Analytic signal

In mathematics and signal processing, an analytic signal is a complex-valued function that has no negative frequency components.  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-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 concept: while the phasor is restricted to time-invariant amplitude, phase, and frequency, the analytic signal allows for time-variable parameters.

If ${\displaystyle s(t)}$ is a real-valued function with Fourier transform ${\displaystyle S(f)}$, then the transform has Hermitian symmetry about the ${\displaystyle f=0}$ axis:

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