# Phase modulation

Phase modulation (PM) is a modulation pattern for conditioning communication signals for transmission. It encodes a message signal as variations in the instantaneous phase of a carrier wave. Phase modulation is one of the two principal forms of angle modulation, together with frequency modulation.

The phase of a carrier signal is modulated to follow the changing signal level (amplitude) of the message signal. The peak amplitude and the frequency of the carrier signal are maintained constant, but as the amplitude of the message signal changes, the phase of the carrier changes correspondingly.

Phase modulation is widely used for transmitting radio waves and is an integral part of many digital transmission coding schemes that underlie a wide range of technologies like Wi-Fi, GSM and satellite television.

PM is used for signal and waveform generation in digital synthesizers, such as the Yamaha DX7, to implement FM synthesis. A related type of sound synthesis called phase distortion is used in the Casio CZ synthesizers.

## Theory

The modulating wave (blue) is modulating the carrier wave (red), resulting the PM signal (green). g(t) = π/2 * sin(2*2πt+ π/2*sin(3*2πt))

PM changes the phase angle of the complex envelope in direct proportion to the message signal.

If m(t) is the message signal to be transmitted and the carrier onto which the signal is modulated is

${\displaystyle c(t)=A_{c}\sin \left(\omega _{\mathrm {c} }t+\phi _{\mathrm {c} }\right).}$ ,

then the modulated signal is

${\displaystyle y(t)=A_{c}\sin \left(\omega _{\mathrm {c} }t+m(t)+\phi _{\mathrm {c} }\right).}$

This shows how ${\displaystyle m(t)}$  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 modulation signal could here be

${\displaystyle m(t)=\cos \left(\omega _{\mathrm {c} }t+h\omega _{\mathrm {m} }(t)\right)\ }$

The mathematics of the spectral behavior reveals that there are two regions of particular interest:

${\displaystyle 2\left(h+1\right)f_{\mathrm {M} }}$ ,
where ${\displaystyle f_{\mathrm {M} }=\omega _{\mathrm {m} }/2\pi }$  and ${\displaystyle h}$  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:

${\displaystyle h\,=\Delta \theta \,}$ ,

where ${\displaystyle \Delta \theta }$  is the peak phase deviation. Compare to the modulation index for frequency modulation.