function [Fcoef1, Fcoef2, f1, f2]=co_fexp2(theta1,theta2,or,safr,ngrid)
% DAMOCO Toolbox, function CO_FEXP2, version 17.01.11
% Given two protophases, the function yields the coupling functions
% f1(theta1,theta2) and f2(theta2,theta1) via fitting a Fourier series.
%
% Form of call:
% [Fcoef1,Fcoef2,f1,f2]=co_fexp2(theta1,theta2,or,safr,ngrid)
% [Fcoef1,Fcoef2]=co_fexp2(theta1,theta2,or,safr,ngrid)
% [Fcoef1,Fcoef2]=co_fexp2(theta1,theta2,or,safr)
% Input: theta1: protophase of the 1st system,
% theta2: protophase of the 2nd system ('external'),
% or: maximal order of Fourier expansion,
% safr: sampling frequency,
% ngrid: size of the grid for function computation,
% by default ngrid = 100
% Output: Fcoef1,Fcoef2 are Fourier coefficients of the coupling functions
% f1,f2 are the functions, computed on a grid
%
theta1 = unwrap(theta1); theta2 = unwrap(theta2);
A = zeros(4*or+1, 4*or*1); % This matrix contains the coefficients A(n+k),(m+l)
% for the linear system of equations for the coefficients Fn,m.
or2=2*or; or21=or2+1; or1=or+1;
Dtheta1 = safr*( theta1(3:end)-theta1(1:end-2) )/2; % Derivative of theta1
Dtheta2 = safr*( theta2(3:end)-theta2(1:end-2) )/2; % Derivative of theta2
% Elimination of the first and last points
theta1=theta1(2:end-1); theta2=theta2(2:end-1);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% Computing the coefficients of the matrix A using symmetries of the cofficients
for n = -or2 : or2
for m = -or2 : n
A(n+or21, m+or21) = mean(exp(1i*(n*theta1 + m*theta2) ));
A(-n+or21, -m+or21)=conj(A(n+or21, m+or21));
end
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%% Computing the coefficients of the matrices Bnm
B1 = zeros(or21*or21); % This vector contains the coefficients B1n,m for the linear equation system for
% the coefficients F2n,m of the protophase theta1
B2 = B1; % This vector contains the coefficients B2n,m for the linear equation system for
% the coefficients F1n,m of the protophase theta2
C = B1; % The elements of the matrix A are reorganized in C to match the requirements of the
% MATLAB function to solve systems of linear equations
ind=1;
for n = -or : or
i1_1=(n+or)*or21;
for m = -or : or
i1=i1_1+m+or1; i4=m+or21;
tmp=exp(-1i*( n*theta1 + m*theta2) );
B1(ind)= mean(Dtheta1.* tmp);
B2(ind)= mean(Dtheta2.* tmp);
ind=ind+1;
for r = -or : or
i3=(r+or)*or21 +or1; i2=(n-r)+or21;
for s = -or : or; % Elements of the matrix A are reorganized in C
C(i1,i3 + s) = A(i2,i4-s);
end
end
end
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%% Solving the system of linear equations to obtain the coefficients Fnm
fc1 = conj(C) \ B1;
fc2 = conj(C) \ B2;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% Reorganizing the Fourier cofficients fc in the matrix Fcoef
Fcoef1 = zeros(or21,or21); Fcoef2 = Fcoef1;
for n = 1 : or21
k=(n-1)*or21;
for m = 1 : or21
Fcoef1(n, m)=fc1(k+m);
Fcoef2(n, m)=fc2(k+m);
end
end
Fcoef2 = Fcoef2.'; % Reorganizing the matrix Fcoef2 to match our convention
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%% Computing the coupling functions f1,f2 on a grid, if required
if nargout == 4;
if nargin == 4; ngrid=100; end; %Default value
[Y,X]=meshgrid(2*pi*(0:ngrid-1)/(ngrid-1),2*pi*(0:ngrid-1)/(ngrid-1));
f1 = zeros(ngrid,ngrid); f2=f1;
for n = -or : or
for m = -or : n
tmp=exp(1i*n*X + 1i*m*Y);
f1 = f1 + 2*real( Fcoef1(n+or1, m+or1) * tmp);
f2 = f2 + 2*real( Fcoef2(n+or1, m+or1) * tmp);
end
end
f1 = f1-real(Fcoef1(or1, or1)); % Fcoef1(0,0) was added twice in the loop
f2 = f2-real(Fcoef2(or1, or1)); % Fcoef2(0,0) was added twice in the loop
end
end