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function [out,nlen] = ref_pbspline(splinetype,L,order,a,centering)
%-*- texinfo -*-
%@deftypefn {Function} ref_pbspline
%@verbatim
%PBSPLINE Periodized B-spline.
% Usage: out=pbspline(splinetype,L,order,a);
% [out,nlen]=pbspline(splinetype,L,order,a);
%
% Input parameters:
% splinetype : type of spline
% L : Length of window.
% order : Order of B-spline.
% a : Time-shift parameter for partition of unity.
% Output parameters:
% out : Almost B-spline.
% nlen : Number of non-zero elements in out.
%
%
% Types are:
% 0 - as pspline, real formed from DFT product
% 1 - forced even by taking ABS value of fft.
% 2 - New even
% 3 - computed from continous case
%@end verbatim
%@strong{Url}: @url{http://ltfat.github.io/doc/reference/ref_pbspline.html}
%@end deftypefn
% Copyright (C) 2005-2016 Peter L. Soendergaard <peter@sonderport.dk>.
% This file is part of LTFAT version 2.2.0
%
% This program 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 3 of the License, or
% (at your option) any later version.
%
% This program 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 this program. If not, see <http://www.gnu.org/licenses/>.
% AUTHOR : Peter L. Soendergaard.
error(nargchk(4,5,nargin));
if nargin==4
centering=0;
end;
% Matlab and Octave uses the following definition of a fractional
% power of a complex number. This is consistant with what Unser uses.
% zp=abs(z).^alpha.*exp(i*alpha*angle(z));
% This is always WPE
s1=middlepad([ones(a,1)],L);
nlen=0;
switch splinetype
case 0
% (Possibly) unsymmeteric spline
% If a=3,7,11,... then the Nyquest frequency will have a negative
% coefficient, and generate a complex spline.
if centering==0
% Method 1
out = real(ifft(fft(s1).^(order+1)));
% Method 2
% Create FFT of spline. Flip over top part, to make it strickly real.
% This method is more robust against a bad FFT implementation.
%sf=fft(s1).^(order+1);
%if rem(L,2)==0
% sf(L/2+2:L)=conj(flipud(sf(2:L/2)));
%else
% sf((L+3)/2:L)=conj(flipud(sf(2:(L+1)/2)));
%end;
%sf(L/2+1)
%out = real(ifft(sf));
else
s2=middlepad([ones(a,1)],L,.5);
out = real(ifft(fft(s1).^order.*fft(s2)));
end;
case 1
% Symmetric spline
if centering==0
out = real(ifft(abs(fft(s1)).^(order+1)));
else
s2=middlepad([ones(a,1)],L,.5);
out = real(ifft(abs(fft(s1)).^order.*abs(fft(s2))));
end;
case 2
% Mild symmetric
intorder=floor(order);
fracorder=order-intorder;
if centering==0
out = real(ifft(abs(fft(s1)).^order.*fft(s1)));
else
s2=middlepad([ones(a,1)],L,.5);
out = real(ifft(abs(fft(s1)).^order.*fft(s2)));
end;
case 3
Llong=ceil(a*(order+2)/L)*L;
x=((0:Llong-1).')/a;
x
out=zeros(Llong,1);
for k=0:order+1
out=out+(-1)^k*ref_bincoeff(order+1,k)*onesidedpower(x-k,order);
out
end;
out=out/factorial(order);
end;
% Scale such that the elements will form a partition of unity.
out=out./a.^order;
% nlen cannot be larger that L
nlen=min(L,nlen);
% If order is a fraction nlen==L
if rem(order,1)~=0
nlen=L;
end;
if use_row_layout
out=out.';
end;
% Normalize
out=out/sqrt(a);
% This code verifies that we have obtained a partition of unity.
%pu=zeros(L,1);
%for ii=0:L/a-1
% pu=pu+circshift(out,a*ii);
%end;
%pu
% Verify middlepad claim
%norm(out-middlepad(middlepad(out,nlen),L))
function xa=onesidedpower(x,a)
% Compute a one-sided power. See Unser and Blu "Fractional Spline and
% Wavelets", section 1.1.2
% Zero all negative values
x=x.*(x>=0)
% Raise
xa=x.^a;
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