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Biomedical signal processing
Ch3.2 快速傅里叶变换(FFT) DFT: N2次的复数乘法, N(N-1)次的复数加法, N很大时, 计算量相当可观, N=1024, 复乘次数: 1,048,576 1965年, JW Cooley & JW Tukey, 在计算数学<<Mathematics of computation>>发表了著名的“机器计算Fourier series的一种算法”的文章, 提出快速傅立叶变换，成为DSP发展史上的里程碑。 FFT brief history. 一次复数乘法需要四次实数乘, 两次实数加, 一次复数加需要两次实数加, Nankai University, CY LI,
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Biomedical signal processing
快速傅里叶变换(FFT) FFT只是DFT的一种算法, 不是新的变换, 使得DFT的乘法计算量由N2降低到 , N=1024, 乘法计算量为5120, 仅为原来的4.88% Useful links on FFT （or） 一次复数乘法需要四次实数乘, 两次实数加, 一次复数加需要两次实数加, Nankai University, CY LI,
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Biomedical signal processing
FFT的原理 DFT运算中含有大量的重复运算, W因子的性质, Nankai University, CY LI,
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Biomedical signal processing
4点DFT运算 此时, 只需一次复数乘法 Nankai University, CY LI,
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Biomedical signal processing
基2FFT算法的推导(1) Nankai University, CY LI,
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Biomedical signal processing
基2FFT算法的推导(2) Nankai University, CY LI,
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Biomedical signal processing
基2FFT算法的推导(3) FFT的蝶形运算单元 -1 X2(k) WNk X1(k)+WNk X2(k) X1(k)-WNk X2(k) +1 X1(k) Nankai University, CY LI,
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Biomedical signal processing
基2FFT算法的思路 把一个序列分为长度减半的偶序列和奇序列, 原序列的DFT就由这两个N/2序列求得. 进一步把N/2序列分解成两个N/4序列, 一直分解到单点序列. 以L=3, N=8为例，需要三级分解, Nankai University, CY LI,
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Biomedical signal processing
N=8的FFT 1, 3, 5, 7 0, 2, 4, 6 0, 4 2, 6 1, 5 3, 7 x(0) x(4) x(2) x(6) x(1) x(5) x(3) x(7) x(0) x(1) x(2) x(3) x(4) x(5) x(6) x(7) 第一级 第二级 第三级 运算时, 是从下往上进行的,1->2->4->8 Nankai University, CY LI,
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Biomedical signal processing
FFT算法的特点 要使得X(k)正常排序, 输入序列的顺序打乱, 以二进制数码倒置顺序排列, 即码位倒置, 计算量: x(000) x(001) x(010) x(011) x(100) x(101) x(110) x(111) x(000) x(100) x(010) x(110) x(001) x(101) x(011) x(111) x(0) x(4) x(2) x(6) x(1) x(5) x(3) x(7) Nankai University, CY LI,
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Biomedical signal processing
频率抽取的FFT算法 上述的方法为按时间分组的FFT。 频率抽取的FFT算法：把X(k)按序列号k的奇偶分组分解后再计算DFT的方法。 Further reading for more information. Nankai University, CY LI,
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Biomedical signal processing
FFT的频率换算 已知Ts, x(n), length: N, X(k)=FFT[x(n)], k频率？ Nankai University, CY LI,
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Biomedical signal processing
举例 x(n)={1,1,0,0}, (x(n-2))N={0,0,1,1} Nankai University, CY LI,
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Biomedical signal processing
Zero padding + N F=0.48Hz, 0.52Hz, N=10 N=10+90, zero padding N=90, effective signal Nankai University, CY LI,
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Biomedical signal processing
阶跃函数的DTFT和DFT的区别 Nankai University, CY LI,
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Biomedical signal processing
N in DFT Nankai University, CY LI,
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Biomedical signal processing
FFT of sin(x) Nankai University, CY LI,
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Biomedical signal processing
fft in MATLAB fft and ifft, for one dimension fft2, ifft2, fftn, two and multi-dimensions, fftshift, ifftshift, shift the DC component to the middle of the spectrum, Nankai University, CY LI,
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Biomedical signal processing
fft FFT(X) is the discrete Fourier transform (DFT) of vector X. For matrices, the FFT operation is applied to each column. For N-D arrays, the FFT operation operates on the first non-singleton dimension. FFT(X,N) is the N-point FFT, padded with zeros if X has less than N points and truncated if it has more. FFT(X,[],DIM) or FFT(X,N,DIM) applies the FFT operation across the dimension DIM. For length N input vector x, the DFT is a length N vector X, with elements N X(k) = sum x(n)*exp(-j*2*pi*(k-1)*(n-1)/N), 1 <= k <= N. n=1 The inverse DFT (computed by IFFT) is given by x(n) = (1/N) sum X(k)*exp( j*2*pi*(k-1)*(n-1)/N), 1 <= n <= N. k=1 Nankai University, CY LI,
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