(* Content-type: application/vnd.wolfram.cdf.text *) (*** Wolfram CDF File ***) (* http://www.wolfram.com/cdf *) (* CreatedBy='Mathematica 8.0' *) (*************************************************************************) (* *) (* The Mathematica License under which this file was created prohibits *) (* restricting third parties in receipt of this file from republishing *) (* or redistributing it by any means, including but not limited to *) (* rights management or terms of use, without the express consent of *) (* Wolfram Research, Inc. *) (* *) (*************************************************************************) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 835, 17] NotebookDataLength[ 8884, 226] NotebookOptionsPosition[ 9303, 222] NotebookOutlinePosition[ 9661, 238] CellTagsIndexPosition[ 9618, 235] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[TextData[{ "Discrete-time complex sinusoids and complex exponentials", "\n", StyleBox["A continuous time complex sinusoid of the form ", FontSize->10], Cell[BoxData[ FormBox[ SuperscriptBox["\[ExponentialE]", "\[ImaginaryI]\[Omega]n"], TraditionalForm]], FontSize->10, FontWeight->"Bold"], ".", StyleBox[" Vary \[Omega] with the slider. What is the period of the \ function when \[Omega] = 1? 2? \[Pi]? 2\[Pi]?", FontSize->10], "\n", Cell[BoxData[ TagBox[ StyleBox[ DynamicModuleBox[{$CellContext`sw$$ = 0, $CellContext`\[Omega]$$ = 6.994952392758524, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{{ Hold[$CellContext`sw$$], 0, "Plot"}, { 0 -> "Real part", 1 -> "Imaginary part", 2 -> "Both"}}, { Hold[$CellContext`\[Omega]$$], 0, 10, Rational[1, 128] Pi}}, Typeset`size$$ = {360., {113., 118.}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = True, $CellContext`sw$20159$$ = False, $CellContext`\[Omega]$20160$$ = 0}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> {$CellContext`sw$$ = 0, $CellContext`\[Omega]$$ = 0}, "ControllerVariables" :> { Hold[$CellContext`sw$$, $CellContext`sw$20159$$, False], Hold[$CellContext`\[Omega]$$, $CellContext`\[Omega]$20160$$, 0]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> Switch[$CellContext`sw$$, 0, ListPlot[{ Re[ Exp[(I $CellContext`\[Omega]$$) (Range[32] - 1)]]}, Filling -> None, PlotStyle -> {Red}, PlotMarkers -> {Automatic, 10}, PlotLabel -> Row[{ "Real component of \!\(\*SuperscriptBox[\(\[ExponentialE]\), \(\ \[ImaginaryI]\\ \[Omega]\\ n\)]\), \[Omega] = ", Round[$CellContext`\[Omega]$$, 0.001], " radians/sample"}]], 1, ListPlot[{ Im[ Exp[(I $CellContext`\[Omega]$$) (Range[32] - 1)]]}, Filling -> None, PlotStyle -> {Blue}, PlotMarkers -> {Automatic, 10}, PlotLabel -> Row[{"Imaginary component of \!\(\*SuperscriptBox[\(\[ExponentialE]\ \), \(\[ImaginaryI]\\ \[Omega]\\ n\)]\), \[Omega] = ", Round[$CellContext`\[Omega]$$, 0.001], " radians/sample"}]], 2, ListPlot[{ Re[ Exp[(I $CellContext`\[Omega]$$) (Range[32] - 1)]], Im[ Exp[(I $CellContext`\[Omega]$$) (Range[32] - 1)]]}, Filling -> None, PlotStyle -> {Red, Blue}, PlotMarkers -> {Automatic, 10}, PlotLabel -> Row[{"Real and imaginary components of \!\(\*SuperscriptBox[\(\ \[ExponentialE]\), \(\[ImaginaryI]\\ \[Omega]\\ n\)]\), \[Omega] = ", Round[$CellContext`\[Omega]$$, 0.001], " radians/sample"}]]], "Specifications" :> {{{$CellContext`sw$$, 0, "Plot"}, { 0 -> "Real part", 1 -> "Imaginary part", 2 -> "Both"}}, {$CellContext`\[Omega]$$, 0, 10, Rational[1, 128] Pi}}, "Options" :> {}, "DefaultOptions" :> {}], ImageSizeCache->{407., {171., 179.}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, SynchronousInitialization->True, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], "Input", CellChangeTimes->{{3.5239661476127977`*^9, 3.5239661504676933`*^9}}], "\n", StyleBox["A continuous time complex exponential of the form ", FontSize->10], Cell[BoxData[ FormBox[ SuperscriptBox["z", "n"], TraditionalForm]], FontSize->10, FontWeight->"Bold"], StyleBox[". Vary ", FontSize->10], Cell[BoxData[ FormBox["z", TraditionalForm]], FontSize->10, FontWeight->"Bold"], StyleBox[" by using the crosshair locator on the left part of the plot. The \ right side shows the resulting function. What happens when you vary the ", FontSize->10], Cell[BoxData[ FormBox[ RowBox[{"Re", "[", "z", "]"}], TraditionalForm]], FontSize->10, FontWeight->"Bold"], StyleBox[" (horizontal) component? The ", FontSize->10], Cell[BoxData[ FormBox[ RowBox[{"Im", "[", "z", "]"}], TraditionalForm]], FontSize->10, FontWeight->"Bold"], StyleBox[" (vertical) component? What happens when the locator is outside \ the ", FontSize->10], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"|", "z", "|"}], "=", "1"}], TraditionalForm]], FontSize->10], StyleBox[" disk? Experiment with what happens when you flip the locator \ across each axis.", FontSize->10], "\n", Cell[BoxData[ TagBox[ StyleBox[ DynamicModuleBox[{$CellContext`z$$ = {-0.5270000000000001, 0.97}, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{{ Hold[$CellContext`z$$], {0.5, 0.5}}, Automatic}}, Typeset`size$$ = { 510., {112., 116.}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = True}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> {$CellContext`z$$ = {0.5, 0.5}}, "ControllerVariables" :> {}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> Row[{ Framed[ Graphics[{Gray, Disk[], Black, Line[{{-1.2, 0}, {1.2, 0}}], Black, Line[{{0, -1.2}, {0, 1.2}}], Black, Text["Re[z]", {1.2, 0.1}], Black, Text["Im[z]", {0.25, 1.1}], Black, Thick, Line[{{1, -0.1}, {1, 0.1}}], Thick, Line[{{-1, -0.1}, {-1, 0.1}}], Thick, Line[{{-0.1, 1}, {0.1, 1}}], Thick, Line[{{-0.1, -1}, {0.1, -1}}], Text[ Style[1, Bold], {1.1, -0.2}], Text[ Style[-1, Bold], {-1.1, -0.2}], Text[ Style[-1, Bold], {-0.1, -1.2}], Text[ Style[1, Bold], {-0.1, 1.2}]}, PlotRange -> 1.4, ImageSize -> 150]], ListPlot[{ Re[(Part[$CellContext`z$$, 1] + I Part[$CellContext`z$$, 2])^( Range[32] - 1)], Im[(Part[$CellContext`z$$, 1] + I Part[$CellContext`z$$, 2])^( Range[32] - 1)]}, Filling -> Axis, PlotStyle -> {Red, Blue}, PlotMarkers -> {Automatic, 10}, PlotLabel -> Row[{"Real (red) and imaginary (blue) components of \ \!\(\*SuperscriptBox[\(z\), \(n\)]\)"}], PlotRange -> {{0, 32}, {-10, 10}}, ImageSize -> 350]}], "Specifications" :> {{{$CellContext`z$$, {0.5, 0.5}}, Automatic, ControlType -> Locator}}, "Options" :> {}, "DefaultOptions" :> {}], ImageSizeCache->{563., {143., 150.}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, SynchronousInitialization->True, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], "Output", CellChangeTimes->{3.50792007821875*^9}] }], "Title", CellChangeTimes->{{3.5068806080625*^9, 3.506880677078125*^9}, { 3.506880707703125*^9, 3.50688071540625*^9}, 3.523966268236225*^9, { 3.52396812774599*^9, 3.5239681288785353`*^9}, 3.5239681932062073`*^9}, FontSize->16] }, WindowSize->{740, 752}, WindowMargins->{{Automatic, 232}, {Automatic, 50}}, FrontEndVersion->"8.0 for Mac OS X x86 (32-bit, 64-bit Kernel) (February 23, \ 2011)", StyleDefinitions->"Default.nb" ] (* End of Notebook Content *) (* Internal cache information *) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[1235, 30, 8064, 190, 802, "Title"] } ] *) (* End of internal cache information *) (* NotebookSignature EuD3RkLJdtG6JCw8qwQoYjmt *)