复旦大学：《Matlab Math》（双语版）CHAPTER 7 Touch-tone Dialing

Matlab Math Fourier变换 Cleve morler著 陈文斌(wbchen@fudan.edu.cn) 复旦大学2002

Matlab Math Cleve Morler著 陈文斌（wbchen@fudan.edu.cn) 复旦大学2002 Fourier 变换

Touch-tone Dialing 我们每天都在使用 Fourier变换,比如移动电话,CD,DVD,JPEG ABC DEF 697 Dtmfanalyze m 2 Dtmfsynthesizem GHI JKL MNO 770 4 PRS TUV WXY 852 7 ⊥OPER 0 1209 13361477

Touch-tone Dialing 我们每天都在使用Fourier变换，比如移动电话，CD,DVD, JPEG 1 ABC 2 DEF 3 GHI 4 JKL 5 MNO 6 PRS 7 TUV 8 WXY 9 * OPER 0 # 697 770 852 941 1209 1336 1477 Dtmfanalyze.m Dtmfsynthesize.m

Finite fourier transform 离散 Fourier换=∑o 2mi/n Y=F k+1,j+1=t FF=nl 离散反 Fourier变换 y=FYy=∑1DkD=c-2 MATLAB omega=exp(-2pi*1/n);j=0: n-1;i F=fft(eye(n) k-i: F=omega. k

Finite Fourier Transform       1 0 1 1 n j j jk k Y  y i n e 2 /   离散  Fourier变换 Y  Fy jk k j f 1, 1   F F nI H  F Y n y 1 H        1 0 1 1 1 n j jk j Yk n y  i n e 2 /    离散反Fourier变换 omega=exp(-2*pi*i/n); j=0:n-1; k=j'; F=omega.^(k*j); F=fft(eye(n));

real(y) imag (y) ●●●●●●●●●命●●@●●● real(fft(y)) imag〔fft(y)) ●●●●●●●●●●命●●●●●●●●●●●●●●●●●●●●● reset fitgui

fftgui

real(y) imag (y) ●●●●●●●●●命●●@●●● real(fft(y)) imag〔fft(y)) reset fitgui

fftgui

real(y) imag (y) ●●●●●●●●●命●●@●●● real(fft(y)) imag〔fft(y)) reset fitgui

fftgui

real(y) imag (y) nyquist point ●●●●●●●●●0●●●●●●●●● real(fft(y)) imag〔fft(y)) ●●●●●●●●●●●●●●●● ●●●●●●● ●●●●●●●● reset fitgui

fftgui Nyquist point

real(y) g (y) ●●●●●●●●●●● ●●●● eal(fft(y) mag(fft(y)) fitgui te

fftgui

对称性 如果y是长度为n的实向量,Y=ft(y) eal(YD ∑ imag (1=0 real(Y2+i)=real(Ym-),j=0,,n/2-1 imag(Y2+)=-imag(Y-)j=0,…,n/2-1

对称性   j real(Y ) y 1 如果y是长度为n的实向量，Y=fft(y) imag( ) 0 Y1  real( ) real( ), 0,..., / 2 1 Y2 j  Yn j j  n  imag( ) imag( ), 0,..., / 2 1 Y2 j   Yn j j  n 

Sunspots For centuries people have noted that the face of the sun is not constant or uniform in appearance, but that correlated with weather and other economically significant terrestrial phenomena. In 1848 Rudolf Wolfer proposed a rule that combined the number and size of these sunspots into a single index. Using archival records astronomers have applied wolfer's rule to determine sunspot activity back to the year 1700 today the sunspot index is measured by many astronomers and the worldwide distributioin of the data is coordinated by the sunspot index data Center at royal Observatory of Belgium http:/sidc.oma.be/html/sunspot.html

Sunspots For centuries people have noted that the face of the sun is not constant or uniform in appearance, but that correlated with weather and other economically significant terrestrial phenomena. In 1848, Rudolf Wolfer proposed a rule that combined the number and size of these sunspots into a single index. Using archival records, astronomers have applied Wolfer's rule to determine sunspot activity back to the year 1700. Today the sunspot index is measured by many astronomers and the worldwide distributioin of the data is coordinated by the Sunspot Index Data Center at Royal Observatory of Belgium. http://sidc.oma.be/html/sunspot.html