(************** 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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*******************************************************************)