function c=ref_dwiltiii(f,g,a,M)
%REF_DWILTIII   Reference DWILT type III
%   Usage:  c=ref_dwiltiii(f,g,a,M);
%

L=size(f,1);
W=size(f,2);

% Possibly zero-extend the window if necessary.
g=fir2long(g,L);

N=L/a;

F=zeros(L,M*N);

l=(0:L-1)';

  
pif=pi/4;

if 0
  % This is the definition where the odd and even indices are split  
  for n=0:floor(N/2)-1
    for m=0:2:M-1
      F(:,1+m+2*n*M)=sqrt(2)*circshift(g,2*n*a).*cos((m+.5)*pi*l/M+pif);
      F(:,1+m+(2*n+1)*M)=sqrt(2)*circshift(g,(2*n+1)*a).*sin((m+.5)*pi*l/M+pif);
    end;
    for m=1:2:M-1
      F(:,1+m+2*n*M)=sqrt(2)*circshift(g,2*n*a).*sin((m+.5)*pi*l/M+pif);
      F(:,1+m+(2*n+1)*M)=sqrt(2)*circshift(g,(2*n+1)*a).*cos((m+.5)*pi*l/M+pif);
    end;
  end;
else
  % Combined definition

    % This is the definition where the odd and even indices are split  
  for n=0:N-1
    for m=0:M-1
      if rem(m+n,2)==0
        F(:,1+m+n*M)=sqrt(2)*circshift(g,n*a).*cos((m+.5)*pi*l/M+1/4*pi+(m+n)*pi);
      else
        F(:,1+m+n*M)=sqrt(2)*circshift(g,n*a).*cos((m+.5)*pi*l/M+3/4*pi+(m+n)*pi);
      end;
    end;
  end;
end;

c=F.'*f;



