Editing Phase modulation

{{short description|Electronic method of transmitting information with a carrier wave}}
{{about|the analog modulation|the digital version|Phase-shift keying}}
{{Modulation techniques}}

'''Phase modulation''' ('''PM''') is a [[signal modulation]] method for conditioning communication signals for [[Transmission (telecommunications)|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]].

In phase modulation, the instantaneous amplitude of the baseband signal modifies the phase of the carrier signal keeping its amplitude and frequency constant. 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 an integral part of many digital transmission coding schemes that underlie a wide range of technologies like [[Wi-Fi]], [[GSM]] and [[satellite television]]. However it is not widely used for transmitting [[Analog signal|analog]] audio signals via [[radio]] waves.{{Why?|date=April 2024}} It is also used for signal and [[waveform]] generation in [[digital synthesizer]]s, such as the [[Yamaha DX7]], to implement [[FM synthesis]]. A related type of sound synthesis called [[phase distortion synthesis|phase distortion]] is used in the [[Casio CZ synthesizers]].

== Foundation ==
In general form, an analog modulation process of a sinusoidal carrier wave may be described by the following equation:<ref>{{cite book |last=Klie |first=Robert H. |author2=Bell Telephone Laboratories |author3=AT&T |isbn=0-932764-13-4 |oclc=894686224 |series=Telecommunication Transmission Engineering |volume=1 |title=Principles |edition=2nd |publisher=Bell Center for Technical Education |date=1977}}</ref>

:<math>m(t) = A(t) \cdot \cos(\omega t + \phi(t))\,</math>.

''A(t)'' represents the time-varying amplitude of the sinusoidal carrier wave and the cosine-term is the carrier at its [[angular frequency]] <math>\omega</math>, and the  instantaneous phase deviation <math>\phi(t)</math>. This description directly provides the two major groups of modulation, [[amplitude modulation]] and [[angle modulation]].  In amplitude modulation, the angle term is held constant, while in angle modulation the term ''A(t)'' is constant and the second term of the equation has a functional relationship to the modulating message signal.

The functional form of the cosine term, which contains the expression of the [[instantaneous phase]] <math>\omega t + \phi(t)</math> as its argument, provides the distinction of the two types of angle modulation, [[frequency modulation]] (FM) and phase modulation (PM).<ref name=haykin>{{cite book |first=Simon |last=Haykin |title=Communication Systems |publisher=Wiley |date=2001 |isbn=0-471-17869-1 |page=107}}</ref>

[[File:Phase-modulation.gif|290px|thumb|right|The modulating wave ({{font color|blue|'''blue'''}}) is modulating the carrier wave ({{font color|red|'''red'''}}), resulting the PM signal ({{font color|green|'''green'''}}). {{center| {{nobr|{{math|&nbsp; ''g''(''t'') {{=}} {{sfrac| ''π'' |2}}×sin[ 2×2''π t''  +  {{sfrac| ''π'' |2}}×sin( 3×2''π t'' ) ] &nbsp;}} }} }}]]

In FM the message signal causes a functional variation of the [[carrier frequency]]. These variations are controlled by both the frequency and the amplitude of the modulating wave.

In phase modulation, the instantaneous phase deviation <math>\phi(t)</math> ([[phase (waves)|phase angle]]) of the carrier is controlled by the modulating waveform, such that the principal frequency remains constant.

In principle, the modulating signal in both frequency and phase modulation may either be analog in nature, or it may be digital.

The mathematics of the [[spectral density|spectral]] behaviour reveals that there are two regions of particular interest:
{{unordered list
 | For small [[amplitude]] signals, PM is similar to [[amplitude modulation]] (AM) and exhibits its unfortunate doubling of [[baseband]] [[bandwidth (signal processing)|bandwidth]] and poor efficiency.
 | For a single large [[sinusoidal]] signal, PM is similar to FM, and its [[bandwidth (signal processing)|bandwidth]] is approximately
: <math>2\left(h + 1\right)f_\text{M}</math>,
Where <math>f_\text{M} = \omega_\text{m}/2\pi</math> and <math>h</math> is the modulation index defined below. This is also known as [[Carson bandwidth rule|Carson's Rule]] for PM.
}}

== Modulation index ==
As with other [[modulation index|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:
: <math>h = \Delta \theta ,</math>
where <math>\Delta \theta</math> is the peak phase deviation. Compare to the modulation index for [[frequency modulation#Modulation index|frequency modulation]].
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== See also ==
* [[Automatic frequency control]] 
* [[Modulation]] for a list of other modulation techniques
* [[Modulation sphere]]
* [[Polar modulation]]
* [[Electro-optic modulator]] for Pockel's Effect phase modulation for applying sidebands to a monochromatic wave

==References==
{{Reflist}}

{{Authority control}}

{{DEFAULTSORT:Phase Modulation}}
[[Category:Radio modulation modes]]