Editing Phase modulation (section)

== 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.
}}