Editing Analytic signal (section)

{{Short description|Particular representation of a signal}}
{{Distinguish|analytic expression|analytic function}}

In [[mathematics]] and [[signal processing]], an '''analytic signal''' is a [[complex-valued function]] that has no [[negative frequency]] components.<ref>Smith, J.O. "Analytic Signals and Hilbert Transform Filters", in Mathematics of the Discrete Fourier Transform (DFT) with Audio Applications, Second Edition, https://ccrma.stanford.edu/~jos/r320/Analytic_Signals_Hilbert_Transform.html, or https://www.dsprelated.com/freebooks/mdft/Analytic_Signals_Hilbert_Transform.html, online book, 2007 edition, accessed 2021-04-29.</ref>&nbsp; The real and imaginary parts of an analytic signal are real-valued functions related to each other by the [[Hilbert transform]].

The '''analytic representation''' of a [[real number|real-valued]] function is an ''analytic signal'', comprising the original function and its Hilbert transform.  This representation facilitates many mathematical manipulations. The basic idea is that the negative frequency components of the [[Fourier transform]] (or [[spectrum]]) of a real-valued function are superfluous, due to the [[Hermitian symmetry]] of such a spectrum. These negative frequency components can be discarded with no loss of information, provided one is willing to deal with a complex-valued function instead. That makes certain attributes of the function more accessible and facilitates the derivation of modulation and demodulation techniques, such as single-sideband.

As long as the manipulated function has no negative frequency components (that is, it is still ''analytic''), the conversion from complex back to real is just a matter of discarding the imaginary part. The analytic representation is a generalization of the [[phasor (sine waves)|phasor]] concept:<ref name="Bracewell">Bracewell, Ron. ''The Fourier Transform and Its Applications''. McGraw-Hill, 2000. pp. 361-362</ref> while the phasor is restricted to time-invariant amplitude, phase, and frequency, the analytic signal allows for time-variable parameters.