hformos.m

function Xk=hformos(S,Hk,A,xver)
% Xk=HFORMOS(S,Hk,A,xver)
%
% Computes a certain quadratic form used by Olhede & Simons, namely
% the function (1/S) [Hk^H * A * Hk] where ^H is the Hermitian transpose
%
% INPUT:
%
% S     A spectral density, which will be inverted
% Hk    A complex 2-column matrix of Fourier-domain observations, [Nx2], or
%       a complex 1-column vector of Fourier-domain observations, [Nx1]
% A     A wavenumber-dependent SYMMETRIC matrix, e.g. Tinv, [Nx3], or
%       a wavenumber-dependent vector, [Nx1]
% xver  1 or 2 Extra verification for the univariate case, when A=1
%
% OUTPUT:
%
% Xk   A column vector with the wavenumbers unwrapped, [N]
%
% NOTE: 
%
% In our formalism, this should result in a real-valued function. 
%
% Last modified by fjsimons-at-alum.mit.edu, 12/19/2023

defval('A',1)
defval('xver',1)

% The Inf's at k=0 get turned into NaN's here
Xk=    1./S.*[  conj(Hk(:,1)).*A(:,1).*Hk(:,1)];
if size(A,2)==3 && size(Hk,2)==2
  Xk=Xk+1./S.*[2*real(conj(Hk(:,2)).*A(:,2).*Hk(:,1))+...
		 conj(Hk(:,2)).*A(:,3).*Hk(:,2)];
end

% The 2*real is a shortcut to write the sum of the cross terms which
% should cancel any imaginary parts... it's a magnitude after all
% Still may have a tiny imaginary part, get rid of it if it is small 
Xk=realize(Xk);

if xver==1 || xver==2
    if A==1
        diferm(Xk,abs(Hk).^2./S,7);
    end
end