function [Qcoef1, Qcoef2, q1, q2]=co_fcplfct2(phi1,phi2,dphi1,dphi2,N,ngrid)
% DAMOCO Toolbox, function CO_FCPLFCT2, version 03.03.14
%
% Given two protophases and their derivatives, the function yields
% the coupling functions q1(phi1,phi2) and q2(phi2,phi1)
% via fitting a Fourier series.
%
% Form of call:
% [Qcoef1,Qcoef2,q1,q2]=co_fcplfct2(phi1,phi2,dphi1,dphi2,N,ngrid)
% [Qcoef1,Qcoef2]=co_fcplfct2(phi1,phi2,dphi1,dphi2,N,ngrid)
% [Qcoef1,Qcoef2,q1,q2]=co_fcplfct2(phi1,phi2,dphi1,dphi2,N)
% Input: phi1: phase of the 1st system,
% phi2: phase of the 2nd system ('external'),
% dphi1: derivative of the phase of the 1st system
% dphi2: derivative of the phase of the 2nd system
% N: maximal order of Fourier expansion,
% ngrid: size of the grid for function computation,
% by default ngrid = 100
% Output: Qcoef1,Qcoef2 are Fourier coefficients of the coupling functions
% q1,q2 are the functions, computed on a grid
%
phi1 = unwrap(phi1); phi2 = unwrap(phi2);
A = zeros(4*N+1, 4*N+1); % This matrix contains the coefficients A(n+k),(m+l)
% for the linear system of equations for the coefficients Qn,m.
or2=2*N; or21=or2+1; or1=N+1;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% Computing the coefficients of the matrix A using symmetries of the coefficients
for n = -or2 : or2
for m = -or2 : n
A(n+or21, m+or21) = mean(exp(1i*(n*phi1 + m*phi2) ));
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 Q1n,m of the phase phi1
B2 = B1; % This vector contains the coefficients B2n,m for the linear equation system for
% the coefficients Q2n,m of the phase phi2
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 = -N : N
i1_1=(n+N)*or21;
for m = -N : N
i1=i1_1+m+or1; i4=m+or21;
tmp=exp(-1i*( n*phi1 + m*phi2) );
B1(ind)= mean(dphi1.* tmp);
B2(ind)= mean(dphi2.* tmp);
ind=ind+1;
for r = -N : N
i3=(r+N)*or21 +or1; i2=(n-r)+or21;
for s = -N : N; % 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 qc_nm
qc1 = conj(C) \ B1;
qc2 = conj(C) \ B2;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% Reorganizing the Fourier cofficients qc in the matrix Qcoef
Qcoef1 = zeros(or21,or21); Qcoef2 = Qcoef1;
for n = 1 : or21
k=(n-1)*or21;
for m = 1 : or21
Qcoef1(n, m)=qc1(k+m);
Qcoef2(n, m)=qc2(k+m);
end
end
Qcoef2 = Qcoef2.'; % Reorganizing the matrix Qcoef2 to match our convention
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%% Computing the coupling functions q1,q2 on a grid, if required
if nargout == 4;
if nargin == 5; ngrid=100; end; %Default value
[Y,X]=meshgrid(2*pi*(0:ngrid-1)/(ngrid-1),2*pi*(0:ngrid-1)/(ngrid-1));
q1 = zeros(ngrid,ngrid); q2=q1;
for n = -N : N
for m = -N : N
tmp=exp(1i*n*X + 1i*m*Y);
q1 = q1 + Qcoef1(n+or1, m+or1) * tmp;
q2 = q2 + Qcoef2(n+or1, m+or1) * tmp;
end
end
q1 = real(q1); % Eliminating imaginary rests (numerical)
q2 = real(q2);
end
end