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function [G,covG] = dvar2frd(P,w,h,varargin)
%DVAR2FRD System and its covariance estimation in frequency domain
% [G,covG]=dvar2frd(P,w,h,p,VARX) returns the system and covariance in the
% frequency domain for the frequencies w and sample time h.
%
% [G,covG]=dvar2frd(P,w,h,p,A,B,C,K) returns the system and covariance in
% the frequency domain for the frequencies w and sample time h.
%
% [G,covG]=dvar2frd(P,w,h,p,A,B,C,D,K) returns the system and covariance
% in the frequency domain for the frequencies w and sample time h.
% Ivo Houtzager
% Delft Center of Systems and Control
% Delft University of Technology
% The Netherlands, 2010
if nargin == 5
p = varargin{1};
VARX = varargin{2};
m = floor(size(VARX,2)/p);
l = size(VARX,1);
r = m-l;
noD = ~(size(VARX,2)/m > p);
N = length(w);
G = zeros(l,r+l,N);
covG = zeros(l,r+l,N,2,2);
for q = 1:N
for j = 0:p
if j == 0
if noD ~= 1
G(:,:,q) = [VARX(:,p*m+1:p*m+r) eye(l)];
else
G(:,:,q) = [zeros(l,r) eye(l)];
end
else
G(:,:,q) = G(:,:,q) + [exp(-j*w(q)*1i*h).*VARX(:,(p-j)*m+1:(p-j)*m+r) -exp(-j*w(q)*1i*h).*VARX(:,(p-j)*m+r+1:(p-j)*m+r+l)];
end
end
G(:,:,q) = [pinv(G(:,r+1:r+l,q))*G(:,1:r,q) pinv(G(:,r+1:r+l,q))];
%T = kron([eye(r) zeros(r,l); -G(:,r+1:r+l,q)*G(:,r+1:r+l,q)*G(:,1:r,q) -G(:,r+1:r+l,q)].',G(:,r+1:r+l,q));
T = kron([eye(r) zeros(r,l); zeros(l,r) -G(:,r+1:r+l,q)].',G(:,r+1:r+l,q));
gradVARX = zeros(l*(l+r),p*l*(l+r)+~noD*r*l);
for c = 1:l
for v = 1:(r+l)
gradc = zeros(l,1);
gradc(c) = 1;
gradv = zeros(l*(l+r),1);
gradv((v-1)*l+1:v*l,1) = gradc;
gradVARX((v-1)*l+c,1:p*l*(l+r)) = kron(exp(-(p:-1:1).*w(q)*1i*h)',gradv)';
if ~noD
if v <= r
gradv = zeros(l*r,1);
gradv((v-1)*l+1:v*l,1) = gradc;
gradVARX((v-1)*l+c,p*l*(l+r)+(1:l*r)) = gradv(1:l*r);
else
gradVARX((v-1)*l+c,p*l*(l+r)+(1:l*r)) = zeros(l*r,1);
end
end
end
end
gradVARX = T*gradVARX;
for c = 1:l
for v = 1:(r+l)
covG(c,v,q,1,1) = real(gradVARX((v-1)*l+c,:))*P*real(gradVARX((v-1)*l+c,:))';
covG(c,v,q,1,2) = real(gradVARX((v-1)*l+c,:))*P*imag(gradVARX((v-1)*l+c,:))';
covG(c,v,q,2,1) = imag(gradVARX((v-1)*l+c,:))*P*real(gradVARX((v-1)*l+c,:))';
covG(c,v,q,2,2) = imag(gradVARX((v-1)*l+c,:))*P*imag(gradVARX((v-1)*l+c,:))';
end
end
end
elseif nargin == 7
A = varargin{1};
B = varargin{2};
C = varargin{3};
K = varargin{4};
l = size(C,1);
n = size(C,2);
r = size(B,2);
B = [B K];
D = [zeros(l,r) eye(l)];
s = exp(w*1i*h);
N = length(s);
gradABKC = zeros(n^2+n+n,N,l,r);
G = zeros(l,r+l,N);
covG = zeros(l,r+l,N,2,2);
for q = 1:N
Ainv = (s(q).*eye(n) - A)^(-1);
dCr = Ainv*B;
dBr = (C*Ainv)';
dAr = zeros(n^2,l,l+r);
for j = 1:n
for b = 1:n
sigma = zeros(n,n);
sigma(b,j) = 1;
for c = 1:l
for v = 1:(r+l)
dAr(b+(j-1)*n,c,v) = C(c,:)*Ainv*sigma*Ainv*B(:,v);
end
end
end
end
for c = 1:l
for v = 1:(r+l)
gradABKC(:,q,c,v) = conj([dAr(:,c,v); dBr(:,c); dCr(:,v)]);
ind = [1:n^2 n^2+(v-1)*n+1:n^2+n*v n^2+(r+l)*n+c:l:n^2+(r+l)*n+n*l];
covG(c,v,q,1,1) = real(gradABKC(:,q,c,v))'*P(ind,ind)*real(gradABKC(:,q,c,v));
covG(c,v,q,1,2) = real(gradABKC(:,q,c,v))'*P(ind,ind)*imag(gradABKC(:,q,c,v));
covG(c,v,q,2,1) = imag(gradABKC(:,q,c,v))'*P(ind,ind)*real(gradABKC(:,q,c,v));
covG(c,v,q,2,2) = imag(gradABKC(:,q,c,v))'*P(ind,ind)*imag(gradABKC(:,q,c,v));
G(c,v,q) = D(c,v) + C(c,:)*Ainv*B(:,v);
end
end
end
elseif nargin == 8
A = varargin{1};
B = varargin{2};
C = varargin{3};
D = varargin{4};
K = varargin{5};
l = size(C,1);
n = size(C,2);
r = size(B,2);
B = [B K];
D = [D eye(l)];
s = exp(w*1i*h);
N = length(s);
gradABKCD = zeros(n^2+n+n+1,N,l,r);
G = zeros(l,r+l,N);
covG = zeros(l,r+l,N,2,2);
for q = 1:N
Ainv = (s(q).*eye(n) - A)^(-1);
dCr = Ainv*B;
dBr = (C*Ainv)';
dAr = zeros(n^2,l,l+r);
for j = 1:n
for b = 1:n
sigma = zeros(n,n);
sigma(b,j) = 1;
for c = 1:l
for v = 1:(r+l)
dAr(b+(j-1)*n,c,v) = C(c,:)*Ainv*sigma*Ainv*B(:,v);
end
end
end
end
for c = 1:l
for v = 1:(r+l)
if v > r
gradABKCD(:,q,c,v) = conj([dAr(:,c,v); dBr(:,c); dCr(:,v); 0]);
ind = [1:n^2 n^2+(v-1)*n+1:n^2+n*v n^2+(r+l)*n+c:l:n^2+(r+l)*n+n*l];
covG(c,v,q,1,1) = real(gradABKCD(:,q,c,v))'*blkdiag(P(ind,ind),0)*real(gradABKCD(:,q,c,v));
covG(c,v,q,1,2) = real(gradABKCD(:,q,c,v))'*blkdiag(P(ind,ind),0)*imag(gradABKCD(:,q,c,v));
covG(c,v,q,2,1) = imag(gradABKCD(:,q,c,v))'*blkdiag(P(ind,ind),0)*real(gradABKCD(:,q,c,v));
covG(c,v,q,2,2) = imag(gradABKCD(:,q,c,v))'*blkdiag(P(ind,ind),0)*imag(gradABKCD(:,q,c,v));
else
gradABKCD(:,q,c,v) = conj([dAr(:,c,v); dBr(:,c); dCr(:,v); 1]);
ind = [1:n^2 n^2+(v-1)*n+1:n^2+n*v n^2+(r+l)*n+c:l:n^2+(r+l)*n+n*l n^2+(r+l)*n+n*l+(v-1)*l+c];
covG(c,v,q,1,1) = real(gradABKCD(:,q,c,v))'*P(ind,ind)*real(gradABKCD(:,q,c,v));
covG(c,v,q,1,2) = real(gradABKCD(:,q,c,v))'*P(ind,ind)*imag(gradABKCD(:,q,c,v));
covG(c,v,q,2,1) = imag(gradABKCD(:,q,c,v))'*P(ind,ind)*real(gradABKCD(:,q,c,v));
covG(c,v,q,2,2) = imag(gradABKCD(:,q,c,v))'*P(ind,ind)*imag(gradABKCD(:,q,c,v));
end
G(c,v,q) = D(c,v) + C(c,:)*Ainv*B(:,v);
end
end
end
else
error('DVAR2RFD requires five, seven or eight input arguments.')
end
end