GSP_DESIGN_ITERSINE - Create a itersine filterbanks

function [ g,mu ] = gsp_design_itersine( G, Nf, param )
%GSP_DESIGN_ITERSINE Create a itersine filterbanks
%   Usage: g = gsp_design_itersine( G, Nf );
%          g = gsp_design_itersine( G, Nf, param );
%   
%   Inputs parameters:
%       G       : Graph structure or lmax
%       Nf      : Number of filter 
%       param   : Structure of optional parameters
%
%   Outputs parameters:
%       g       : filterbanks
%       mu      : centers of the filters
%
%   This function creates an itersine half overlap filterbank of Nf filters
%   Going from 0 to lambda_{max}
%
%   This filterbank is tight for an overlap of 2 and other particular
%   values. The function normalizes the window such that the framebound is
%   1.
%
%   The itersine window between -0.5 and 0.5 is defined as
%
%       g(t) = sin( 0.5 pi (cos pi t)^2 )
%
%   This function will compute the maximum eigenvalue of the laplacian. To
%   be more efficient, you can precompute it using:
%
%       G = gsp_estimate_lmax(G);
%
%   Example:
%
%         Nf = 20;
%         G = gsp_sensor(100);
%         G = gsp_estimate_lmax(G);
%         g = gsp_design_itersine(G, Nf);   
%         gsp_plot_filter(G,g);  
%         [A,B] = gsp_filterbank_bounds(G,g)
%
%   param is an optional structure containing the following fields
%
%    param.verbose*: verbosity level. 0 no log - 1 display warnings.
%     (default 1) 
%    param.overlap*: Overlap : default 2
%
%
%   Url: https://epfl-lts2.github.io/gspbox-html/doc/filters/gsp_design_itersine.html

% Copyright (C) 2013-2016 Nathanael Perraudin, Johan Paratte, David I Shuman.
% This file is part of GSPbox version 0.7.5
%
% 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/>.

% If you use this toolbox please kindly cite
%     N. Perraudin, J. Paratte, D. Shuman, V. Kalofolias, P. Vandergheynst,
%     and D. K. Hammond. GSPBOX: A toolbox for signal processing on graphs.
%     ArXiv e-prints, Aug. 2014.
% http://arxiv.org/abs/1408.5781

% Author: Nathanael Perraudin
% Date  : 18 June 2014
% Testing: test_filter

if nargin < 3
    param = struct;
end

if ~isfield(param,'verbose'), param.verbose = 1; end
if ~isfield(param,'overlap'), param.overlap = 2; end

if isstruct(G)
    if ~isfield(G,'lmax')
        if param.verbose
            fprintf('GSP_DESIGN_ITERSINE has to compute lmax \n')
        end
        G = gsp_estimate_lmax(G);
    end
   lmax = G.lmax;
else
   lmax = G;
end


k = @(x) sin(0.5 * pi * (cos(pi*x)).^2) .* (x>=-0.5 & x<= 0.5);

g = cell(Nf,1);

scale = lmax/(Nf-param.overlap+1)*(param.overlap);

mu = zeros(Nf,1);

for ii=1:Nf
    g{ii} = @(x) k(x/scale-(ii-param.overlap/2)/param.overlap)...
                    ./sqrt(param.overlap)*sqrt(2);
    mu(ii) = (ii-param.overlap/2)/param.overlap * scale;
end


end