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function X = fft_2N(x)
% 計算長度爲2N的實序列的FFT
% John G.Proakis,數字信號處理,第四版
% Define variables:
% x --- Input real vector of length 2*N(power of 2)
% X --- Output vector,FFT result
% Record of revisions:
% Date Programmer Description of change
% ==== ========== =====================
% 2014/0421 GOU DH Original code
%
% test
clear,clc
x = randn(64,1);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
x = x(:);
nx = length(x);
t = reshape(x,2,nx/2);
x1 = t(1,:)';
x2 = t(2,:)'; % 時域抽取,奇偶分離
[X1,X2] = fft_2_N(x1,x2,nx/2);
k = [0:nx/2-1]';
W2Nk = exp(-j*2*pi*k/(nx));
T = X2.*W2Nk;
G1 = X1 + T;
G2 = X1 - T;
X = [G1;G2];
% test
sum(abs(X - fft(x,nx)))
return;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Subfunction
function [X1,X2] = fft_2_N(x1,x2,N)
% 計算2個N點實序列的FFT
if ~isreal(x1) | ~isreal(x2)
return;
end
x = x1 + j*x2;
X = fft_dif2(x,-1);
T = [X(1); flipud(X(2:end))];
T = conj(T);
X1 = (X + T)/2;
X2 = (X - T)/(2*j);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Subfunction
function [X]=fft_dif2(x,f)
% 頻域抽取基2-FFT(基2-DIF-FFT)
nx = length(x);
M = fix(log2(nx));
if 2^M < nx, M = M + 1; end % 級數M
x = [x(:);zeros(2^M-nx,1)];
N = length(x);
%
C = j*pi*f;
for m = 0:1:M-1 % 級循環序號
B = 2^(M-1-m); % 蝶形因子序號間距=蝶形因子個數=2^(M-1-m)
for r = 0:1:B-1 % 蝶形因子循環序號
Wnp = exp(C*r/B);
for k = r:2*B:N-1
tmp1 = x(k+1);
tmp2 = x(k+B+1);
x(k+1) = tmp1+tmp2;
x(k+B+1) = (tmp1-tmp2)*Wnp;
end
end
end
% 倒位序
% X = x;
LH = N/2;
J = LH;
N1 = N-2;
for I = 1:1:N1
if I < J
T = x(I+1);
x(I+1) = x(J+1);
x(J+1) = T;
end
K = LH;
while J >= K
J = J-K;
K = K/2;
end
J = J+K;
end
if f == 1
x = x/N;
end
X = x;
收藏於 2014-04-21
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