Fast Fourier Transform

1. Fast Fourier Transform

2. Fourier 变换: 存在的条件: Jean Baptiste Joseph Fourier (1768 - 1830) 反变换:

3. 当 g(x) 为实函数

4. 1 d(t) w t delta 函数 TopHat 函数

5. cos(w0t) w t +w0 -w0 0 0 cosine 函数 sine 函数

6. 位移性质: 相似性质: -a a

7. energy theorem, Rayleigh’s theorem The zero frequency point 反变换: 代入

8. 常用的Fourier变换

9. 常用的其他定义 连续傅立叶变换 (Continuous Fourier Transform) 离散傅立叶变换 (Discrete Fourier Transform) 连续傅立叶变换(Continuous Fourier Transform) where 离散傅立叶变换(Discrete Fourier Transform) For u=0,1,2,…,N-1 For x=0,1,2,…,N-1

10. 连续Fourier变换(Continuous Fourier Transform) 反变换 DFT: IDFT:

11. 矩阵形式的DFT

12. omega = exp(-2*pi*i/n); j = 0:n-1; k = j'; F = omega.^(k*j); % an easier,and quicker, way to generate F is F = fft(eye(n));

13. Fast Fourier Transform • The Fast Fourier Transform (FFT) is a very efficient algorithm for performing a discrete Fourier transform • FFT principle first used by Gauss in 1805? • FFT algorithm published by Cooley & Tukey in 1965 • In 1969, the 2048 point analysis of a seismic trace took 13 ½ hours. Using the FFT, the same task on the same machine took 2.4 seconds!

14. 离散Fourier变换(DFT) (此处定义与教材和MATLAB保持一致) Requires N2 complex multiplies and N(N-1) complex additions WN=e-j2π/N 对称性: 周期性:

15. 两个长度为N/2的DFT之和

16. N/2 DFT x[0,2,4,6] X[0…7] • Cross feed of G[k] and H[k] in flow diagram is called a “butterfly”, due to the shape N/2 DFT x[1,3,5,7] or simplify: -1

18. 对N=8， 因为WN/2 = -1, Ｘk0和Ｘk1具有周期N/2, Diagrammatically (butterfly), There are N/2 butterflies for this stage of the FFT, and each butterfly requires one multiplication

19. The splitting of {Xk} into two half-size DFTs can be repeated on Ｘk0 and Ｘk1 themselves,

20. {Ｘk00} is the N/4-point DFT of {x0, x4,…, xN-4}, • {Ｘk01} is the N/4-point DFT of {x2, x6,…, xN-2}, • {Ｘk10} is the N/4-point DFT of {x1, x5,…, xN-3}, • {Ｘk11} is the N/4-point DFT of {x3, x7,…, xN-1}.

21. bit reversal 0, 1, 2, 3, 4, 5, 6, 7 is reordered to 0, 4, 2, 6, 1, 5, 3, 7

22. 定义 c=[0 2 4 6 1 3 5 7]

25. function y=fft(x,n) if n=1 y=x else m=n/2; w=e-i2πn yT=fft(x(0:2:n),m) yB=fft(x(1:2:n),m) d=[1,w,…,wm-1]T z=d.*yB y=[yT+z; yB+z] end 设 n=2t

26. fftgui(y)产生4个plots: real(y),imag(y), real(fft(y)),imag(fft(y)) print -deps FftGui.eps print –depsc2 FftGui.eps

27. y0=1,y1=…=yn-1=0,

28. y0=0,y1=1,y2=…=yn-1=0,

29. FFT is the sum of two sinusoids

30. Nyquist point

31. 若y是长度为n的实向量,Y=fft(y),则 real(Y0)=∑yj imag(Y0)=0 real(Yj)=real(Yn-j), j=1,…,n/2 imag(Yj)=-imag(Yn-j),j=1,…,n/2

32. % the sampling rate. Fs = 32768; t = 0:1/Fs:0.25; % the button in position (k,j) for k=1:4 for j=1:3 y1 = sin(2*pi*fr(k)*t); y2 = sin(2*pi*fc(j)*t); y = (y1 + y2)/2; input('Press any key:)'); sound(y,Fs) end end 697 770 852 941 1209 1336 1477

35. load sunspot.dat t = sunspot(:,1)'; wolfer = sunspot(:,2)'; n = length(wolfer); c = polyfit(t,wolfer,1); trend = polyval(c,t); plot(t,[wolfer; trend],'-', t,wolfer,'k.')

37. y = wolfer - trend; Y = fft(y); Fs = 1; % Sample rate f = (0:n/2)*Fs/n; pow = abs(Y(1:n/2+1)); plot([f; f],[0*pow; pow],'c-', … f,pow,'b.', ... 'linewidth',2, 'markersize',16)

40. plot(fft(eye(17))) axis square

41. Chebyshev Polynomial 递推关系: 满足微分方程: 扩充到复平面 扩充到|z|>1 第二类Chebyshev多项式

42. shg,hold on fplot('x',[-1,1]) fplot('2*x^2-1',[-1,1]) fplot('4*x^3-3*x',[-1,1]) fplot('8*x^4-8*x^2+1',[-1,1]) fplot('16*x^5-20*x^3+5*x',[-1,1])

43. “We do not make things, We make things better.”

44. The price (in euros) of a magazine has changed as follows Estimate the price in November 2002 by extrapolating these data.