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

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Presentation on theme: "Fast Fourier Transform. Fourier 变换 : 存在的条件 : 反变换 : Jean Baptiste Joseph Fourier (1768 - 1830)"— Presentation transcript:

1 Fast Fourier Transform

2 Fourier 变换 : 存在的条件 : 反变换 : Jean Baptiste Joseph Fourier ( )

3 当 g(x) 为实函数

4 delta 函数 t (t)(t)   TopHat 函数

5       cos(  0 t) t  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 连续傅立叶变换 (Continuous Fourier Transform) 离散傅立叶变换 (Discrete Fourier Transform) where For u=0,1,2,…,N-1 For x=0,1,2,…,N-1 连续傅立叶变换 (Continuous Fourier Transform) 离散傅立叶变换 (Discrete Fourier Transform) 常用的其他定义

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 Requires N 2 complex multiplies and N(N-1) complex additions 离散 Fourier 变换 (DFT) ( 此处定义与教材和 MATLAB 保持一致 ) 对称性 : 周期性 : W N =e -j2π/N

15 两个长度为 N/2 的 DFT 之和

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

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18 因为 W N/2 = -1, X k 0 和 X k 1 具有周期 N/2, 对 N=8 , There are N/2 butterflies for this stage of the FFT, and each butterfly requires one multiplication Diagrammatically (butterfly),

19 The splitting of {X k } into two half-size DFTs can be repeated on X k 0 and X k 1 themselves,

20 –{ X k 00 } is the N/4-point DFT of {x 0, x 4, …, x N-4 }, –{ X k 01 } is the N/4-point DFT of {x 2, x 6, …, x N-2 }, –{ X k 10 } is the N/4-point DFT of {x 1, x 5, …, x N-3 }, –{ X k 11 } is the N/4-point DFT of {x 3, x 7, …, x N-1 }.

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

22 定义 c=[ ]

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

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 y 0 =1,y 1 =…=y n-1 =0,

28 y 0 =0,y 1 =1,y 2 =…=y n-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(Y 0 )=∑y j imag(Y 0 )=0 real(Y j )=real(Y n-j ), j=1,…,n/2 imag(Y j )=-imag(Y n-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

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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.')

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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)

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

41 Chebyshev Polynomial 扩充到复平面 扩充到 |z|>1 递推关系 : 满足微分方程 : 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. Nov. 87Dec. 88Nov. 90Jan. 93Jan. 95Jan. 96Nov. 96Nov ?


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