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Notebook[{

Cell[CellGroupData[{
Cell["Biorthogonal Filters", "Subtitle"],

Cell["\<\
Patrick J. Van Fleet
Minicourse #5, Joint Mathematics Meetings
New Orleans, LA\
\>", "Subsubtitle"],

Cell["Objective", "Section"],

Cell[CellGroupData[{

Cell["\<\
In this notebook, we will construct a biorthogonal filter pair.\
\>", "Subsubtitle"],

Cell["Conventions", "Section"]
}, Open  ]],

Cell[CellGroupData[{

Cell[TextData[{
  "This notebook uses the package ",
  StyleBox["WaveletFunctions",
    FontColor->RGBColor[0.501961, 0, 0]],
  " (written by Patrick Van Fleet).  All commands from the WaveletFunctions \
library will be denoted in ",
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    FontColor->RGBColor[0.501961, 0, 0]],
  ".  Help is available for almost every command in ",
  StyleBox["WaveletFunctions",
    FontColor->RGBColor[0.501961, 0, 0]],
  ".  Click on Help and then Help Browser (or Shift F1).  Next click the \
Add-ons & Links tab.  Scroll down until you find ",
  StyleBox["WaveletFunctions",
    FontColor->RGBColor[0.501961, 0, 0]],
  ".  Like all ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " help screens, the help is \"live\" - you can either execute the commands \
in the help to see the effects of the command or cut and paste them into your \
own notebook.\n\nValues that can be changed by the user are defined using ",
  StyleBox["blue",
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  ".\n\nComments are useful within cells of code.  Any code enclosed by (*  \
*) is a comment and ignored by the ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " kernel."
}], "Subsubtitle"],

Cell[CellGroupData[{

Cell["Load WaveletFunctions", "Section"],

Cell[BoxData[
    \(<< WaveletFunctions`WaveletFunctions`\)], "Input"]
}, Open  ]],

Cell["Spelling Off", "Section"]
}, Open  ]],

Cell[CellGroupData[{

Cell["\<\
This turns off the annoying error messages you get when the spelling of two \
variable names is close.\
\>", "Subsubtitle"],

Cell[BoxData[{
    \(\(Off[General::spell];\)\), "\[IndentingNewLine]", 
    \(\(Off[General::spell1];\)\)}], "Input"],

Cell["Solving for the Length-5 Filter", "Section"]
}, Open  ]],

Cell[CellGroupData[{

Cell[TextData[{
  "We need two Fourier series ",
  Cell[BoxData[
      \(TraditionalForm\`\(H\& \[Tilde] \)\ \((\[Omega])\)\ and\ \(\(H(\
\[Omega])\)\(.\)\)\)]],
  "  The first is known while the second has unknown coefficients ",
  Cell[BoxData[
      \(TraditionalForm\`h\_\(-2\)\  = \ h\_2, \ h\_\(-1\)\  = \ h\_1, \ 
      and\ \(\(h\_0\)\(.\)\)\)]]
}], "Subsubtitle"],

Cell[BoxData[{
    \(\(Clear[h];\)\), "\[IndentingNewLine]", 
    \(\(htilde = {Sqrt[2]/4, Sqrt[2]/2, 
          Sqrt[2]/4};\)\), "\[IndentingNewLine]", 
    \(\(Ht[w_] := 
        htilde . 
          Table[E^\((I*k*w)\), {k, \(-1\), 1}];\)\), "\[IndentingNewLine]", 
    \(\(f\  = \ {h[2], h[1], h[0], h[1], h[2]};\)\), "\[IndentingNewLine]", 
    \(\(H[w_] := 
        f . Table[
            E^\((I*k*w)\), {k, \(-2\), 2}];\)\), "\[IndentingNewLine]", 
    \(\(Print[\*"\"\<The Fourier series are \!\(\(H\& \[Tilde] \)\)(\[Omega]) \
= \>\"", Ht[w], "\< and H(\[Omega]) = \>", H[w], "\<.\>"];\)\)}], "Input"]
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Cell[CellGroupData[{

Cell["Next, let's define the Fourier series associated with h. ", \
"Subsubtitle"],

Cell[BoxData[{
    \(\(P[w_] := 
        Ht[w]*H[\(-w\)]\  + \ 
          Ht[w + Pi]*H[\(-w\) + Pi];\)\), "\[IndentingNewLine]", 
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}, Open  ]],

Cell[CellGroupData[{

Cell["\<\
Now we equate coefficients.  The above expression must be 2.  This gives rise \
to two equations:\
\>", "Subsubtitle"],

Cell[BoxData[{
    \(\(eqs = 
        Table[Coefficient[Expand[P[w]], E^\((I*w)\), 2*k] \[Equal] 
            2*DiscreteDelta[k], {k, 0, 1}];\)\), "\[IndentingNewLine]", 
    \(TableForm[eqs]\)}], "Input"]
}, Open  ]],

Cell[CellGroupData[{

Cell["\<\
The first equation is one of the lowpass conditions.  Since the above cell is \
an underdetermined system of equations, we'll add the other lowpass \
condition.\
\>", "Subsubtitle"],

Cell[BoxData[{
    \(\(equations = 
        Append[eqs, 
          h[2] - h[1] + h[0] - h[1] + h[2] \[Equal] 
            0];\)\), "\[IndentingNewLine]", 
    \(TableForm[equations]\), "\[IndentingNewLine]", 
    \(\)}], "Input"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Now we can solve the system!", "Subsubtitle"],

Cell[BoxData[{
    \(\(s = 
        Solve[equations, {h[0], h[1], h[2]}];\)\), "\[IndentingNewLine]", 
    \(TableForm[s]\)}], "Input"]
}, Open  ]],

Cell[CellGroupData[{

Cell[TextData[{
  "Here is some ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " code to strip out the arrows from the solutions.  The ",
  StyleBox["/.",
    FontWeight->"Bold"],
  " simply replaces all the h[0], h[1], h[2] with their values"
}], "Subsubtitle"],

Cell[BoxData[{
    \(\(v = Flatten[{h[0], h[1], h[2]} /. s];\)\), "\[IndentingNewLine]", 
    \(\(Print["\<The solution is (h[0], h[1], h[2]) = \>", 
        v, "\<.\>"];\)\)}], "Input"]
}, Open  ]],

Cell[CellGroupData[{

Cell[TextData[{
  "In practice, the short filter is multiplied by ",
  Cell[BoxData[
      \(TraditionalForm\`\@2\ and\ the\ long\ filter\ is\ divided\ by\ \
\@2\)]],
  ".  The result is the LeGall filter pair:"
}], "Subsubtitle"],

Cell[BoxData[{
    \(\(Clear[h];\)\), "\[IndentingNewLine]", 
    \(h = Join[Take[Reverse[v], 2], v]/Sqrt[2]\), "\[IndentingNewLine]", 
    \(ht = Sqrt[2]*htilde\)}], "Input"],

Cell[BoxData[""], "Input"]
}, Open  ]]
}, Open  ]]
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