📄 cdpei.mht
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Date: Fri, 10 Sep 2004 12:03:16 +0800
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<BODY><PRE>function Fa=3DDFPei(f,a)
% Compute discrete fractional Fourier transform
% of order a of vector f according to Pei/Yeh/Tseng
N=3Dlength(f); f=3Df(:);
shft =3D rem((0:N-1)+fix(N/2),N)+1;
global hn_saved p_saved
if (nargin=3D=3D2), p =3D 2; end;
p =3D min(max(2,p),N-1);
if (length(hn_saved) ~=3D N | p_saved ~=3D p),
hn =3D make_hn(N,p);
hn_saved =3D hn; p_saved =3D p;
else
hn =3D hn_saved;
end;
Fa(shft,1)=3Dhn*(exp(-j*pi/2*a*[0:N-1]).'.*(hn'*f(shft)));
function hn =3D make_hn(N,p)
even =3D rem(N,2)=3D=3D0;
shft =3D rem((0:N-1)+fix(N/2),N)+1;
% Gauss-Hermite samples
u =3D (-N/2:(N/2)-1)'/sqrt(N/2/pi);
ex =3D exp(-u.^2/2);=20
hn(:,1) =3D ex; r =3D norm(hn(:,1)); hn(:,1) =3D hn(:,1)/r;
hn(:,2)=3D2*u.*ex; s =3D norm(hn(:,2)); hn(:,2) =3D hn(:,2)/s;
r =3D s/r;
for k =3D 3:N+1
hn(:,k)=3D2*r*u.*hn(:,k-1)-2*(k-2)*hn(:,k-2);
s =3D norm(hn(:,k)); hn(:,k) =3D hn(:,k)/s;
r=3Ds/r;
end
if (even), hn(:,N)=3D[]; else, hn(:,N+1)=3D[]; end
hn(shft,:) =3D hn;
% eigenvectors of DFT matrix
E =3D make_E(N,N/2);
for k =3D 1:4
if even % N even
switch k
case {1,3}
indx =3D k:4:N+1;
if (rem(N,4) ~=3D 0 && k=3D=3D3) || (rem(N,4) =3D=3D 0 =
&& k=3D=3D1),
indx(end) =3D indx(end)-1;
end
case {2,4}
indx =3D k:4:N-1;
end
else % N odd
indx =3D k:4:N;
end
hn(:,k:4:N) =3D orth(E(:,indx)*E(:,indx)'*hn(:,indx));
end
function E =3D make_E(N,p)
%Returns sorted eigenvectors and eigenvalues of corresponding vectors
%Construct matrix H, use approx order ord
d2 =3D [1 -2 1]; d_p =3D 1; s =3D 0; st =3D zeros(1,N);
for k =3D 1:p/2,
d_p =3D conv(d2,d_p);
st([N-k+1:N,1:k+1]) =3D d_p; st(1) =3D 0;
s =3D s + (-1)^(k-1)*prod(1:(k-1))^2/prod(1:2*k)*2*st; =20
end;
% H =3D circulant + diagonal
col =3D (0:N-1)'; row =3D (N:-1:1);
idx =3D col(:,ones(N,1)) + row(ones(N,1),:);
st =3D [s(N:-1:2).';s(:)];
H =3D st(idx)+diag(real(fft(s)));
%Construct transformation matrix V
r =3D floor(N/2);
even =3D ~rem(N,2);
V1 =3D (eye(N-1)+flipud(eye(N-1)))/sqrt(2);
V1(N-r:end,N-r:end) =3D -V1(N-r:end,N-r:end);
if (even), V1(r,r)=3D1; end
V =3D eye(N); V(2:N,2:N) =3D V1;
% Compute eigenvectors
VHV =3D V*H*V';
E =3D zeros(N);
Ev =3D VHV(1:r+1,1:r+1); Od =3D VHV(r+2:N,r+2:N);
[ve,ee]=3Deig(Ev); [vo,eo]=3Deig(Od);=20
%malab eig returns sorted eigenvalues
%if different routine gives unsorted eigvals, then sort first
%[d,inde] =3D sort(diag(ee)); [d,indo] =3D sort(diag(eo));
%ve =3D ve(:,inde'); vo =3D vo(:,indo');
E(1:r+1,1:r+1) =3D fliplr(ve); E(r+2:N,r+2:N) =3D fliplr(vo);
E =3D V*E;
%shuffle eigenvectors
ind =3D [1:r+1;r+2:2*r+2]; ind =3D ind(:);
if (even), ind([N,N+2])=3D[]; else ind(N+1)=3D[]; end
E =3D E(:,ind');
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