PTPFUNDUAL - Sampled, periodized dual TP function of finite type

Usage

gd=ptpfundual(w,a,M,L)
gd=ptpfundual({w,width},a,M,L)
gd=ptpfundual(...,inc)
[gd,nlen]=ptpfundual(...)

Input parameters

Output parameters

Description

ptpfundual(w,a,M,L) computes samples of a dual window of totally positive function of finite type >=2 with weights w. Please see ptpfun for definition of totally positive functions. The lattice parameters \(a,M\) must satisfy \(M > a\) to ensure the system is a frame.

ptpfundual({w,width},a,M,L) works as above but in addition the width parameter determines the integer stretching factor of the original TP function. For explanation see help of ptpfun.

ptpfundual(...,inc) or ptpfundual(...,'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.

[gd,nlen]=ptpfundual(...) as gd might have a compact support, nlen contains a number of non-zero elements in gd. This is the case when gd is symmetric. If gd is not symmetric, nlen is extended to twice the length of the longer tail.

If \(nlen = L\), gd has a 'full' support meaning it is a periodization of a dual TP function.

If \(nlen < L\), additional zeros can be removed by calling gd=middlepad(gd,nlen).

Examples:

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

w = [-3,-1,1,3];a = 25; M = 31; inc = 10;
L = 1e6; L = dgtlength(L,a,M);
width = M;

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

% Compute a dual window using pebfundual
tic
[gd,nlen] = ptpfundual({w,width},a,M,L,inc);
ttpfundual=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('PTPFUNDUAL elapsed time %f s\n',ttpfundual);
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 PTPFUNDUAL: %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));