[{"data":1,"prerenderedAt":191},["ShallowReactive",2],{"content:\u002Fen\u002Farchive\u002Fdirac-delta":3},{"id":4,"title":5,"ambience":6,"articleStatus":7,"assets":8,"body":9,"breadcrumb":180,"chunks":8,"completed":6,"def":8,"description":181,"disclaimer":6,"extension":182,"hasMath":30,"image":8,"indexed":6,"key":183,"language":184,"meta":185,"metaTitle":8,"navigation":30,"nextArticle":8,"nextArticleTransition":8,"path":186,"previousArticle":8,"previousArticleTransition":8,"publishedAt":187,"review":6,"seo":188,"seriesPart":8,"seriesTitle":8,"stem":189,"subtitle":8,"tags":8,"tocEnabled":30,"__hash__":190},"legacy_en\u002Fen\u002Farchive\u002Fdirac-delta.md","Dirac delta function",false,"archived",null,{"type":10,"value":11,"toc":173},"minimark",[12,16,32,35,43,46,51,54,57,60,63,66,69,73,96,104,107,115,118,133,141,165],[13,14,15],"p",{},"The Dirac delta function is a rather special function defined as follows.",[17,18,22],"div",{"className":19},[20,21],"math-display-scroll","edge-fade-scroll",[23,24],"math",{"display":25,"className":26,"style":28,"dataCopyLatex":29,"dataCopyDisplay":30,"innerHTML":31},"block",[27],"tml-display","display:block math;","\\delta(x) = \\underset{u \\to 0}{\\lim} {1 \\over {\\left|u\\right|\\sqrt{\\pi}}} e^{-{x^2 \\over u^2}}",true,"\u003Cmrow>\u003Cmi>δ\u003C\u002Fmi>\u003Cmrow>\u003Cmo fence=\"true\" form=\"prefix\" stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>x\u003C\u002Fmi>\u003Cmo fence=\"true\" form=\"postfix\" stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cmo>=\u003C\u002Fmo>\u003Cmrow>\u003Cmunder>\u003Cmrow>\u003Cmi>lim\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003Cmspace width=\"0.1667em\">\u003C\u002Fmspace>\u003C\u002Fmrow>\u003Cmrow>\u003Cmi>u\u003C\u002Fmi>\u003Cmo>→\u003C\u002Fmo>\u003Cmn>0\u003C\u002Fmn>\u003C\u002Fmrow>\u003C\u002Fmunder>\u003C\u002Fmrow>\u003Cmfrac>\u003Cmn>1\u003C\u002Fmn>\u003Cmrow>\u003Cmrow>\u003Cmo fence=\"true\" form=\"prefix\" stretchy=\"true\">|\u003C\u002Fmo>\u003Cmi>u\u003C\u002Fmi>\u003Cmo fence=\"true\" form=\"postfix\" stretchy=\"true\">|\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cmsqrt>\u003Cmrow>\u003Cmi>π\u003C\u002Fmi>\u003Cmspace width=\"0pt\" height=\"0.5em\">\u003C\u002Fmspace>\u003C\u002Fmrow>\u003C\u002Fmsqrt>\u003C\u002Fmrow>\u003C\u002Fmfrac>\u003Cmsup>\u003Cmi>e\u003C\u002Fmi>\u003Cmrow class=\"tml-sml-pad\">\u003Cmo form=\"prefix\" stretchy=\"false\" lspace=\"0em\" rspace=\"0em\">−\u003C\u002Fmo>\u003Cmfrac>\u003Cmsup scriptlevel=\"2\" style=\"math-depth:2;\">\u003Cmi>x\u003C\u002Fmi>\u003Cmn scriptlevel=\"2\" class=\"tml-sml-pad\">2\u003C\u002Fmn>\u003C\u002Fmsup>\u003Cmsup scriptlevel=\"2\" style=\"math-depth:2;\">\u003Cmi>u\u003C\u002Fmi>\u003Cmn scriptlevel=\"2\" class=\"tml-sml-pad\">2\u003C\u002Fmn>\u003C\u002Fmsup>\u003C\u002Fmfrac>\u003C\u002Fmrow>\u003C\u002Fmsup>\u003C\u002Fmrow>",[13,33,34],{},"Or, less formally,",[17,36,38],{"className":37},[20,21],[23,39],{"display":25,"className":40,"style":28,"dataCopyLatex":41,"dataCopyDisplay":30,"innerHTML":42},[27],"\\delta(x) = \\begin{cases} \\infty \\quad \\text{si } x = 0 \\\\ 0 \\ \\ \\quad \\text{sinon} \\end{cases}","\u003Cmrow>\u003Cmi>δ\u003C\u002Fmi>\u003Cmrow>\u003Cmo fence=\"true\" form=\"prefix\" stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>x\u003C\u002Fmi>\u003Cmo fence=\"true\" form=\"postfix\" stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cmo>=\u003C\u002Fmo>\u003Cmrow>\u003Cmo fence=\"true\" form=\"prefix\" stretchy=\"true\">{\u003C\u002Fmo>\u003Cmtable>\u003Cmtr>\u003Cmtd class=\"tml-left\" style=\"padding-left:0em;padding-right:0em;\">\u003Cmrow>\u003Cmi>∞\u003C\u002Fmi>\u003Cmspace width=\"1em\">\u003C\u002Fmspace>\u003Cmtext>si \u003C\u002Fmtext>\u003Cmi>x\u003C\u002Fmi>\u003Cmo>=\u003C\u002Fmo>\u003Cmn>0\u003C\u002Fmn>\u003C\u002Fmrow>\u003C\u002Fmtd>\u003C\u002Fmtr>\u003Cmtr>\u003Cmtd class=\"tml-left\" style=\"padding-left:0em;padding-right:0em;\">\u003Cmrow>\u003Cmn>0\u003C\u002Fmn>\u003Cmtext> \u003C\u002Fmtext>\u003Cmtext> \u003C\u002Fmtext>\u003Cmspace width=\"1em\">\u003C\u002Fmspace>\u003Cmtext>sinon\u003C\u002Fmtext>\u003C\u002Fmrow>\u003C\u002Fmtd>\u003C\u002Fmtr>\u003C\u002Fmtable>\u003Cmo fence=\"true\" form=\"postfix\" stretchy=\"true\">\u003C\u002Fmo>\u003C\u002Fmrow>\u003C\u002Fmrow>",[13,44,45],{},"It is the identity of the convolution operator.",[47,48,50],"h2",{"id":49},"link-with-image-processing","Link with image processing",[13,52,53],{},"In image processing, the convolution operator is a powerful tool for applying filters to images.",[13,55,56],{},"An example of the use of this filtering is the creation of what is called a Gaussian blur (or Gaussian smoothing).",[13,58,59],{},"Intuitively, when we convolve an image with a Gaussian filter (whose values follow a Gaussian function of center the center of the filter), we obtain a new image whose each pixel is worth a weighted average of the pixels of the original image.",[13,61,62],{},"This makes the image appear blurred. By changing the parameters of the Gaussian function used, the smoothing of the image is more or less accentuated.",[13,64,65],{},"Let's imagine that the filter follows the delta function of Dirac (\"normalized\"). In this case we only average a single pixel, that is to say that the blur becomes null and the result of the convolution is simply the original image.",[13,67,68],{},"This is a way to understand that the delta function is the identity of the convolution operator.",[47,70,72],{"id":71},"link-with-probability-theory","Link with probability theory",[13,74,75],{},[76,77,80,81,85,86,90,91,95],"span",{"className":78},[79],"inline-math","Let ",[23,82],{"dataCopyLatex":83,"dataCopyDisplay":6,"innerHTML":84},"X_n \\sim \\mathcal{N}(\\mu, {1 \\over n})","\u003Cmrow>\u003Cmsub>\u003Cmi>X\u003C\u002Fmi>\u003Cmi>n\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmo>∼\u003C\u002Fmo>\u003Cmi class=\"mathcal\">𝒩\u003C\u002Fmi>\u003Cmrow>\u003Cmo fence=\"true\" form=\"prefix\" stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>μ\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmfrac>\u003Cmn>1\u003C\u002Fmn>\u003Cmi>n\u003C\u002Fmi>\u003C\u002Fmfrac>\u003Cmo fence=\"true\" form=\"postfix\" stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003C\u002Fmrow>",". Denoting ",[23,87],{"dataCopyLatex":88,"dataCopyDisplay":6,"innerHTML":89},"f_n","\u003Cmsub>\u003Cmi>f\u003C\u002Fmi>\u003Cmi>n\u003C\u002Fmi>\u003C\u002Fmsub>"," the density function of ",[23,92],{"dataCopyLatex":93,"dataCopyDisplay":6,"innerHTML":94},"X_n","\u003Cmsub>\u003Cmi>X\u003C\u002Fmi>\u003Cmi>n\u003C\u002Fmi>\u003C\u002Fmsub>",".",[17,97,99],{"className":98},[20,21],[23,100],{"display":25,"className":101,"style":28,"dataCopyLatex":102,"dataCopyDisplay":30,"innerHTML":103},[27],"f_n(x) = {\\sqrt{n} \\over \\sqrt{2\\pi}}e^{-{n(x - \\mu)^2\\over 2}}","\u003Cmrow>\u003Cmsub>\u003Cmi>f\u003C\u002Fmi>\u003Cmi>n\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmrow>\u003Cmo fence=\"true\" form=\"prefix\" stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>x\u003C\u002Fmi>\u003Cmo fence=\"true\" form=\"postfix\" stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cmo>=\u003C\u002Fmo>\u003Cmfrac>\u003Cmsqrt>\u003Cmrow>\u003Cmi>n\u003C\u002Fmi>\u003Cmspace width=\"0pt\" height=\"0.5em\">\u003C\u002Fmspace>\u003C\u002Fmrow>\u003C\u002Fmsqrt>\u003Cmsqrt>\u003Cmrow>\u003Cmn>2\u003C\u002Fmn>\u003Cmi>π\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsqrt>\u003C\u002Fmfrac>\u003Cmsup>\u003Cmi>e\u003C\u002Fmi>\u003Cmrow class=\"tml-sml-pad\">\u003Cmo form=\"prefix\" stretchy=\"false\" lspace=\"0em\" rspace=\"0em\">−\u003C\u002Fmo>\u003Cmfrac>\u003Cmrow scriptlevel=\"2\" style=\"math-depth:2;\">\u003Cmi>n\u003C\u002Fmi>\u003Cmsup>\u003Cmrow>\u003Cmo fence=\"true\" form=\"prefix\" stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>x\u003C\u002Fmi>\u003Cmo>−\u003C\u002Fmo>\u003Cmi>μ\u003C\u002Fmi>\u003Cmo fence=\"true\" form=\"postfix\" stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cmn scriptlevel=\"2\">2\u003C\u002Fmn>\u003C\u002Fmsup>\u003C\u002Fmrow>\u003Cmn scriptlevel=\"2\" style=\"math-depth:2;\">2\u003C\u002Fmn>\u003C\u002Fmfrac>\u003C\u002Fmrow>\u003C\u002Fmsup>\u003C\u002Fmrow>",[13,105,106],{},"Then,",[17,108,110],{"className":109},[20,21],[23,111],{"display":25,"className":112,"style":28,"dataCopyLatex":113,"dataCopyDisplay":30,"innerHTML":114},[27],"f_n(x) \\overset{n \\to \\infty}{\\to} \\delta(x - \\mu)","\u003Cmrow>\u003Cmsub>\u003Cmi>f\u003C\u002Fmi>\u003Cmi>n\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmrow>\u003Cmo fence=\"true\" form=\"prefix\" stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>x\u003C\u002Fmi>\u003Cmo fence=\"true\" form=\"postfix\" stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cmrow>\u003Cmover>\u003Cmo>→\u003C\u002Fmo>\u003Cmrow>\u003Cmi>n\u003C\u002Fmi>\u003Cmo>→\u003C\u002Fmo>\u003Cmi>∞\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmover>\u003C\u002Fmrow>\u003Cmi>δ\u003C\u002Fmi>\u003Cmrow>\u003Cmo fence=\"true\" form=\"prefix\" stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>x\u003C\u002Fmi>\u003Cmo>−\u003C\u002Fmo>\u003Cmi>μ\u003C\u002Fmi>\u003Cmo fence=\"true\" form=\"postfix\" stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003C\u002Fmrow>",[13,116,117],{},"This function fulfills the properties of a probability density function.",[13,119,120,95],{},[76,121,123,124,128,129],{"className":122},[79],"Using the central limit theorem, let ",[23,125],{"dataCopyLatex":126,"dataCopyDisplay":6,"innerHTML":127},"X_1, X_2, ..., X_n \\overset{\\text{iid}}{\\sim} \\mathcal{N}(\\mu, 1)","\u003Cmrow>\u003Cmsub>\u003Cmi>X\u003C\u002Fmi>\u003Cmn>1\u003C\u002Fmn>\u003C\u002Fmsub>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmsub>\u003Cmi>X\u003C\u002Fmi>\u003Cmn>2\u003C\u002Fmn>\u003C\u002Fmsub>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi>.\u003C\u002Fmi>\u003Cmi>.\u003C\u002Fmi>\u003Cmi>.\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmsub>\u003Cmi>X\u003C\u002Fmi>\u003Cmi>n\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmrow>\u003Cmover>\u003Cmo>∼\u003C\u002Fmo>\u003Cmtext>iid\u003C\u002Fmtext>\u003C\u002Fmover>\u003C\u002Fmrow>\u003Cmi class=\"mathcal\">𝒩\u003C\u002Fmi>\u003Cmrow>\u003Cmo fence=\"true\" form=\"prefix\" stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>μ\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmn>1\u003C\u002Fmn>\u003Cmo fence=\"true\" form=\"postfix\" stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003C\u002Fmrow>",", and denoting their average by ",[23,130],{"dataCopyLatex":131,"dataCopyDisplay":6,"innerHTML":132},"\\overline{X}","\u003Cmenclose notation=\"top\" class=\"tml-overline\">\u003Cmi>X\u003C\u002Fmi>\u003C\u002Fmenclose>",[17,134,136],{"className":135},[20,21],[23,137],{"display":25,"className":138,"style":28,"dataCopyLatex":139,"dataCopyDisplay":30,"innerHTML":140},[27],"\\overline{X} \\sim \\mathcal{N}(\\mu,{1 \\over n}), \\ n \\to \\infty","\u003Cmrow>\u003Cmenclose notation=\"top\" class=\"tml-overline\">\u003Cmi>X\u003C\u002Fmi>\u003C\u002Fmenclose>\u003Cmo>∼\u003C\u002Fmo>\u003Cmi class=\"mathcal\">𝒩\u003C\u002Fmi>\u003Cmrow>\u003Cmo fence=\"true\" form=\"prefix\" stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>μ\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmfrac>\u003Cmn>1\u003C\u002Fmn>\u003Cmi>n\u003C\u002Fmi>\u003C\u002Fmfrac>\u003Cmo fence=\"true\" form=\"postfix\" stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmtext> \u003C\u002Fmtext>\u003Cmi>n\u003C\u002Fmi>\u003Cmo>→\u003C\u002Fmo>\u003Cmi>∞\u003C\u002Fmi>\u003C\u002Fmrow>",[13,142,143],{},[76,144,146,147,151,152,154,155,159,160,164],{"className":145},[79],"Now, let ",[23,148],{"dataCopyLatex":149,"dataCopyDisplay":6,"innerHTML":150},"Y","\u003Cmi>Y\u003C\u002Fmi>"," be a random variable with the same support as ",[23,153],{"dataCopyLatex":131,"dataCopyDisplay":6,"innerHTML":132},", and let ",[23,156],{"dataCopyLatex":157,"dataCopyDisplay":6,"innerHTML":158},"S = Y + \\overline{X}","\u003Cmrow>\u003Cmi>S\u003C\u002Fmi>\u003Cmo>=\u003C\u002Fmo>\u003Cmi>Y\u003C\u002Fmi>\u003Cmo>+\u003C\u002Fmo>\u003Cmenclose notation=\"top\" class=\"tml-overline\">\u003Cmi>X\u003C\u002Fmi>\u003C\u002Fmenclose>\u003C\u002Fmrow>",". We then have when ",[23,161],{"dataCopyLatex":162,"dataCopyDisplay":6,"innerHTML":163},"n","\u003Cmi>n\u003C\u002Fmi>"," goes to infinity:",[17,166,168],{"className":167},[20,21],[23,169],{"display":25,"className":170,"style":28,"dataCopyLatex":171,"dataCopyDisplay":30,"innerHTML":172},[27],"f_S(s) = (f_Y * f_{\\overline{X}})(s) = (f_Y * \\delta(\\cdot - \\mu))(s) = f_Y(s - \\mu)","\u003Cmrow>\u003Cmsub>\u003Cmi>f\u003C\u002Fmi>\u003Cmi>S\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmrow>\u003Cmo fence=\"true\" form=\"prefix\" stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>s\u003C\u002Fmi>\u003Cmo fence=\"true\" form=\"postfix\" stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cmo>=\u003C\u002Fmo>\u003Cmrow>\u003Cmo fence=\"true\" form=\"prefix\" stretchy=\"false\">(\u003C\u002Fmo>\u003Cmsub>\u003Cmi>f\u003C\u002Fmi>\u003Cmi>Y\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmo>∗\u003C\u002Fmo>\u003Cmsub>\u003Cmi>f\u003C\u002Fmi>\u003Cmenclose notation=\"top\" class=\"tml-overline\">\u003Cmi>X\u003C\u002Fmi>\u003C\u002Fmenclose>\u003C\u002Fmsub>\u003Cmo fence=\"true\" form=\"postfix\" stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cmrow>\u003Cmo fence=\"true\" form=\"prefix\" stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>s\u003C\u002Fmi>\u003Cmo fence=\"true\" form=\"postfix\" stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cmo>=\u003C\u002Fmo>\u003Cmrow>\u003Cmo fence=\"true\" form=\"prefix\" stretchy=\"false\">(\u003C\u002Fmo>\u003Cmsub>\u003Cmi>f\u003C\u002Fmi>\u003Cmi>Y\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmo>∗\u003C\u002Fmo>\u003Cmi>δ\u003C\u002Fmi>\u003Cmrow>\u003Cmo fence=\"true\" form=\"prefix\" stretchy=\"false\">(\u003C\u002Fmo>\u003Cmo form=\"prefix\" stretchy=\"false\">⋅\u003C\u002Fmo>\u003Cmo form=\"prefix\" stretchy=\"false\">−\u003C\u002Fmo>\u003Cmi>μ\u003C\u002Fmi>\u003Cmo fence=\"true\" form=\"postfix\" stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cmo fence=\"true\" form=\"postfix\" stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cmrow>\u003Cmo fence=\"true\" form=\"prefix\" stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>s\u003C\u002Fmi>\u003Cmo fence=\"true\" form=\"postfix\" stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cmo>=\u003C\u002Fmo>\u003Cmsub>\u003Cmi>f\u003C\u002Fmi>\u003Cmi>Y\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmrow>\u003Cmo fence=\"true\" form=\"prefix\" stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>s\u003C\u002Fmi>\u003Cmo>−\u003C\u002Fmo>\u003Cmi>μ\u003C\u002Fmi>\u003Cmo fence=\"true\" form=\"postfix\" stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003C\u002Fmrow>",{"title":174,"searchDepth":175,"depth":175,"links":176},"",5,[177,179],{"id":49,"depth":178,"text":50},2,{"id":71,"depth":178,"text":72},"\u002Fen \u002Farchive \u002Fdirac-delta","Some notes on the Dirac delta function","md","dirac-delta","en",{},"\u002Fen\u002Farchive\u002Fdirac-delta","2022-05-04",{"title":5,"description":181},"en\u002Farchive\u002Fdirac-delta","ZXF_Yaq_8WQYmf4AIvl2RUJsW4FCCQx56aSnJRlmA3k",1792158877473]