SetDirectory[$HomeDirectory<>"/Documents/handouts/math467/FourierHigherDim"]

Higher-Dimensional Fourier Transform

f[x1_,x2_]:=Piecewise[{{1,-1<=x1<=1&&-1<=x2<=1},{0,True}}]

Plot3D[f[x1,x2],{x1,-3,3},{x2,-3,3},ExclusionsStyle->Opacity[1/2],Mesh->None,AxesLabel->{"\!\(\*TemplateBox[<|"boxes" -> FormBox[
SubscriptBox[
StyleBox["x", "TI"], "1"], TraditionalForm], "errors" -> {}, "input" -> "x_{1}", "state" -> "Boxes"|>,"TeXAssistantTemplate"]\)","\!\(\*TemplateBox[<|"boxes" -> FormBox[
SubscriptBox[
StyleBox["x", "TI"], "2"], TraditionalForm], "errors" -> {}, "input" -> "x_{2}", "state" -> "Boxes"|>,"TeXAssistantTemplate"]\)","\!\(\*TemplateBox[<|"boxes" -> FormBox[
RowBox[{
StyleBox["f", "TI"], "(", 
RowBox[{
SubscriptBox[
StyleBox["x", "TI"], "1"], ",", 
SubscriptBox[
StyleBox["x", "TI"], "2"]}], ")"}], TraditionalForm], "errors" -> {}, "input" -> "f(x_{1}, x_{2})", "state" -> "Boxes"|>,"TeXAssistantTemplate"]\)"}]

Export["unitstep.eps",%,"EPS"]

Export["unitstep.pdf",%%,"PDF"]

FourierTransform[f[x1,x2],{x1,x2},{ω1,ω2},FourierParameters->{0,-1}]

Plot3D[((2 Sin[ω1] Sin[ω2])/(π ω1 ω2)),{ω1,-3π,3π},{ω2,-3π,3π},PlotRange->All,Mesh->None,AxesLabel->{"\!\(\*TemplateBox[<|"boxes" -> FormBox[
SubscriptBox["ω", "1"], TraditionalForm], "errors" -> {}, "input" -> "\\omega_{1}", "state" -> "Boxes"|>,"TeXAssistantTemplate"]\)","\!\(\*TemplateBox[<|"boxes" -> FormBox[
SubscriptBox["ω", "2"], TraditionalForm], "errors" -> {}, "input" -> "\\omega_{2}", "state" -> "Boxes"|>,"TeXAssistantTemplate"]\)","\!\(\*TemplateBox[<|"boxes" -> FormBox[
RowBox[{
OverscriptBox[
StyleBox["f", "TI"], "^"], "(", 
RowBox[{
SubscriptBox["ω", "1"], ",", 
SubscriptBox["ω", "2"]}], ")"}], TraditionalForm], "errors" -> {}, "input" -> "\\hat{f}(\\omega_{1}, \\omega_{2})", "state" -> "Boxes"|>,"TeXAssistantTemplate"]\)"},Exclusions->None]

Export["unitsteptransform.eps",%,"EPS"]

Export["unitsteptransform.pdf",%%,"PDF"]