Editing Analytic signal (section)

==Properties==

===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.

=== {{anchor|Complex envelope}} Complex envelope/baseband===
Analytic signals are often shifted in frequency (down-converted) toward 0&nbsp;Hz, possibly creating [non-symmetrical] negative frequency components:
<math display="block">{s_\mathrm{a}}_{\downarrow}(t) \triangleq s_\mathrm{a}(t)e^{-j\omega_0 t} = s_\mathrm{m}(t)e^{j(\phi(t) - \omega_0 t)},</math>
where <math>\omega_0</math> is an arbitrary reference angular frequency.<ref name=Bracewell/>

This function goes by various names, such as ''complex envelope'' and ''complex [[baseband]]''. The complex envelope is not unique; it is determined by the choice of <math>\omega_0</math>. This concept is often used when dealing with [[passband signal]]s. If <math>s(t)</math> is a modulated signal, <math>\omega_0</math> might be equated to its [[carrier frequency]].

In other cases, <math>\omega_0</math> is selected to be somewhere in the middle of the desired passband. Then a simple [[low-pass filter]] with real coefficients can excise the portion of interest. Another motive is to reduce the highest frequency, which reduces the minimum rate for alias-free sampling. A frequency shift does not undermine the mathematical tractability of the complex signal representation. So in that sense, the down-converted signal is still ''analytic''. However, restoring the real-valued representation is no longer a simple matter of just extracting the real component. Up-conversion may be required, and if the signal has been [[Sampling (signal processing)|sampled]] (discrete-time), [[interpolation]] ([[upsampling]]) might also be necessary to avoid [[aliasing]].

If <math>\omega_0</math> is chosen larger than the highest frequency of <math>s_\mathrm{a}(t),</math> then <math>{s_\mathrm{a}}_{\downarrow}(t)</math> has no positive frequencies. In that case, extracting the real component restores them, but in reverse order; the low-frequency components are now high ones and vice versa. This can be used to demodulate a type of [[single-sideband modulation|single-sideband]] signal called ''lower sideband'' or ''inverted sideband''.

Other choices of reference frequency are sometimes considered:
* Sometimes <math>\omega_0</math> is chosen to minimize <math display="block">\int_0^{+\infty}(\omega - \omega_0)^2|S_\mathrm{a}(\omega)|^2\, d\omega.</math>
* Alternatively,<ref>{{Cite journal|last=Justice|first=J.|date=1979-12-01|title=Analytic signal processing in music computation| journal=IEEE Transactions on Acoustics, Speech, and Signal Processing|volume=27|issue=6|pages=670–684| doi=10.1109/TASSP.1979.1163321| issn=0096-3518}}</ref> <math>\omega_0</math> can be chosen to minimize the mean square error in linearly approximating the ''unwrapped'' instantaneous phase <math>\phi(t)</math>: <math display="block">\int_{-\infty}^{+\infty}[\omega(t) - \omega_0]^2 |s_\mathrm{a}(t)|^2\, dt</math>
* or another alternative (for some optimum <math>\theta</math>): <math display="block">\int_{-\infty}^{+\infty}[\phi(t) - (\omega_0 t + \theta)]^2\, dt.</math>

In the field of time-frequency signal processing, it was shown that the analytic signal was needed in the definition of the [[Wigner–Ville distribution]] so that the method can have the desirable properties needed for practical applications.<ref>B. Boashash, "Notes on the use of the Wigner distribution for time frequency signal analysis", IEEE Trans. on Acoustics, Speech, and Signal Processing, vol. 26, no. 9, 1987</ref>

Sometimes the phrase "complex envelope" is given the simpler meaning of the [[complex amplitude]] of a (constant-frequency) phasor;{{efn|"the complex envelope (or complex amplitude)"<ref>{{Cite book|url=https://books.google.com/books?id=tOeeJyP95IQC |title=Time-Frequency Analysis|last1=Hlawatsch|first1=Franz|last2=Auger|first2=François|date=2013-03-01|publisher=John Wiley & Sons |isbn=9781118623831|language=en}}</ref>}}{{efn|"the complex envelope (or complex amplitude)", p.&nbsp;586 <ref>{{Cite book |url=https://books.google.com/books?id=kxICp6t-CDAC&q=%2522complex%2520amplitude%2522%2520%2522complex%2520envelope%2522&pg=RA1-PA586 |title=Encyclopedia of Optical Engineering: Abe-Las, pages 1-1024|last=Driggers|first=Ronald G.|date=2003-01-01 |publisher=CRC Press|isbn=9780824742508|language=en}}</ref>}}
other times the complex envelope <math> s_m(t)</math> as defined above is interpreted as a time-dependent generalization of the complex amplitude.{{efn|"Complex envelope is an extended interpretation of complex amplitude as a function of time." p.&nbsp;85<ref>{{Cite book|url=https://books.google.com/books?id=tXQy5JdQyZoC|title=Global Environment Remote Sensing| last=Okamoto|first=Kenʼichi|date=2001-01-01|publisher=IOS Press|isbn=9781586031015|language=en}}</ref>}} Their relationship is not unlike that in the real-valued case: varying [[Envelope (waves)|envelope]] generalizing constant [[amplitude]].