Editing Analytic signal (section)

===Instantaneous amplitude and phase===
[[Image:analytic.svg|thumb|300px|A function in blue and the magnitude of its analytic representation in red, showing the envelope effect.]]
An analytic signal can also be expressed in [[polar coordinates]]:
:<math>s_\mathrm{a}(t) = s_\mathrm{m}(t)e^{j\phi(t)},</math>
where the following time-variant quantities are introduced:
*<math>s_\mathrm{m}(t) \triangleq |s_\mathrm{a}(t)|</math> is called the ''instantaneous amplitude'' or the ''[[envelope (waves)|envelope]]'';
*<math>\phi(t) \triangleq \arg\!\left[s_\mathrm{a}(t)\right]</math> is called the ''[[instantaneous phase]]'' or ''phase angle''.

In the accompanying diagram, the blue curve depicts <math>s(t)</math> and the red curve depicts the corresponding <math>s_\mathrm{m}(t)</math>.

The time derivative of the [[phase wrapping|unwrapped]] instantaneous phase has units of ''radians/second'', and is called the ''instantaneous angular frequency'':
:<math>\omega(t) \triangleq \frac{d\phi}{dt}(t).</math>

The ''[[Instantaneous phase#Instantaneous frequency|instantaneous frequency]]'' (in [[hertz]]) is therefore:
:<math>f(t)\triangleq \frac{1}{2\pi}\omega(t).</math> &nbsp;<ref>B. Boashash, "Estimating and Interpreting the Instantaneous Frequency of a Signal-Part I: Fundamentals", Proceedings of the IEEE, Vol. 80, No. 4, pp. 519–538, April 1992</ref>

The instantaneous amplitude, and the instantaneous phase and frequency are in some applications used to measure and detect local features of the signal. Another application of the analytic representation of a signal relates to demodulation of [[modulation|modulated signals]]. The polar coordinates conveniently separate the effects of [[amplitude modulation]] and phase (or frequency) modulation, and effectively demodulates certain kinds of signals.