分数阶傅立叶变换程序汇总（转载）
转载链接http://forum.vibunion.com/forum.php?mod=viewthread&action=printable&tid=41561
声振论坛
标题: 分数阶傅立叶变换程序汇总（个人收集自网上） [打印本页]
作者: simon21 时间: 2007-4-26 19:46
标题: 分数阶傅立叶变换程序汇总（个人收集自网上）
都是从网上收集来的，由于时间比较久，处处都忘记了，如果是谁的原创请和我联系，我在帖子中标出来的
第二楼：the fast fractional Fourier transform
程序参考下文中的算法
H.M. Ozaktas, M.A. Kutay, and G. Bozdagi.
Digital computation of the fractional Fourier transform.
IEEE Trans. Sig. Proc., 44:2141–2150, 1996.
第三楼：分数阶fourior变换的源码，主要用来处理线性调频信号
第四楼：离散分数阶傅立叶变换，算法参考
C. Candan, M.A. Kutay, and H.M. Ozaktas.
The discrete Fractional Fourier Transform.
IEEE Trans. Sig. Proc., 48:1329–1337, 2000
第五楼：the discrete fractional Fourier transform，算法参考：
S.-C. Pei, M.-H. Yeh, and C.-C. Tseng.
Discrete fractional Fourier-transform based on orthogonal projections.
IEEE Trans. Sig. Proc., 47(5):1335–1348, 1999.
第六楼：a demonstration programfor the previous three routines
第七楼： the discrete fractional Cosine transform，算法参考：
S.-C. Pei, M.-H. Yeh.
Discrete fractional Cosine and Sine transforms.
IEEE Trans. Sig. Proc., 49(6):1198–1207, 2001
第八楼：the discrete fractional Sine transform，算法参考：
S.-C. Pei, M.-H. Yeh.
Discrete fractional Cosine and Sine transforms.
IEEE Trans. Sig. Proc., 49(6):1198–1207, 2001
第九楼：test for Disfrct.m，算法参考：
S.-C. Pei, M.-H. Yeh.
Discrete fractional Cosine and Sine transforms.
IEEE Trans. Sig. Proc., 49(6):1198–1207, 2001
第十楼：the rescaling preprocessing for the frft routine，详细情况参考
H.M. Ozaktas, M.A. Kutay, and G. Bozdagi.
Digital computation of the fractional Fourier transform.
IEEE Trans. Sig. Proc., 44:2141–2150, 1996.
[ 本帖最后由 simon21 于 2007-4-26 20:14 编辑 ]
作者: simon21 时间: 2007-4-26 19:48
程序参考下文中的算法
H.M. Ozaktas, M.A. Kutay, and G. Bozdagi.
Digital computation of the fractional Fourier transform.
IEEE Trans. Sig. Proc., 44:2141–2150, 1996.
function Faf = frft(f, a)
% The fast Fractional Fourier Transform
% input: f = samples of the signal
% a = fractional power
% output: Faf = fast Fractional Fourier transform
error(nargchk(2, 2, nargin));
f = f(:);
N = length(f);
shft = rem((0:N-1)+fix(N/2),N)+1;
sN = sqrt(N);
a = mod(a,4);
% do special cases
if (a==0), Faf = f; return; end;
if (a==2), Faf = flipud(f); return; end;
if (a==1), Faf(shft,1) = fft(f(shft))/sN; return; end
if (a==3), Faf(shft,1) = ifft(f(shft))*sN; return; end
% reduce to interval 0.5 < a < 1.5
if (a>2.0), a = a-2; f = flipud(f); end
if (a>1.5), a = a-1; f(shft,1) = fft(f(shft))/sN; end
if (a<0.5), a = a+1; f(shft,1) = ifft(f(shft))*sN; end
% the general case for 0.5 < a < 1.5
alpha = a*pi/2;
tana2 = tan(alpha/2);
sina = sin(alpha);
f = [zeros(N-1,1) ; interp(f) ; zeros(N-1,1)];
% chirp premultiplication
chrp = exp(-i*pi/N*tana2/4*(-2*N+2:2*N-2)'.^2);
f = chrp.*f;
% chirp convolution
c = pi/N/sina/4;
Faf = fconv(exp(i*c*(-(4*N-4):4*N-4)'.^2),f);
Faf = Faf(4*N-3:8*N-7)*sqrt(c/pi);
% chirp post multiplication
Faf = chrp.*Faf;
% normalizing constant
Faf = exp(-i*(1-a)*pi/4)*Faf(N:2:end-N+1);
function xint=interp(x)
% sinc interpolation
N = length(x);
y = zeros(2*N-1,1);
y(1:2:2*N-1) = x;
xint = fconv(y(1:2*N-1), sinc([-(2*N-3):(2*N-3)]'/2));
xint = xint(2*N-2:end-2*N+3);function z = fconv(x,y)
% convolution by fft
N = length([x(:);y(:)])-1;
P = 2^nextpow2(N);
z = ifft( fft(x,P) .* fft(y,P));
z = z(1:N);
程序来自于http://www.cs.kuleuven.ac.be/cwi … oftware/FRFT/frft.m
作者: simon21 时间: 2007-4-26 19:51
分数阶fourior变换的源码，主要用来处理线性调频信号
%DISCRETE FRACTIONAL FOURIER TRANSFORM MATRIX GENERATOR
%by Cagatay Candan <[email]candan@ieee.org[/email]>, July 1998, Ankara
%Copyright 1998 Cagatay Candan
%This code may be used for scientific and educational purposes
%provided credit is given to the publications below:
%
%This Matlab function generates the discrete fractional
%Fourier transform matrix originally described in:
%Cagatay Candan, M. Alper Kutay, and Haldun M. Ozaktas,
%The discrete fractional Fourier Transform,
%IEEE Transactions on Signal Processing, 48:1329-1337, May 2000,
%(also in Proc ICASSP’99, pages 1713-1716, IEEE, 1999);
%and further described in:
%Haldun M. Ozaktas, Zeev Zalevsky, and M. Alper Kutay,
%The Fractional Fourier Transform with Applications in Optics and
%Signal Processing, Wiley, 2000, chapter 6.
function F=dFRT(N,a,ord)
%function F=dFRT(N,a,ord) returns the NxN discrete fractional
%Fourier transform matrix with fractional order 'a'.
%The optional argument 'ord' is the order of approximation
%of the S matrix (default 2).%Note: This Matlab file has some subfunctions for generating S_{2k}
% matrices, eigenvector ordering etc. These functions are not
% visible from the Matlab workspace.global Evec Eval ordpif nargin==2, ord=2;end;if (length(Evec)~=N | ordp~=ord),[Evec,Eval]=dis_s(N,ord);ordp=ord;
end;even=~rem(N,2);
F=Evec*diag(exp(-j*pi/2*a*([0:N-2 N-1+even])))*Evec';
function M=cconvm(v);
%Generates circular Convm matrixv=v(:);N=length(v);
M=zeros(N,N);dum=v;
for k=1:N,M(:,k)=dum;dum=[dum(N); dum(1:N-1)];
end;function S=creates(N,ord)
%Creates S matrix of approximation order ord
%When ord=1, elementary S matrix is returnedord=ord/2;
dum=[1 -2 1];s=0;
for k=1:ord,s=(-1)^(k-1)*prod(1:(k-1))^2/prod(1:2*k)*2*[0 dum(k+2:2*k+1) zeros(1,N-1-2*k) dum(1:k)]+s;dum=conv(dum,[1 -2 1]);
end;
S=cconvm(s)+diag(real(fft(s)));
function [Evec,Eval]=dis_s(N,ord)
%function [Evec,Eval]=dis_s(N)
%Returns sorted eigenvectors and eigenvalues of corresponding vectorsif nargin==1, ord=2;end;%%Construct S Matrix
%S=diag(2*cos(2*pi/N*([0:N-1])))+diag(ones(1,N-1),1)+diag(ones(1,N-1),-1);
%S(1,N)=1;S(N,1)=1;
%%
S=creates(N,ord);%%%%%%%Construct P matrixp=N;
r=floor(p/2);
P=zeros(p,p);P(1,1)=1;
even=~rem(p,2);
for k=1:r-even,P(k+1,k+1)=1/sqrt(2);P(k+1,p-(k+1)+2)=1/sqrt(2);
end;if (even), P(r+1,r+1)=1; end;for k=r+1:p-1,P(k+1,k+1)=-1/sqrt(2);P(k+1,p-(k+1)+2)=1/sqrt(2);
end;%%%%%%CS=P*S*P';C2=CS(floor(1:N/2+1),floor(1:N/2+1));S2=CS(floor(N/2+2):N,floor(N/2+2):N);[vc,ec]=eig(C2);[vs,es]=eig(S2);
qvc=[vc ;zeros(ceil(N/2-1),floor(N/2+1))];
SC2=P*qvc; %Even Eigenvector of Sqvs=[zeros(floor(N/2+1),ceil(N/2-1));vs];
SS2=P*qvs; %Odd Eigenvector of Ses=diag(es);ec=diag(ec);
[d,in]=sort(-ec);
SC2=SC2(:,in);
ec=ec(in);[d,in]=sort(-es);
SS2=SS2(:,in);
es=es(in);if rem(N,2)==0,S2C2=zeros(N,N+1);SS2(:,size(SS2,2)+1)=zeros(N,1);S2C2(:,[0:2:N]+1)=SC2;S2C2(:,[1:2:N]+1)=SS2;S2C2(:,N)=[];
elseS2C2=zeros(N,N);S2C2(:,[0:2:N]+1)=SC2;S2C2(:,[1:2:N-1]+1)=SS2;
end;Evec=S2C2;
Eval=(-j).^[ 0:N-2 (N-1)+even];%END
复制代码
作者: simon21 时间: 2007-4-26 19:56
离散分数阶傅立叶变换- Separate score step Fourier transforms
算法参考
C. Candan, M.A. Kutay, and H.M. Ozaktas.
The discrete Fractional Fourier Transform.
IEEE Trans. Sig. Proc., 48:1329–1337, 2000
function Fa=DFPei(f,a)% Compute discrete fractional Fourier transform
% of order a of vector f according to Pei/Yeh/TsengN=length(f); f=f(:);
shft = rem((0:N-1)+fix(N/2),N)+1;global hn_saved p_saved
if (nargin==2), p = 2; end;
p = min(max(2,p),N-1);if (length(hn_saved) ~= N | p_saved ~= p),hn = make_hn(N,p);hn_saved = hn; p_saved = p;
elsehn = hn_saved;
end;
Fa(shft,1)=hn*(exp(-j*pi/2*a*[0:N-1]).'.*(hn'*f(shft)));
function hn = make_hn(N,p)
even = rem(N,2)==0;
shft = rem((0:N-1)+fix(N/2),N)+1;
% Gauss-Hermite samples
u = (-N/2:(N/2)-1)'/sqrt(N/2/pi);
ex = exp(-u.^2/2);
hn(:,1) = ex; r = norm(hn(:,1)); hn(:,1) = hn(:,1)/r;
hn(:,2)=2*u.*ex; s = norm(hn(:,2)); hn(:,2) = hn(:,2)/s;
r = s/r;
for k = 3:N+1hn(:,k)=2*r*u.*hn(:,k-1)-2*(k-2)*hn(:,k-2);s = norm(hn(:,k)); hn(:,k) = hn(:,k)/s;r=s/r;
end
if (even), hn(:,N)=[]; else, hn(:,N+1)=[]; end
hn(shft,:) = hn;% eigenvectors of DFT matrix
E = make_E(N,N/2);for k = 1:4
if even % N evenswitch kcase {1,3}indx = k:4:N+1;if (rem(N,4) ~= 0 && k==3) || (rem(N,4) == 0 && k==1),indx(end) = indx(end)-1;endcase {2,4}indx = k:4:N-1;end
else % N oddindx = k:4:N;
end
hn(:,k:4:N) = orth(E(:,indx)*E(:,indx)'*hn(:,indx));
endfunction E = make_E(N,p)
%Returns sorted eigenvectors and eigenvalues of corresponding vectors%Construct matrix H, use approx order ordd2 = [1 -2 1]; d_p = 1; s = 0; st = zeros(1,N);
for k = 1:p/2,d_p = conv(d2,d_p);st([N-k+1:N,1:k+1]) = d_p; st(1) = 0;s = s + (-1)^(k-1)*prod(1:(k-1))^2/prod(1:2*k)*2*st;
end;
% H = circulant + diagonal
col = (0:N-1)'; row = (N:-1:1);
idx = col(:,ones(N,1)) + row(ones(N,1),:);
st = [s(N:-1:2).';s(:)];
H = st(idx)+diag(real(fft(s)));%Construct transformation matrix Vr = floor(N/2);
even = ~rem(N,2);
V1 = (eye(N-1)+flipud(eye(N-1)))/sqrt(2);
V1(N-r:end,N-r:end) = -V1(N-r:end,N-r:end);
if (even), V1(r,r)=1; end
V = eye(N); V(2:N,2:N) = V1;% Compute eigenvectorsVHV = V*H*V';
E = zeros(N);
Ev = VHV(1:r+1,1:r+1); Od = VHV(r+2:N,r+2:N);
[ve,ee]=eig(Ev); [vo,eo]=eig(Od);
%malab eig returns sorted eigenvalues
%if different routine gives unsorted eigvals, then sort first
%[d,inde] = sort(diag(ee)); [d,indo] = sort(diag(eo));
%ve = ve(:,inde'); vo = vo(:,indo');
E(1:r+1,1:r+1) = fliplr(ve); E(r+2:N,r+2:N) = fliplr(vo);
E = V*E;
%shuffle eigenvectors
ind = [1:r+1;r+2:2*r+2]; ind = ind(:);
if (even), ind([N,N+2])=[]; else ind(N+1)=[]; end
E = E(:,ind');
复制代码
本程序来自http://www.cs.kuleuven.ac.be/cwi … ftware/FRFT/cdpei.m
[ 本帖最后由 simon21 于 2007-4-26 19:57 编辑 ]
作者: simon21 时间: 2007-4-26 20:00
算法参考
S.-C. Pei, M.-H. Yeh, and C.-C. Tseng.
Discrete fractional Fourier-transform based on orthogonal projections.
IEEE Trans. Sig. Proc., 47(5):1335–1348, 1999.
function y=Disfrft(f,a,p)
% Computes discrete fractional Fourier transform
% of order a of vector x
% p (optional) is order of approximation, default N/2
N = length(f); even = ~rem(N,2);
shft = rem((0:N-1)+fix(N/2),N)+1;
f = f(:);
if (nargin==2), p = N/2; end;
p = min(max(2,p),N-1);
E = dFRFT(N,p);
y(shft,1)=E*(exp(-j*pi/2*a*([0:N-2 N-1+even])).'.*(E'*f(shft)));
function E=dFRFT(N,p)
%function E=dFRFT(N,a,p) returns the NxN eigenvectors of the
%Fourier transform matrix
%The optional argument p is the order of approximation
global E_saved p_saved
if (length(E_saved) ~= N | p_saved ~= p),E = make_E(N,p);E_saved = E; p_saved = p;
elseE = E_saved;
end;
function E = make_E(N,p)
%Returns sorted eigenvectors and eigenvalues of corresponding vectors%Construct matrix H, use approx order ord
d2 = [1 -2 1]; d_p = 1; s = 0; st = zeros(1,N);
for k = 1:p/2,d_p = conv(d2,d_p);st([N-k+1:N,1:k+1]) = d_p; st(1) = 0;temp = [1:k;1:k]; temp=temp(:)'./[1:2*k];s = s + (-1)^(k-1)*prod(temp)*2*st;
end;
% H = circulant + diagonal
col = (0:N-1)'; row = (N:-1:1);
idx = col(:,ones(N,1)) + row(ones(N,1),:);
st = [s(N:-1:2).';s(:)];
H = st(idx)+diag(real(fft(s)));
%Construct transformation matrix V
r = floor(N/2);
even = ~rem(N,2);
V1 = (eye(N-1)+flipud(eye(N-1)))/sqrt(2);
V1(N-r:end,N-r:end) = -V1(N-r:end,N-r:end);
if (even), V1(r,r)=1; end
V = eye(N); V(2:N,2:N) = V1;
% Compute eigenvectors
VHV = V*H*V';
E = zeros(N);
Ev = VHV(1:r+1,1:r+1); Od = VHV(r+2:N,r+2:N);
[ve,ee]=eig(Ev); [vo,eo]=eig(Od);
%malab eig returns sorted eigenvalues
%if different routine gives unsorted eigvals, then sort first
%[d,inde] = sort(diag(ee)); [d,indo] = sort(diag(eo));
%ve = ve(:,inde'); vo = vo(:,indo');
E(1:r+1,1:r+1) = fliplr(ve); E(r+2:N,r+2:N) = fliplr(vo);
E = V*E;
%shuffle eigenvectors
ind = [1:r+1;r+2:2*r+2]; ind = ind(:);
if (even), ind([N,N+2])=[]; else ind(N+1)=[]; end
E = E(:,ind');
复制代码
[ 本帖最后由 simon21 于 2007-4-26 20:08 编辑 ]
作者: simon21 时间: 2007-4-26 20:09
a demonstration programfor the previous three routines
x=0.0:0.02:2*pi; y =cos(x);
clear p_saved hn_saved E_saved
for a=0:0.05:4fy=Disfrft(y,a);fys=cdpei(y,a);fyss=frft(y,a);% blue,green,red,cyanfigure(1)subplot(311);plot(x,real([fy,fys,fyss]));title(['a = ',num2str(a)]);subplot(312);plot(x,imag([fy,fys,fyss]));subplot(313);plot(x,abs([fy,fys,fyss]));pause(0.7);
end
复制代码
作者: simon21 时间: 2007-4-26 20:10
the discrete fractional Cosine
算法参考：
S.-C. Pei, M.-H. Yeh.
Discrete fractional Cosine and Sine transforms.
IEEE Trans. Sig. Proc., 49(6):1198–1207, 2001
function y = Disfrct(f,a,p)
% Computes discrete fractional cosine transform
% of order a of vector f
% p (optional) is order of approximation, default N/2
% S-C Pei, M-H Yeh, IEEE Tr SP 49(6)2001, pp.1198-1207
N = length(f);
shft = rem((0:N-1)+fix(N/2),N)+1;
f = f(:);
if (nargin==2), p = N/2; end;
p = min(max(2,p),N-1);
E = dFRCT(N,p);
y=E*(exp(-j*pi*a*([0:N-1])).'.*(E'*f));function E=dFRCT(N,p)
%function E=dFRCT(N,p) returns the NxN eigenvectors of the
%Fourier Cosine transform matrixglobal EC_saved pC_savedif (length(EC_saved) ~= N | pC_saved ~= p),E = make_EC(N,p);EC_saved = E; pC_saved = p;
elseE = EC_saved;
end;
function E = make_EC(N,p)
% Returns sorted eigenvectors and eigenvalues of corresponding vectors
% Construct matrix H, use approx order p
N1=2*N-2;
d2 = [1 -2 1]; d_p = 1; s = 0; st = zeros(1,N1);
for k = 1:p/2,d_p = conv(d2,d_p);st([N1-k+1:N1,1:k+1]) = d_p; st(1) = 0;temp = [1:k;1:k]; temp = temp(:)'./[1:2*k];s = s + (-1)^(k-1)*prod(temp)*2*st;
end;
H = toeplitz(s(:),s)+diag(real(fft(s)));
% Construct transformation matrix V
V = [zeros(N-2,1),eye(N-2),zeros(N-2,1),flipud(eye(N-2))]/sqrt(2);
V = [1,zeros(1,N1-1);V;zeros(1,N-1),1,zeros(1,N-2)];
% Compute eigenvectors
Ev = V*H*V';
[ve,ee]=eig(Ev);
% malab eig returns sorted eigenvalues
% if different routine gives unsorted eigvals, then sort first
% [d,inde] = sort(diag(ee));
% ve = ve(:,inde');
E = fliplr(ve);
E(end,:) = E(end,:)/sqrt(2);
复制代码
[ 本帖最后由 simon21 于 2007-4-26 20:11 编辑 ]
作者: simon21 时间: 2007-4-26 20:11
the discrete fractional Sine transform
算法参考：
S.-C. Pei, M.-H. Yeh.
Discrete fractional Cosine and Sine transforms.
IEEE Trans. Sig. Proc., 49(6):1198–1207, 2001
function y = Disfrst(f,a,p)
% Computes discrete fractional sine transform
% of order a of vector f
% p (optional) is order of approximation, default N/2
% S-C Pei, M-H Yeh, IEEE Tr SP 49(6)2001, pp.1198-1207
N = length(f);
shft = rem((0:N-1)+fix(N/2),N)+1;
f = f(:);
if (nargin==2), p = N/2; end;
p = min(max(2,p),N-1);
E = dFRST(N,p);
y=E*(exp(-j*pi*a*([0:N-1])).'.*(E'*f));
function E=dFRST(N,p)
%function E=dFRST(N,p) returns the NxN eigenvectors of the
%Fourier Sine transform matrixglobal ES_saved pS_savedif (length(ES_saved) ~= N | pS_saved ~= p),E = make_ES(N,p);ES_saved = E; pS_saved = p;
elseE = ES_saved;
end;
function E = make_ES(N,p)
% Returns sorted eigenvectors and eigenvalues of corresponding vectors
% Construct matrix H, use approx order p
N1=2*N+2;
d2 = [1 -2 1]; d_p = 1; s = 0; st = zeros(1,N1);
for k = 1:p/2,d_p = conv(d2,d_p);st([N1-k+1:N1,1:k+1]) = d_p; st(1) = 0;temp = [1:k;1:k]; temp = temp(:)'./[1:2*k];s = s + (-1)^(k-1)*prod(temp)*2*st;
end;
H = toeplitz(s(:),s)+diag(real(fft(s)));
% Construct transformation matrix V
r = N;
V = [zeros(N,1),flipud(eye(N)),zeros(N,1),-eye(N)]/sqrt(2);
% Compute eigenvectors
Od = V*H*V';
[vo,eo]=eig(Od);
% malab eig returns sorted eigenvalues
% if different routine gives unsorted eigvals, then sort first
% [d,inde] = sort(diag(eo));
% vo = vo(:,inde');
E = flipud(fliplr(vo));
复制代码
作者: simon21 时间: 2007-4-26 20:13
test for Disfrct.m
算法参考：
S.-C. Pei, M.-H. Yeh.
Discrete fractional Cosine and Sine transforms.
IEEE Trans. Sig. Proc., 49(6):1198–1207, 2001
% Disfrftfunction testfrct(x)
x=[-35:35];
y = zeros(size(x));
y(fix(length(x)/2)+1)=1;
subplot(321),
plot(x,y);
title('symmetric delta')yf=Disfrft(y,5/6);
subplot(322),
plot(x,real(yf),'b',x,imag(yf),'r');
axis([-32,32,-0.2,0.2]);
title('FrFT for a = 5/6 of symmetric delta')% Disfrctx=[0:35];
y1(1:1)=1;
y1(2:36)=zeros(1,35);
subplot(323),
plot(x,y1)
axis([0,35,0,1])
title('half delta')yc=Disfrct(y1,5/6);
subplot(324),
plot(x,real(yc),'b',x,imag(yc),'r');
axis([0,35,0-.2,0.2]);
title('FrCT for a = 5/6 of half delta')% Disfrcstx=[-35:0];
y1(1:36)=zeros(1,36);
y1(1:1)=-1;
subplot(325),
plot(x,y1)
axis([-35,0,-1,0])
title('half delta')ys=Disfrst(y1,5/6);
subplot(326),
plot(x,real(ys),'b',x,imag(ys),'r');
axis([-35,0,-.2,0.2]);
title('FrST for a = 5/6 of half delta')作者: simon21 时间: 2007-4-26 20:14
the rescaling preprocessing for the frft routine
算法参考：
H.M. Ozaktas, M.A. Kutay, and G. Bozdagi.
Digital computation of the fractional Fourier transform.
IEEE Trans. Sig. Proc., 44:2141–2150, 1996.
function[signal,a_new,fact]=rescale(signal,a,delta_x1);
% This routine calculates the parameters to transform the input for the
% fractional Fourier transform when the support of the input
% input is not N^2 with N=sqrt(length(signal))
% Parameters: signal = the signal to be transformed
% a = the order of the transform
% delta_x1 = the length of the support of the input signal.
% To compute the frft with these data use
% Output: signal = possibly flipped signal to avoid infinite factor
% a_new = compute the frft of order a_new
% fact = and elementwise multiply the result with fact
N = length(signal);
delta_x2 = sqrt(N);
k = delta_x2/delta_x1;a_new = mod(a,4);if k == 1 || a == 0a_new = a;fact = 1;return
elseif a == 2signal = flipud(signal)
elsephi = a*pi/2;psi = acot(cot(phi)/k^2);c = csc(phi)/(k*csc(psi));x = linspace(-delta_x1/2,delta_x1/2,N);u = x;nu = c*u;a_new = 2*psi/pi;A_phi = exp(-i*(pi*sign(sin(phi))/4-phi/2))/sqrt(abs(sin(phi)));A_psi = exp(-i*(pi*sign(sin(psi))/4-psi/2))/sqrt(abs(sin(psi)));fact = A_phi/(k*A_psi)*exp(i*pi*(cot(phi)/c^2-cot(psi))*(nu.^2))';
end;
分数阶傅立叶变换程序汇总（转载）相关推荐
- 分数阶傅立叶变换 matlab,【综述】分数阶傅里叶变换(FRFT)
原标题:[综述]分数阶傅里叶变换(FRFT) 作者:WTT整理 傅里叶级数(傅里叶变换)几乎在所有科学和工程领域发挥着重要作用.黎曼积分和勒贝格积分均起源于对傅里叶级数的研究,傅里叶级数(傅里叶变换) ...
- 分数阶傅立叶变换中午matlab,怎么做短时分数阶傅里叶变换
我的分数阶傅里叶变换如下,怎么将它和短时傅里叶变换结合起来,变成短时分数阶傅里叶变换,有没有大佬指点一下下!!! frft: function Faf = frft(f, a) % The fast ...
- 线性代数28——复矩阵和快速傅立叶变换
原文 | https://mp.weixin.qq.com/s/YzPoPnRb-gEm_EiV9et0TA 实矩阵也可能碰到复特征值,因此无可避免地在矩阵运算中碰到复数. 矩阵当然也有可能包含复数, ...
- 数字图像处理：第九章 线性系统、卷积、傅立叶变换
第九章 线性系统.卷积.傅立叶变换 目录 1. 线性系统 2. 二维卷积 3. Fourier变换 作业 1. 线性系统 线性系统理论在"系统分析与控制"及&qu ...
- 数字图像处理 线性系统、卷积、傅立叶变换
第九章 线性系统.卷积.傅立叶变换 目录 1. 线性系统 2. 二维卷积 3. Fourier变换 作业 1. 线性系统 线性系统理论在"系统分析与控制"及&qu ...
- 论文总结2：分数阶傅里叶变换在信号检测与图像处理中的应用研究
分数阶傅里叶变换在信号检测与图像处理中的应用研究(李琼) 发展历史 特性 二维分数阶傅里叶变换 基于分数阶傅里叶变换的图像分析 分数阶变换域中图像的能量分布 分数阶傅里叶变换域中的图像的幅度和相位信息 ...
- dsp实现快速傅里叶的C语言程序,DSP-快速傅立叶变换(FFT)算法实验
<DSP-快速傅立叶变换(FFT)算法实验>由会员分享,可在线阅读,更多相关<DSP-快速傅立叶变换(FFT)算法实验(10页珍藏版)>请在人人文库网上搜索. 1.中 南 大 ...
- 分数傅立叶变换的性质
主要内容: 回顾--分数傅立叶变换基于积分核形式的推导 当P＝1的时候,可以把分数傅里叶变换的积分核退化为傅里叶变换的积分核形式. 在坐标系当中,当旋转角度到90°的时候就退化为傅里叶变换,当旋转角度 ...
- [转载]傅立叶变换在图像处理中的作用
原文地址:傅立叶变换在图像处理中的作用作者:白屋顶黑乌鸦 从现代数学的眼光来看,傅里叶变换是一种特殊的积分变换.它能将满足一定条件的某个函数表示成正弦基函数的线性组合或者积分.在不同的研究领域,傅里叶 ...
最新文章
- python中删除对象方法的区别_python中delattr删除对象方法的代码分析
- md5sum命令详解
- Go——从文件路径解析解析GAVC坐标解决方案
- 删除weblogic域
- mysql调查问卷_mysql – 解密数据库ER图用于问卷调查
- STM32F429与CC2530 ZigBee模块通信
- Poi HSSFCellStyle.ALIGN_CENTER VERTICAL_CENTER 等爆红的解决办法
- 建文高考成绩查询2021,深圳市建文外国语学校2020年高考喜报
- 模型建模流程及逻辑回归案例
- golang学习之negroni/gizp源码分析
- 面试经历---UC（2016年01月11日下午面试）
- JavaCV 实现照片打马赛克功能
- 三分钟读懂外汇保证金交易，告诉你我为什么要做外汇交易！（开篇）
- c语言中有余数的除法,《有余数的除法》课堂实录
- 聚焦2021年‘人工智能’产业三大发展趋势有哪些？
- 服务器修复 dns,AD 林恢复 - 配置 DNS 服务器服务
- Python实现多层感知器网络
- Error: Duplicate key name ‘PCS_STATS_IDX‘ (state=42000,code=1061)
- 提交软件至Mac App Store前必读
- android更新后忘记密码,教您6种方案解锁LG手机，如果您忘记了密码