(*^ ::[paletteColors = 128; automaticGrouping; currentKernel; fontset = title, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, L1, e8, 24, "Times"; ; fontset = subtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, L1, e6, 18, "Times"; ; fontset = subsubtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, italic, L1, e6, 14, "Times"; ; fontset = section, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, grayBox, M22, bold, L1, a20, 18, "Times"; ; fontset = subsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, blackBox, M19, bold, L1, a15, 14, "Times"; ; fontset = subsubsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, whiteBox, M18, bold, L1, a12, 12, "Times"; ; fontset = text, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = smalltext, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 10, "Times"; ; fontset = input, noPageBreakBelow, nowordwrap, preserveAspect, groupLikeInput, M42, N23, bold, L1, 12, "Courier"; ; fontset = output, output, inactive, noPageBreakBelow, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-5, 12, "Courier"; ; fontset = message, inactive, noPageBreakBelow, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L1, 12, "Courier"; ; fontset = print, inactive, noPageBreakBelow, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L1, 12, "Courier"; ; fontset = info, inactive, noPageBreakBelow, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L1, 12, "Courier"; ; fontset = postscript, PostScript, formatAsPostScript, output, inactive, noPageBreakBelow, nowordwrap, preserveAspect, groupLikeGraphics, M7, l34, w282, h287, L1, 12, "Courier"; ; fontset = name, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, italic, L1, 10, "Times"; ; fontset = header, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = Left Header, nohscroll, cellOutline, 12; fontset = footer, inactive, nohscroll, noKeepOnOnePage, preserveAspect, center, M7, L1, 12; fontset = Left Footer, cellOutline, blackBox, 12; fontset = help, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 10, "Times"; ; fontset = clipboard, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = completions, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12, "Courier"; ; fontset = special1, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special2, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special3, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special4, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special5, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12;] :[font = title; inactive; preserveAspect; startGroup; ] Daubechies Minimum Phase Wavelets :[font = subsubtitle; inactive; preserveAspect; ] Jack K. Cohen Colorado School of Mines, 1993 jkc@keller.mines.colorado.edu :[font = input; preserveAspect; ] <{0, 1.1}, PlotStyle -> {GrayLevel[0], GrayLevel[0.5]}, DefaultFont -> {"Courier", 12}, PlotLabel -> label ] ] :[font = text; inactive; preserveAspect; ] The degree of overlap between M0 and M1 is significant because the overlap represents aliased energy. This is an issue if we decompose into wavelets and do a non-trivial filtering operation before recomposition. Naturally enough, the smaller support Daubechies wavelets have more overlap than the larger support ones. 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DisplayFunction :> $DisplayFunction, ColorOutput -> Automatic, Axes -> Automatic, AxesOrigin -> Automatic, PlotLabel -> FontForm[StringForm["Daubechies' M0, M1: n =\ `1`", 8], {"Helvetica-Bold", 16}], AxesLabel -> None, Ticks -> Automatic, GridLines -> None, Prolog -> {}, Epilog -> {}, AxesStyle -> Automatic, Background -> Automatic, DefaultColor -> Automatic, DefaultFont :> {"Courier", 12}, RotateLabel -> True, Frame -> False, FrameStyle -> Automatic, FrameTicks -> Automatic, FrameLabel -> None, PlotRegion -> Automatic}]} ;[o] {-Graphics-, -Graphics-, -Graphics-} :[font = section; inactive; Cclosed; preserveAspect; startGroup; ] Daubechies PhiHat and PsiHat :[font = text; inactive; preserveAspect; ] The idea is to use the infinite product representation. It turns out that for most purposes, sufficiently good convergence occurs with just a few factors. Recall that PhiHat is just the product of the M0's with successively halved arguments, while PsiHat has one initial factor of M1. A reference for this is Ten Lectures, page 175, top formula. I don't need the fractional power of 2Pi because of my putting a 2PI in the Fourier exponent. The root result for the product formula for PhiHat is the fundamental equation 5.1.17 cited above. Note: we can construct frequency domain wavelet packets in a similar way. Note: the third argument specifies how many factors to take---these functions are just approximations that get better for large `fac'. Empirically though, `fac' doesn't have to be very large and the convergence is superb once n is 4 or larger. Caution: M0 and M1 were defined above. :[font = input; preserveAspect; ] PhiHat[n_, xi_, facs_] := Product[M0[n, xi/2^k], {k, facs}] PsiHat[n_, xi_, facs_] := M1[n, xi/2] * Product[M0[n, xi/2^(k+1)], {k, facs}] :[font = subsection; inactive; Cclosed; preserveAspect; startGroup; ] Basic usage :[font = input; preserveAspect; startGroup; ] Clear[phihat]; n = 3; facs = 8; phihat[xi_] = N@Abs@PhiHat[n, xi, facs]; Table[{xi, phihat[xi]}, {xi, 0, 1.0, 0.1}]//TableForm :[font = output; output; inactive; preserveAspect; endGroup; endGroup; ] TableForm[{{0, 1.000000000000002}, {0.1, 0.999928210408839}, {0.2, 0.996167450738771}, {0.3, 0.967399498096586}, {0.4, 0.875148249695441}, {0.5, 0.6980584999875913}, {0.6, 0.4621406617153238}, {0.7, 0.2332662156863307}, {0.8, 0.0758300379325094}, {0.9, 0.00946926794401827}, {1., 1.040922256204261*10^-16}}] ;[o] 0 1. 0.1 0.999928 0.2 0.996167 0.3 0.967399 0.4 0.875148 0.5 0.698058 0.6 0.462141 0.7 0.233266 0.8 0.07583 0.9 0.00946927 -16 1. 1.04092 10 :[font = subsection; inactive; Cclosed; preserveAspect; startGroup; ] Plotting the transforms :[font = text; inactive; preserveAspect; ] Warning: this isn't a fully general plotter. In particular, the plot range is hard wired to something that seemed nice for the small number of tests that I did. :[font = input; preserveAspect; ] PlotPhiHat[n_, facs_] := Module[{string, label}, string = StringForm["Daubechies' |Phi^|: n = `1`", n]; label = FontForm[string, {"Helvetica-Bold", 16}]; Plot[Evaluate@Abs@PhiHat[n, xi, facs], {xi, -16/n, 16/n}, PlotRange->{0, 1.1}, DefaultFont -> {"Courier", 12}, PlotLabel -> label ] ] PlotPsiHat[n_, facs_] := Module[{string, label}, string = StringForm["Daubechies' |Psi^|: n = `1`", n]; label = FontForm[string, {"Helvetica-Bold", 16}]; Plot[Evaluate@Abs@PsiHat[m, xi, facs], {xi, -16/n, 16/n}, PlotRange->{0, 1.1}, DefaultFont -> {"Courier", 12}, PlotLabel -> label ] ] :[font = text; inactive; preserveAspect; ] Experiment with PhiHat[2] to get suitable `fac' value. 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0.97247 0 lineto 0.97371 0 lineto 0.97495 0 lineto 0.97619 0 lineto Mfstroke grestore grestore % End of Graphics MathPictureEnd :[font = text; inactive; preserveAspect; endGroup; endGroup; ] LEFT TO DO: Work out the number of factors needed for some definite accuracy criterion and put the resulting functions into the Daubechies.m package. :[font = section; inactive; Cclosed; preserveAspect; startGroup; ] Daubechies Minimum Phase Phi and Psi :[font = text; inactive; preserveAspect; ] I'm using the algorithm outlines in Strang: Wavelets and Dilation Equations: A Brief Introduction, SIAM Review, V 31, N 4, pp614-627, Dec 1989. There is some dancing around between Lists and Subscripted variables--no doubt a wizard could do better. Also note that the package supplies the h's (Ten Lectures, eq. 5.1.18), and I needed the cn's (first eq. on page 145), which differ by sqrt(2). Strang's construction is closely related to the refinement construction on p 202ff of Ten Lectures. But I understood Strang's version better. Note: The Daubechies scaling functions and wavelets are defined recursively, so it's up to you to decide how densely to compute them. The variable `jmax' controls this. LEFT TO DO: I didn't get around to writing a general refinement function: the n = 2 and n = 4 subsections differ only by the initialization of n and the jmax chosen. I tried once, but failed, for reasons I don't remember now. ;[s] 3:0,0;540,1;541,2;941,-1; 3:1,11,8,Times,0,12,0,0,0;1,13,10,Times,0,14,0,0,0;1,11,8,Times,0,12,0,0,0; :[font = subsection; inactive; Cclosed; preserveAspect; startGroup; ] Function to compute phi on the integers in its support :[font = input; preserveAspect; endGroup; ] integerPhi[n_Integer?Positive] := Module[ {m = 2n, sqrt2 = N@Sqrt[2], jlo, jhi, c, e, p}, hn = dblh[n]; (* Tong's function for the hn *) Do[ c[k-1] = hn[[k]]*sqrt2, {k, m}]; For[i = 1, i < m - 1, ++i, jlo = Max[1, 2i - m + 1]; jhi = Min[2i, m - 2]; For[j = 1, j < m - 1, ++j, If[ jlo <= j <= jhi, (* then *) l[i][j] = c[2i - j], (* else *) l[i][j] = 0 ] ] ]; e = Eigensystem@ Table[l[i][j], {i, 1, m - 2}, {j, 1, m - 2}]; p = e[[2]][[1]]; p /= (Plus @@ p) (* comps have to sum to one *) ] :[font = subsection; inactive; Cclosed; preserveAspect; startGroup; ] n = 2 :[font = text; inactive; preserveAspect; ] Set number of vanishing moments in the wavelet.. :[font = input; preserveAspect; ] n = 2; :[font = text; inactive; preserveAspect; ] Compute d[n] and psi[n]. :[font = input; preserveAspect; startGroup; ] jmax = 7; (* number of levels i/2^j to compute *) m = 2n; (* convenience constant *) imax = 2^jmax; (* convenience constant *) dx = 1/imax; (* convenience constant *) hn = dblh[n]; (* Oops, repeated from integerPhi *) Do[ c[k-1] = hn[[k]]*N@Sqrt[2], {k, m}] Do[ (* wasteful, but clear *) d[k] = (-1)^k c[m-k-1], {k, 0, m-1}] p = integerPhi[n] Do[ phi[i] = p[[i]], {i, 1, m-2}] phi[x_] := 0.0 /; (x <= 0 || x >= m-1) :[font = output; output; inactive; preserveAspect; endGroup; ] {1.366025403784439, -0.3660254037844386} ;[o] {1.36603, -0.366025} :[font = text; inactive; preserveAspect; ] Refine using the duplication formula. :[font = input; preserveAspect; ] Do[ Do[ phi[i/2^j] = Sum[c[k] phi[i/2^(j-1) - k], {k, 0, m-1}], {i, 1, (m-1) imax, 2}], {j, 1, jmax}] philist = {}; Do[ AppendTo[philist, {N[i], phi[i]}], {i, 0, m-1, dx}] Clear[psi]; Do[ Do[ 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Use options in the Actions Settings dialog box to control when Unformatted text is generated. ;[o] -Graphics- :[font = subsection; inactive; Cclosed; preserveAspect; startGroup; ] n = 4 :[font = text; inactive; preserveAspect; ] Set number of vanishing moments in the wavelet.. :[font = input; preserveAspect; ] n = 4; :[font = text; inactive; preserveAspect; ] Compute d[n] and psi[n]. :[font = input; preserveAspect; startGroup; ] jmax = 5; (* number of levels i/2^j to compute *) m = 2n; (* convenience constant *) imax = 2^jmax; (* convenience constant *) dx = 1/imax; (* convenience constant *) hn = dblh[n]; (* Oops, repeated from integerPhi *) Do[ c[k-1] = hn[[k]]*N@Sqrt[2], {k, m}] Do[ (* wasteful, but clear *) d[k] = (-1)^k c[m-k-1], {k, 0, m-1}] p = integerPhi[n] Do[ phi[i] = p[[i]], {i, 1, m-2}] phi[x_] := 0.0 /; (x <= 0 || x >= m-1) :[font = output; output; inactive; preserveAspect; endGroup; ] {1.007169977725601, -0.03383695405283359, 0.03961046271590284, -0.01176435820572664, 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