(*^ ::[ Information = "This is a Mathematica Notebook file. It contains ASCII text, and can be transferred by email, ftp, or other text-file transfer utility. It should be read or edited using a copy of Mathematica or MathReader. If you received this as email, use your mail application or copy/paste to save everything from the line containing (*^ down to the line containing ^*) into a plain text file. On some systems you may have to give the file a name ending with ".ma" to allow Mathematica to recognize it as a Notebook. The line below identifies what version of Mathematica created this file, but it can be opened using any other version as well."; FrontEndVersion = "Macintosh Mathematica Notebook Front End Version 2.2"; MacintoshStandardFontEncoding; fontset = title, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, e8, 24, "Times"; fontset = subtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, e6, 18, "Times"; fontset = subsubtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, italic, e6, 14, "Times"; fontset = section, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, grayBox, M22, bold, a20, 18, "Times"; fontset = subsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, blackBox, M19, bold, a15, 14, "Times"; fontset = subsubsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, whiteBox, M18, bold, a12, 12, "Times"; fontset = text, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = smalltext, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 10, "Times"; fontset = input, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeInput, M42, N23, bold, L-5, 12, "Courier"; fontset = output, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-5, 12, "Courier"; fontset = message, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, R65535, L-5, 12, "Courier"; fontset = print, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-5, 12, "Courier"; fontset = info, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, B65535, L-5, 12, "Courier"; fontset = postscript, PostScript, formatAsPostScript, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeGraphics, M7, l34, w282, h287, 12, "Courier"; fontset = name, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, italic, 10, "Geneva"; fontset = header, inactive, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = leftheader, inactive, L2, 12, "Times"; fontset = footer, inactive, noKeepOnOnePage, preserveAspect, center, M7, 12, "Times"; fontset = leftfooter, inactive, L2, 12, "Times"; fontset = help, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 10, "Times"; fontset = clipboard, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = completions, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special1, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special2, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special3, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special4, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special5, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; paletteColors = 128; currentKernel; ] :[font = title; inactive; preserveAspect] The Discrete Periodic Wavelet Transform in 1D: :[font = subtitle; inactive; preserveAspect] A Demonstration of the Routine Wavelet1Dv1.m :[font = subsubtitle; inactive; preserveAspect] by James F. Scholl Mathematical Physics Department Rockwell Science Center 1049 Camino Dos Rios Thousand Oaks, CA 91360 e-mail: jfs@risc.rockwell.com :[font = section; inactive; Cclosed; preserveAspect; startGroup] Calling the Routine, With Some Wavelet Bases to Try :[font = input; preserveAspect] <<"Wavelet1Dv1.m" :[font = subsection; inactive; preserveAspect] the main functions: :[font = input; preserveAspect; startGroup] ?DPWT1D :[font = print; inactive; preserveAspect; endGroup] DPWT1D[dat,lfil,lev] performs a 1D discrete wavelet transform of the input signal dat w.r.t the wavelet basis system represented by the input low pass filter lfil. The high pass filter hfil is automatically calculated from lfil. The transform is performed by doing lev levels of decomposition using a recursive pyramid filtering / downsampling scheme. It is preferred that the input data length be a power of 2, but even-length data can be used, but the number of levels of decomposition may be limited. For J levels of decomposition and signal length 2^L, the output is in the form {AJ,{DJ,D(J-1),...,D2,D1}}, where AJ is the approximation subsignal of length 2^(L-J) and Di is the ith detail subsignal of length 2^(L-i). :[font = input; preserveAspect; startGroup] ?IDPWT1D :[font = print; inactive; preserveAspect; endGroup] IDPWT1D[wdat,lfil] performs the inverse 1D discrete wavelet transform on the input coefficients wdat. The input wdat must in the form {AJ,{DJ,D(J-1),...D2,D1}}, where AJ and Di are the approximation subsignal and the ith detail subsignal for a J-level forward decomposition. The number of levels in the recursive pyramid upsampling / filter procedure is automatically determined from the length of wdat. The wavelet basis system is represented by the low pass filter coefficients lfil; the high pass filter is calculated within the routine. :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] Some Basis system filters: :[font = input; dontPreserveAspect] haar = {0.5, 0.5}; :[font = input; preserveAspect] D4 = 0.5 * {N[(1 + Sqrt[3])/4], N[(3 + Sqrt[3])/4], N[(3 - Sqrt[3])/4], N[(1 - Sqrt[3])/4]}; :[font = input; preserveAspect] Coif2 = {-0.051429728471,0.238929728471, 0.602859456942, 0.272140543058, -0.051429972847, -0.011070271529}; :[font = input; dontPreserveAspect; endGroup; endGroup] Coif4 = {0.011587596739, -0.02932013798, -0.04763959031, 0.273021046535, 0.574682393857, 0.294867193696, -0.054085607092, -0.042026480461, 0.016744410163, 0.003967883613, -0.001289203356, -0.000509505399}; :[font = section; inactive; Cclosed; preserveAspect; startGroup] Test Data :[font = subsection; inactive; preserveAspect] Random data of length 32 to demonstrate perfect reconstruction: :[font = input; preserveAspect; startGroup] testdata1 = Table[Random[Real, {-1, 1}], {32}] :[font = output; output; inactive; preserveAspect; endGroup] {-0.6346532678744840391, -0.3071969094076378506, -0.2033912777260766982, 0.2213810663261303262, 0.9507932181061825722, 0.2085242677125683319, -0.5637332512582795736, 0.1440591930963853762, -0.5425866167121139671, 0.5881928188328302988, -0.7715635433912494481, -0.3239351864249406648, 0.8225385933246565773, -0.9606154624831813682, 0.1426703734444883896, 0.3015070611912512962, -0.1089180070210235529, -0.3758660932948300503, 0.4377928564157725284, -0.8022768716426036869, -0.1179072930888229491, -0.5963285256578223782, -0.7397836013190662781, -0.9419649032114396694, 0.3231517716388123772, -0.8066138979300451995, -0.5508360821364066784, 0.7043424924754274211, -0.7414507590328510166, 0.9203321599158241325, 0.1880695985994535685, -0.6580029248288824773} ;[o] {-0.634653, -0.307197, -0.203391, 0.221381, 0.950793, 0.208524, -0.563733, 0.144059, -0.542587, 0.588193, -0.771564, -0.323935, 0.822539, -0.960615, 0.14267, 0.301507, -0.108918, -0.375866, 0.437793, -0.802277, -0.117907, -0.596329, -0.739784, -0.941965, 0.323152, -0.806614, -0.550836, 0.704342, -0.741451, 0.920332, 0.18807, -0.658003} :[font = subsection; inactive; preserveAspect] A Hermite Function Modulated Gaussian with noise to Demonstrate subpictures and scale: :[font = input; preserveAspect] signal = (1./250.) Table[(HermiteH[6,x] Exp[-x^2/2])^2, {x, -6.35, 6.35, 0.1}]; :[font = input; preserveAspect; startGroup] Length[signal] :[font = output; output; inactive; preserveAspect; endGroup] 128 ;[o] 128 :[font = input; preserveAspect] noise = Table[Random[Real, {-10, 10}], {128}]; :[font = input; preserveAspect] testdata2 = signal + noise; :[font = input; preserveAspect; startGroup] {Min[testdata2], Max[testdata2]} :[font = output; output; inactive; preserveAspect; endGroup] {-9.652576895328187823, 109.5984389455244226} ;[o] {-9.65258, 109.598} :[font = subsubsection; inactive; preserveAspect] Signal plot: :[font = input; preserveAspect; startGroup] ListPlot[signal, PlotJoined->True, PlotRange->{-20.,120.}, PlotLabel->"(Hermite[6,x] Exp[-x^2/2])^2"] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 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.89435 .06044 L .90179 .1266 L .90923 .08574 L .91667 .10526 L .92411 .09563 L .93155 .07411 L .93899 .08267 L .94643 .0632 L .95387 .09876 L .96131 .12649 L .96875 .08499 L .97619 .09362 L Mfstroke % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = subsubsection; inactive; preserveAspect] Total Test Signal Plot: :[font = input; preserveAspect; startGroup] ListPlot[testdata2, PlotJoined->True, PlotRange->{-20.,120.}, PlotLabel->"signal and noise"] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.00744048 0.0882906 0.00441453 [ [(20)] .17262 .08829 0 2 Msboxa [(40)] .32143 .08829 0 2 Msboxa [(60)] .47024 .08829 0 2 Msboxa [(80)] .61905 .08829 0 2 Msboxa [(100)] .76786 .08829 0 2 Msboxa [(120)] .91667 .08829 0 2 Msboxa [(signal and noise)] .5 .61803 0 -2 Msboxa [(-20)] .01131 0 1 0 Msboxa [(20)] .01131 .17658 1 0 Msboxa [(40)] .01131 .26487 1 0 Msboxa [(60)] .01131 .35316 1 0 Msboxa [(80)] .01131 .44145 1 0 Msboxa [(100)] .01131 .52974 1 0 Msboxa [(120)] .01131 .61803 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .17262 .08829 m .17262 .09454 L s P [(20)] .17262 .08829 0 2 Mshowa p .002 w .32143 .08829 m .32143 .09454 L s P [(40)] .32143 .08829 0 2 Mshowa p .002 w .47024 .08829 m .47024 .09454 L s P [(60)] .47024 .08829 0 2 Mshowa p .002 w .61905 .08829 m .61905 .09454 L s P [(80)] .61905 .08829 0 2 Mshowa p .002 w .76786 .08829 m .76786 .09454 L s P [(100)] .76786 .08829 0 2 Mshowa p .002 w .91667 .08829 m .91667 .09454 L s P [(120)] .91667 .08829 0 2 Mshowa p .001 w .05357 .08829 m .05357 .09204 L s P p .001 w .08333 .08829 m .08333 .09204 L s P p .001 w .1131 .08829 m .1131 .09204 L s P p .001 w .14286 .08829 m .14286 .09204 L s P p .001 w .20238 .08829 m .20238 .09204 L s P p .001 w .23214 .08829 m .23214 .09204 L s P p .001 w .2619 .08829 m .2619 .09204 L s P p .001 w .29167 .08829 m .29167 .09204 L s P p .001 w .35119 .08829 m .35119 .09204 L s P p .001 w .38095 .08829 m .38095 .09204 L s P p .001 w .41071 .08829 m .41071 .09204 L s P p .001 w .44048 .08829 m .44048 .09204 L s P p .001 w .5 .08829 m .5 .09204 L s P p .001 w .52976 .08829 m .52976 .09204 L s P p .001 w .55952 .08829 m .55952 .09204 L s P p .001 w .58929 .08829 m .58929 .09204 L s P p .001 w .64881 .08829 m .64881 .09204 L s P p .001 w .67857 .08829 m .67857 .09204 L s P p .001 w .70833 .08829 m .70833 .09204 L s P p .001 w .7381 .08829 m .7381 .09204 L s P p .001 w .79762 .08829 m .79762 .09204 L s P p .001 w .82738 .08829 m .82738 .09204 L s P p .001 w .85714 .08829 m .85714 .09204 L s P p .001 w .8869 .08829 m .8869 .09204 L s P p .001 w .94643 .08829 m .94643 .09204 L s P p .001 w .97619 .08829 m .97619 .09204 L s P p .002 w 0 .08829 m 1 .08829 L s P [(signal and noise)] .5 .61803 0 -2 Mshowa p .002 w .02381 0 m .03006 0 L s P [(-20)] .01131 0 1 0 Mshowa p .002 w .02381 .17658 m .03006 .17658 L s P [(20)] .01131 .17658 1 0 Mshowa p .002 w .02381 .26487 m .03006 .26487 L s P [(40)] .01131 .26487 1 0 Mshowa p .002 w .02381 .35316 m .03006 .35316 L s P [(60)] .01131 .35316 1 0 Mshowa p .002 w .02381 .44145 m .03006 .44145 L s P [(80)] .01131 .44145 1 0 Mshowa p .002 w .02381 .52974 m .03006 .52974 L s P [(100)] .01131 .52974 1 0 Mshowa p .002 w .02381 .61803 m .03006 .61803 L s P [(120)] .01131 .61803 1 0 Mshowa p .001 w .02381 .01766 m .02756 .01766 L s P p .001 w .02381 .03532 m .02756 .03532 L s P p .001 w .02381 .05297 m .02756 .05297 L s P p .001 w .02381 .07063 m .02756 .07063 L s P p .001 w .02381 .10595 m .02756 .10595 L s P p .001 w .02381 .12361 m .02756 .12361 L s P p .001 w .02381 .14126 m .02756 .14126 L s P p .001 w .02381 .15892 m .02756 .15892 L s P p .001 w .02381 .19424 m .02756 .19424 L s P p .001 w .02381 .2119 m .02756 .2119 L s P p .001 w .02381 .22956 m .02756 .22956 L s P p .001 w .02381 .24721 m .02756 .24721 L s P p .001 w .02381 .28253 m .02756 .28253 L s P p .001 w .02381 .30019 m .02756 .30019 L s P p .001 w .02381 .31785 m .02756 .31785 L s P p .001 w .02381 .3355 m .02756 .3355 L s P p .001 w .02381 .37082 m .02756 .37082 L s P p .001 w .02381 .38848 m .02756 .38848 L s P p .001 w .02381 .40614 m .02756 .40614 L s P p .001 w .02381 .42379 m .02756 .42379 L s P p .001 w .02381 .45911 m .02756 .45911 L s P p .001 w .02381 .47677 m .02756 .47677 L s P p .001 w .02381 .49443 m .02756 .49443 L s P p .001 w .02381 .51209 m .02756 .51209 L s P p .001 w .02381 .5474 m .02756 .5474 L s P p .001 w .02381 .56506 m .02756 .56506 L s P p .001 w .02381 .58272 m .02756 .58272 L s P p .001 w .02381 .60038 m .02756 .60038 L s P p .002 w .02381 0 m .02381 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath .004 w .03125 .0474 m .03869 .09864 L .04613 .06188 L .05357 .12129 L .06101 .08501 L .06845 .09999 L .07589 .05993 L .08333 .1226 L .09077 .04864 L .09821 .09176 L .10565 .05505 L .1131 .07907 L .12054 .06006 L .12798 .09015 L .13542 .06083 L .14286 .11421 L .1503 .11276 L .15774 .08085 L .16518 .10975 L .17262 .05603 L .18006 .12043 L .1875 .10669 L .19494 .14012 L .20238 .08401 L .20982 .13007 L .21726 .1866 L .2247 .22391 L .23214 .22746 L .23958 .33318 L .24702 .33212 L .25446 .40905 L .2619 .4584 L .26935 .557 L .27679 .53992 L .28423 .52221 L .29167 .45002 L .29911 .33525 L .30655 .24064 L .31399 .19828 L .32143 .1317 L .32887 .04568 L .33631 .07516 L .34375 .16085 L .35119 .22585 L .35863 .34506 L .36607 .41469 L .37351 .38876 L .38095 .28219 L .38839 .24031 L .39583 .11874 L Mistroke .40327 .13233 L .41071 .11601 L .41815 .17874 L .4256 .24432 L .43304 .37864 L .44048 .31651 L .44792 .31991 L .45536 .22536 L .4628 .17011 L .47024 .08331 L .47768 .09945 L .48512 .20918 L .49256 .28824 L .5 .35456 L .50744 .33969 L .51488 .24687 L .52232 .17135 L .52976 .09492 L .5372 .12451 L .54464 .10212 L .55208 .2334 L .55952 .26184 L .56696 .38573 L .5744 .32214 L .58185 .30465 L .58929 .19691 L .59673 .10743 L .60417 .0605 L .61161 .10066 L .61905 .22838 L .62649 .31758 L .63393 .40549 L .64137 .36081 L .64881 .35255 L .65625 .24511 L .66369 .20919 L .67113 .09675 L .67857 .07119 L .68601 .12637 L .69345 .21524 L .70089 .26555 L .70833 .34795 L .71577 .43279 L .72321 .47677 L .73065 .57212 L .7381 .52794 L .74554 .52342 L .75298 .39018 L .76042 .37758 L .76786 .34512 L Mistroke .7753 .23637 L .78274 .23759 L .79018 .19998 L .79762 .12436 L .80506 .12057 L .8125 .12707 L .81994 .12272 L .82738 .11863 L .83482 .11731 L .84226 .05903 L .8497 .10052 L .85714 .0914 L .86458 .11956 L .87202 .08183 L .87946 .11149 L .8869 .11956 L .89435 .06046 L .90179 .12661 L .90923 .08575 L .91667 .10526 L .92411 .09563 L .93155 .07411 L .93899 .08267 L .94643 .0632 L .95387 .09876 L .96131 .12649 L .96875 .08499 L .97619 .09362 L Mfstroke % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = section; inactive; Cclosed; preserveAspect; startGroup] Test Calculation for Perfect Reconstruction :[font = subsection; inactive; preserveAspect] Random signals make very robust tests for perfect reconstruction: :[font = subsubsection; inactive; preserveAspect] Wavelet Transform will be done w.r.t the Daubechies D4 system :[font = text; inactive; preserveAspect] :[font = text; inactive; preserveAspect] First I will do the calculation with too many levels to illustrate this error handling condition: :[font = input; preserveAspect; startGroup] DPWT1D[testdata1, D4, 7] :[font = output; output; inactive; preserveAspect; endGroup] "lev too large" ;[o] lev too large :[font = text; inactive; preserveAspect] :[font = text; inactive; preserveAspect] Next will be the full decomposition (lev=5); this will also demonstrate how the output data is arranged: :[font = input; preserveAspect; startGroup] wcof1 = DPWT1D[testdata1, D4, 5] :[font = output; output; inactive; preserveAspect; endGroup] {{-0.1498209063550616984}, {{-0.08074387269326284749}, {0.213250262496435598, -0.05298105553243019281}, {0.1314726371613773447, -0.09782937764175061115, -0.148432014507318204, -0.0573301987476109327}, {-0.2994279881103769917, -0.129823457803742948, 0.2251503747466968425, -0.01775093076646859454, -0.03954153439505471955, 0.1685108000368382235, -0.4044169592006808479, 0.3045660692878218569}, {0.08914670882230204578, -0.4747119658015780425, 0.5027023177126195069, 0.4930623073998575972, 0.3893335174519671757, -0.9365377172370791323, -0.05840780053142295773, -0.003092765474419131657, -0.602478979908637947, -0.2210020282548677079, 0.01059615148404197357, -0.6836004467855069868, 0.4680877346994515224, 0.8141009224433381903, -0.257937190760253253, 0.1834120210902081599}}} ;[o] {{-0.149821}, {{-0.0807439}, {0.21325, -0.0529811}, {0.131473, -0.0978294, -0.148432, -0.0573302}, {-0.299428, -0.129823, 0.22515, -0.0177509, -0.0395415, 0.168511, -0.404417, 0.304566}, {0.0891467, -0.474712, 0.502702, 0.493062, 0.389334, -0.936538, -0.0584078, -0.00309277, -0.602479, -0.221002, 0.0105962, -0.6836, 0.468088, 0.814101, -0.257937, 0.183412}}} :[font = text; inactive; preserveAspect] :[font = text; inactive; preserveAspect] Next I will reconstruct the data: :[font = input; preserveAspect; startGroup] testdata1R = IDPWT1D[wcof1, D4] :[font = message; inactive; preserveAspect] General::spell1: Possible spelling error: new symbol name "testdata1R" is similar to existing symbol "testdata1". :[font = output; output; inactive; preserveAspect; endGroup] {-0.6346532678744840392, -0.3071969094076378507, -0.2033912777260766982, 0.2213810663261303263, 0.9507932181061825723, 0.208524267712568332, -0.5637332512582795736, 0.1440591930963853762, -0.5425866167121139671, 0.5881928188328302988, -0.7715635433912494481, -0.3239351864249406648, 0.8225385933246565773, -0.9606154624831813682, 0.1426703734444883896, 0.3015070611912512963, -0.1089180070210235529, -0.3758660932948300504, 0.4377928564157725284, -0.8022768716426036871, -0.1179072930888229492, -0.5963285256578223784, -0.7397836013190662784, -0.9419649032114396696, 0.3231517716388123771, -0.8066138979300451996, -0.5508360821364066784, 0.7043424924754274212, -0.7414507590328510166, 0.9203321599158241327, 0.1880695985994535685, -0.6580029248288824773} ;[o] {-0.634653, -0.307197, -0.203391, 0.221381, 0.950793, 0.208524, -0.563733, 0.144059, -0.542587, 0.588193, -0.771564, -0.323935, 0.822539, -0.960615, 0.14267, 0.301507, -0.108918, -0.375866, 0.437793, -0.802277, -0.117907, -0.596329, -0.739784, -0.941965, 0.323152, -0.806614, -0.550836, 0.704342, -0.741451, 0.920332, 0.18807, -0.658003} :[font = text; inactive; preserveAspect] :[font = text; inactive; preserveAspect] The residual between the reconstructed and original signal; note the perfect reconstruction!! :[font = input; preserveAspect; startGroup] testdata1 - testdata1R :[font = output; output; inactive; preserveAspect; endGroup; endGroup] {1.084202172485504434*10^-19, 2.710505431213761085*10^-20, 0., -(1.084202172485504434*10^-19), -(1.084202172485504434*10^-19), -(1.084202172485504434*10^-19), 5.42101086242752217*10^-20, 0., 0., 0., 5.42101086242752217*10^-20, 8.131516293641283255*10^-20, 5.42101086242752217*10^-20, 0., -(1.355252715606880543*10^-20), -(8.131516293641283255*10^-20), 1.355252715606880543*10^-20, 5.42101086242752217*10^-20, -(5.42101086242752217*10^-20), 1.084202172485504434*10^-19, 8.131516293641283255*10^-20, 1.626303258728256651*10^-19, 2.710505431213761085*10^-19, 2.710505431213761085*10^-19, 5.42101086242752217*10^-20, 1.084202172485504434*10^-19, 5.42101086242752217*10^-20, -(1.084202172485504434*10^-19), 0., -(1.626303258728256651*10^-19), 0., 5.42101086242752217*10^-20} ;[o] -19 -20 -19 -19 {1.0842 10 , 2.71051 10 , 0., -1.0842 10 , -1.0842 10 , -19 -20 -20 -1.0842 10 , 5.42101 10 , 0., 0., 0., 5.42101 10 , -20 -20 -20 -20 8.13152 10 , 5.42101 10 , 0., -1.35525 10 , -8.13152 10 , -20 -20 -20 -19 1.35525 10 , 5.42101 10 , -5.42101 10 , 1.0842 10 , -20 -19 -19 -19 8.13152 10 , 1.6263 10 , 2.71051 10 , 2.71051 10 , -20 -19 -20 -19 5.42101 10 , 1.0842 10 , 5.42101 10 , -1.0842 10 , 0., -19 -20 -1.6263 10 , 0., 5.42101 10 } :[font = section; inactive; Cclosed; preserveAspect; startGroup] Test Calculation Illustrating Subsignal Manipulation and Simple-Minded Noise Removal :[font = subsection; inactive; preserveAspect] We can separate signal from noise by looking at the different scale subsignals. We will try to remove the noise. :[font = subsubsection; inactive; preserveAspect] Wavelet Transform will be done w.r.t the 4th order Coifman basis (Coif4) with 4 levels of decomposition :[font = input; preserveAspect] Wcof2 = DPWT1D[testdata2, Coif4, 4]; :[font = text; inactive; preserveAspect] :[font = text; inactive; preserveAspect] These are the component subsignals in the decomposition: :[font = input; preserveAspect; startGroup] A4 = Wcof2[[1]]; Length[A4] :[font = output; output; inactive; preserveAspect; endGroup] 8 ;[o] 8 :[font = input; preserveAspect; startGroup] D4 = Wcof2[[2,1]]; Length[D4] :[font = output; output; inactive; preserveAspect; endGroup] 8 ;[o] 8 :[font = input; preserveAspect; startGroup] D3 = Wcof2[[2,2]]; Length[D3] :[font = output; output; inactive; preserveAspect; endGroup] 16 ;[o] 16 :[font = input; preserveAspect; startGroup] D2 = Wcof2[[2,3]]; Length[D2] :[font = output; output; inactive; preserveAspect; endGroup] 32 ;[o] 32 :[font = input; preserveAspect; startGroup] D1 = Wcof2[[2,4]]; Length[D1] :[font = output; output; inactive; preserveAspect; endGroup] 64 ;[o] 64 :[font = text; inactive; preserveAspect] :[font = text; inactive; preserveAspect] We will plot the subsignals to illustrate how the output is arranged as well as the multiresolution aspect of wavelets. We will plot by increasing frequency. :[font = input; preserveAspect; startGroup] ListPlot[A4, PlotJoined->True, PlotRange->All, PlotLabel->"A4 subsignal / coefficients of W.T.", PlotStyle->RGBColor[1,0,0]] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations -0.112245 0.136054 0.0688735 0.00898545 [ [(2)] .15986 .06887 0 2 Msboxa [(3)] .29592 .06887 0 2 Msboxa [(4)] .43197 .06887 0 2 Msboxa [(5)] .56803 .06887 0 2 Msboxa [(6)] .70408 .06887 0 2 Msboxa [(7)] .84014 .06887 0 2 Msboxa [(8)] .97619 .06887 0 2 Msboxa [(A4 subsignal / coefficients of W.T.)] .5 .61803 0 -2 Msboxa [(10)] .01131 .15873 1 0 Msboxa [(20)] .01131 .24858 1 0 Msboxa [(30)] .01131 .33844 1 0 Msboxa [(40)] .01131 .42829 1 0 Msboxa [(50)] .01131 .51815 1 0 Msboxa [(60)] .01131 .608 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .15986 .06887 m .15986 .07512 L s P [(2)] .15986 .06887 0 2 Mshowa p .002 w .29592 .06887 m .29592 .07512 L s P [(3)] .29592 .06887 0 2 Mshowa p .002 w .43197 .06887 m .43197 .07512 L s P [(4)] .43197 .06887 0 2 Mshowa p .002 w .56803 .06887 m .56803 .07512 L s P [(5)] .56803 .06887 0 2 Mshowa p .002 w .70408 .06887 m .70408 .07512 L s P [(6)] .70408 .06887 0 2 Mshowa p .002 w .84014 .06887 m .84014 .07512 L s P [(7)] .84014 .06887 0 2 Mshowa p .002 w .97619 .06887 m .97619 .07512 L s P [(8)] .97619 .06887 0 2 Mshowa p .001 w .05102 .06887 m .05102 .07262 L s P p .001 w .07823 .06887 m .07823 .07262 L s P p .001 w .10544 .06887 m .10544 .07262 L s P p .001 w .13265 .06887 m .13265 .07262 L s P p .001 w .18707 .06887 m .18707 .07262 L s P p .001 w .21429 .06887 m .21429 .07262 L s P p .001 w .2415 .06887 m .2415 .07262 L s P p .001 w .26871 .06887 m .26871 .07262 L s P p .001 w .32313 .06887 m .32313 .07262 L s P p .001 w .35034 .06887 m .35034 .07262 L s P p .001 w .37755 .06887 m .37755 .07262 L s P p .001 w .40476 .06887 m .40476 .07262 L s P p .001 w .45918 .06887 m .45918 .07262 L s P p .001 w .48639 .06887 m .48639 .07262 L s P p .001 w .51361 .06887 m .51361 .07262 L s P p .001 w .54082 .06887 m .54082 .07262 L s P p .001 w .59524 .06887 m .59524 .07262 L s P p .001 w .62245 .06887 m .62245 .07262 L s P p .001 w .64966 .06887 m .64966 .07262 L s P p .001 w .67687 .06887 m .67687 .07262 L s P p .001 w .73129 .06887 m .73129 .07262 L s P p .001 w .7585 .06887 m .7585 .07262 L s P p .001 w .78571 .06887 m .78571 .07262 L s P p .001 w .81293 .06887 m .81293 .07262 L s P p .001 w .86735 .06887 m .86735 .07262 L s P p .001 w .89456 .06887 m .89456 .07262 L s P p .001 w .92177 .06887 m .92177 .07262 L s P p .001 w .94898 .06887 m .94898 .07262 L s P p .002 w 0 .06887 m 1 .06887 L s P [(A4 subsignal / coefficients of W.T.)] .5 .61803 0 -2 Mshowa p .002 w .02381 .15873 m .03006 .15873 L s P [(10)] .01131 .15873 1 0 Mshowa p .002 w .02381 .24858 m .03006 .24858 L s P [(20)] .01131 .24858 1 0 Mshowa p .002 w .02381 .33844 m .03006 .33844 L s P [(30)] .01131 .33844 1 0 Mshowa p .002 w .02381 .42829 m .03006 .42829 L s P [(40)] .01131 .42829 1 0 Mshowa p .002 w .02381 .51815 m .03006 .51815 L s P [(50)] .01131 .51815 1 0 Mshowa p .002 w .02381 .608 m .03006 .608 L s P [(60)] .01131 .608 1 0 Mshowa p .001 w .02381 .08684 m .02756 .08684 L s P p .001 w .02381 .10482 m .02756 .10482 L s P p .001 w .02381 .12279 m .02756 .12279 L s P p .001 w .02381 .14076 m .02756 .14076 L s P p .001 w .02381 .1767 m .02756 .1767 L s P p .001 w .02381 .19467 m .02756 .19467 L s P p .001 w .02381 .21264 m .02756 .21264 L s P p .001 w .02381 .23061 m .02756 .23061 L s P p .001 w .02381 .26655 m .02756 .26655 L s P p .001 w .02381 .28452 m .02756 .28452 L s P p .001 w .02381 .3025 m .02756 .3025 L s P p .001 w .02381 .32047 m .02756 .32047 L s P p .001 w .02381 .35641 m .02756 .35641 L s P p .001 w .02381 .37438 m .02756 .37438 L s P p .001 w .02381 .39235 m .02756 .39235 L s P p .001 w .02381 .41032 m .02756 .41032 L s P p .001 w .02381 .44626 m .02756 .44626 L s P p .001 w .02381 .46423 m .02756 .46423 L s P p .001 w .02381 .4822 m .02756 .4822 L s P p .001 w .02381 .50018 m .02756 .50018 L s P p .001 w .02381 .53612 m .02756 .53612 L s P p .001 w .02381 .55409 m .02756 .55409 L s P p .001 w .02381 .57206 m .02756 .57206 L s P p .001 w .02381 .59003 m .02756 .59003 L s P p .001 w .02381 .0509 m .02756 .0509 L s P p .001 w .02381 .03293 m .02756 .03293 L s P p .001 w .02381 .01496 m .02756 .01496 L s P p .002 w .02381 0 m .02381 .61803 L s P P p 1 0 0 r .004 w .02381 .34675 m .15986 .31092 L .29592 .60332 L .43197 .16282 L .56803 .07787 L .70408 .01472 L .84014 .4896 L .97619 .40513 L s P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = input; preserveAspect; startGroup] ListPlot[D4, PlotJoined->True, PlotRange->All, PlotLabel->"D4 subsignal / coefficients of W.T.", PlotStyle->RGBColor[0,0,1]] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations -0.112245 0.136054 0.439519 0.025687 [ [(2)] .15986 .43952 0 2 Msboxa [(3)] .29592 .43952 0 2 Msboxa [(4)] .43197 .43952 0 2 Msboxa [(5)] .56803 .43952 0 2 Msboxa [(6)] .70408 .43952 0 2 Msboxa [(7)] .84014 .43952 0 2 Msboxa [(8)] .97619 .43952 0 2 Msboxa [(D4 subsignal / coefficients of W.T.)] .5 .61803 0 -2 Msboxa [(-15)] .01131 .05421 1 0 Msboxa [(-10)] .01131 .18265 1 0 Msboxa [(-5)] .01131 .31108 1 0 Msboxa [(5)] .01131 .56795 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .15986 .43952 m .15986 .44577 L s P [(2)] .15986 .43952 0 2 Mshowa p .002 w .29592 .43952 m .29592 .44577 L s P [(3)] .29592 .43952 0 2 Mshowa p .002 w .43197 .43952 m .43197 .44577 L s P [(4)] .43197 .43952 0 2 Mshowa p .002 w .56803 .43952 m .56803 .44577 L s P [(5)] .56803 .43952 0 2 Mshowa p .002 w .70408 .43952 m .70408 .44577 L s P [(6)] .70408 .43952 0 2 Mshowa p .002 w .84014 .43952 m .84014 .44577 L s P [(7)] .84014 .43952 0 2 Mshowa p .002 w .97619 .43952 m .97619 .44577 L s P [(8)] .97619 .43952 0 2 Mshowa p .001 w .05102 .43952 m .05102 .44327 L s P p .001 w .07823 .43952 m .07823 .44327 L s P p .001 w .10544 .43952 m .10544 .44327 L s P p .001 w .13265 .43952 m .13265 .44327 L s P p .001 w .18707 .43952 m .18707 .44327 L s P p .001 w .21429 .43952 m .21429 .44327 L s P p .001 w .2415 .43952 m .2415 .44327 L s P p .001 w .26871 .43952 m .26871 .44327 L s P p .001 w .32313 .43952 m .32313 .44327 L s P p .001 w .35034 .43952 m .35034 .44327 L s P p .001 w .37755 .43952 m .37755 .44327 L s P p .001 w .40476 .43952 m .40476 .44327 L s P p .001 w .45918 .43952 m .45918 .44327 L s P p .001 w .48639 .43952 m .48639 .44327 L s P p .001 w .51361 .43952 m .51361 .44327 L s P p .001 w .54082 .43952 m .54082 .44327 L s P p .001 w .59524 .43952 m .59524 .44327 L s P p .001 w .62245 .43952 m .62245 .44327 L s P p .001 w .64966 .43952 m .64966 .44327 L s P p .001 w .67687 .43952 m .67687 .44327 L s P p .001 w .73129 .43952 m .73129 .44327 L s P p .001 w .7585 .43952 m .7585 .44327 L s P p .001 w .78571 .43952 m .78571 .44327 L s P p .001 w .81293 .43952 m .81293 .44327 L s P p .001 w .86735 .43952 m .86735 .44327 L s P p .001 w .89456 .43952 m .89456 .44327 L s P p .001 w .92177 .43952 m .92177 .44327 L s P p .001 w .94898 .43952 m .94898 .44327 L s P p .002 w 0 .43952 m 1 .43952 L s P [(D4 subsignal / coefficients of W.T.)] .5 .61803 0 -2 Mshowa p .002 w .02381 .05421 m .03006 .05421 L s P [(-15)] .01131 .05421 1 0 Mshowa p .002 w .02381 .18265 m .03006 .18265 L s P [(-10)] .01131 .18265 1 0 Mshowa p .002 w .02381 .31108 m .03006 .31108 L s P [(-5)] .01131 .31108 1 0 Mshowa p .002 w .02381 .56795 m .03006 .56795 L s P [(5)] .01131 .56795 1 0 Mshowa p .001 w .02381 .0799 m .02756 .0799 L s P p .001 w .02381 .10559 m .02756 .10559 L s P p .001 w .02381 .13128 m .02756 .13128 L s P p .001 w .02381 .15696 m .02756 .15696 L s P p .001 w .02381 .20834 m .02756 .20834 L s P p .001 w .02381 .23402 m .02756 .23402 L s P p .001 w .02381 .25971 m .02756 .25971 L s P p .001 w .02381 .2854 m .02756 .2854 L s P p .001 w .02381 .33677 m .02756 .33677 L s P p .001 w .02381 .36246 m .02756 .36246 L s P p .001 w .02381 .38815 m .02756 .38815 L s P p .001 w .02381 .41383 m .02756 .41383 L s P p .001 w .02381 .46521 m .02756 .46521 L s P p .001 w .02381 .49089 m .02756 .49089 L s P p .001 w .02381 .51658 m .02756 .51658 L s P p .001 w .02381 .54227 m .02756 .54227 L s P p .001 w .02381 .02853 m .02756 .02853 L s P p .001 w .02381 .00284 m .02756 .00284 L s P p .001 w .02381 .59364 m .02756 .59364 L s P p .002 w .02381 0 m .02381 .61803 L s P P p 0 0 1 r .004 w .02381 .08921 m .15986 .60332 L .29592 .47991 L .43197 .51832 L .56803 .01472 L .70408 .57838 L .84014 .42403 L .97619 .45324 L s P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = input; preserveAspect; startGroup] ListPlot[D3, PlotJoined->True, PlotRange->All, PlotLabel->"D3 subsignal / coefficients of W.T.", PlotStyle->RGBColor[0,0,1]] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations -0.0396825 0.0634921 0.354209 0.00887148 [ [(4)] .21429 .35421 0 2 Msboxa [(6)] .34127 .35421 0 2 Msboxa [(8)] .46825 .35421 0 2 Msboxa [(10)] .59524 .35421 0 2 Msboxa [(12)] .72222 .35421 0 2 Msboxa [(14)] .84921 .35421 0 2 Msboxa [(16)] .97619 .35421 0 2 Msboxa [(D3 subsignal / coefficients of W.T.)] .5 .61803 0 -2 Msboxa [(-30)] .0748 .08806 1 0 Msboxa [(-20)] .0748 .17678 1 0 Msboxa [(-10)] .0748 .26549 1 0 Msboxa [(10)] .0748 .44292 1 0 Msboxa [(20)] .0748 .53164 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .21429 .35421 m .21429 .36046 L s P [(4)] .21429 .35421 0 2 Mshowa p .002 w .34127 .35421 m .34127 .36046 L s P [(6)] .34127 .35421 0 2 Mshowa p .002 w .46825 .35421 m .46825 .36046 L s P [(8)] .46825 .35421 0 2 Mshowa p .002 w .59524 .35421 m .59524 .36046 L s P [(10)] .59524 .35421 0 2 Mshowa p .002 w .72222 .35421 m .72222 .36046 L s P [(12)] .72222 .35421 0 2 Mshowa p .002 w .84921 .35421 m .84921 .36046 L s P [(14)] .84921 .35421 0 2 Mshowa p .002 w .97619 .35421 m .97619 .36046 L s P [(16)] .97619 .35421 0 2 Mshowa p .001 w .1127 .35421 m .1127 .35796 L s P p .001 w .1381 .35421 m .1381 .35796 L s P p .001 w .16349 .35421 m .16349 .35796 L s P p .001 w .18889 .35421 m .18889 .35796 L s P p .001 w .23968 .35421 m .23968 .35796 L s P p .001 w .26508 .35421 m .26508 .35796 L s P p .001 w .29048 .35421 m .29048 .35796 L s P p .001 w .31587 .35421 m .31587 .35796 L s P p .001 w .36667 .35421 m .36667 .35796 L s P p .001 w .39206 .35421 m .39206 .35796 L s P p .001 w .41746 .35421 m .41746 .35796 L s P p .001 w .44286 .35421 m .44286 .35796 L s P p .001 w .49365 .35421 m .49365 .35796 L s P p .001 w .51905 .35421 m .51905 .35796 L s P p .001 w .54444 .35421 m .54444 .35796 L s P p .001 w .56984 .35421 m .56984 .35796 L s P p .001 w .62063 .35421 m .62063 .35796 L s P p .001 w .64603 .35421 m .64603 .35796 L s P p .001 w .67143 .35421 m .67143 .35796 L s P p .001 w .69683 .35421 m .69683 .35796 L s P p .001 w .74762 .35421 m .74762 .35796 L s P p .001 w .77302 .35421 m .77302 .35796 L s P p .001 w .79841 .35421 m .79841 .35796 L s P p .001 w .82381 .35421 m .82381 .35796 L s P p .001 w .8746 .35421 m .8746 .35796 L s P p .001 w .9 .35421 m .9 .35796 L s P p .001 w .9254 .35421 m .9254 .35796 L s P p .001 w .95079 .35421 m .95079 .35796 L s P p .001 w .0619 .35421 m .0619 .35796 L s P p .001 w .03651 .35421 m .03651 .35796 L s P p .001 w .01111 .35421 m .01111 .35796 L s P p .002 w 0 .35421 m 1 .35421 L s P [(D3 subsignal / coefficients of W.T.)] .5 .61803 0 -2 Mshowa p .002 w .0873 .08806 m .09355 .08806 L s P [(-30)] .0748 .08806 1 0 Mshowa p .002 w .0873 .17678 m .09355 .17678 L s P [(-20)] .0748 .17678 1 0 Mshowa p .002 w .0873 .26549 m .09355 .26549 L s P [(-10)] .0748 .26549 1 0 Mshowa p .002 w .0873 .44292 m .09355 .44292 L s P [(10)] .0748 .44292 1 0 Mshowa p .002 w .0873 .53164 m .09355 .53164 L s P [(20)] .0748 .53164 1 0 Mshowa p .001 w .0873 .01709 m .09105 .01709 L s P p .001 w .0873 .03484 m .09105 .03484 L s P p .001 w .0873 .05258 m .09105 .05258 L s P p .001 w .0873 .07032 m .09105 .07032 L s P p .001 w .0873 .10581 m .09105 .10581 L s P p .001 w .0873 .12355 m .09105 .12355 L s P p .001 w .0873 .14129 m .09105 .14129 L s P p .001 w .0873 .15904 m .09105 .15904 L s P p .001 w .0873 .19452 m .09105 .19452 L s P p .001 w .0873 .21227 m .09105 .21227 L s P p .001 w .0873 .23001 m .09105 .23001 L s P p .001 w .0873 .24775 m .09105 .24775 L s P p .001 w .0873 .28324 m .09105 .28324 L s P p .001 w .0873 .30098 m .09105 .30098 L s P p .001 w .0873 .31872 m .09105 .31872 L s P p .001 w .0873 .33647 m .09105 .33647 L s P p .001 w .0873 .37195 m .09105 .37195 L s P p .001 w .0873 .3897 m .09105 .3897 L s P p .001 w .0873 .40744 m .09105 .40744 L s P p .001 w .0873 .42518 m .09105 .42518 L s P p .001 w .0873 .46067 m .09105 .46067 L s P p .001 w .0873 .47841 m .09105 .47841 L s P p .001 w .0873 .49615 m .09105 .49615 L s P p .001 w .0873 .5139 m .09105 .5139 L s P p .001 w .0873 .54938 m .09105 .54938 L s P p .001 w .0873 .56712 m .09105 .56712 L s P p .001 w .0873 .58487 m .09105 .58487 L s P p .001 w .0873 .60261 m .09105 .60261 L s P p .002 w .0873 0 m .0873 .61803 L s P P p 0 0 1 r .004 w .02381 .01472 m .0873 .33335 L .15079 .48473 L .21429 .53629 L .27778 .58611 L .34127 .49836 L .40476 .06276 L .46825 .53666 L .53175 .27657 L .59524 .37412 L .65873 .3445 L .72222 .33837 L .78571 .34902 L .84921 .38778 L .9127 .26363 L .97619 .60332 L s P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = input; preserveAspect; startGroup] ListPlot[D2, PlotJoined->True, PlotRange->All, PlotLabel->"D2 subsignal / coefficients of W.T.", PlotStyle->RGBColor[0,0,1]] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations -0.00691244 0.030722 0.299353 0.0189137 [ [(5)] .1467 .29935 0 2 Msboxa [(10)] .30031 .29935 0 2 Msboxa [(15)] .45392 .29935 0 2 Msboxa [(20)] .60753 .29935 0 2 Msboxa [(25)] .76114 .29935 0 2 Msboxa [(30)] .91475 .29935 0 2 Msboxa [(D2 subsignal / coefficients of W.T.)] .5 .61803 0 -2 Msboxa [(-15)] -0.01941 .01565 1 0 Msboxa [(-10)] -0.01941 .11022 1 0 Msboxa [(-5)] -0.01941 .20478 1 0 Msboxa [(5)] -0.01941 .39392 1 0 Msboxa [(10)] -0.01941 .48849 1 0 Msboxa [(15)] -0.01941 .58306 1 0 Msboxa [ -0.00791 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .1467 .29935 m .1467 .3056 L s P [(5)] .1467 .29935 0 2 Mshowa p .002 w .30031 .29935 m .30031 .3056 L s P [(10)] .30031 .29935 0 2 Mshowa p .002 w .45392 .29935 m .45392 .3056 L s P [(15)] .45392 .29935 0 2 Mshowa p .002 w .60753 .29935 m .60753 .3056 L s P [(20)] .60753 .29935 0 2 Mshowa p .002 w .76114 .29935 m .76114 .3056 L s P [(25)] .76114 .29935 0 2 Mshowa p .002 w .91475 .29935 m .91475 .3056 L s P [(30)] .91475 .29935 0 2 Mshowa p .001 w .02381 .29935 m .02381 .3031 L s P p .001 w .05453 .29935 m .05453 .3031 L s P p .001 w .08525 .29935 m .08525 .3031 L s P p .001 w .11598 .29935 m .11598 .3031 L s P p .001 w .17742 .29935 m .17742 .3031 L s P p .001 w .20814 .29935 m .20814 .3031 L s P p .001 w .23886 .29935 m .23886 .3031 L s P p .001 w .26959 .29935 m .26959 .3031 L s P p .001 w .33103 .29935 m .33103 .3031 L s P p .001 w .36175 .29935 m .36175 .3031 L s P p .001 w .39247 .29935 m .39247 .3031 L s P p .001 w .4232 .29935 m .4232 .3031 L s P p .001 w .48464 .29935 m .48464 .3031 L s P p .001 w .51536 .29935 m .51536 .3031 L s P p .001 w .54608 .29935 m .54608 .3031 L s P p .001 w .5768 .29935 m .5768 .3031 L s P p .001 w .63825 .29935 m .63825 .3031 L s P p .001 w .66897 .29935 m .66897 .3031 L s P p .001 w .69969 .29935 m .69969 .3031 L s P p .001 w .73041 .29935 m .73041 .3031 L s P p .001 w .79186 .29935 m .79186 .3031 L s P p .001 w .82258 .29935 m .82258 .3031 L s P p .001 w .8533 .29935 m .8533 .3031 L s P p .001 w .88402 .29935 m .88402 .3031 L s P p .001 w .94547 .29935 m .94547 .3031 L s P p .001 w .97619 .29935 m .97619 .3031 L s P p .002 w 0 .29935 m 1 .29935 L s P [(D2 subsignal / coefficients of W.T.)] .5 .61803 0 -2 Mshowa p .002 w -0.00691 .01565 m -0.00066 .01565 L s P [(-15)] -0.01941 .01565 1 0 Mshowa p .002 w -0.00691 .11022 m -0.00066 .11022 L s P [(-10)] -0.01941 .11022 1 0 Mshowa p .002 w -0.00691 .20478 m -0.00066 .20478 L s P [(-5)] -0.01941 .20478 1 0 Mshowa p .002 w -0.00691 .39392 m -0.00066 .39392 L s P [(5)] -0.01941 .39392 1 0 Mshowa p .002 w -0.00691 .48849 m -0.00066 .48849 L s P [(10)] -0.01941 .48849 1 0 Mshowa p .002 w -0.00691 .58306 m -0.00066 .58306 L s P [(15)] -0.01941 .58306 1 0 Mshowa p .001 w -0.00691 .03456 m -0.00316 .03456 L s P p .001 w -0.00691 .05347 m -0.00316 .05347 L s P p .001 w -0.00691 .07239 m -0.00316 .07239 L s P p .001 w -0.00691 .0913 m -0.00316 .0913 L s P p .001 w -0.00691 .12913 m -0.00316 .12913 L s P p .001 w -0.00691 .14804 m -0.00316 .14804 L s P p .001 w -0.00691 .16696 m -0.00316 .16696 L s P p .001 w -0.00691 .18587 m -0.00316 .18587 L s P p .001 w -0.00691 .2237 m -0.00316 .2237 L s P p .001 w -0.00691 .24261 m -0.00316 .24261 L s P p .001 w -0.00691 .26153 m -0.00316 .26153 L s P p .001 w -0.00691 .28044 m -0.00316 .28044 L s P p .001 w -0.00691 .31827 m -0.00316 .31827 L s P p .001 w -0.00691 .33718 m -0.00316 .33718 L s P p .001 w -0.00691 .35609 m -0.00316 .35609 L s P p .001 w -0.00691 .37501 m -0.00316 .37501 L s P p .001 w -0.00691 .41284 m -0.00316 .41284 L s P p .001 w -0.00691 .43175 m -0.00316 .43175 L s P p .001 w -0.00691 .45066 m -0.00316 .45066 L s P p .001 w -0.00691 .46958 m -0.00316 .46958 L s P p .001 w -0.00691 .5074 m -0.00316 .5074 L s P p .001 w -0.00691 .52632 m -0.00316 .52632 L s P p .001 w -0.00691 .54523 m -0.00316 .54523 L s P p .001 w -0.00691 .56415 m -0.00316 .56415 L s P p .001 w -0.00691 .60197 m -0.00316 .60197 L s P p .002 w -0.00691 0 m -0.00691 .61803 L s P P p 0 0 1 r .004 w .02381 .2534 m .05453 .32887 L .08525 .31604 L .11598 .18883 L .1467 .45832 L .17742 .3088 L .20814 .17944 L .23886 .55459 L .26959 .0675 L .30031 .60332 L .33103 .11864 L .36175 .53214 L .39247 .13522 L .4232 .26343 L .45392 .4623 L .48464 .01472 L .51536 .52411 L .54608 .07213 L .5768 .33304 L .60753 .46753 L .63825 .23378 L .66897 .32231 L .69969 .36291 L .73041 .25539 L .76114 .32695 L .79186 .32909 L .82258 .23085 L .8533 .34551 L .88402 .30451 L .91475 .28985 L .94547 .28997 L .97619 .2842 L s P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = input; preserveAspect; startGroup] ListPlot[D1, PlotJoined->True, PlotRange->All, PlotLabel->"D1 subsignal / coefficients of W.T.", PlotStyle->RGBColor[0,0,1]] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.00869237 0.0151172 0.312884 0.0341803 [ [(10)] .15986 .31288 0 2 Msboxa [(20)] .31104 .31288 0 2 Msboxa [(30)] .46221 .31288 0 2 Msboxa [(40)] .61338 .31288 0 2 Msboxa [(50)] .76455 .31288 0 2 Msboxa [(60)] .91572 .31288 0 2 Msboxa [(D1 subsignal / coefficients of W.T.)] .5 .61803 0 -2 Msboxa [(-7.5)] -0.00381 .05653 1 0 Msboxa [(-5)] -0.00381 .14198 1 0 Msboxa [(-2.5)] -0.00381 .22743 1 0 Msboxa [(2.5)] -0.00381 .39833 1 0 Msboxa [(5)] -0.00381 .48379 1 0 Msboxa [(7.5)] -0.00381 .56924 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .15986 .31288 m .15986 .31913 L s P [(10)] .15986 .31288 0 2 Mshowa p .002 w .31104 .31288 m .31104 .31913 L s P [(20)] .31104 .31288 0 2 Mshowa p .002 w .46221 .31288 m .46221 .31913 L s P [(30)] .46221 .31288 0 2 Mshowa p .002 w .61338 .31288 m .61338 .31913 L s P [(40)] .61338 .31288 0 2 Mshowa p .002 w .76455 .31288 m .76455 .31913 L s P [(50)] .76455 .31288 0 2 Mshowa p .002 w .91572 .31288 m .91572 .31913 L s P [(60)] .91572 .31288 0 2 Mshowa p .001 w .03893 .31288 m .03893 .31663 L s P p .001 w .06916 .31288 m .06916 .31663 L s P p .001 w .0994 .31288 m .0994 .31663 L s P p .001 w .12963 .31288 m .12963 .31663 L s P p .001 w .1901 .31288 m .1901 .31663 L s P p .001 w .22033 .31288 m .22033 .31663 L s P p .001 w .25057 .31288 m .25057 .31663 L s P p .001 w .2808 .31288 m .2808 .31663 L s P p .001 w .34127 .31288 m .34127 .31663 L s P p .001 w .3715 .31288 m .3715 .31663 L s P p .001 w .40174 .31288 m .40174 .31663 L s P p .001 w .43197 .31288 m .43197 .31663 L s P p .001 w .49244 .31288 m .49244 .31663 L s P p .001 w .52268 .31288 m .52268 .31663 L s P p .001 w .55291 .31288 m .55291 .31663 L s P p .001 w .58314 .31288 m .58314 .31663 L s P p .001 w .64361 .31288 m .64361 .31663 L s P p .001 w .67385 .31288 m .67385 .31663 L s P p .001 w .70408 .31288 m .70408 .31663 L s P p .001 w .73432 .31288 m .73432 .31663 L s P p .001 w .79478 .31288 m .79478 .31663 L s P p .001 w .82502 .31288 m .82502 .31663 L s P p .001 w .85525 .31288 m .85525 .31663 L s P p .001 w .88549 .31288 m .88549 .31663 L s P p .001 w .94596 .31288 m .94596 .31663 L s P p .001 w .97619 .31288 m .97619 .31663 L s P p .002 w 0 .31288 m 1 .31288 L s P [(D1 subsignal / coefficients of W.T.)] .5 .61803 0 -2 Mshowa p .002 w .00869 .05653 m .01494 .05653 L s P [(-7.5)] -0.00381 .05653 1 0 Mshowa p .002 w .00869 .14198 m .01494 .14198 L s P [(-5)] -0.00381 .14198 1 0 Mshowa p .002 w .00869 .22743 m .01494 .22743 L s P [(-2.5)] -0.00381 .22743 1 0 Mshowa p .002 w .00869 .39833 m .01494 .39833 L s P [(2.5)] -0.00381 .39833 1 0 Mshowa p .002 w .00869 .48379 m .01494 .48379 L s P [(5)] -0.00381 .48379 1 0 Mshowa p .002 w .00869 .56924 m .01494 .56924 L s P [(7.5)] -0.00381 .56924 1 0 Mshowa p .001 w .00869 .07362 m .01244 .07362 L s P p .001 w .00869 .09071 m .01244 .09071 L s P p .001 w .00869 .1078 m .01244 .1078 L s P p .001 w .00869 .12489 m .01244 .12489 L s P p .001 w .00869 .15907 m .01244 .15907 L s P p .001 w .00869 .17616 m .01244 .17616 L s P p .001 w .00869 .19325 m .01244 .19325 L s P p .001 w .00869 .21034 m .01244 .21034 L s P p .001 w .00869 .24452 m .01244 .24452 L s P p .001 w .00869 .26161 m .01244 .26161 L s P p .001 w .00869 .2787 m .01244 .2787 L s P p .001 w .00869 .29579 m .01244 .29579 L s P p .001 w .00869 .32997 m .01244 .32997 L s P p .001 w .00869 .34706 m .01244 .34706 L s P p .001 w .00869 .36415 m .01244 .36415 L s P p .001 w .00869 .38124 m .01244 .38124 L s P p .001 w .00869 .41542 m .01244 .41542 L s P p .001 w .00869 .43251 m .01244 .43251 L s P p .001 w .00869 .44961 m .01244 .44961 L s P p .001 w .00869 .4667 m .01244 .4667 L s P p .001 w .00869 .50088 m .01244 .50088 L s P p .001 w .00869 .51797 m .01244 .51797 L s P p .001 w .00869 .53506 m .01244 .53506 L s P p .001 w .00869 .55215 m .01244 .55215 L s P p .001 w .00869 .03944 m .01244 .03944 L s P p .001 w .00869 .02235 m .01244 .02235 L s P p .001 w .00869 .00526 m .01244 .00526 L s P p .001 w .00869 .58633 m .01244 .58633 L s P p .001 w .00869 .60342 m .01244 .60342 L s P p .002 w .00869 0 m .00869 .61803 L s P P p 0 0 1 r .004 w .02381 .60332 m .03893 .46386 L .05404 .39369 L .06916 .43324 L .08428 .43358 L .0994 .18066 L .11451 .07214 L .12963 .25011 L .14475 .09229 L .15986 .41483 L .17498 .09114 L .1901 .16861 L .20522 .19161 L .22033 .24399 L .23545 .38241 L .25057 .19901 L .26568 .4261 L .2808 .26305 L .29592 .17508 L .31104 .47519 L .32615 .13201 L .34127 .08244 L .35639 .22368 L .3715 .16881 L .38662 .11008 L .40174 .25252 L .41686 .14245 L .43197 .41396 L .44709 .42009 L .46221 .24524 L .47732 .1622 L .49244 .04468 L .50756 .10136 L .52268 .15424 L .53779 .30432 L .55291 .19315 L .56803 .416 L .58314 .51464 L .59826 .45075 L .61338 .51202 L .6285 .17771 L .64361 .45257 L .65873 .29745 L .67385 .16029 L .68896 .19858 L .70408 .01472 L .7192 .51049 L .73432 .41039 L .74943 .15517 L .76455 .36635 L Mistroke .77967 .31637 L .79478 .09855 L .8099 .26745 L .82502 .14711 L .84014 .45802 L .85525 .53912 L .87037 .35881 L .88549 .25764 L .9006 .18771 L .91572 .46203 L .93084 .40823 L .94596 .4978 L .96107 .50209 L .97619 .39115 L Mfstroke P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = text; inactive; preserveAspect] :[font = text; inactive; preserveAspect] Note the noise manifests itself in the highest frequency subsignal; lets try to remove it by setting these coefficients to zero: :[font = input; preserveAspect; startGroup] D1prime = Table[0.0, {64}] :[font = output; output; inactive; preserveAspect; endGroup] {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.} ;[o] {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.} :[font = text; inactive; preserveAspect] :[font = text; inactive; preserveAspect] The revised coefficients for the inverse wavelet transform: This also illustrates how we can do some signal processing. :[font = input; preserveAspect; startGroup] Wcof2a = {A4, {D4, D3, D2, D1prime}}; :[font = message; inactive; preserveAspect; endGroup] General::spell1: Possible spelling error: new symbol name "Wcof2a" is similar to existing symbol "Wcof2". :[font = input; preserveAspect; startGroup] testdata2R = IDPWT1D[Wcof2a, Coif4]; :[font = message; inactive; preserveAspect; endGroup] General::spell1: Possible spelling error: new symbol name "testdata2R" is similar to existing symbol "testdata2". :[font = text; inactive; preserveAspect] :[font = text; inactive; preserveAspect] A plot of the reconstructed signal: :[font = input; preserveAspect; startGroup] recon1 = ListPlot[testdata2R, PlotRange->All, PlotJoined->True, PlotStyle->RGBColor[1,0,0], PlotLabel->"signal with some noise removed"] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0163105 0.00749906 0.0566297 0.00515787 [ [(20)] .16629 .05663 0 2 Msboxa [(40)] .31627 .05663 0 2 Msboxa [(60)] .46625 .05663 0 2 Msboxa [(80)] .61624 .05663 0 2 Msboxa [(100)] .76622 .05663 0 2 Msboxa [(120)] .9162 .05663 0 2 Msboxa [(signal with some noise removed)] .5 .61803 0 -2 Msboxa [(20)] .00381 .15979 1 0 Msboxa [(40)] .00381 .26294 1 0 Msboxa [(60)] .00381 .3661 1 0 Msboxa [(80)] .00381 .46926 1 0 Msboxa [(100)] .00381 .57242 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .16629 .05663 m .16629 .06288 L s P [(20)] .16629 .05663 0 2 Mshowa p .002 w .31627 .05663 m .31627 .06288 L s P [(40)] .31627 .05663 0 2 Mshowa p .002 w .46625 .05663 m .46625 .06288 L s P [(60)] .46625 .05663 0 2 Mshowa p .002 w .61624 .05663 m .61624 .06288 L s P [(80)] .61624 .05663 0 2 Mshowa p .002 w .76622 .05663 m .76622 .06288 L s P [(100)] .76622 .05663 0 2 Mshowa p .002 w .9162 .05663 m .9162 .06288 L s P [(120)] .9162 .05663 0 2 Mshowa p .001 w .04631 .05663 m .04631 .06038 L s P p .001 w .0763 .05663 m .0763 .06038 L s P p .001 w .1063 .05663 m .1063 .06038 L s P p .001 w .1363 .05663 m .1363 .06038 L s P p .001 w .19629 .05663 m .19629 .06038 L s P p .001 w .22628 .05663 m .22628 .06038 L s P p .001 w .25628 .05663 m .25628 .06038 L s P p .001 w .28628 .05663 m .28628 .06038 L s P p .001 w .34627 .05663 m .34627 .06038 L s P p .001 w .37627 .05663 m .37627 .06038 L s P p .001 w .40626 .05663 m .40626 .06038 L s P p .001 w .43626 .05663 m .43626 .06038 L s P p .001 w .49625 .05663 m .49625 .06038 L s P p .001 w .52625 .05663 m .52625 .06038 L s P p .001 w .55624 .05663 m .55624 .06038 L s P p .001 w .58624 .05663 m .58624 .06038 L s P p .001 w .64623 .05663 m .64623 .06038 L s P p .001 w .67623 .05663 m .67623 .06038 L s P p .001 w .70622 .05663 m .70622 .06038 L s P p .001 w .73622 .05663 m .73622 .06038 L s P p .001 w .79621 .05663 m .79621 .06038 L s P p .001 w .82621 .05663 m .82621 .06038 L s P p .001 w .85621 .05663 m .85621 .06038 L s P p .001 w .8862 .05663 m .8862 .06038 L s P p .001 w .94619 .05663 m .94619 .06038 L s P p .001 w .97619 .05663 m .97619 .06038 L s P p .002 w 0 .05663 m 1 .05663 L s P [(signal with some noise removed)] .5 .61803 0 -2 Mshowa p .002 w .01631 .15979 m .02256 .15979 L s P [(20)] .00381 .15979 1 0 Mshowa p .002 w .01631 .26294 m .02256 .26294 L s P [(40)] .00381 .26294 1 0 Mshowa p .002 w .01631 .3661 m .02256 .3661 L s P [(60)] .00381 .3661 1 0 Mshowa p .002 w .01631 .46926 m .02256 .46926 L s P [(80)] .00381 .46926 1 0 Mshowa p .002 w .01631 .57242 m .02256 .57242 L s P [(100)] .00381 .57242 1 0 Mshowa p .001 w .01631 .07726 m .02006 .07726 L s P p .001 w .01631 .09789 m .02006 .09789 L s P p .001 w .01631 .11852 m .02006 .11852 L s P p .001 w .01631 .13916 m .02006 .13916 L s P p .001 w .01631 .18042 m .02006 .18042 L s P p .001 w .01631 .20105 m .02006 .20105 L s P p .001 w .01631 .22168 m .02006 .22168 L s P p .001 w .01631 .24231 m .02006 .24231 L s P p .001 w .01631 .28358 m .02006 .28358 L s P p .001 w .01631 .30421 m .02006 .30421 L s P p .001 w .01631 .32484 m .02006 .32484 L s P p .001 w .01631 .34547 m .02006 .34547 L s P p .001 w .01631 .38673 m .02006 .38673 L s P p .001 w .01631 .40737 m .02006 .40737 L s P p .001 w .01631 .428 m .02006 .428 L s P p .001 w .01631 .44863 m .02006 .44863 L s P p .001 w .01631 .48989 m .02006 .48989 L s P p .001 w .01631 .51052 m .02006 .51052 L s P p .001 w .01631 .53115 m .02006 .53115 L s P p .001 w .01631 .55179 m .02006 .55179 L s P p .001 w .01631 .036 m .02006 .036 L s P p .001 w .01631 .01537 m .02006 .01537 L s P p .001 w .01631 .59305 m .02006 .59305 L s P p .001 w .01631 .61368 m .02006 .61368 L s P p .002 w .01631 0 m .01631 .61803 L s P P p 1 0 0 r .004 w .02381 .02949 m .03131 .0403 L .03881 .05633 L .04631 .06457 L .05381 .0702 L .0613 .06281 L .0688 .05227 L .0763 .04891 L .0838 .04608 L .0913 .03916 L .0988 .03349 L .1063 .0343 L .1138 .03803 L .1213 .04111 L .1288 .04589 L .1363 .06647 L .14379 .08534 L .15129 .06848 L .15879 .04883 L .16629 .0585 L .17379 .07288 L .18129 .08182 L .18879 .09298 L .19629 .09256 L .20379 .09955 L .21129 .14792 L .21879 .2074 L .22628 .25816 L .23378 .31219 L .24128 .36139 L .24878 .41166 L .25628 .50728 L .26378 .58839 L .27128 .59677 L .27878 .56679 L .28628 .46419 L .29378 .33991 L .30127 .25767 L .30877 .18566 L .31627 .08577 L .32377 .01472 L .33127 .04881 L .33877 .1218 L .34627 .24362 L .35377 .36219 L .36127 .40621 L .36877 .40903 L .37627 .314 L .38376 .19839 L .39126 .1282 L Mistroke .39876 .08435 L .40626 .10045 L .41376 .1467 L .42126 .26037 L .42876 .36729 L .43626 .35661 L .44376 .3087 L .45126 .22159 L .45876 .13273 L .46625 .08129 L .47375 .06374 L .48125 .18014 L .48875 .31034 L .49625 .35074 L .50375 .3539 L .51125 .25365 L .51875 .13692 L .52625 .08579 L .53375 .06576 L .54124 .11472 L .54874 .1886 L .55624 .29016 L .56374 .37493 L .57124 .35577 L .57874 .29907 L .58624 .18109 L .59374 .06795 L .60124 .04607 L .60874 .06804 L .61624 .2034 L .62373 .35055 L .63123 .39534 L .63873 .4003 L .64623 .348 L .65373 .26903 L .66123 .16171 L .66873 .06791 L .67623 .06477 L .68373 .10069 L .69123 .17856 L .69873 .27692 L .70622 .36314 L .71372 .44412 L .72122 .53599 L .72872 .60332 L .73622 .58216 L .74372 .52806 L .75122 .46263 L .75872 .38722 L .76622 .32096 L Mistroke .77372 .25931 L .78121 .2152 L .78871 .17874 L .79621 .12761 L .80371 .08495 L .81121 .09118 L .81871 .10566 L .82621 .08946 L .83371 .07164 L .84121 .05963 L .84871 .05143 L .85621 .06169 L .8637 .07499 L .8712 .07917 L .8787 .08059 L .8862 .06921 L .8937 .05704 L .9012 .06581 L .9087 .07563 L .9162 .07103 L .9237 .06382 L .9312 .04675 L .9387 .03216 L .94619 .0501 L .95369 .07129 L .96119 .07448 L .96869 .07253 L .97619 .051 L Mfstroke P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = text; inactive; preserveAspect] :[font = text; inactive; preserveAspect] A plot of the original signal without the noise: :[font = input; preserveAspect; startGroup] original = ListPlot[signal, PlotRange->All, PlotJoined->True, PlotStyle->RGBColor[0,0,1], PlotLabel->"original signal w/o noise"] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0163105 0.00749906 0.0147151 0.00588346 [ [(20)] .16629 .01472 0 2 Msboxa [(40)] .31627 .01472 0 2 Msboxa [(60)] .46625 .01472 0 2 Msboxa [(80)] .61624 .01472 0 2 Msboxa [(100)] .76622 .01472 0 2 Msboxa [(120)] .9162 .01472 0 2 Msboxa [(original signal w/o noise)] .5 .61803 0 -2 Msboxa [(20)] .00381 .13238 1 0 Msboxa [(40)] .00381 .25005 1 0 Msboxa [(60)] .00381 .36772 1 0 Msboxa [(80)] .00381 .48539 1 0 Msboxa [(100)] .00381 .60306 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .16629 .01472 m .16629 .02097 L s P [(20)] .16629 .01472 0 2 Mshowa p .002 w .31627 .01472 m .31627 .02097 L s P [(40)] .31627 .01472 0 2 Mshowa p .002 w .46625 .01472 m .46625 .02097 L s P [(60)] .46625 .01472 0 2 Mshowa p .002 w .61624 .01472 m .61624 .02097 L s P [(80)] .61624 .01472 0 2 Mshowa p .002 w .76622 .01472 m .76622 .02097 L s P [(100)] .76622 .01472 0 2 Mshowa p .002 w .9162 .01472 m .9162 .02097 L s P [(120)] .9162 .01472 0 2 Mshowa p .001 w .04631 .01472 m .04631 .01847 L s P p .001 w .0763 .01472 m .0763 .01847 L s P p .001 w .1063 .01472 m .1063 .01847 L s P p .001 w .1363 .01472 m .1363 .01847 L s P p .001 w .19629 .01472 m .19629 .01847 L s P p .001 w .22628 .01472 m .22628 .01847 L s P p .001 w .25628 .01472 m .25628 .01847 L s P p .001 w .28628 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preserveAspect] Here is how much noise we did remove: :[font = input; preserveAspect; startGroup] ListPlot[testdata2-testdata2R, PlotRange->{-20., 120.}, PlotJoined->True, PlotStyle->RGBColor[0,0.5,0], PlotLabel->"noise removed"] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.00744048 0.0882906 0.00441453 [ [(20)] .17262 .08829 0 2 Msboxa [(40)] .32143 .08829 0 2 Msboxa [(60)] .47024 .08829 0 2 Msboxa [(80)] .61905 .08829 0 2 Msboxa [(100)] .76786 .08829 0 2 Msboxa [(120)] .91667 .08829 0 2 Msboxa [(noise removed)] .5 .61803 0 -2 Msboxa [(-20)] .01131 0 1 0 Msboxa [(20)] .01131 .17658 1 0 Msboxa [(40)] .01131 .26487 1 0 Msboxa [(60)] .01131 .35316 1 0 Msboxa [(80)] .01131 .44145 1 0 Msboxa [(100)] .01131 .52974 1 0 Msboxa [(120)] .01131 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.09204 L s P p .001 w .38095 .08829 m .38095 .09204 L s P p .001 w .41071 .08829 m .41071 .09204 L s P p .001 w .44048 .08829 m .44048 .09204 L s P p .001 w .5 .08829 m .5 .09204 L s P p .001 w .52976 .08829 m .52976 .09204 L s P p .001 w .55952 .08829 m .55952 .09204 L s P p .001 w .58929 .08829 m .58929 .09204 L s P p .001 w .64881 .08829 m .64881 .09204 L s P p .001 w .67857 .08829 m .67857 .09204 L s P p .001 w .70833 .08829 m .70833 .09204 L s P p .001 w .7381 .08829 m .7381 .09204 L s P p .001 w .79762 .08829 m .79762 .09204 L s P p .001 w .82738 .08829 m .82738 .09204 L s P p .001 w .85714 .08829 m .85714 .09204 L s P p .001 w .8869 .08829 m .8869 .09204 L s P p .001 w .94643 .08829 m .94643 .09204 L s P p .001 w .97619 .08829 m .97619 .09204 L s P p .002 w 0 .08829 m 1 .08829 L s P [(noise removed)] .5 .61803 0 -2 Mshowa p .002 w .02381 0 m .03006 0 L s P [(-20)] .01131 0 1 0 Mshowa p .002 w .02381 .17658 m .03006 .17658 L s P [(20)] .01131 .17658 1 0 Mshowa p .002 w .02381 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