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  "body": "\n## Definition\n\nLet $f \\in L^2(\\R)$ be real-valued, an element of the [[$L^2$ space|SASP14MA1Z]].\nLet $Hf$ be its [[Hilbert transform|NSCEBXM7ZM]].\nThe *analytic signal* of $f$ is the complex-valued function\n$$f_a = f + i\\, Hf.$$\n\n## Remarks\n\nWith the [[Fourier transform|DPVM34ADZV]] $\\hat f(\\xi) = \\int_\\R f(x) e^{-2\\pi i x \\xi}\\, dx$, one has $\\widehat{Hf}(\\xi) = -i \\sgn(\\xi) \\hat f(\\xi)$, so\n$$\\widehat{f_a}(\\xi) = \\big(1 + \\sgn(\\xi)\\big) \\hat f(\\xi),$$\nwhich is $2 \\hat f(\\xi)$ for $\\xi \u003e 0$ and $0$ for $\\xi \u003c 0$.\nSince $f$ is real, $f = \\operatorname{Re} f_a$, and $f_a$ is determined by $f$.\nOne writes $f_a(t) = A(t) e^{i\\phi(t)}$ with $A = \\abs{f_a}$ the instantaneous amplitude and $\\phi$ the instantaneous phase.\n\nExample: $f(x) = \\frac{1}{1 + x^2}$.\nIts Hilbert transform is $\\frac{x}{1 + x^2}$, so $f_a(x) = \\frac{1 + ix}{1 + x^2} = \\frac{1}{1 - ix} = \\frac{i}{x + i}$.\nThe function $z \\mapsto \\frac{i}{z + i}$ is holomorphic on the upper half-plane, as the definition predicts.\n\nNon-example: the clause \"real-valued\" matters.\nFor the complex function $g = \\frac{i}{x + i}$ one has $Hg = -i g$, so $g + i Hg = 2g \\ne g$, and the real part of $g + iHg$ is $2 \\operatorname{Re} g$, not $\\operatorname{Re} g$.\n",
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