Analytic signal

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a> and <a href="/facts/Signal_processing/Npaja6zb">signal processing</a>, an analytic signal is a <a href="/facts/Complex-valued_function/hgCoQlH0">complex-valued function</a> that has no <a href="/facts/Negative_frequency/YasYp0L0">negative frequency</a> components.  The real and imaginary parts of an analytic signal are real-valued functions related to each other by the <a href="/facts/Hilbert_transform/RoQaC8MS">Hilbert transform</a>.
The analytic representation of a <a href="/facts/Real_number/R02gw5Pb">real-valued</a> 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 <a href="/facts/Fourier_transform/XXhKJtc8">Fourier transform</a> (or <a href="/facts/Spectrum/wPUjkpqI">spectrum</a>) of a real-valued function are superfluous, due to the <a href="/facts/Hermitian_symmetry/3U40nuaK">Hermitian symmetry</a> 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 <a href="/facts/Phasor_(sine_waves)/LyjSSUDo">phasor</a> concept: while the phasor is restricted to time-invariant amplitude, phase, and frequency, the analytic signal allows for time-variable parameters.

Analytic signal open-in-new

Analytic signal