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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[ 652165, 17687]*) (*NotebookOutlinePosition[ 652854, 17711]*) (* CellTagsIndexPosition[ 652810, 17707]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[TextData[{ "Notebook accompanying the paper\n", StyleBox["How were the MP3 filter coeffients produced?", FontSlant->"Italic"] }], "Subtitle", FontSize->36], Cell["\<\ by Tom H. Koornwinder, version of 5 February 2007, \ http://www.science.uva.nl/~thk/art/misc/\ \>", "Subsubtitle", FontSize->24], Cell[TextData[StyleBox["Korteweg-de Vries Institute, University of Amsterdam, \ thk@science.uva.nl", FontSize->14]], "Subsubtitle", FontSize->24], Cell[TextData[{ StyleBox["The analysis filter coefficients ", FontSize->14], Cell[BoxData[ FormBox[ StyleBox[\(C[i]\), FontSlant->"Italic"], TraditionalForm]], FontSize->14], StyleBox[" (", FontSize->14], StyleBox["i ", FontSize->14, FontSlant->"Italic"], StyleBox["= 1,\[Ellipsis],255) taken from Annex C, Table C.1 in:\n", FontSize->14], StyleBox["Information technology --- Coding of moving pictures and \ associated audio for digital storage media at up to about 1,5 Mbit/s --- Part \ 3: Audio,", FontSize->14, FontSlant->"Italic"], StyleBox["\nISO/IEC 11172-3:1993, International Organization for \ Standardization, 1993;\n\ http://www.iso.org/iso/en/prods-services/popstds/mpeg.html or\n\ http://www.cise.ufl.edu/~jfd/mp3links.html.\nThe coefficients are here \ written as ", FontSize->14], StyleBox["c[[i]]", FontSize->14, FontWeight->"Bold"], StyleBox[".", FontSize->14] }], "Text", FontSize->16], Cell[BoxData[ \(\(c = {\(-0.000000477\), \(-0.000000477\), \(-0.000000477\), \ \(-0.000000477\), \(-0.000000477\), \(-0.000000477\), \(-0.000000954\), \ \(-0.000000954\), \(-0.000000954\), \(-0.000000954\), \(-0.000001431\), \ \(-0.000001431\), \(-0.000001907\), \(-0.000001907\), \(-0.000002384\), \ \(-0.000002384\), \(-0.000002861\), \(-0.000003338\), \(-0.000003338\), \ \(-0.000003815\), \(-0.000004292\), \(-0.000004768\), \(-0.000005245\), \ \(-0.000006199\), \(-0.000006676\), \(-0.000007629\), \(-0.000008106\), \ \(-0.000009060\), \(-0.000010014\), \(-0.000011444\), \(-0.000012398\), \ \(-0.000013828\), \(-0.000014782\), \(-0.000016689\), \(-0.000018120\), \ \(-0.000019550\), \(-0.000021458\), \(-0.000023365\), \(-0.000025272\), \ \(-0.000027657\), \(-0.000030041\), 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Cell[BoxData[ TagBox[GridBox[{ {"1", "6.74029351870436`*^-8"}, {"2", \(-1.7507238093182666`*^-8\)}, {"3", "7.559471893649487`*^-8"}, {"4", \(-9.637692525247977`*^-9\)}, {"5", \(-6.419067538035`*^-7\)}, {"6", "2.8849014507499992`*^-8"}, {"7", "1.8688340690000005`*^-9"} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], Function[ BoxForm`e$, TableForm[ BoxForm`e$]]]], "Output"] }, Open ]], Cell[TextData[{ "We still have to add a parity condition:\n", StyleBox["\n", FontSize->9], "(5a) \[CurlyPhi](0) > 0", StyleBox["\n", FontSize->14], StyleBox["\nor equivalently", FontSize->10], StyleBox["\n", FontSize->14], StyleBox["\n", FontSize->10], "(5b) ", StyleBox[" ", FontSize->14], StyleBox[" ", FontSize->14], StyleBox[" ", FontSize->16], Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(\[Sum]\)\)\+\(\(\ \ \)\(n = 1\)\)\%\(\(\ \ \ \)\(511\)\)\)], FontSize->16], StyleBox[" ", FontSize->16], Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(h\_\(\(n\)\(\ \)\)\)\)\)], 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T - T . K;\)\), "\[IndentingNewLine]", \(Sum[Abs[KT[\([i, j]\)]], {i, 1, 511}, {j, 1, 511}]\)}], "Input"], Cell[BoxData[ \(5.96702129374016`*^-9\)], "Output"] }, Open ]], Cell[TextData[{ "We pass to 256 \[Times] 256 versions ", StyleBox["K0", FontSlant->"Italic"], " and ", StyleBox[" T0", FontSlant->"Italic"], " of ", StyleBox["K", FontSlant->"Italic"], " resp. 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K0;\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Sum[Abs[K0T0[\([i, j]\)]], {i, 1, 256}, {j, 1, 256}]\)], "Input"], Cell[BoxData[ \(2.6521608589846757`*^-9\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Chop[Eigenvalues[K0]]\)], "Input"], Cell[BoxData[ \({1.`, 0.9999999999999999`, 0.9999999999997267`, 0.9999999997510958`, 0.9999998989247901`, 0.9999795753031352`, 0.9979070789762648`, 0.9097871834088815`, 0.3088869114679921`, 0.014431804870469536`, 0.00022688322752147005`, 2.038768659245381`*^-6, 1.189540518331783`*^-8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 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256]\)], Norm[K0 . v0[\([m]\)] - \(\((K0 . v0[\([m]\)])\)[\([256]\)]\/v0[\([m, 256]\)]\) v0[\([m]\)]]}, {m, 1, 5}];\)\), "\[IndentingNewLine]", \(TableForm[%]\)}], "Input"], Cell[BoxData[ TagBox[GridBox[{ {\(-6.661338147750939`*^-16\), "9.175877739407641`*^-16"}, {"1.1102230246251565`*^-16", "1.6489257795958483`*^-15"}, {"2.73003841755326`*^-13", "2.7128132453015703`*^-15"}, {"2.489031203367631`*^-10", "2.823251761570284`*^-15"}, {"1.0107521097868499`*^-7", "1.7478523023126877`*^-15"} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], Function[ BoxForm`e$, TableForm[ BoxForm`e$]]]], "Output"] }, Open ]], Cell[TextData[{ "We extend these five eigenvectors ", StyleBox["v0[[m]]", FontWeight->"Bold"], " to symmetric eigenvectors ", StyleBox["v[[m]] ", FontWeight->"Bold"], "of ", StyleBox["K", FontSlant->"Italic"], " and ", StyleBox[" T", FontSlant->"Italic"], " :" }], "Text"], Cell[BoxData[ \(\(v = Table[Which[n < 256, v0[\([i, n]\)]/Sqrt[2], n > 256, v0[\([i, 512 - n]\)]/Sqrt[2], True, v0[\([i, 256]\)]], {i, 1, 5}, {n, 1, 511}];\)\)], "Input"], Cell[TextData[{ "The five vectors ", StyleBox["v[[m]] ", FontWeight->"Bold"], " form an orthonormal system:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Table[ Chop[v[\([m]\)] . v[\([n]\)]], {m, 1, 5}, {n, 1, 5}];\)\), "\[IndentingNewLine]", \(TableForm[%]\)}], "Input"], Cell[BoxData[ TagBox[GridBox[{ {"0.9999999999999999`", "0", "0", "0", "0"}, {"0", "0.9999999999999998`", "0", "0", "0"}, {"0", "0", "0.9999999999999998`", "0", "0"}, {"0", "0", "0", "0.9999999999999998`", "0"}, {"0", "0", "0", "0", "1.`"} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], Function[ BoxForm`e$, TableForm[ BoxForm`e$]]]], "Output"] }, Open ]], Cell[TextData[{ "Now we look for coefficients ", StyleBox["a[i] ", FontWeight->"Bold"], " such that\n", StyleBox["h1 =", FontSize->14, FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(\[Sum]\)\)\+\(\(\ \ 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TraditionalForm]]], " is minimal.\n\nWith the aid of the quadratic form" }], "Text"], Cell[BoxData[ \(\(q = Table[Sum[ v[\([i, m - 32 l]\)] v[\([j, m + 32 l]\)], {m, 1 + 32 l, 511 - 32 l}], {l, 1, 7}, {i, 1, 5}, {j, 1, 5}];\)\)], "Input"], Cell["we have to dind the following minimum:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(FindMinimum[ Sum[Sum[q[\([n, i, j]\)] a[i] a[j], {i, 1, 5}, {j, 1, 5}]^2, {n, 1, 7}] + \((Sum[a[i]^2, {i, 1, 5}] - 1/64)\)^2, {{a[1], 0.1}, {a[2], 0}, {a[3], 0}, {a[4], 0}, {a[5], 0}}]\)], "Input"], Cell[BoxData[ \({8.399098142082387`*^-13, {a[1] \[Rule] 0.10338409413932634`, a[2] \[Rule] \(-0.062034880200050393`\), a[3] \[Rule] 0.03178064058497127`, a[4] \[Rule] \(-0.008834586558885827`\), a[5] \[Rule] 0.0005861981908168957`}}\)], "Output"] }, Open ]], Cell[TextData[{ "Then ", StyleBox["h1", FontWeight->"Bold"], " below is the desired vector:" }], "Text"], Cell[BoxData[{ \(\(h1 = Sum[a[j]\ v[\([j]\)], {j, 1, 5}] /. %[\([2]\)];\)\), "\[IndentingNewLine]", 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