Chapter 4 Fourier Analysis 中國文化大學4章中國文化大學傅叶分析.docVIP

Chapter 5 Fourier Analysis 5-1 Fourier Series of a Periodical Function Fourier series: f(x) is a periodical function with period=2L and defined on an interval: -L≦x≦L. f(x+2L)=f(x), and then f(x)=, where , , and In case f(x) is Parseval’s Identity for Fourier series: Orthogonalities: and Eg. Expand f(x)=, f(x+4)=f(x) into Fourier series and [交大資科所] (Sol.) , f(x+4)=f(x), , ∵ Even function, ∴ bn=0, =1 ∴ In Matlab language, we can use the following instructions to obtain the finite sum of =1-. x = 0:0.001:4*pi; y=1; for i=1:5 y=y-8*cos((2*i-1)*pi*x/2)/(2*i-1)^2/pi^2 end plot (x,y) 1- Eg. Find the Fourier series of f(x)=|x| for –πxπ. [台大電研] (Sol.) 2L=2π, L=π, ∵ Even function, ∴ bn=0, n. , , ∴ Eg. Expand f(x)= and f(x+4)=f(x) into Fourier series. Find (a) and (b) . [文化電機轉學考] (Sol.) ∵ Odd function, ∴ an=0, n Set x=1, (b) , In Matlab language, we can use the following instructions to obtain the finite sum of =. x = 0:0.001:4*pi; y=0; for i=1:5 y=y+4*sin((2*i-1)*pi*x/2)/(2*i-1)/pi end plot (x,y) Eg. Find the Fourier series of f(x)= and use the results to show that . [台大電研] (Sol.) ∴ ∴ In Matlab language, we can use the following instructions to obtain the finite sum of . x = 0:0.001:4*pi; y=1/pi+sin(x)/2; for n=1:20 y=y-2*cos(2*n*x)/pi/(4*n^2-1) end plot (x,y) Eg. Find the Fourier series of |cos(2x)| and calculate [交大材研、(Sol.) (a), , ∵ Even function, ∴ (b) In Matlab language, we can use the following instructions to obtain the finite sum of . x = 0:0.001:4*pi; y=2/pi; for n=1:6 y=y+4*(-1)^(n+1)*cos(4*n*x)/(4*n^2-1)/pi end plot (x,y) Discrete spectrum of f(t): 5-2 Fourier Transforms and Inverse Fourier Transforms Fourier Transform pair defined in Engineering: , where ω=2πν. Parseval’s Iden