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

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Fast Fourier Transform

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

delta 函数 t (t)(t)   TopHat 函数

      cos(  0 t) t  cosine 函数 sine 函数

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

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));

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!

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

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

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

–{ 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 }.

bit reversal 0, 1, 2, 3, 4, 5, 6, 7 is reordered to 0, 4, 2, 6, 1, 5, 3, 7 DecimalBinary Decimal 0000 0 1001 1004 2010 2 3011 1106 4100 0011 5101 5 6110 0113 7111 7

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

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

y 0 =1,y 1 =…=y n-1 =0,

y 0 =0,y 1 =1,y 2 =…=y n-1 =0,

FFT is the sum of two sinusoids

Nyquist point

697 770 852 941 1209 1336 1477 % 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

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.')

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)

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

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

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])

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

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. 00 4.5566.577.588 11437638799109157180 4.5566.577.588?

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