Editing Angle 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 |author=AT&T Bell Laboratories Staff |title=Telecommunication Transmission Engineering |volume=1—''Principles''|edition=2|publisher=AT&T Bell Center for Technical Education|year=1977}}</ref>

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

<math>A(t)</math> 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 <math>A(t)</math> 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 /> In FM the message signal causes a functional variation of the [[instantaneous 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> of the carrier is controlled by the modulating waveform, such that the principal frequency remains constant.

For angle modulation, the [[instantaneous frequency]] of an angle-modulated carrier wave is given by the first derivative with respect to time of the instantaneous phase:
: <math> \omega_I = \frac{d}{dt} [ \omega t + \phi(t) ] = \omega + \phi'(t) ,</math>
in which <math>\phi'(t)</math> may be defined as the instantaneous frequency deviation, measured in rad/s.

For frequency modulation (FM), the modulating signal <math> s(t)</math> is related linearly to the instantaneous frequency deviation, that is <math> \phi_{FM}' = K_{FM} s(t),</math> which gives the FM modulated waveform as<blockquote><math> m_{FM}(t) = A \cos \left( \omega t + K_{FM} \int s(\tau) d\tau \right).</math></blockquote>For phase modulation (PM), the modulating signal <math> s(t)</math> is related linearly to the instantaneous phase deviation, that is <math> \phi_{PM}(t) = K_{PM}s(t),</math> which gives the PM modulated waveform as<blockquote><math> m_{PM}(t) = A \cos \left( \omega t + K_{PM} s(t) \right). </math></blockquote>In principle, the modulating signal in both frequency and phase modulation may either be analog in nature, or it may be digital. In general, however, when using digital signals to modify the carrier wave, the method is called ''[[Keying (telecommunications)|keying]]'', rather than modulation.<ref>Whitham D. Reeve, ''Subscriber Loop Signaling and Transmission Handbook- Digital'', IEEE Press (1995), {{ISBN|0-7803-0440-3}}, p. 5.</ref> Thus, telecommunications [[modem]]s use [[frequency-shift keying]] (FSK), [[phase-shift keying]] (PSK), or [[amplitude and phase-shift keying|amplitude-phase keying]] (APK), or various combinations. Furthermore, another digital modulation is [[line coding]], which uses a [[baseband]] carrier, rather than a [[passband]] wave.

The methods of angle modulation can provide better discrimination against interference and noise than amplitude modulation.<ref name=haykin>Simon Haykin, ''Communication Systems'', John Wiley & Sons (2001), {{ISBN|0-471-17869-1}}, p. 107</ref> These improvements, however, are a tradeoff against increased bandwidth requirements.