(* Content-type: application/mathematica *)

(*** Wolfram Notebook File ***)
(* http://www.wolfram.com/nb *)

(* CreatedBy='Mathematica 6.0' *)

(*CacheID: 234*)
(* Internal cache information:
NotebookFileLineBreakTest
NotebookFileLineBreakTest
NotebookDataPosition[       145,          7]
NotebookDataLength[     14389,        479]
NotebookOptionsPosition[     12419,        411]
NotebookOutlinePosition[     12837,        427]
CellTagsIndexPosition[     12794,        424]
WindowFrame->Normal
ContainsDynamic->False*)

(* Beginning of Notebook Content *)
Notebook[{

Cell[CellGroupData[{
Cell["Daubechies Filters", "Title",
 CellChangeTimes->{{3.4217479401079364`*^9, 3.4217479429048653`*^9}}],

Cell["\<\
Wavelet Workshop
June 4-7, 2008
University of St. Thomas\
\>", "Subtitle",
 CellChangeTimes->{{3.421426484890625*^9, 3.42142648821875*^9}}],

Cell["\<\
Catherine Beneteau
Caroline Haddad
David Ruch
Patrick Van Fleet\
\>", "Subsubtitle",
 CellChangeTimes->{{3.42144927371875*^9, 3.42144928703125*^9}}],

Cell[CellGroupData[{

Cell["Objective", "Section",
 CellChangeTimes->{{3.4217479766243973`*^9, 3.421747978296315*^9}}],

Cell["\<\
The purpose of this notebook is to introduce you to Daubechies filters and \
see how they can be used in applications.\
\>", "Text"]
}, Open  ]],

Cell[CellGroupData[{

Cell["DiscreteWavelets", "Section"],

Cell[TextData[{
 StyleBox["You should run this cell each time you open this notebook!!",
  FontColor->RGBColor[1, 0, 0]],
 "  It loads the ",
 StyleBox["Mathematica",
  FontSlant->"Italic"],
 " package ",
 StyleBox["DiscreteWavelets",
  FontFamily->"Courier"],
 " for use in subsequent computations."
}], "Text"],

Cell[BoxData[
 RowBox[{"<<", "DiscreteWavelets`DiscreteWavelets`"}]], "Input",
 CellChangeTimes->{{3.421749074496252*^9, 3.421749074590005*^9}},
 FontColor->GrayLevel[0.500008]]
}, Open  ]],

Cell[CellGroupData[{

Cell["Help on DiscreteWavelets", "Section"],

Cell[TextData[{
 "If you ever need help with ",
 StyleBox["DiscreteWavelets",
  FontFamily->"Courier"],
 ", go to the ",
 StyleBox["Documentation Center",
  FontSlant->"Italic"],
 ", then look at the lower right for a link called ",
 StyleBox["Installed Add-Ons",
  FontSlant->"Italic"],
 ".  Click this and then click ",
 StyleBox["DiscreteWavelets",
  FontSlant->"Italic"],
 ". "
}], "Text",
 CellChangeTimes->{{3.421748017312939*^9, 3.4217480658454313`*^9}}]
}, Open  ]],

Cell[CellGroupData[{

Cell["Solving for the D4 Filter", "Section",
 CellChangeTimes->{{3.4217480856584387`*^9, 3.421748091408586*^9}}],

Cell["We first set up the orthogonality conditions", "Text",
 CellChangeTimes->{{3.421748098877527*^9, 3.4217481194249277`*^9}}],

Cell[BoxData[{
 RowBox[{
  RowBox[{"Clear", "[", 
   RowBox[{"h", ",", "h0", ",", "h1", ",", "h2", ",", "h3"}], "]"}], 
  ";"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{"h", "=", 
   RowBox[{"{", 
    RowBox[{"h0", ",", "h1", ",", "h2", ",", "h3"}], "}"}]}], 
  ";"}], "\[IndentingNewLine]", 
 RowBox[{"orth", "=", 
  RowBox[{"{", 
   RowBox[{
    RowBox[{
     RowBox[{"Total", "[", 
      RowBox[{"h", "^", "2"}], "]"}], "\[Equal]", "1"}], ",", 
    RowBox[{
     RowBox[{
      RowBox[{"h0", "*", "h2"}], "+", 
      RowBox[{"h1", "*", "h3"}]}], "\[Equal]", "0"}]}], "}"}]}]}], "Input",
 CellChangeTimes->{{3.421748125471958*^9, 3.42174818355157*^9}, {
  3.4217489035856276`*^9, 3.4217489036950045`*^9}, {3.4217490726680803`*^9, 
  3.4217490727930837`*^9}}],

Cell["\<\
Next the lowpass conditions - we do this with a Fourier series!  The last \
argument of FiniteFourier is the starting index for the series.\
\>", "Text",
 CellChangeTimes->{{3.42174818494223*^9, 3.421748239146743*^9}}],

Cell[BoxData[{
 RowBox[{
  RowBox[{
   RowBox[{
    RowBox[{"H", "[", "w_", "]"}], ":=", 
    RowBox[{
     StyleBox["FiniteFourier",
      FontColor->RGBColor[1, 0, 0]], "[", 
     RowBox[{"w", ",", "h", ",", "0"}], "]"}]}], ";"}], 
  " "}], "\[IndentingNewLine]", 
 RowBox[{"lowpass", "=", 
  RowBox[{"{", 
   RowBox[{
    RowBox[{
     RowBox[{"H", "[", "0", "]"}], "\[Equal]", 
     RowBox[{"Sqrt", "[", "2", "]"}]}], ",", 
    RowBox[{
     RowBox[{"H", "[", "Pi", "]"}], "\[Equal]", "0"}], ",", 
    RowBox[{
     RowBox[{
      RowBox[{"H", "'"}], "[", "Pi", "]"}], "\[Equal]", "0"}]}], 
   "}"}]}]}], "Input",
 CellChangeTimes->{{3.4217482055365076`*^9, 3.421748258475363*^9}, {
  3.421748904804408*^9, 3.421748904945037*^9}, {3.4217490716211786`*^9, 
  3.421749071746182*^9}}],

Cell["Now we solve the system", "Text",
 CellChangeTimes->{{3.421748276100814*^9, 3.421748281772834*^9}}],

Cell[BoxData[{
 RowBox[{
  RowBox[{"eqs", "=", 
   RowBox[{"Join", "[", 
    RowBox[{"orth", ",", "lowpass"}], "]"}]}], ";"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{"s", "=", 
   RowBox[{"Solve", "[", 
    RowBox[{"eqs", ",", "h"}], "]"}]}], ";"}], "\[IndentingNewLine]", 
 RowBox[{"Print", "[", 
  RowBox[{"\"\<There are \>\"", ",", 
   RowBox[{"Length", "[", "s", "]"}], ",", "\"\< solutions.\>\""}], 
  "]"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{"sln1", "=", 
   RowBox[{"h", "/.", 
    RowBox[{"s", "[", 
     RowBox[{"[", "1", "]"}], "]"}]}]}], ";"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{"sln2", "=", 
   RowBox[{"h", "/.", 
    RowBox[{"s", "[", 
     RowBox[{"[", "2", "]"}], "]"}]}]}], ";"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{"Print", "[", 
   RowBox[{"\"\<The first solution is \>\"", ",", 
    RowBox[{"TableForm", "[", "sln1", "]"}], ",", "\"\<.\\n\>\""}], "]"}], 
  ";"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{
   RowBox[{"Print", "[", 
    RowBox[{"\"\<The second solution is \>\"", ",", 
     RowBox[{"TableForm", "[", "sln2", "]"}], ",", "\"\<.\>\""}], "]"}], 
   ";"}], "\[IndentingNewLine]"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{"Print", "[", "\"\<Numerical versions:\\n\>\"", "]"}], 
  ";"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{"Print", "[", 
   RowBox[{"\"\<The first solution is \>\"", ",", 
    RowBox[{"TableForm", "[", 
     RowBox[{"N", "[", "sln1", "]"}], "]"}], ",", "\"\<.\\n\>\""}], "]"}], 
  ";"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{"Print", "[", 
   RowBox[{"\"\<The second solution is \>\"", ",", 
    RowBox[{"TableForm", "[", 
     RowBox[{"N", "[", "sln2", "]"}], "]"}], ",", "\"\<.\>\""}], "]"}], 
  ";"}]}], "Input",
 CellChangeTimes->{{3.421748291179325*^9, 3.4217484861530657`*^9}, {
  3.4217489021637154`*^9, 3.421748902304344*^9}, {3.4217490693398705`*^9, 
  3.421749069480499*^9}}],

Cell["Note that the second solution is a reflection of the first.", "Text",
 CellChangeTimes->{{3.421748489090641*^9, 3.42174849782524*^9}}]
}, Open  ]],

Cell[CellGroupData[{

Cell["The Fourier Series Associated with the Daub4 Filter.", "Section",
 CellChangeTimes->{{3.4217485035910125`*^9, 3.42174851521631*^9}}],

Cell["Here is the Fourier series for the D4 filter.", "Text",
 CellChangeTimes->{{3.4217485239040327`*^9, 3.4217485290135384`*^9}}],

Cell[BoxData[{
 RowBox[{
  RowBox[{
   RowBox[{"H4", "[", "w_", "]"}], ":=", 
   RowBox[{
    StyleBox["FiniteFourier",
     FontColor->RGBColor[1, 0, 0]], "[", 
    RowBox[{"w", ",", "sln2", ",", "0"}], "]"}]}], 
  ";"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{
   RowBox[{"H2", "[", "w_", "]"}], ":=", 
   RowBox[{
    StyleBox["FiniteFourier",
     FontColor->RGBColor[1, 0, 0]], "[", 
    RowBox[{"w", ",", 
     RowBox[{"Haar", "[", "]"}], ",", "0"}], "]"}]}], 
  ";"}], "\[IndentingNewLine]", 
 RowBox[{"Plot", "[", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{
     RowBox[{"Abs", "[", 
      RowBox[{"H4", "[", "w", "]"}], "]"}], ",", 
     RowBox[{"Abs", "[", 
      RowBox[{"H2", "[", "w", "]"}], "]"}]}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{"w", ",", "0", ",", "Pi"}], "}"}], ",", 
   RowBox[{"PlotStyle", "\[Rule]", 
    RowBox[{"{", 
     RowBox[{"Red", ",", "Blue"}], "}"}]}]}], "]"}]}], "Input",
 CellChangeTimes->{{3.4217485360293427`*^9, 3.4217485967965236`*^9}, 
   3.421748634422487*^9, {3.421748900069912*^9, 3.4217489001636643`*^9}, {
   3.421749067324194*^9, 3.421749067449197*^9}}]
}, Open  ]],

Cell[CellGroupData[{

Cell["The Highpass Filter", "Section",
 CellChangeTimes->{{3.42174876719151*^9, 3.4217487693634415`*^9}}],

Cell["\<\
We obtain the highpass filter by reversing the solution and multiplying each \
coefficient alternately  by 1 and - 1.\
\>", "Text",
 CellChangeTimes->{{3.421748777098014*^9, 3.421748813177063*^9}}],

Cell[BoxData[{
 RowBox[{
  RowBox[{"hp", "=", 
   RowBox[{"sln2", "*", 
    RowBox[{"{", 
     RowBox[{"1", ",", 
      RowBox[{"-", "1"}], ",", "1", ",", 
      RowBox[{"-", "1"}]}], "}"}]}]}], ";"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{
   RowBox[{"G", "[", "w_", "]"}], ":=", 
   RowBox[{
    StyleBox["FiniteFourier",
     FontColor->RGBColor[1, 0, 0]], "[", 
    RowBox[{"w", ",", "hp", ",", "0"}], "]"}]}], 
  ";"}], "\[IndentingNewLine]", 
 RowBox[{"Plot", "[", 
  RowBox[{
   RowBox[{"Abs", "[", 
    RowBox[{"G", "[", "w", "]"}], "]"}], ",", 
   RowBox[{"{", 
    RowBox[{"w", ",", "0", ",", "Pi"}], "}"}]}], "]"}]}], "Input",
 CellChangeTimes->{{3.421748818786581*^9, 3.4217488986948767`*^9}, {
  3.421749065777279*^9, 3.4217490659491587`*^9}}]
}, Open  ]],

Cell[CellGroupData[{

Cell["The Wavelet Matrix", "Section",
 CellChangeTimes->{{3.421748660438778*^9, 3.4217486630169687`*^9}}],

Cell[BoxData[{
 RowBox[{
  RowBox[{"n", "=", "8"}], ";"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{"g", "=", 
   RowBox[{"{", 
    RowBox[{"g0", ",", "g1", ",", "g2", ",", "g3"}], "}"}]}], 
  ";"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{"r1", "=", 
   RowBox[{"Join", "[", 
    RowBox[{"h", ",", 
     RowBox[{"Table", "[", 
      RowBox[{"0", ",", 
       RowBox[{"{", 
        RowBox[{"n", "-", "4"}], "}"}]}], "]"}]}], "]"}]}], 
  ";"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{"top", "=", 
   RowBox[{"Table", "[", 
    RowBox[{
     RowBox[{"RotateRight", "[", 
      RowBox[{"r1", ",", 
       RowBox[{"2", "*", "k"}]}], "]"}], ",", 
     RowBox[{"{", 
      RowBox[{"k", ",", "0", ",", 
       RowBox[{
        RowBox[{"n", "/", "2"}], "-", "1"}]}], "}"}]}], "]"}]}], 
  ";"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{"r2", "=", 
   RowBox[{"Join", "[", 
    RowBox[{"g", ",", 
     RowBox[{"Table", "[", 
      RowBox[{"0", ",", 
       RowBox[{"{", 
        RowBox[{"n", "-", "4"}], "}"}]}], "]"}]}], "]"}]}], 
  ";"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{"bot", "=", 
   RowBox[{"Table", "[", 
    RowBox[{
     RowBox[{"RotateRight", "[", 
      RowBox[{"r2", ",", 
       RowBox[{"2", "*", "k"}]}], "]"}], ",", 
     RowBox[{"{", 
      RowBox[{"k", ",", "0", ",", 
       RowBox[{
        RowBox[{"n", "/", "2"}], "-", "1"}]}], "}"}]}], "]"}]}], 
  ";"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{"W", "=", 
   RowBox[{"Join", "[", 
    RowBox[{"top", ",", "bot"}], "]"}]}], ";"}], "\[IndentingNewLine]", 
 RowBox[{"MatrixForm", "[", "W", "]"}]}], "Input",
 CellChangeTimes->{{3.4217486744235106`*^9, 3.421748763066405*^9}, {
  3.421748896147937*^9, 3.4217488965229464`*^9}, {3.4217490393547277`*^9, 
  3.421749064011609*^9}}]
}, Open  ]],

Cell[CellGroupData[{

Cell["Daubechies Filters in DiscreteWavelets", "Section",
 CellChangeTimes->{{3.4217489075076027`*^9, 3.4217489150859213`*^9}}],

Cell[TextData[{
 "We can generate Daubechies filters in DiscreteWavelets using the ",
 StyleBox["Daub",
  FontColor->RGBColor[1, 0, 0]],
 " command."
}], "Text",
 CellChangeTimes->{{3.421748919429783*^9, 3.421748933633271*^9}}],

Cell[BoxData[
 RowBox[{"Daub", "[", "2", "]"}]], "Input",
 CellChangeTimes->{{3.4217489493367987`*^9, 3.4217489510712175`*^9}}],

Cell[BoxData[
 RowBox[{"Daub", "[", "4", "]"}]], "Input",
 CellChangeTimes->{{3.421748953165021*^9, 3.4217489539150405`*^9}}],

Cell[BoxData[
 RowBox[{"Daub", "[", "6", "]"}]], "Input",
 CellChangeTimes->{{3.4217489573526287`*^9, 3.4217489590245466`*^9}}],

Cell[BoxData[
 RowBox[{"Daub", "[", "12", "]"}]], "Input",
 CellChangeTimes->{{3.4217489615089846`*^9, 3.4217489635715375`*^9}}]
}, Open  ]],

Cell[CellGroupData[{

Cell["Exercise", "Section",
 CellChangeTimes->{{3.421748966024726*^9, 3.421748967524764*^9}}],

Cell["Can you write code to find the D6 filter?", "Text",
 CellChangeTimes->{{3.4217489728530254`*^9, 3.4217489812282395`*^9}}],

Cell[BoxData[
 RowBox[{"(*", " ", 
  RowBox[{"Put", " ", "code", " ", "here"}], " ", "*)"}]], "Input",
 CellChangeTimes->{{3.42174898672838*^9, 3.421748990009714*^9}}]
}, Open  ]]
}, Open  ]]
},
WindowSize->{1272, 683},
WindowMargins->{{0, Automatic}, {Automatic, 0}},
FrontEndVersion->"6.0 for Microsoft Windows (32-bit) (April 20, 2007)",
StyleDefinitions->FrontEnd`FileName[{"Creative"}, "NaturalColor.nb", 
  CharacterEncoding -> "WindowsANSI"]
]
(* End of Notebook Content *)

(* Internal cache information *)
(*CellTagsOutline
CellTagsIndex->{}
*)
(*CellTagsIndex
CellTagsIndex->{}
*)
(*NotebookFileOutline
Notebook[{
Cell[CellGroupData[{
Cell[590, 23, 105, 1, 73, "Title"],
Cell[698, 26, 149, 5, 96, "Subtitle"],
Cell[850, 33, 158, 6, 89, "Subsubtitle"],
Cell[CellGroupData[{
Cell[1033, 43, 96, 1, 75, "Section"],
Cell[1132, 46, 142, 3, 29, "Text"]
}, Open  ]],
Cell[CellGroupData[{
Cell[1311, 54, 35, 0, 75, "Section"],
Cell[1349, 56, 312, 10, 29, "Text"],
Cell[1664, 68, 177, 3, 41, "Input"]
}, Open  ]],
Cell[CellGroupData[{
Cell[1878, 76, 43, 0, 75, "Section"],
Cell[1924, 78, 461, 15, 29, "Text"]
}, Open  ]],
Cell[CellGroupData[{
Cell[2422, 98, 112, 1, 75, "Section"],
Cell[2537, 101, 128, 1, 29, "Text"],
Cell[2668, 104, 764, 22, 82, "Input"],
Cell[3435, 128, 228, 4, 29, "Text"],
Cell[3666, 134, 785, 24, 62, "Input"],
Cell[4454, 160, 105, 1, 29, "Text"],
Cell[4562, 163, 1890, 51, 242, "Input"],
Cell[6455, 216, 140, 1, 29, "Text"]
}, Open  ]],
Cell[CellGroupData[{
Cell[6632, 222, 138, 1, 75, "Section"],
Cell[6773, 225, 131, 1, 29, "Text"],
Cell[6907, 228, 1111, 33, 82, "Input"]
}, Open  ]],
Cell[CellGroupData[{
Cell[8055, 266, 105, 1, 75, "Section"],
Cell[8163, 269, 207, 4, 29, "Text"],
Cell[8373, 275, 762, 23, 82, "Input"]
}, Open  ]],
Cell[CellGroupData[{
Cell[9172, 303, 105, 1, 75, "Section"],
Cell[9280, 306, 1768, 57, 182, "Input"]
}, Open  ]],
Cell[CellGroupData[{
Cell[11085, 368, 127, 1, 75, "Section"],
Cell[11215, 371, 227, 6, 29, "Text"],
Cell[11445, 379, 127, 2, 41, "Input"],
Cell[11575, 383, 125, 2, 41, "Input"],
Cell[11703, 387, 127, 2, 41, "Input"],
Cell[11833, 391, 128, 2, 41, "Input"]
}, Open  ]],
Cell[CellGroupData[{
Cell[11998, 398, 93, 1, 75, "Section"],
Cell[12094, 401, 127, 1, 29, "Text"],
Cell[12224, 404, 167, 3, 41, "Input"]
}, Open  ]]
}, Open  ]]
}
]
*)

(* End of internal cache information *)