% AO Phase Reconstruction Problem in 1D

% Computational grid.
n = 200;
h = 1/(n+1);
x = [h:h:1-h];

% First and second derivative matrices
e = ones(n,1);
A = (1/h^2)*spdiags([e, -2*e, e],[-1 0 1], n,n); 
A = full(A);  
D = (1/h)*spdiags([-e e], [0 1], n,n);
D = full(D);

% Generate true phase using biharmonic approximation of the Kolmogorov
% covariance matrix.
xx = 100*randn(n,1);
phase = A\xx;
sig = 0.1;
data = D*phase + sig*rand(n,1);
SNR = norm(data)/sqrt(n*sig^2) 

% Compute phase reconstruction
recon = A\(D*data);

% Plote results
figure(1)
 plot(x,data)
 title('Derivative Data Plot')
figure(2)
 plot(x,phase)
 hold on
 plot(x,recon,'r--')
 hold off
 title('True phase (-) and reconstructed phase (--)')