ is the peak phase deviation. Compare to the modulation index for frequency modulation.