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  "body": "\n## Definition\n\nLet $f \\in L^2(\\R)$ (see [[Lp space|SASP14MA1Z]]).\nThe *ambiguity function* of $f$ is the function $Af \\colon \\R^2 \\to \\C$,\n$$Af(x, \\omega) = \\int_{\\R} f\\!\\left(t + \\tfrac{x}{2}\\right) \\overline{f\\!\\left(t - \\tfrac{x}{2}\\right)} \\, e^{-2\\pi i \\omega t} \\, dt,$$\nfor a time shift $x \\in \\R$ and a frequency shift $\\omega \\in \\R$.\n\n## Remarks\n\nThe substitution $t = s - x/2$ gives $Af(x, \\omega) = e^{\\pi i x \\omega} V_f f(x, \\omega)$, where $V_f f$ is the [[short-time Fourier transform|3DHNP5NEWH]] of $f$ with window $f$.\nHence $\\abs{Af(x, \\omega)} \\le Af(0, 0) = \\norm{f}_2^2$, by the Cauchy–Schwarz inequality.\nIn radar texts the ambiguity function is often written as $\\chi(\\tau, \\nu) = \\int s(t) \\overline{s(t - \\tau)} e^{2\\pi i \\nu t} \\, dt$.\nIt differs from $Af$ by a sign convention and a phase factor, and $\\abs{\\chi(\\tau, \\nu)} = \\abs{Af(\\tau, -\\nu)}$ for $s = f$.\n\nExample: let $f(t) = 2^{1/4} e^{-\\pi t^2}$.\nThen $f(t + x/2) \\overline{f(t - x/2)} = \\sqrt2 \\, e^{-\\pi x^2/2} e^{-2\\pi t^2}$, and the Fourier transform of $e^{-2\\pi t^2}$ is $2^{-1/2} e^{-\\pi \\omega^2/2}$.\nSo $Af(x, \\omega) = e^{-\\pi (x^2 + \\omega^2)/2}$, a single peak of height $1 = \\norm{f}_2^2$ at the origin.\n\nNon-example: the autocorrelation $R(x) = \\int_\\R f(t) \\overline{f(t - x)} \\, dt$ is not the ambiguity function.\nThe substitution $t = s + x/2$ shows $R(x) = Af(x, 0)$, so $R$ is only the slice $\\omega = 0$ of $Af$ and does not see frequency shifts.\n",
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