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function Faf = fRFt(f, a)! t0 K% ~" X" \1 }: t
% The fast Fractional Fourier Transform
3 ?! A$ [6 _. W8 w1 e) a% I1 d% input: f = samples of the signal2 i( _ `7 P% M! T
% a = fractional power2 S' X. s4 j+ V% m" z# I7 J
% output: Faf = fast Fractional Fourier transform error(nargchk(2, 2, nargin)); f = f( ;
}2 {2 f2 k1 l6 y d. jN = length(f);" p4 t! \' M; X9 v. q5 A* Z
shft = rem((0:N-1)+fix(N/2),N)+1;- O3 k7 D9 {, y8 d, H) k" L
sN = sqrt(N);/ k X8 c O3 ^* O! ^
a = mod(a,4); % do special cases0 U/ i |- T+ O6 {
if (a==0), Faf = f; return; end;/ h1 j: [. }& ?
if (a==2), Faf = flipud(f); return; end;
# r4 b5 X' ?6 @5 r4 Oif (a==1), Faf(shft,1) = fft(f(shft))/sN; return; end
6 R9 V* v1 N: k0 P7 Tif (a==3), Faf(shft,1) = ifft(f(shft))*sN; return; end % reduce to interval 0.5 < a < 1.5
' O9 Y$ A. P2 G+ {if (a>2.0), a = a-2; f = flipud(f); end5 R: A0 V& n% l, n: C5 S
if (a>1.5), a = a-1; f(shft,1) = fft(f(shft))/sN; end( _. y9 e. h' y! Q" O4 e8 t1 I
if (a<0.5), a = a+1; f(shft,1) = ifft(f(shft))*sN; end % the general case for 0.5 < a < 1.55 y- `7 @& w& ~
alpha = a*pi/2;
+ ]; M* @! g7 T/ ytana2 = tan(alpha/2);
; N+ Y8 R4 d* i! Y5 H9 rsina = sin(alpha);
4 m5 Y' P5 n- ^& D. ]: {6 m! V" gf = [zeros(N-1,1) ; interp(f) ; zeros(N-1,1)]; % chirp premultiplication
( g3 U: h& z1 M* _# ]! Ochrp = exp(-i*pi/N*tana2/4*(-2*N+2:2*N-2)'.^2);( L( `8 h M1 U5 R* i* J) ^
f = chrp.*f; % chirp convolution' j; A$ D+ I" x4 r4 A1 B
c = pi/N/sina/4;. q# ^7 h/ R# k3 w
Faf = fconv(exp(i*c*(-(4*N-4):4*N-4)'.^2),f);* B5 G* t0 S2 J! w' b$ [ v+ H4 R
Faf = Faf(4*N-3:8*N-7)*sqrt(c/pi); % chirp post multiplication
( w# ~/ s1 m! p' Z! Z3 IFaf = chrp.*Faf; % normalizing constant0 Q' W& W) C. f2 d8 r* O
Faf = exp(-i*(1-a)*pi/4)*Faf(N:2:end-N+1); %%%%%%%%%%%%%%%%%%%%%%%%%8 a& w; F: Z" u, o7 }5 a
function xint=interp(x)6 I! `2 J+ Y+ T
% sinc interpolation N = length(x);# o9 B$ C9 U( k) Q" O* H( U
y = zeros(2*N-1,1);. X* B* A5 B8 C! U% Q& q: \6 f0 e
y(1:2:2*N-1) = x;
9 F0 O; \. U, { \xint = fconv(y(1:2*N-1), sinc([-(2*N-3) 2*N-3)]'/2));0 t c8 b, Z$ H5 f$ l
xint = xint(2*N-2:end-2*N+3); %%%%%%%%%%%%%%%%%%%%%%%%%
, t: x+ @3 U* O/ S _6 }1 wfunction z = fconv(x,y)
( i( G& u3 t. l* A# P$ L% convolution by fft N = length([x( ;y( ])-1;
! K; o9 U) K% x0 M: sP = 2^nextpow2(N);1 _8 \ e2 ]$ K4 | [. N
z = ifft( fft(x,P) .* fft(y,P));
9 H/ Q2 z: K4 v# f" a+ fz = z(1:N); |