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  "body": "\n## Summary\n\nThe analytic signal of a real function $f$ is the complex function $f + i\\, Hf$, where $H$ is the [[Hilbert transform|NSCEBXM7ZM]].\nIts Fourier transform vanishes at negative frequencies and is twice that of $f$ at positive frequencies.\nIt is the boundary value of a [[holomorphic function|0NAHXR1FRH]] on the upper half-plane, which explains the name.\nIn signal processing it defines the instantaneous amplitude and phase of a real signal.\n\n## Explanations\n\n### L1 human\n\nA real signal, such as a sound wave, contains each frequency twice, once with a positive and once with a negative sign, and the two halves carry the same information.\nThe analytic signal throws away the negative frequencies and doubles the positive ones.\nThe result is a complex signal that turns around the origin of the plane.\nIts distance from the origin is the envelope of the signal and its angle is the phase.\nFor example, the analytic signal of $\\cos t$ is $e^{it} = \\cos t + i \\sin t$.\n",
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