(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 123209, 4138]*) (*NotebookOutlinePosition[ 124263, 4174]*) (* CellTagsIndexPosition[ 124219, 4170]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{Cell[TextData[ "Filter Design by Pole-Zero Placement"], "Title", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["by"], "Subsubtitle", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "H. Joel Trussell\nDept. of Electrical and Computer Engineering\nNorth \ Carolina State University\nRaleigh, NC 27695-7911"], "Subtitle", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["e-mail", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Plain"], StyleBox[": hjt@ecehjt.ncsu.edu", Evaluatable->False, AspectRatioFixed->True] }], "Subsubtitle", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["Initialization"], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "zeros = {}; poles = {};\n\nplotRoots[ roots_, text_ ] :=\n\tGraphics[ Map[ \ Text[text, #]&, roots ] ]\n\nunitcircle = Graphics[ Circle[{0,0}, 1], \ AspectRatio -> 1 ];\n\ncomplexRoots[ roots_] := Block[{newroots,k,theroot },\n\ \tnewroots = {};\n\tFor [k=1,k<=Length[roots],k++,\n\t\tIf [Abs[roots[[k,2]]] \ < 0.05,\n\t\t\tnewroots = Append[ newroots, roots[[k,1]] ],\n\t\t\t(* else *)\ \n\t\t\ttheroot = roots[[k,1]]+I*roots[[k,2]];\n\t\t\tnewroots = Append[ \ newroots, theroot ];\n\t\t\tnewroots = Append[ newroots, Conjugate[theroot] \ ];\n\t\t\t]\n\t];\n\tReturn[newroots]]\n\nplotPoleZero[poles_,zeros_] := \ Block[{temp,PlotPZ },\n\ttemp =Transpose[{Re[poles],Im[poles]}];\n\tpolePlot \ = plotRoots[ temp, \"X\" ];\n\ttemp =Transpose[{Re[zeros],Im[zeros]}];\n\t\ zeroPlot = plotRoots[ temp, \"O\" ];\n\tPlotPZ = Show[ unitcircle, polePlot, \ zeroPlot,\n AxesLabel -> { \"Re\", \"Im\" }, Axes -> True,\n \ PlotLabel -> \"Poles/Zeros\",DisplayFunction->Identity];\n Return[PlotPZ];\ \n ]\n\t\nPlotPairs[x_,y_] := Transpose[{x,y}];\nFreqAxis := \ Table[i*Pi/128,{i,0,128}];"], "Input", InitializationCell->True, AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["Motivation and Guidelines"], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[StyleBox[ "The purpose of this exercise is to give the student insight into the \ relation between the physical location of poles and zeros in the Z-plane and \ the frequency response of the system represented by those poles and zeros.", Evaluatable->False, AspectRatioFixed->True, FontSize->14]], "Text", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["Pole-Zero Plot"], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Recall that ", Evaluatable->False, AspectRatioFixed->True, FontSize->14], StyleBox["\[Pi]", Evaluatable->False, AspectRatioFixed->True, FontSize->14], StyleBox[ " on the unit circle corresponds to the Nyquist frequency for a sampled \ signal. 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Use options in the Actions Settings dialog box to control when Unformatted text is generated.\ \>", "\<\ -Graphics-\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[TextData[ "Now, to obtain the frequency response evaluate the following"], "Text", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["Compute Frequency Response"], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData[{ StyleBox[ "The frequecy response of the filter is obtained by evaluating the rational \ polynomial at points around the unit circle. This is computed efficiently by \ using the Fast Fourier Transform. Expand the numerator and denominator by \ multiplying the terms (1 - p", Evaluatable->False, PageBreakWithin->False, AspectRatioFixed->True, FontFamily->"Times", FontSize->14], StyleBox["k", Evaluatable->False, PageBreakWithin->False, AspectRatioFixed->True, FontFamily->"Times", FontSize->14, FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[" ", Evaluatable->False, PageBreakWithin->False, AspectRatioFixed->True, FontFamily->"Times", FontSize->14, FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox["z", Evaluatable->False, PageBreakWithin->False, AspectRatioFixed->True, FontFamily->"Times", FontSize->14], StyleBox["-1", Evaluatable->False, PageBreakWithin->False, AspectRatioFixed->True, FontFamily->"Times", FontSize->14, FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox[ "). The polynomials in z evaluated at 256 points on the unit circle can be \ computed efficiently by taking the FFT of a sequence of the polynomial \ coefficients which has been padded with zeros to a length of 256. Division of \ the FFTs yields the frequency response. We plot the magnitude response in dB \ and amplitude as well as the phase response. 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If you get a very small plot, you can \ enlarge it by clicking on the plot to bring up the frame, then positioning \ the cursor at the lower right hand corner and dragging the corner to a larger \ position."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData[ "Show[ GraphicsArray[{ {\n Plotmag = ListPlot[\n \t\ PlotPairs[FreqAxis,Take[NmagResponse,129]],\n PlotJoined->True,\n\t\ PlotLabel -> \"Magnitude Response (Absolute)\",\n\tDisplayFunction->Identity,\ \n\tPlotRange->All],\n Plotdbmag = ListPlot[\n \t\ PlotPairs[FreqAxis,Take[dbmagResponse,129]],\n PlotJoined->True,\n\t\ PlotLabel -> \"Magnitude Response (dB)\",\n\tAxesOrigin -> \ {0,Min[Take[dbmagResponse,129]]},\n\tDisplayFunction->Identity]},{\n Plotarg \ = ListPlot[\n \tPlotPairs[FreqAxis,Take[phaseResponse,129]],\n \ PlotJoined->True,\n\tPlotLabel -> \"Phase Response\",\n\t\ DisplayFunction->Identity],\n PlotPZ = plotPoleZero[poles,zeros]}}],\n \t\ DisplayFunction :> 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