Analytic signal

Summary

The analytic signal of a real function $f$ is the complex function $f + i\, Hf$, where $H$ is the Hilbert transform. Its Fourier transform vanishes at negative frequencies and is twice that of $f$ at positive frequencies. It is the boundary value of a holomorphic function on the upper half-plane, which explains the name. In signal processing it defines the instantaneous amplitude and phase of a real signal.