[{"data":1,"prerenderedAt":188},["ShallowReactive",2],{"content-notes\u002Fdirac-delta":3},{"id":4,"title":5,"ambience":6,"assets":7,"body":8,"breadcrumb":178,"chunks":7,"completed":6,"creation_date":7,"def":7,"description":179,"disclaimer":6,"extension":180,"featured":6,"hasMath":28,"image":7,"indexed":6,"key":181,"language":182,"meta":183,"metaTitle":7,"navigation":28,"path":184,"review":6,"seo":185,"stem":186,"subtitle":7,"tags":7,"tocEnabled":28,"__hash__":187},"content\u002Fen\u002Fnotes\u002Fdirac-delta.md","Dirac delta function",false,null,{"type":9,"value":10,"toc":171},"minimark",[11,15,30,33,41,44,49,52,55,58,61,64,67,71,94,102,105,113,116,131,139,163],[12,13,14],"p",{},"The Dirac delta function is a rather special function defined as follows.",[16,17,20],"div",{"className":18},[19],"math-display-scroll",[21,22],"math",{"display":23,"className":24,"style":26,"dataCopyLatex":27,"dataCopyDisplay":28,"innerHTML":29},"block",[25],"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>",[12,31,32],{},"Or, less formally,",[16,34,36],{"className":35},[19],[21,37],{"display":23,"className":38,"style":26,"dataCopyLatex":39,"dataCopyDisplay":28,"innerHTML":40},[25],"\\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>",[12,42,43],{},"It is the identity of the convolution operator.",[45,46,48],"h2",{"id":47},"link-with-image-processing","Link with image processing",[12,50,51],{},"In image processing, the convolution operator is a powerful tool for applying filters to images.",[12,53,54],{},"An example of the use of this filtering is the creation of what is called a Gaussian blur (or Gaussian smoothing).",[12,56,57],{},"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.",[12,59,60],{},"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.",[12,62,63],{},"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.",[12,65,66],{},"This is a way to understand that the delta function is the identity of the convolution operator.",[45,68,70],{"id":69},"link-with-probability-theory","Link with probability theory",[12,72,73],{},[74,75,78,79,83,84,88,89,93],"span",{"className":76},[77],"inline-math","Let ",[21,80],{"dataCopyLatex":81,"dataCopyDisplay":6,"innerHTML":82},"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 ",[21,85],{"dataCopyLatex":86,"dataCopyDisplay":6,"innerHTML":87},"f_n","\u003Cmsub>\u003Cmi>f\u003C\u002Fmi>\u003Cmi>n\u003C\u002Fmi>\u003C\u002Fmsub>"," the density function of ",[21,90],{"dataCopyLatex":91,"dataCopyDisplay":6,"innerHTML":92},"X_n","\u003Cmsub>\u003Cmi>X\u003C\u002Fmi>\u003Cmi>n\u003C\u002Fmi>\u003C\u002Fmsub>",".",[16,95,97],{"className":96},[19],[21,98],{"display":23,"className":99,"style":26,"dataCopyLatex":100,"dataCopyDisplay":28,"innerHTML":101},[25],"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>",[12,103,104],{},"Then,",[16,106,108],{"className":107},[19],[21,109],{"display":23,"className":110,"style":26,"dataCopyLatex":111,"dataCopyDisplay":28,"innerHTML":112},[25],"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>",[12,114,115],{},"This function fulfills the properties of a probability density function.",[12,117,118,93],{},[74,119,121,122,126,127],{"className":120},[77],"Using the central limit theorem, let ",[21,123],{"dataCopyLatex":124,"dataCopyDisplay":6,"innerHTML":125},"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 ",[21,128],{"dataCopyLatex":129,"dataCopyDisplay":6,"innerHTML":130},"\\overline{X}","\u003Cmenclose notation=\"top\" class=\"tml-overline\">\u003Cmi>X\u003C\u002Fmi>\u003C\u002Fmenclose>",[16,132,134],{"className":133},[19],[21,135],{"display":23,"className":136,"style":26,"dataCopyLatex":137,"dataCopyDisplay":28,"innerHTML":138},[25],"\\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>",[12,140,141],{},[74,142,144,145,149,150,152,153,157,158,162],{"className":143},[77],"Now, let ",[21,146],{"dataCopyLatex":147,"dataCopyDisplay":6,"innerHTML":148},"Y","\u003Cmi>Y\u003C\u002Fmi>"," be a random variable with the same support as ",[21,151],{"dataCopyLatex":129,"dataCopyDisplay":6,"innerHTML":130},", and let ",[21,154],{"dataCopyLatex":155,"dataCopyDisplay":6,"innerHTML":156},"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 ",[21,159],{"dataCopyLatex":160,"dataCopyDisplay":6,"innerHTML":161},"n","\u003Cmi>n\u003C\u002Fmi>"," goes to infinity:",[16,164,166],{"className":165},[19],[21,167],{"display":23,"className":168,"style":26,"dataCopyLatex":169,"dataCopyDisplay":28,"innerHTML":170},[25],"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":172,"searchDepth":173,"depth":173,"links":174},"",5,[175,177],{"id":47,"depth":176,"text":48},2,{"id":69,"depth":176,"text":70},"\u002Fen \u002Fnotes \u002Fdirac-delta","Some notes on the Dirac delta function","md","dirac-delta","en",{},"\u002Fen\u002Fnotes\u002Fdirac-delta",{"title":5,"description":179},"en\u002Fnotes\u002Fdirac-delta","qKJH6cGO94RPE1jNkMZyxw3NCeeiSpnBhiT-BGT8oVA",1786809143424]