(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of 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). 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I show that selecting the most significant wavelets from a burst light \ curve is an effective way to obtain a template for cross correlation. One \ attractive feature of this method is that it requires the exchange of very \ little information to accurately compare burst arrival times across \ interplanetary distances."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["On the Format"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "This report is a ", StyleBox["Mathematica", FontSlant->"Italic"], " notebook. If you are reading a text copy, the notebook itself may be \ obtained at . Although ", StyleBox["Mathematica", FontSlant->"Italic"], " notation is, in some cases, a bit harder to read than standard \ mathematical notation, it allows me to present the algorithmic content of the \ report in totality. Indeed, the electronic version, together with the \ electronic references, may be used as data reduction code as is." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "One hazard of this format is that the thread of the exposition may be lost \ in programming detail. I have hidden ", StyleBox["Mathematica", FontSlant->"Italic"], " definitions that are not central to the exposition in \"closed cells\". \ Users of the electronic version may open and inspect these at will, and I can \ provide a fully opened reprint on request." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["General Toolkit (hidden)"], "Subsubsection", Evaluatable->False, InitializationCell->True, AspectRatioFixed->True], Cell[TextData[ "Off[General::spell1]\nSetOptions[ListPlot,PlotRange->All];\n\ <True, AspectRatioFixed->True], Cell[TextData[{ "The direct cross correlation function ", StyleBox["ccf", FontFamily->"Courier", FontWeight->"Bold"], " will use the ", StyleBox["shift", FontFamily->"Courier", FontWeight->"Bold"], " function to shift the data past the template, truncating it to the \ template length." }], "Text", Evaluatable->False, InitializationCell->True, AspectRatioFixed->True], Cell[TextData[ "shift[data_,delay_,length_]:=Take[data,{delay+1,delay+length}]"], "Input", InitializationCell->True, AspectRatioFixed->True], Cell[TextData[{ "The Fourier cross correlation method ", StyleBox["ftccf", FontFamily->"Courier", FontWeight->"Bold"], " will use ", StyleBox["pad", FontFamily->"Courier", FontWeight->"Bold"], " to extend the template with zeros to match the length of the data." }], "Text", Evaluatable->False, InitializationCell->True, AspectRatioFixed->True], Cell[TextData[ "pad[template_,length_]:=\n\t\ Join[template,Table[0.,{length-Length[template]}]]"], "Input", InitializationCell->True, AspectRatioFixed->True], Cell[TextData[{ "The function ", StyleBox["freqsq", FontFamily->"Courier", FontWeight->"Bold"], " tabulates the square of the frequency in radians/sample period associated \ with the result of ", StyleBox["Fourier", FontFamily->"Courier", FontWeight->"Bold"], ". The equivalent bandwidth calculation needs this." }], "Text", Evaluatable->False, InitializationCell->True, AspectRatioFixed->True], Cell[TextData[ "freqsq[ len_ ] := freqsq[ len ] =\n\tModule[\n\t\t{ w1 = N[ 2 Pi / len ]},\n\ \tJoin[ \n\t\tTable[ (w1 k)^2, { k, 0, len/2 }], \n\t\tTable[ (w1 (k-len))^2, \ \n\t\t\t{ k, len/2 + 1, len - 1 }]\n\t]\n]"], "Input", InitializationCell->True, AspectRatioFixed->True], Cell[TextData[{ "The function ", StyleBox["peakdisplacement", FontFamily->"Courier", FontWeight->"Bold"], " finds the displacement of the global peak of a smooth function \ (represented as a list) relative to the first position. Quadratic \ interpolation allows fractional displacements to be accurately determined. \ The function is assumed to be cyclic and the result corresponds to the branch \ of minimum absolute value. 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It also removes the constant wavelet. The result is smoothed \ slightly with a convolutional filter. The smoothing step is necessary to make \ interpolation to continuous time possible. 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Here, I'm using the \"cross \ correlation function\", which involves displacing one function of time past \ another, and integrating their product. "], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "In the digital representation, measured functions are represented by \ samples, so integration is replaced by summation. The cross correlation \ function is just the \"dot product\" of one set of samples with another:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "ccf[template_,data_,delay_]:=\n\t\ template.shift[data,delay,Length[template]]"], "Input", InitializationCell->True, AspectRatioFixed->True], Cell[TextData[{ "The function ", StyleBox["ccf", FontFamily->"Courier", FontWeight->"Bold"], " computes an element of the non-cyclic discrete cross correlation of a \ template with a longer data series." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "If the whole function is needed, it is often more efficient to use FFT \ methods:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "ftccf[template_,data_]:=\n Chop[InverseFourier[\n Fourier[data] \ Conjugate[Fourier[pad[template,Length[data]]]]]]"], "Input", InitializationCell->True, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["The Radar Problem"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "In radar range determination, the shape of the signal is known (although \ its lag is not), and the measurement errors in samples of the received \ signal may often be reasonably assumed to be of constant variance and \ uncorrelated (at least over a wider bandwidth than the signal). In this case, \ it is known", StyleBox["1", FontVariations->{"CompatibilityType"->"Superscript"}], " that cross-correlating the data stream against the known pulse shape and \ finding the peak in the cross correlation minimizes the effect of measurement \ noise to first order." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "The lag estimate is just the location of the peak of the cross correlation \ (see the definition of ", StyleBox["peakdisplacement", FontFamily->"Courier", FontWeight->"Bold"], " in the toolkit):" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "estlag[ template_List, data_List ]:= \n\tpeakdisplacement[ \ ftccf[template,data]]"], "Input", InitializationCell->True, AspectRatioFixed->True], Cell[TextData[ "To first order, the error in the estimated lag is inversely proportional to \ the product of the signal to noise ratio and the \"effective bandwidth\" of \ the measurement. For a derivation in continuous time, see reference 1. "], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "For the following discussion, I will assume that the signal is normalized \ so that the measurement variance of each sample is unity. The received \ signal is presumed to be the product of an amplitude (", StyleBox["amp", FontFamily->"Courier", FontWeight->"Bold"], ") and a ", StyleBox["shape", FontFamily->"Courier", FontWeight->"Bold"], ". If ", StyleBox["shape", FontFamily->"Courier", FontWeight->"Bold"], " is normalized to represent unity signal to noise ratio (however defined), \ ", StyleBox["amp", FontFamily->"Courier", FontWeight->"Bold"], " is then the (amplitude) signal to noise ratio (units are \"sigmas\"). The \ most theoretically reasonable normalization is probably ", StyleBox["shape.shape==1", FontFamily->"Courier", FontWeight->"Bold"], ", but the lag error calculation is insensitive to this normalization. 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Here, I'm normalizing the template to make the amplitude \ equal to the signal to noise ratio, as determined by a matched filter", StyleBox["1", FontVariations->{"CompatibilityType"->"Superscript"}], ". This is reasonable because the pulse shape is known ahead of time, and a \ matched filter is therefore possible." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "template = pad[sburst, 256]/Sqrt[sburst.sburst];\nsnr = 10.;"], "Input", AspectRatioFixed->True], Cell[TextData[ "To save time, I'm using uniformly distributed noise. I can do this because \ the correlation method minimizes the variance of the result (to first order) \ for any uncorrelated noise of uniform variance. The special properties of the \ Gaussian distribution are not required. 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Nevertheless, it is still a reasonably \ effective template, only slightly worse than the true profile."], "Text", Evaluatable->False, AspectRatioFixed->True]}, Open]]}, Open]], Cell[CellGroupData[{Cell[TextData["The Gamma Ray Burst Problem"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "In the gamma ray burst problem, the pulse shape is not known beforehand, \ and the Poisson noise in the data is correlated with the pulse. For data in \ which the expected count rate exceeds one per bin, Poisson correlations may \ be effectively removed by a \"square root transformation", StyleBox["2", FontVariations->{"CompatibilityType"->"Superscript"}], "\":" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["sqtrans[x_] := N[Sqrt[x] + Sqrt[1 + x]]"], "Input", InitializationCell->True, AspectRatioFixed->True], Cell[TextData[{ "The function ", StyleBox["sqtrans", FontFamily->"Courier", FontWeight->"Bold"], " yields a list of data points with uncorrelated noise of unit variance." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "To estimate the burst shape, I take a wavelet transform", StyleBox["4", FontVariations->{"CompatibilityType"->"Superscript"}], " of the measured reference light curve, zero all wavelet coefficients with \ a significance of less than a given threshold, and transform back to the time \ domain. This yields a relatively smooth template, representing the \"true\" \ burst shape without noise. This representation is ", StyleBox["very", FontSlant->"Italic"], " crude when the signal to noise ratio is low, but the cross correlation \ method is so forgiving that it is nevertheless an effective template." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "If the template actually represented the true noiseless light curve, the \ lag errors between the template and any measured light curve could be \ estimated as in the radar problem. For a quick estimate, I'll assume that the \ template is acuurate below. In actual fact, the template is usually better \ correlated with the reference light curve than with an independent \ measurement, and there are also systematic offsets due to the transform \ truncation, so the errors are unsymmetrical. 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error relative to the true lag is less than 1% of the burst duration, \ even at this rather poor signal to noise. This was with an arbitrarily chosen \ random seed (0), and further simulation confirms that this result is \ typical."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "I can also estimate the error using the radar error estimate. I estimate \ the signal to noise ratio by taking the dot product of a normalized template \ with the reference, and then use the radar method. A factor of ", StyleBox["Sqrt[2.]", FontFamily->"Courier", FontWeight->"Bold"], " is needed because noise in both the reference and the lagged curve \ contributes." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData[ "normtemp = template/Sqrt[template.template];\nestsnr = normtemp.sqtrans[ref]\ \nSqrt[2.] deltat[ normtemp, estsnr]"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ 11.20399125535061\ \>", "\<\ 11.204\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True], Cell[OutputFormData["\<\ 0.7026656340707611\ \>", "\<\ 0.702666\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[TextData[ "By this estimate, the burst is thus ~11 sigma, and the RMS (1 sigma) error \ estimate is a bit less than observed in the example above."], "Text", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["More Simulations (hidden in proceedings)"], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "runsim[ bcount_ ] := Module[ {", StyleBox["\t", FontSize->12], "ref,template,reflag,lagged,lag,\n\t\tlagerror,normtemp,estsnr,estrms},\n\t\ ref = simulate[0.,bcount,500.];\n\ttemplate = \ waveletfilter[sqtrans[ref],4.5];\n\treflag = estlag[ template,sqtrans[ref]];\n\ \tlagged = simulate[55.55,bcount,500.];\n\tlag = estlag[ \ template,sqtrans[lagged]];\n\tlagerror = lag-reflag-55.55;\n\tnormtemp = \ template/Sqrt[template.template];\n\testsnr = normtemp.sqtrans[ref];\n\t\ estrms = Sqrt[2.] deltat[ normtemp, estsnr];\n\t{lagerror,estrms}\n]" }], "Input", AspectRatioFixed->True], Cell[TextData[ "Make a matrix containing the results of 25 simulations: "], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["sims = Transpose[ Table[ runsim[300.], {25}]];"], "Input", AspectRatioFixed->True], Cell[TextData[ "The first row is the actual errors in the lag calculation:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["stat[sims[[1]]]"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ {StringForm[\"Mean=``\", NumberForm[0.06373121716396782, 4]], StringForm[\"Sigma=``\", NumberForm[1.435300857303858, 4]], StringForm[\"Range=``\", NumberForm[{-4.315199647554536, 2.23530241743164}, 4]]}\ \>", "\<\ {Mean=0.06373, Sigma=1.435, Range={-4.315, 2.235}}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[TextData["The second row is the corresponding error estimates:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["stat[sims[[2]]]"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ {StringForm[\"Mean=``\", NumberForm[1.264420270770814, 4]], StringForm[\"Sigma=``\", NumberForm[1.664197184801212, 4]], StringForm[\"Range=``\", NumberForm[{0.6409173794561071, 7.057870717787004}, 4]]}\ \>", "\<\ {Mean=1.264, Sigma=1.664, Range={0.6409, 7.058}}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[TextData[ "The error estimates cover a wide range (not surprising at this low signal to \ noise ratio). Note, however, that the mean estimated error is not far from \ the observed value of sigma for the actual errors. "], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Increasing the burst amplitude improves the precision, as expected:"], "Text",\ Evaluatable->False, AspectRatioFixed->True], Cell[TextData["sims = Transpose[ Table[ runsim[600.], {25}]];"], "Input", AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["stat[sims[[1]]]\nstat[sims[[2]]]"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ {StringForm[\"Mean=``\", NumberForm[0.2413833441357036, 4]], StringForm[\"Sigma=``\", NumberForm[0.4791440586527229, 4]], StringForm[\"Range=``\", NumberForm[{-1.09610520341711, 1.34090395564381}, 4]]}\ \>", "\<\ {Mean=0.2414, Sigma=0.4791, Range={-1.096, 1.341}}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True], Cell[OutputFormData["\<\ {StringForm[\"Mean=``\", NumberForm[0.4158845253118196, 4]], StringForm[\"Sigma=``\", NumberForm[0.06616036703817435, 4]], StringForm[\"Range=``\", NumberForm[{0.3252754412771674, 0.5742044246817891}, 4]]}\ \>", "\<\ {Mean=0.4159, Sigma=0.06616, Range={0.3253, 0.5742}}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[TextData["sims = Transpose[ Table[ runsim[1200.], {25}]];"], "Input", AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["stat[sims[[1]]]\nstat[sims[[2]]]"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ {StringForm[\"Mean=``\", NumberForm[-0.02962223188751579, 4]], StringForm[\"Sigma=``\", NumberForm[0.2858387398436671, 4]], StringForm[\"Range=``\", NumberForm[{-0.4958937271378546, 0.4588010371628997}, 4]]}\ \>", "\<\ {Mean=-0.02962, Sigma=0.2858, Range={-0.4959, 0.4588}}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True], Cell[OutputFormData["\<\ {StringForm[\"Mean=``\", NumberForm[0.2563449680813281, 4]], StringForm[\"Sigma=``\", NumberForm[0.02518341322955562, 4]], StringForm[\"Range=``\", NumberForm[{0.2173039823161007, 0.3089697905273454}, 4]]}\ \>", "\<\ {Mean=0.2563, Sigma=0.02518, Range={0.2173, 0.309}}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[TextData[{ "I've also compared the wavelet method with direct correlation of the \ observed light curves", StyleBox["3", FontVariations->{"CompatibilityType"->"Superscript"}], ":" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "rundirect[ bcount_ ] := Module[ {template,lagged,lag,},\n\ttemplate = \ simulate[0.,bcount,500.];\n\tlagged = simulate[55.55,bcount,500.];\n\tlag = \ estlag[ N[template],N[lagged]];\n\tlag-55.55\n\t]"], "Input", AspectRatioFixed->True], Cell[TextData["sims = Table[ rundirect[1200.], {25}];"], "Input", AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["stat[sims]"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ {StringForm[\"Mean=``\", NumberForm[-0.05784567340399633, 4]], StringForm[\"Sigma=``\", NumberForm[0.3175283888678222, 4]], StringForm[\"Range=``\", NumberForm[{-0.5102122015915001, 0.821951219512201}, 4]]}\ \>", "\<\ {Mean=-0.05785, Sigma=0.3175, Range={-0.5102, 0.822}}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[TextData[ "The accuracy of the two methods is comparable, but the direct method \ requires the exchange of about 10 times as much information in this case."], "Text", Evaluatable->False, AspectRatioFixed->True]}, Open]]}, Open]]}, Open]], Cell[CellGroupData[{Cell[TextData["Conclusions"], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Wavelet transform filtering is an effective technique for generating \ correlation templates for burst time lag analysis. The templates are \ characterized by a small number of coefficients, and may therefore be \ exchanged across interplanetary distances at low cost."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "This method is not fully developed. I do not know what the best wavelet \ basis for this method might be. In the simulations I have run, this method \ shows very similar accuracy to direct correlation of the full profiles", StyleBox["3", FontVariations->{"CompatibilityType"->"Superscript"}], ", but I find it difficult to believe that this is a general feature. " }], "Text", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["Acknowledgements"], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "I acknowledge stimulating conversations with Kevin Hurley and George Ricker. \ This work was supported by NASA grant NAGW4763."], "Text", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["Bibliography"], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "1. Skolnik, Merrill I., Introduction to Radar Systems, 462-474 (McGraw-Hill \ 1962)."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "2. Pollard J. H., ", StyleBox["Numerical and Statistical Techniques", FontSlant->"Italic"], ", 243 (Cambridge University Press 1977)." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "3. Hurley, K., AIP Conference Proceedings ", StyleBox["307", FontWeight->"Bold"], ", 687 (1994)." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "4. Scholl, James F., \ ."], "Text", Evaluatable->False, AspectRatioFixed->True]}, Open]]}, Open]] }, FrontEndVersion->"NeXT 3.0", ScreenRectangle->{{0, 1053}, {0, 832}}, AutoGeneratedPackage->None, WindowToolbars->{}, CellGrouping->Automatic, WindowSize->{520, 600}, WindowMargins->{{259, Automatic}, {Automatic, 32}}, PrivateNotebookOptions->{"ColorPalette"->{RGBColor, -1}}, ShowCellLabel->True, ShowCellTags->False, RenderingOptions->{"ObjectDithering"->True, "RasterDithering"->False}, CharacterEncoding->"NeXTAutomaticEncoding", Magnification->1.25 ] (*********************************************************************** Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. 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