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3.480874699751749*^9}], Cell[CellGroupData[{ Cell[TextData[{ "Se muestrea la se\[NTilde]al ", Cell[BoxData[ FormBox[ RowBox[{"x", "(", "t", ")"}], TraditionalForm]]], " para obtener ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SuperscriptBox["x", "*"], "(", "n", ")"}], "=", FormBox[ RowBox[{"x", "(", RowBox[{"n", " ", "T"}], ")"}], TraditionalForm]}], TraditionalForm]]], ", donde ", Cell[BoxData[ FormBox["T", TraditionalForm]], FormatType->"TraditionalForm"], " es el intervalo de muestreo. ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["f", "s"], "=", FractionBox["1", "T"]}], TraditionalForm]], FormatType->"TraditionalForm"], " es la frecuencia de muestreo en ", Cell[BoxData[ FormBox["Hz", TraditionalForm]], FormatType->"TraditionalForm"], " y ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["\[Omega]", "S"], "=", RowBox[{"2", " ", "\[Pi]", " ", SubscriptBox["f", "s"]}]}], TraditionalForm]], FormatType->"TraditionalForm"], " es la frecuencia angular en ", Cell[BoxData[ FormBox[ FractionBox["rad", "s"], TraditionalForm]], FormatType->"TraditionalForm"], "." }], "Text", CellGroupingRules->{GroupTogetherGrouping, 10000.}, CellChangeTimes->{{3.480872754819388*^9, 3.4808728518939404`*^9}, { 3.4808733800981536`*^9, 3.4808733984052005`*^9}, {3.4808734699922953`*^9, 3.480873490870489*^9}, {3.480881702821387*^9, 3.480881759771118*^9}}], Cell[BoxData[ TagBox[ StyleBox[ DynamicModuleBox[{$CellContext`T$$ = 0.498, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{{ Hold[$CellContext`T$$], 1}, 0.1, 2}, { Hold[ Dynamic[ Style[ StringJoin[ "\!\(\*SubscriptBox[\(f\), \(s\)]\)=\!\(\*FractionBox[\(1\), \ \(T\)]\)=", ToString[ NumberForm[1/$CellContext`T$$, {4, 2}]]]]]], Manipulate`Dump`ThisIsNotAControl}}, Typeset`size$$ = { 442., {83., 88.}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = False, $CellContext`T$601$$ = 0}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> {$CellContext`T$$ = 1}, "ControllerVariables" :> { Hold[$CellContext`T$$, $CellContext`T$601$$, 0]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> Grid[{{ Plot[ $CellContext`xCont[$CellContext`t], {$CellContext`t, -2, 2}, Filling -> Axis, PlotLabel -> "x(t)", Epilog -> { PointSize[Large], Table[ Point[{$CellContext`i, $CellContext`xCont[$CellContext`i]}], {$CellContext`i, Round[-2, $CellContext`T$$], 2, $CellContext`T$$}]}], DiscretePlot[ $CellContext`xCont[$CellContext`T$$ $CellContext`n], \ {$CellContext`n, Ceiling[(-2)/$CellContext`T$$], Floor[2/$CellContext`T$$]}, PlotRange -> {0, Full}, Filling -> Axis, PlotStyle -> PointSize[Large], PlotLabel -> "\!\(\*SuperscriptBox[\(x\), \(*\)]\)(n)"]}}], "Specifications" :> {{{$CellContext`T$$, 1}, 0.1, 2, Exclusions -> {0.}, Appearance -> "Labeled"}, Dynamic[ Style[ StringJoin[ "\!\(\*SubscriptBox[\(f\), \(s\)]\)=\!\(\*FractionBox[\(1\), \(T\)]\ \)=", ToString[ NumberForm[1/$CellContext`T$$, {4, 2}]]]]]}, "Options" :> {ControlPlacement -> Top}, "DefaultOptions" :> {}], ImageSizeCache->{500., {155., 160.}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, Initialization:>({$CellContext`xCont[ Pattern[$CellContext`t, Blank[]]] := Sinc[Pi $CellContext`t]^2, Attributes[PlotRange] = {ReadProtected}}; Typeset`initDone$$ = True), SynchronousInitialization->True, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], "Input", CellGroupingRules->{GroupTogetherGrouping, 10000.}, CellChangeTimes->{3.4808733800991535`*^9, 3.4808741113479805`*^9, 3.480881809233899*^9}] }, Closed]], Cell[TextData[{ "El teorema de Nyquist establece que la frecuencia de muestreo ", Cell[BoxData[ FormBox[ SubscriptBox["f", "s"], TraditionalForm]]], " debe ser ", StyleBox["mayor", FontWeight->"Bold"], " al ", StyleBox["doble", FontWeight->"Bold"], " de la frecuencia m\[AAcute]xima ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["f", "m\[AAcute]x"], "=", FractionBox[ SubscriptBox["\[Omega]", "m\[AAcute]x"], RowBox[{"2", "\[Pi]"}]]}], TraditionalForm]]], " para que la se\[NTilde]al continua pueda ser reconstruida posteriormente. \ Es decir, si se muestrea a m\[AAcute]s del doble de la frecuencia \ m\[AAcute]xima de la se\[NTilde]al, se tiene en la se\[NTilde]al discreta \ toda la informaci\[OAcute]n que conten\[IAcute]a la cont\[IAcute]nua y por lo \ tanto es posible volver a la se\[NTilde]al continua partiendo de la discreta." }], "Text", CellChangeTimes->{{3.48087341029088*^9, 3.4808734485610695`*^9}, { 3.480874120883526*^9, 3.4808741585556808`*^9}, {3.480874239580385*^9, 3.4808743610183744`*^9}, {3.480874725794238*^9, 3.4808747359658203`*^9}, { 3.48178816254211*^9, 3.481788173278724*^9}}] }, Closed]], Cell[CellGroupData[{ Cell["Aliasing", "Section", CellChangeTimes->{{3.4808744754559193`*^9, 3.480874476283967*^9}}], Cell["\<\ Se llama aliasing al fen\[OAcute]meno que ocurre cuando no se respeta el \ teorema de Nyquist y grupos de distintas frecuencias que conten\[IAcute]a la \ se\[NTilde]al continua se hacen inseparables e indistinguibles en la se\ \[NTilde]al muestreada. Es decir, algunas frecuencias aparentan ser otras.\ \>", "Text", CellChangeTimes->{{3.4808744780110655`*^9, 3.4808745930906477`*^9}, 3.480876096775674*^9, {3.481787415103359*^9, 3.481787415162362*^9}}], Cell[CellGroupData[{ Cell["\<\ Siguiendo con el ejemplo anterior, vemos c\[OAcute]mo la transformada de \ Fourier empieza a perder su forma cuando la frecuencia de muestreo es menor a \ la de Nyquist:\ \>", "Text", CellChangeTimes->{{3.480876401384096*^9, 3.4808764054713306`*^9}, { 3.4808773412208605`*^9, 3.4808773723376403`*^9}}], Cell[BoxData[ TagBox[ StyleBox[ DynamicModuleBox[{$CellContext`T$$ = 0.5, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{{ Hold[$CellContext`T$$], 0.5}, 0.25, 1.25}, { Hold[ Dynamic[ Style[ StringJoin[ "\!\(\*SubscriptBox[\(f\), \(s\)]\)=\!\(\*FractionBox[\(1\), \ \(T\)]\)=", ToString[ NumberForm[1/$CellContext`T$$, {4, 2}]]]]]], Manipulate`Dump`ThisIsNotAControl}}, Typeset`size$$ = { 442., {90.5, 95.5}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = False, $CellContext`T$279437$$ = 0}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> {$CellContext`T$$ = 0.5}, "ControllerVariables" :> { Hold[$CellContext`T$$, $CellContext`T$279437$$, 0]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> Grid[{{ Show[ Plot[ $CellContext`xCont[$CellContext`t], {$CellContext`t, -2, 2}, Filling -> Axis, PlotLabel -> "x(t)"], DiscretePlot[ $CellContext`xCont[$CellContext`t], {$CellContext`t, Round[-2, $CellContext`T$$], 2, $CellContext`T$$}, Filling -> Axis, FillingStyle -> Black, PlotStyle -> Directive[Black, PointSize[Large]]], Ticks -> {{{$CellContext`T$$, "T=\!\(\*FractionBox[\(2 \[Pi]\), SubscriptBox[\(\[Omega]\), \ \(s\)]]\)"}}, Automatic}], Plot[ Evaluate[ Join[{ Abs[ $CellContext`FxPer[$CellContext`t, 2 (Pi/$CellContext`T$$)]]}, { $CellContext`FxList[$CellContext`t, 2 (Pi/$CellContext`T$$)]}]], {$CellContext`t, -20, 20}, PlotStyle -> Join[{ Directive[Red]}, ConstantArray[ Directive[Blue, Dotted], 21]], Filling -> {1 -> Axis}, FillingStyle -> Directive[Red, Opacity[0.3]], PlotLabel -> "\[ScriptCapitalF]{x}(\[Omega])", PlotRange -> {0, 1.5}]}}], "Specifications" :> {{{$CellContext`T$$, 0.5}, 0.25, 1.25, Exclusions -> {0.}, Appearance -> "Labeled"}, Dynamic[ Style[ StringJoin[ "\!\(\*SubscriptBox[\(f\), \(s\)]\)=\!\(\*FractionBox[\(1\), \(T\)]\ \)=", ToString[ NumberForm[1/$CellContext`T$$, {4, 2}]]]]]}, "Options" :> {}, "DefaultOptions" :> {}], ImageSizeCache->{500., {163., 168.}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, Initialization:>({$CellContext`xCont[ Pattern[$CellContext`t, Blank[]]] := Sinc[Pi $CellContext`t]^2, $CellContext`FxPer[ Pattern[$CellContext`t, Blank[]], Pattern[$CellContext`per, Blank[]]] := Sum[ UnitTriangle[($CellContext`t - $CellContext`k $CellContext`per)/(2 Pi)], {$CellContext`k, -5, 5}], $CellContext`FxList[$CellContext`t, Pattern[$CellContext`per, Blank[]]] := Table[ $CellContext`FxCont[$CellContext`t - $CellContext`k \ $CellContext`per], {$CellContext`k, -10, 10}], $CellContext`FxList[ Pattern[$CellContext`t, Blank[]], Pattern[$CellContext`per, Blank[]]] := Table[ UnitTriangle[($CellContext`t - $CellContext`k $CellContext`per)/(2 Pi)], {$CellContext`k, -5, 5}], $CellContext`FxCont[ Pattern[$CellContext`\[Omega], Blank[]]] = ((Pi^2 Sign[2 Pi - $CellContext`\[Omega]])/ 2 - ((Pi $CellContext`\[Omega]) Sign[2 Pi - $CellContext`\[Omega]])/ 4 - ((Pi $CellContext`\[Omega]) Sign[$CellContext`\[Omega]])/ 2 + (Pi^2 Sign[2 Pi + $CellContext`\[Omega]])/ 2 + ((Pi $CellContext`\[Omega]) Sign[2 Pi + $CellContext`\[Omega]])/ 4)/Pi^2, Attributes[PlotRange] = {ReadProtected}}; Typeset`initDone$$ = True), SynchronousInitialization->True, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], "Input", CellChangeTimes->{3.4808822960552177`*^9, 3.481783964642004*^9}] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ "En el siguiente ejemplo se puede ver que cuando la frecuencia de muestreo \ es menor al doble de la frecuencia de una se\[NTilde]al senoidal, se puede \ encontrar otra se\[NTilde]al senoidal cont\[IAcute]nua que pasa por esos \ mismos puntos y de frecuencia menor (l\[IAcute]nea punteada). Esa frecuencia \ mayor entonces es un ", StyleBox["alias", FontSlant->"Italic"], " de la menor." }], "Text", CellChangeTimes->{{3.480876101545947*^9, 3.4808761964253736`*^9}, { 3.480877422801527*^9, 3.480877456154436*^9}, {3.481784017830046*^9, 3.4817840223463044`*^9}}], Cell[BoxData[ TagBox[ StyleBox[ DynamicModuleBox[{$CellContext`frecMuestreo$$ = 2, $CellContext`frecSin$$ = 1, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{{ Hold[$CellContext`frecSin$$], 1, "\!\(\*SubscriptBox[\(f\), \(m\[AAcute]x\)]\)"}, 0, 4}, {{ Hold[$CellContext`frecMuestreo$$], 1.5, "\!\(\*SubscriptBox[\(f\), \(s\)]\)"}, 0.1, 4}}, Typeset`size$$ = { 691., {210., 218.}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = False, $CellContext`frecSin$157619$$ = 0, $CellContext`frecMuestreo$157620$$ = 0}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> {$CellContext`frecMuestreo$$ = 1.5, $CellContext`frecSin$$ = 1}, "ControllerVariables" :> { Hold[$CellContext`frecSin$$, $CellContext`frecSin$157619$$, 0], Hold[$CellContext`frecMuestreo$$, $CellContext`frecMuestreo$157620$$, 0]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> Show[ Plot[{ Cos[((2 Pi) $CellContext`t) $CellContext`frecSin$$], Cos[((2 Pi) $CellContext`t) Mod[$CellContext`frecSin$$, $CellContext`frecMuestreo$$, \ (-$CellContext`frecMuestreo$$)/2]]}, {$CellContext`t, -2, 2}, PlotRange -> {-1.1, 1.1}, PlotStyle -> {Automatic, Dashed}, ImageSize -> Large], DiscretePlot[ Cos[((2 Pi) $CellContext`t) $CellContext`frecSin$$], {$CellContext`t, Ceiling[-2, 1/$CellContext`frecMuestreo$$], 2, 1/$CellContext`frecMuestreo$$}, PlotStyle -> Directive[Black, PointSize[Large]]], Ticks -> { Table[{AddOns`i/$CellContext`frecMuestreo$$, StringJoin[ ToString[AddOns`i], "\!\(\*SubscriptBox[\(T\), \(s\)]\)"]}, { AddOns`i, Ceiling[(-2) $CellContext`frecMuestreo$$], Floor[2 $CellContext`frecMuestreo$$]}]}, PlotRangePadding -> Scaled[0.03]], "Specifications" :> {{{$CellContext`frecSin$$, 1, "\!\(\*SubscriptBox[\(f\), \(m\[AAcute]x\)]\)"}, 0, 4, Appearance -> "Labeled"}, {{$CellContext`frecMuestreo$$, 1.5, "\!\(\*SubscriptBox[\(f\), \(s\)]\)"}, 0.1, 4, Appearance -> "Labeled"}}, "Options" :> {}, "DefaultOptions" :> {}], ImageSizeCache->{749., {284., 289.}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, Initialization:>({Attributes[PlotRange] = {ReadProtected}}; Typeset`initDone$$ = True), SynchronousInitialization->True, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], "Input", CellGroupingRules->{GroupTogetherGrouping, 10000.}, CellChangeTimes->{{3.480876175645185*^9, 3.4808762012416487`*^9}, 3.4817844243472977`*^9, 3.481784876809177*^9, 3.4817878835901546`*^9}] }, Closed]], Cell[CellGroupData[{ Cell["\<\ En el siguiente ejemplo tambi\[EAcute]n se puede apreciar lo anterior. Adem\ \[AAcute]s se ve c\[OAcute]mo var\[IAcute]a la transformada de Fourier a \ medida que cambia la frecuencia de la senoide.\ \>", "Text", CellChangeTimes->{{3.4808762594139767`*^9, 3.4808763467809734`*^9}, { 3.48087741324098*^9, 3.480877414404046*^9}, {3.4817848893308935`*^9, 3.481784901746603*^9}, {3.481787914720936*^9, 3.4817879181371307`*^9}}], Cell[TextData[Cell[BoxData[ TagBox[ StyleBox[ DynamicModuleBox[{$CellContext`f$$ = 1000, $CellContext`fs$$ = 16000, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{ Hold[ Style["Frecuencia de la senoide", Bold]], Manipulate`Dump`ThisIsNotAControl}, {{ Hold[$CellContext`f$$], 1000, "f [Hz]"}, 1000, 10000, 1}, { Hold[ Style["Frecuencia de muestreo", Bold]], Manipulate`Dump`ThisIsNotAControl}, {{ Hold[$CellContext`fs$$], 16000, "\!\(\*SubscriptBox[\(f\), \(s\)]\)"}, { 8000 -> "8 kHz", 16000 -> "16 kHz", 48000 -> "48 kHz"}}}, Typeset`size$$ = {455., {227., 235.}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = True, $CellContext`f$1964$$ = 0, $CellContext`fs$1965$$ = False}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> {$CellContext`f$$ = 1000, $CellContext`fs$$ = 16000}, "ControllerVariables" :> { Hold[$CellContext`f$$, $CellContext`f$1964$$, 0], Hold[$CellContext`fs$$, $CellContext`fs$1965$$, False]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> GraphicsColumn[{ Show[ Plot[ Sin[((2 Pi) $CellContext`f$$) $CellContext`t], {$CellContext`t, 0, 0.001}, PlotRange -> {-1.1, 1.1}, PlotStyle -> {{ RGBColor[0.25, 0.43, 0.82]}}], ListPlot[ Table[{$CellContext`t, Sin[((2 Pi) $CellContext`f$$) $CellContext`t]}, {$CellContext`t, 0, 0.001, 1/$CellContext`fs$$}], PlotMarkers -> {Automatic}, Filling -> Axis, FillingStyle -> RGBColor[0.25, 0.43, 0.82]], If[$CellContext`f$$ > $CellContext`fs$$/2, Plot[ Sin[((2 Pi) ($CellContext`f$$ - $CellContext`fs$$)) \ $CellContext`t], {$CellContext`t, 0, 0.001}, PlotRange -> {-1.1, 1.1}, PlotStyle -> { If[$CellContext`f$$ > $CellContext`fs$$, RGBColor[0.6, 0.73, 0.36], RGBColor[0.9, 0.42, 0.17]], Dashing[{0.01, 0.01}]}], Graphics[]], Frame -> True, FrameLabel -> {{None, None}, { TraditionalForm[ Text[ $CellContext`t[$CellContext`s]]], Row[{$CellContext`sin, "(", 2 Pi, Style["f ", Italic], $CellContext`t, ")"}]}}, FrameTicks -> {Automatic, None}, AspectRatio -> 0.35, ImageSize -> {350, 175}, ImagePadding -> {{20, 20}, {35, 35}}], Graphics[{{ RGBColor[0.25, 0.43, 0.82], Arrowheads[Large], Arrow[{{$CellContext`f$$, 0}, {$CellContext`f$$, 1}}], Arrow[{{-$CellContext`f$$, 0}, {-$CellContext`f$$, 1}}]}, { RGBColor[0.9, 0.42, 0.17], Arrowheads[Large], Arrow[{{$CellContext`f$$ + $CellContext`fs$$, 0}, {$CellContext`f$$ + $CellContext`fs$$, 1}}], Arrow[{{-$CellContext`f$$ + $CellContext`fs$$, 0}, {-$CellContext`f$$ + $CellContext`fs$$, 1}}]}, { RGBColor[0.6, 0.73, 0.36], Arrowheads[Large], Arrow[{{$CellContext`f$$ - $CellContext`fs$$, 0}, {$CellContext`f$$ - $CellContext`fs$$, 1}}], Arrow[{{-$CellContext`f$$ - $CellContext`fs$$, 0}, {-$CellContext`f$$ - $CellContext`fs$$, 1}}]}, { RGBColor[0.49, 0, 0], Arrowheads[Large], Arrow[{{$CellContext`f$$ + 2 $CellContext`fs$$, 0}, {$CellContext`f$$ + 2 $CellContext`fs$$, 1}}], Arrow[{{-$CellContext`f$$ + 2 $CellContext`fs$$, 0}, {-$CellContext`f$$ + 2 $CellContext`fs$$, 1}}]}, {Gray, Arrowheads[Large], Arrow[{{$CellContext`f$$ - 2 $CellContext`fs$$, 0}, {$CellContext`f$$ - 2 $CellContext`fs$$, 1}}], Arrow[{{-$CellContext`f$$ - 2 $CellContext`fs$$, 0}, {-$CellContext`f$$ - 2 $CellContext`fs$$, 1}}]}, {Dashed, Line[{{-$CellContext`fs$$, 0}, {-$CellContext`fs$$, 1}}], Line[{{$CellContext`fs$$, 0}, {$CellContext`fs$$, 1}}]}, Text[ "\!\(\*SubscriptBox[\(f\), \(s\)]\)", {$CellContext`fs$$, 1.1}], Text[ "-\!\(\*SubscriptBox[\(f\), \(s\)]\)", {-$CellContext`fs$$, 1.1}]}, Frame -> True, Axes -> True, PlotRange -> {{-$CellContext`fs$$ - 2000, $CellContext`fs$$ + 2000}, {0, 1.2}}, FrameLabel -> {{None, None}, { Row[{ Style["f", Italic], " (Hz)"}], Row[{ Style["S", Italic], "(", Style["f", Italic], ")"}]}}, FrameTicks -> {Automatic, None}, AspectRatio -> 0.35, PlotRangeClipping -> True, ImageSize -> {350, 175}, ImagePadding -> {{20, 20}, {35, 35}}]}], "Specifications" :> { Style[ "Frecuencia de la senoide", Bold], {{$CellContext`f$$, 1000, "f [Hz]"}, 1000, 10000, 1, Appearance -> "Labeled", ImageSize -> Small, ControlType -> Slider}, Delimiter, Style[ "Frecuencia de muestreo", Bold], {{$CellContext`fs$$, 16000, "\!\(\*SubscriptBox[\(f\), \(s\)]\)"}, { 8000 -> "8 kHz", 16000 -> "16 kHz", 48000 -> "48 kHz"}}}, "Options" :> {ControlPlacement -> Left}, "DefaultOptions" :> {}], ImageSizeCache->{807., {265., 270.}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, SynchronousInitialization->True, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], CellChangeTimes->{{3.4817856009445953`*^9, 3.4817856163984795`*^9}}]], "Text", CellChangeTimes->{3.480876348413067*^9, 3.4817856276501226`*^9}] }, Closed]], Cell[CellGroupData[{ Cell["\<\ En el ejemplo siguiente puede ver en l\[IAcute]nea punteada una senoide que a \ la frecuencia de muestreo elegida es siempre muestreada en sus \ ra\[IAcute]ces. Por lo tanto, se obtendr\[AAcute] la misma se\[NTilde]al \ discreta a partir de cualquier se\[NTilde]al si se le suma, o no, dicha \ senoide. En azul, la senoide de frecuencia elegida. En verde, la suma de la azul y la \ punteada.. Observe c\[OAcute]mo, si no se exigen restricciones al ancho de \ banda de la se\[NTilde]al continua, no se puede saber, a partir de la se\ \[NTilde]al discreta, qu\[EAcute] se\[NTilde]al continua representa.\ \>", "Text", CellChangeTimes->{{3.481004050535307*^9, 3.481004338579782*^9}, { 3.4817851065633183`*^9, 3.481785114254758*^9}, {3.4817851631975574`*^9, 3.481785272527811*^9}, {3.481787941106445*^9, 3.481787948959894*^9}, { 3.481788334037919*^9, 3.481788360676443*^9}}], Cell[BoxData[ TagBox[ StyleBox[ DynamicModuleBox[{$CellContext`frecMuestreo$$ = 1.5, $CellContext`frecSeno$$ = 0.68, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{{ Hold[$CellContext`frecSeno$$], 1}, 0, 4}, {{ Hold[$CellContext`frecMuestreo$$], 1}, 0, 4}}, Typeset`size$$ = { 691., {210., 218.}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = False, $CellContext`frecSeno$1132770$$ = 0, $CellContext`frecMuestreo$1132775$$ = 0}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> {$CellContext`frecMuestreo$$ = 1, $CellContext`frecSeno$$ = 1}, "ControllerVariables" :> { Hold[$CellContext`frecSeno$$, $CellContext`frecSeno$1132770$$, 0], Hold[$CellContext`frecMuestreo$$, $CellContext`frecMuestreo$1132775$$, 0]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> Plot[{ Sin[((2 Pi) $CellContext`frecSeno$$) $CellContext`t], Sin[(((2 Pi) 2) $CellContext`frecMuestreo$$) $CellContext`t], Sin[((2 Pi) $CellContext`frecSeno$$) $CellContext`t] + Sin[(((2 Pi) 2) $CellContext`frecMuestreo$$) $CellContext`t]}, \ {$CellContext`t, -2, 2}, PlotRange -> {-2.1, 2.1}, PlotStyle -> {Automatic, Dashed, Automatic}, Epilog -> { PointSize[Large], Point[ Table[{$CellContext`tt, Sin[((2 Pi) $CellContext`tt) $CellContext`frecSeno$$]}, \ {$CellContext`tt, Round[-2, 1/$CellContext`frecMuestreo$$], 2, 1/$CellContext`frecMuestreo$$}]]}, ImageSize -> Large], "Specifications" :> {{{$CellContext`frecSeno$$, 1}, 0, 4, Appearance -> "Labeled"}, {{$CellContext`frecMuestreo$$, 1}, 0, 4, Appearance -> "Labeled"}}, "Options" :> {}, "DefaultOptions" :> {}], ImageSizeCache->{749., {284., 289.}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, Initialization:>({Attributes[PlotRange] = {ReadProtected}}; Typeset`initDone$$ = True), SynchronousInitialization->True, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], "Input"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ "En este puede escuchar el efecto del ", StyleBox["aliasing. 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