PEBFUNDUAL - Dual window of sampled, periodized EB-spline

Usage

g=pebfundual(w,a,M,L)
g=pebfundual({w,width},a,M,L)
g=pebfundual(...,inc)

Input parameters

Output parameters

Description

pebfundual(w,a,M,L) computes samples of a dual window of EB spline with weights w. Please see pebfun for definition of EB splines. The lattice parameters \(a,M\) must satisfy \(M > a\) to ensure the system is a frame.

pebfundual({w,width},a,M,L) works as above but in addition the width parameter determines the integer stretching factor of the original EB spline. For explanation see help of pebfun.

pebfundual(...,inc) or pebfundual(...,'inc',inc) works as above, but integer inc denotes number of additional columns to compute window function gd. 'inc'-many are added at each side. It should be smaller than 100 to have comfortable execution-time. The higher the number the closer gd is to the canonical dual window. The default value is 10.

Examples:

The following example compares dual windows computed using 2 different approaches.:

w = [-3,-1,1,3];a = 25; M = 31; inc = 1;
L = 1e6; L = dgtlength(L,a,M);
width = M;

% Create the window
g = pebfun(L,w,width);

% Compute a dual window using pebfundual
tic
[gd,nlen] = pebfundual({w,width},a,M,L,inc);
tebfundual=toc;

% We know that gd has only nlen nonzero samples, lets shrink it.
gd = middlepad(gd,nlen);

% Compute the canonical window using gabdual
tic
gdLTFAT = gabdual(g,a,M,L);
tgabdual=toc;

fprintf('PEBFUNDUAL elapsed time %f s\n',tebfundual);
fprintf('GABDUAL elapsed time    %f s\n',tgabdual);

% Test on random signal
f = randn(L,1);

fr = idgt(dgt(f,g,a,M),gd,a,numel(f));
fprintf('Reconstruction error PEBFUNDUAL: %e\n',norm(f-fr)/norm(f));

fr = idgt(dgt(f,g,a,M),gdLTFAT,a,numel(f));
fprintf('Reconstruction error GABDUAL:    %e\n',norm(f-fr)/norm(f));