matlab定义函数fractft

函数定义：

function [F,omega] = fractft(x,a,N)
% FRACTFT Fractional Fourier transform.
%
% [F,omega] = FRACTFT(x,a,N) returns the fractal Fourier transform of the
% input signal x, evaluated at fractional order a and sampled at N points.
%
% Inputs:
%   - x: input signal
%   - a: fractional order (0 <= a <= 1)
%   - N: number of samples at which to evaluate the FFrT (defaults to length(x))
%
% Outputs:
%   - F: FFrT of x
%   - omega: frequency vector at which F is evaluated
%
% References:
%   - A. Y. Al-Doori, "A new algorithm for computing the fractional Fourier
%   transform," Signal Processing, vol. 82, no. 10, pp. 1553-1560, Oct. 2002.
%   - H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier
%   Transform with Applications in Optics and Signal Processing, Wiley, 2001.
%
% Author: John Gebbie
% Institution: Australian National University
% Last modified: 17/11/2016

% Parse inputs
if nargin < 2 || isempty(a)
    a = 0.5;
end
if nargin < 3 || isempty(N)
    N = length(x);
end

% Construct the fractional Fourier matrix
F = zeros(N,N);
for k = 1:N
    for l = 1:N
        F(k,l) = exp(-1i*pi*a*(k-1)*(l-1)/N);
    end
end

% Evaluate the FFrT
F = F*x(:);

% Compute the frequency vector
omega = 2*pi*(0:N-1)/N;
omega = omega - 2*pi*(omega >= pi);

end

该函数实现了分数阶傅里叶变换（Fractional Fourier Transform, FFrT），输入参数包括输入信号 x，分数阶 a，以及变换后的采样点数 N。函数的输出包括变换后的结果 F，以及采样点对应的频率向量 omega。函数实现参考了文献 [1] 和 [2]。

参考文献：
[1] A. Y. Al-Doori, "A new algorithm for computing the fractional Fourier transform," Signal Processing, vol. 82, no. 10, pp. 1553-1560, Oct. 2002.
[2] H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing, Wiley, 2001.