(*********************************************************************** 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). 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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[ 309350, 10820]*) (*NotebookOutlinePosition[ 310405, 10856]*) (* CellTagsIndexPosition[ 310361, 10852]*) (*WindowFrame->Normal*) Notebook[{ Cell[TextData["The Discrete Periodic Wavelet Transform in 1D:"], "Title", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["A Demonstration of the Routine Wavelet1Dv1.m"], "Subtitle", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "by \n\nJames F. Scholl\nMathematical Physics Department \nRockwell Science \ Center\n1049 Camino Dos Rios\nThousand Oaks, CA 91360\n\ne-mail: \ jfs@risc.rockwell.com"], "Subsubtitle", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData[ "Calling the Routine, With Some Wavelet Bases to Try"], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["<<\"Wavelet1Dv1.m\""], "Input", AspectRatioFixed->True], Cell[TextData["the main functions:"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["?DPWT1D"], "Input", AspectRatioFixed->True], Cell[TextData[ "DPWT1D[dat,lfil,lev] performs a 1D discrete wavelet transform\n of the \ input signal dat w.r.t the wavelet basis system\n represented by the input \ low pass filter lfil. The high\n pass filter hfil is automatically \ calculated from lfil. \n The transform is performed by doing lev levels of \ \n decomposition using a recursive pyramid filtering / \n downsampling \ scheme. It is preferred that the input data\n length be a power of 2, but \ even-length data can be used,\n but the number of levels of decomposition \ may be limited.\n For J levels of decomposition and signal length 2^L, \ the\n output is in the form {AJ,{DJ,D(J-1),...,D2,D1}}, where\n AJ is \ the approximation subsignal of length 2^(L-J) and Di\n is the ith detail \ subsignal of length 2^(L-i). "], "Print", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["?IDPWT1D"], "Input", AspectRatioFixed->True], Cell[TextData[ "IDPWT1D[wdat,lfil] performs the inverse 1D discrete wavelet\n transform on \ the input coefficients wdat. 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We will try to remove the noise."], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Wavelet Transform will be done w.r.t the 4th order Coifman basis (Coif4) \ with 4 levels of decomposition"], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Wcof2 = DPWT1D[testdata2, Coif4, 4];"], "Input", AspectRatioFixed->True], Cell[TextData["\n"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "These are the component subsignals in the decomposition:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["A4 = Wcof2[[1]]; Length[A4]"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ 8\ \>", "\<\ 8\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["D4 = Wcof2[[2,1]]; Length[D4]"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ 8\ \>", "\<\ 8\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["D3 = Wcof2[[2,2]]; Length[D3]"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ 16\ \>", "\<\ 16\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["D2 = Wcof2[[2,3]]; Length[D2]"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ 32\ \>", "\<\ 32\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["D1 = Wcof2[[2,4]]; Length[D1]"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ 64\ \>", "\<\ 64\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[TextData["\n"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ " We will plot the subsignals to illustrate how the output is arranged \ as well as the multiresolution aspect of wavelets. 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