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function [h,g,a]=wfilt_spline(m,n)
% WFILT_SPLINE Birthogonal spline wavelets
% Usage: [h,g,a]=wfilt_spline(m,n);
%
% Input parameters:
% m : Number of zeros at $z=-1$ of the lowpass filter in `g{1}`
% n : Number of zeros at $z=-1$ of the lowpass filter in
% `h{1}`. $m+n$ must be even.
%
% `[h,g,a]=wfilt_spline(m,n)` returns the analysis and synthesis filters
% corresponding to a biortoghonal scheme with spline wavelets of compact
% support.
%
% Examples:
% ---------
%
% Frequency responses of the analysis filters:::
%
% w = fwtinit({'spline',4,4});
% wtfftfreqz(w.h);
%
% Frequency responses of the synthesis filters:::
%
% w = fwtinit({'spline',4,4});
% wtfftfreqz(w.g);
%
%
% Original copyright goes to:
% Copyright (C) 1994, 1995, 1996, by Universidad de Vigo
% Author: Jose Martin Garcia
% e-mail: Uvi_Wave@tsc.uvigo.es
if(nargin<2)
error('%s: Too few input parameters.',upper(mfilename));
end
if(rem(m+n,2)~=0)
error('%s: M+N must be even.',upper(mfilename));
end
% Calculate rh coefficients, RH(z)=sqrt(2)*((1+z^-1)/2)^m;
rh=sqrt(2)*(1/2)^m*binewton(m);
% Calculate h coefficients, H(-z)=sqrt(2)*((1+z^-1)/2)^n*P(z)
% First calculate P(z) (pol)
if (rem(n,2)==0)
N=n/2+m/2;
else
N=(n+m-2)/2+1;
end
pol=trigpol(N);
% Now calculate ((1+z*-1)/2)^n;
r0=(1/2)^n*binewton(n);
hrev=sqrt(2)*conv(r0,pol);
l=length(hrev);
hh=hrev(l:-1:1);
[h{2}, g{2}]=calhpf(hh,rh);
h{1} = hh;
g{1} = rh;
if(length(h{1})>length(h{2}))
if(rem(length(h{1}),2)~=1)
r0 = (length(h{1})-length(h{2}))/2;
l0 = r0;
else
r0 = (length(h{1})-length(h{2}))/2+1;
l0 = (length(h{1})-length(h{2}))/2-1;
end
h{2} = [zeros(1,l0), h{2}, zeros(1,r0) ];
else
if(rem(length(h{1}),2)~=1)
r0 = (length(h{2})-length(h{1}))/2;
l0 = r0;
else
r0 = (length(h{2})-length(h{1}))/2+1;
l0 = (length(h{2})-length(h{1}))/2-1;
end
h{1} = [zeros(1,l0), h{1}, zeros(1,r0) ];
end
if(length(g{1})>length(g{2}))
if(rem(length(g{1}),2)~=1)
r0 = (length(g{1})-length(g{2}))/2;
l0 = r0;
else
r0 = (length(g{1})-length(g{2}))/2+1;
l0 = (length(g{1})-length(g{2}))/2-1;
end
g{2} = [zeros(1,l0), g{2}, zeros(1,r0) ];
else
if(rem(length(g{1}),2)~=1)
r0 = (length(g{2})-length(g{1}))/2;
l0 = r0;
else
r0 = (length(g{2})-length(g{1}))/2+1;
l0 = (length(g{2})-length(g{1}))/2-1;
end
g{1} = [zeros(1,l0), g{1}, zeros(1,r0) ];
end
% adding "the convenience" zero
if(rem(length(h{1}),2))
h{1}= [0, h{1}];
h{2}= [0, h{2}];
g{1}= [0, g{1}];
g{2}= [0, g{2}];
end
a= [2;2];
function c=binewton(N)
% BINEWTON generate coefficients of Newton binomial.
%
% BINEWTON(N) generates the N+1 coefficients of
% the Nth order Newton binomial.
%
% See also: NUMCOMB
%--------------------------------------------------------
% 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: Nuria Gonzalez Prelcic
% e-mail: Uvi_Wave@tsc.uvigo.es
%--------------------------------------------------------
c=[1];
for j=1:N,
c=[c,numcomb(N,j)];
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)
for j=1:length(x)
if x(j)==0,
y(j)=1;
else
y(j)=x(j)*fact(x(j)-1);
end
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 [g,rg]=calhpf(h,rh)
% CALHPF Obtain high pass analysis and synthesis filters
% in a biortoghonal filterbank.
lrh=length(rh);
if (rem(lrh,2)) % rh has odd length
nrh=(lrh-1)/2; % Support [-nrh,nrh]
else % rh has even length
nrh=lrh/2-1; % Support [-nrh,nrh+1]
end
if (rem(nrh,2)) % nrh is odd
flag=1;
else % nrh is even
flag=0;
end
grev=chsign(rh(lrh:-1:1),flag);
g=grev(lrh:-1:1);
lh=length(h);
if (rem(lh,2)) % h has odd length
nh=(lh-1)/2; % Support [-nh,nh]
else % h has even length
nh=lh/2-1; % Support [-nh,nh+1]
end
if (rem(nh,2))% nh is odd
flag=1;
else
flag=0; % nh is even
end
rg=chsign(h,flag);
function y=chsign(x,flag)
lx=length(x);
if (flag==1)
y=(-1).^(1:lx).*x;
else
y=-(-1).^(1:lx).*x;
end