(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.0' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing 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[ 1144329, 30244]*) (*NotebookOutlinePosition[ 1144993, 30267]*) (* CellTagsIndexPosition[ 1144949, 30263]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["DFT part 2: randomising data and filtering", "Subtitle"], Cell["\<\ I. 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Joining all segments as matchsticks one after the other draws the bigger n \ - gon in the previous Graphics cell.\ \>", "Text"], Cell["\<\ The reason that we end up close to the origin is that errors going \ in opposite directions level out. This is quite similar to the rolling of dice (where sums tend to a multiple \ of 3.5) or tossing a coin if we replace heads by 1, tails by -1 and repeating \ the toss.\ \>", "Text"], Cell["\<\ It can be explained by the analogue of a drunk person :by stumbling \ over some random distance (with distances averaging to 1) in each of the n \ directions 2\[Pi]j/n he will not get far.\ \>", "Text"] }, Closed]], Cell[CellGroupData[{ Cell["List programming for a sum", "Subsection"], Cell["\<\ Obtaining the final sum is easiest with a Dot product command: \ try\ \>", "Text"], Cell[BoxData[{ \(afew = {a, b, c, d}\), "\[IndentingNewLine]", \(m = Length[afew]\), "\[IndentingNewLine]", \(counts = Range[m]\), "\[IndentingNewLine]", \(afew\ counts\), "\[IndentingNewLine]", \(afew . counts\)}], "Input"], Cell["We have performed", "Text"], Cell[BoxData[ \(\(coeffList = Table[ .8 + \ .4\ Random[\ ], {13}];\)\)], "Input"], Cell[BoxData[ \(\(n = \ Length[coeffList];\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(coeffList . directions[n]\)], "Input"], Cell[BoxData[ \(\(\(0.38990401272091746`\)\(\[InvisibleSpace]\)\) + 0.3332749904523752`\ \[ImaginaryI]\)], "Output"] }, Closed]], Cell["\<\ The fact that these values are close to the origin is enhance if we \ divide by the square root of n , which we did to obtain the discrete \ Fourier transform.\ \>", "Text", FontWeight->"Bold"] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell["Discrete Fourier transform of a noisy list.", "Section"], Cell[CellGroupData[{ Cell["Vectors with arbitrary lengths", "Subsection"], Cell["\<\ We recommend that this buildup be first performed with a list of \ positive numbers for lengths. 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{176.938, 0}} -> {-1.5101, -1.32971, \ 0.0552619, 0.0149972}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(coeffList // TableForm\)], "Input"], Cell[BoxData[ InterpretationBox[GridBox[{ {\(-0.917165989753966`\)}, {"0.9820719658197813`"}, {\(-1.04740912438699`\)}, {"0.9457460643615458`"}, {\(-1.1769915541522962`\)}, {"0.8530725653476048`"}, {\(-1.1633484838866741`\)}, {"0.8671708589025762`"}, {\(-1.1354448383334037`\)}, {"1.171156051820835`"}, {\(-0.8810079697640973`\)}, {"0.9397356903385761`"}, {\(-0.9805300260974975`\)}, {"0.9754576962305596`"} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], TableForm[ {-0.91716598975396602, 0.98207196581978129, -1.0474091243869901, 0.94574606436154585, -1.1769915541522962, 0.85307256534760478, -1.1633484838866741, 0.86717085890257617, -1.1354448383334037, 1.171156051820835, -0.88100796976409734, 0.93973569033857607, -0.98053002609749751, 0.97545769623055956}]]], "Output"] }, Closed]], Cell["For later reference, here is their sum:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Plus @@ coeffList\)], "Input"], Cell[BoxData[ \(\(-0.5674870935534461`\)\)], "Output"] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(n = \ Length[coeffList]\)], "Input"], Cell[BoxData[ \(14\)], "Output"] }, Closed]], Cell[BoxData[ \(\(Map[coords, coeffList\ directions[n] // N];\)\)], "Input"], Cell["\<\ We now plot the endpoints of the vectors with lengths in coeffList\ \ \>", "Text"], Cell["\<\ Remark: since some numbers are negative, some of these endpoints \ lie on the other side of the origin, back from the original ray. 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