利用分数阶傅里叶变换对线性调频信号进行参数估计

function Faf = frft(f, a) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Ozakats 算法说明 % % 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 %幅值归一化 ？ if (a<0.5), a = a+1; f(shft,1) = ifft(f(shft))*sN; end %幅值归一化 ？ % 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)]; % 2倍内插，前后各补N个0，数据长度从N变为4N-3 % chirp premultiplication chrp = exp(-1i*pi/N*tana2/4*(-2*N+2:2*N-2)'.^2); %% 4N-3点的chirp信号 f = chrp.*f; % chirp convolution c = pi/N/sina/4; Faf = fconv(exp(1i*c*(-(4*N-4):4*N-4)'.^2),f);%% 8N-7个点的chirp与4N-3点的chirp做卷积，变成12N-11点信号 Faf = Faf(4*N-3:8*N-7)*sqrt(c/pi);%% 取中间4N-3点信号 % chirp post multiplication Faf = chrp.*Faf; % normalizing constant Faf = exp(-1i*(1-a)*pi/4)*Faf(N:2:end-N+1); %% 2倍抽取 function xint=interp(x) % sinc interpolation N = length(x); y = zeros(2*N-1,1); y(1:2:2*N-1) = x; %% 2倍内插，2N-1个点 xint = fconv(y(1:2*N-1), sinc([-(2*N-3):(2*N-3)]'/2)); %% 2N-1个点的y与4N-3个点的sinc做卷 xint = xint(2*N-2:end-2*N+3); %% 取 2N个点 %% 卷积函数 function z = fconv(x,y) % convolution by fft N = length([x(:);y(:)])-1; %% % P = 2^nextpow2(N); %% 离N最近的2的整数次幂 P = 2^nextpow2(N); z = ifft( fft(x,P) .* fft(y,P)); %% x，y卷积运算，P为FFT点数，为2的指数次幂 z = z(1:N); %%