File:Daubechies4-functions.svg
Summary
| Description |
English: Daubechies 4 tap wavelet and scaling functions |
| Date | |
| Source | Own work |
| Author | LutzL |
| Other versions | Daubechies4-functions.png |
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Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled GNU Free Documentation License. |
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file daubechies.magma
see en:Magma computer algebra system, with link for an online calculator.
R<I>:=ComplexField(30);
LP<Z> := LaurentSeriesRing(R);
T:=(-Z+2-Z^-1)/4;
TR<U>:=PowerSeriesRing(R);
Pol<X>:=PolynomialRing(R);
readi A,"What accuracy should the wavelets have?";
PolarFactorization:=function(A)
pp:=Truncate( (1-U+O(U^A))^-A );
pp:=Evaluate(pp,X);
p:=LP!1;
for rr in Roots(pp) do
r:=rr[1];r;
for ff in Roots(X^2+(4*r-2)*X+1) do
f:=ff[1];
if Abs(f) ge 1 then p*:=Z-f; end if;
end for;
end for;
p1:=Evaluate(p,1);
return LP![ Real(c): c in Eltseq(p/p1) ],pp;
end function;
p,p2:=PolarFactorization(A);p;a:=2^(1-A)*(1+Z)^A*p;a2:=(1-X)^A*p2;
"Scaling sequence",Coefficients(a),"product filter", Coefficients(a2);
Decimation:= function(c)
return LP![ Coefficient(c,2*k): k in [0..Degree(c) div 2+2] ];
end function;
// Power iteration for the values of the scaling function at integer locations
shape := LP!(Z^A);
for i := 1 to 10 do
for k := 1 to 10 do
shape := Decimation(a*shape); shape := shape/Evaluate(shape,1);
end for;
shape:=LP![ R!(1+Coefficient(shape,k))-1: k in [0..Degree(shape)] ];
end for;
wave := LP![ (-1)^k*Coefficient(a,2*A-k): k in [1..2*A] ];
scal := shape;
dx := 1; supp := 2*A-1; pow:=1;
for i := 1 to 4 do
scal := scal*LP!Evaluate(a,Z^pow);
dx /:=2; supp *:=2; pow*:=2;
end for;
wave := scal*LP!Evaluate(wave,Z^pow);
scal := scal*LP!Evaluate(a,Z^pow);
dx /:=2; supp *:=2; pow*:=2;
a2;
fp := Open(Sprintf("daub%o-scal.dat",A),"w");
RO:=RealField(12);
for k := 1 to supp do
fprintf fp, "%o\t%o\t%o\t%o\t%o\n",RO!(k*dx),
RO!Coefficient(scal,k),
RO!Coefficient(wave,k),
RO!Abs(Evaluate(scal,Exp(I*2*Pi(RO)*dx^2*k))*dx),
RO!Abs(Evaluate(wave,Exp(I*2*Pi(RO)*dx^2*k))*dx);
end for;
Flush(fp);
set data style lines
set zeroa
set key bottom
set term svg enhanced size 1000,750 fsize 24
set out "Daubechies4-functions.svg"
set title "Daubechies 4 tap wavelet"
pl "daub2-scal.dat" u 1:2 ti "scaling function" w lines lt 3 lw 3,\
"daub2-scal.dat" u 1:3 ti "wavelet function" w lines lt 1 lw 3
set out