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function [h,g,a]=wfilt_sym(N)
%WFILT_SYM Symlet filters
% Usage: [h,g,a]=wfilt_sym(N);
%
% `[h,g,a]=wfilt_sym(N)` generates the "least asymmetric" Daubechies'
% wavelets or "symlets". Zeros of the trigonometrical polynomial the
% filters of are selected alternatingly inside and outside the unit
% circle.
%
% References: daub98tenlectures
%
% Examples:
% ---------
%
% Frequency responses of the analysis filters:::
%
% w = fwtinit({'sym',10});
% wtfftfreqz(w.h);
%
%
% Original copyright goes to:
% Copyright (C) 1994, 1995, 1996, by Universidad de Vigo
% Author: Jose Martin Garcia
% e-mail: Uvi_Wave@tsc.uvigo.es
num_coefs = 2*N;
if num_coefs==2 % Haar filters
g{1} = 0.5*[sqrt(2) sqrt(2)];
g{2} = 0.5*[sqrt(2) -sqrt(2)];
h{1} = 0.5*[sqrt(2) sqrt(2)];
h{2} = 0.5*[-sqrt(2) sqrt(2)];
return
end
N=num_coefs/2;
poly=trigpol(N); %Calculate trigonometric polynomial
ceros=roots(poly); %Calculate roots
realzeros=[];
imagzeros=[];
numrealzeros=0;
numimagzeros=0;
for i=1:2*(N-1)
if (imag(ceros(i))==0)
numrealzeros=numrealzeros+1;
realzeros(numrealzeros)=ceros(i);
else
numimagzeros=numimagzeros+1;
imagzeros(numimagzeros)=ceros(i);
end
end
%% complex zeros are grouped together
i=0;
for cont=1:numimagzeros/4
modulos(cont)=abs(imagzeros(cont+i));
alfa(cont)=angle(imagzeros(cont+i));
i=i+1;
end
%% Calculate phase contribution of complex and real zeros for all the
%% combination of these zeros. Each group of zeros is identified with a binary
%% number.
indice=2^(numimagzeros/4+numrealzeros/2);
fase=zeros(indice,1001);
for cont=0:indice-1,
bin=dec2bina(cont,log2(indice));
for i=1:length(bin)-numrealzeros/2
if bin(i)
R=1/modulos(i);
else
R=modulos(i);
end
alf=alfa(i);
fase(cont+1,:)=fase(cont+1,:)+atang1(R,alf);
end
ind=1;
for i=length(bin)-numrealzeros/2+1:length(bin)
if bin(i)
R=realzeros(ind+1);
R=realzeros(ind+1);
else
R=realzeros(ind);
end
ind=ind+2;
fase(cont+1,:)=fase(cont+1,:)+atang2(R);
end
end
%% To retain only the non linear part of the phase.
fas=linealiz(fase);
imagzeros=[];
zerosreales=[];
%% To see which phase is closer to zero we select the one with minimun variance
[maximo,pos]=min(sum(fas'.^2));
bin=dec2bina(pos-1,log2(indice));
for i=1:length(bin)-numrealzeros/2
if bin(i)
z1=1/modulos(i)*exp(j*alfa(i));
else
z1=modulos(i)*exp(j*alfa(i));
end
imagzeros=[imagzeros z1 conj(z1)];
end
ind=1;
for i=length(bin)-numrealzeros/2+1:length(bin)
if bin(i)
zerosreales=[zerosreales realzeros(ind+1)];
else
zerosreales=[zerosreales realzeros(ind)];
end
ind=ind+2;
end
% Construction of rh from its zeros
numrealzeros=numrealzeros/2;
numimagzeros=numimagzeros/2;
rh=[1 1];
for i=2:N
rh=conv(rh,[1 1]);
end
for i=1:numrealzeros
rh=conv(rh,[1 -zerosreales(i)]);
end
for i=1:2:numimagzeros
rh=conv(rh,[1 -2*real(imagzeros(i)) abs(imagzeros(i))^2]);
end
% Once ho is factorized in its zeros, it must be normalized multiplying by "cte".
cte=sqrt(2)/sum(rh);
rh=cte*rh;
g{1} = rh;
g{2} = -(-1).^(1:length(rh)).*g{1}(end:-1:1);
h{1}=g{1}(length(g{1}):-1:1);
h{2}=g{2}(length(g{2}):-1:1);
a= [2;2];
function bin=dec2bina(num,bits)
%DEC2BINA BIN = DEC2BINA(NUM,BITS) returns a vector which contains
% the decimal number NUM in binary format, with a number of
% digits equal to BITS. It is an auxiliary function used by
% SYMLETS.
%--------------------------------------------------------
% Copyright (C) 1994, 1995, 1996, by Universidad de Vigo
%
%
% Uvi_Wave is free software; you can redistribute it and/or modify it
% under the terms of the GNU General Public License as published by the
% Free Software Foundation; either version 2, or (at your option) any
% later version.
%
% Uvi_Wave is distributed in the hope that it will be useful, but WITHOUT
% ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
% FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
% for more details.
%
% You should have received a copy of the GNU General Public License
% along with Uvi_Wave; see the file COPYING. If not, write to the Free
% Software Foundation, 675 Mass Ave, Cambridge, MA 02139, USA.
%
% Author: Jose Martin Garcia
% e-mail: Uvi_Wave@tsc.uvigo.es
%--------------------------------------------------------
if nargin<2
flag=0;
else
flag=1;
end
bin=[];
coc=num;
while coc>1
bin=[rem(coc,2) bin];
coc=fix(coc/2);
end
bin=[coc bin];
if flag
if length(bin)<bits
bin=[zeros(1,bits-length(bin)) bin];
end
end
function fase=atang1(R,alfa)
%ATANG1 PHASE=ATANG1(R,ALFA) returns the phase contribution
% of a complex pair of zeros. Linear terms have been
% removed. It is an auxiliary function used by SYMLETS.
%--------------------------------------------------------
% Copyright (C) 1994, 1995, 1996, by Universidad de Vigo
%
%
% Uvi_Wave is free software; you can redistribute it and/or modify it
% under the terms of the GNU General Public License as published by the
% Free Software Foundation; either version 2, or (at your option) any
% later version.
%
% Uvi_Wave is distributed in the hope that it will be useful, but WITHOUT
% ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
% FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
% for more details.
%
% You should have received a copy of the GNU General Public License
% along with Uvi_Wave; see the file COPYING. If not, write to the Free
% Software Foundation, 675 Mass Ave, Cambridge, MA 02139, USA.
%
% Author: Jose Martin Garcia
% e-mail: Uvi_Wave@tsc.uvigo.es
%--------------------------------------------------------
w=[0:2*pi/1e3:2*pi]; %frequency axis
fas=atan( (1-R^2)*sin(w)./((1+R^2)*cos(w)-2*R*cos(alfa)) );
zero=acos(2*R*cos(alfa)/(1+R^2));
u1=ceil(zero*1000/(2*pi))+1;
u2=ceil((2*pi-zero)*1000/(2*pi))+1;
if (1-R^2)*sin(zero)<0
cte=pi;
fase=fas+w;
else
fase=fas-w;
cte=-pi;
end
fase(u1:1001)=fase(u1:1001)-cte;
fase(u2:1001)=fase(u2:1001)-cte;
function fase=atang2(R)
%ATANG2 PHASE=ATANG2(R) returns the phase contribution of
% a real zero. Linear terms have been removed. It is
% an auxiliary function used by SYMLETS.
%--------------------------------------------------------
% Copyright (C) 1994, 1995, 1996, by Universidad de Vigo
%
%
% Uvi_Wave is free software; you can redistribute it and/or modify it
% under the terms of the GNU General Public License as published by the
% Free Software Foundation; either version 2, or (at your option) any
% later version.
%
% Uvi_Wave is distributed in the hope that it will be useful, but WITHOUT
% ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
% FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
% for more details.
%
% You should have received a copy of the GNU General Public License
% along with Uvi_Wave; see the file COPYING. If not, write to the Free
% Software Foundation, 675 Mass Ave, Cambridge, MA 02139, USA.
%
% Author: Jose Martin Garcia
% e-mail: Uvi_Wave@tsc.uvigo.es
%--------------------------------------------------------
w=[0:2*pi/1e3:2*pi]; %frequency axis
fas=atan( (1+R)/(1-R)*tan(w/2) );
if R<1
fase=fas-w;
cte=-pi;
else
fase=fas+w;
cte=pi;
end;
u=ceil(pi*1000/(2*pi))+2;
fase(u:1001)=fase(u:1001)-cte;
function fase=linealiz(f)
%LINEALIZ PHASE = LINEALIZ(F) is an auxiliary function used
% by SYMLETS. It eliminates the linearity of the
% phase vector F.
%--------------------------------------------------------
% Copyright (C) 1994, 1995, 1996, by Universidad de Vigo
%
%
% Uvi_Wave is free software; you can redistribute it and/or modify it
% under the terms of the GNU General Public License as published by the
% Free Software Foundation; either version 2, or (at your option) any
% later version.
%
% Uvi_Wave is distributed in the hope that it will be useful, but WITHOUT
% ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
% FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
% for more details.
%
% You should have received a copy of the GNU General Public License
% along with Uvi_Wave; see the file COPYING. If not, write to the Free
% Software Foundation, 675 Mass Ave, Cambridge, MA 02139, USA.
%
% Author: Jose Martin Garcia
% e-mail: Uvi_Wave@tsc.uvigo.es
%--------------------------------------------------------
if abs(sum(f(1,:))) >0.0001
w=[0:2*pi/1e3:2*pi];
[m,n]=size(f);
fase=zeros(m,n);
for cont=1 : m
if sum(f(cont,:)) >0
fase(cont,:)=f(cont,:)-w/2;
else
fase(cont,:)=f(cont,:)+w/2;
end
end
else
fase=f;
end
function polinomio=trigpol(N)
coefs=zeros(N,2*N-1);
coefs(1,N)=1;
for i=1:N-1
fila=[1 -2 1];
for j=2:i
fila=conv(fila,[1 -2 1]);
end;
fila=numcomb(N-1+i,i)*(-0.25)^i*fila;
fila=[ zeros(1,(N-i-1)) fila zeros(1,(N-i-1))];
coefs(i+1,:)=fila;
end
for i=0:(2*(N-1))
polinomio(i+1)=0;
for j=1:N
polinomio(i+1)=polinomio(i+1)+coefs(j,i+1);
end
end;
function y=numcomb(n,k)
if n==k,
y=1;
elseif k==0,
y=1;
elseif k==1,
y=n;
else
y=fact(n)/(fact(k)*fact(n-k));
end
function y=fact(x)
% FACT Factorial.
% FACT(X) is the factorial of the elements in X vector.
for j=1:length(x)
if x(j)==0,
y(j)=1;
else
y(j)=x(j)*fact(x(j)-1);
end
end

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