% 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 = freturnend
if (a==2), Faf = flipud(f)returnend
if (a==1), Faf(shft,1) = fft(f(shft))/sNreturnend
if (a==3), Faf(shft,1) = ifft(f(shft))*sNreturnend
% reduce to interval 0.5 <a <1.5
if (a>2.0), a = a-2f = flipud(f)end
if (a>1.5), a = a-1f(shft,1) = fft(f(shft))/sNend
if (a<0.5), a = a+1f(shft,1) = ifft(f(shft))*sNend
% 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)
fs=1000%采样频率N=1024 %采样点数
n=0:N-1
t=n/fs
f0=100 %信号频率
x=sin(2*pi*f0*t)
y=abs(fft(x,N)) %傅里叶变换后画出幅度谱
plot(y)
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