傅里葉變換的可視化及應用研究(編輯修改稿)

　

【文章內容簡介】 使得其在頻譜分析中得到廣泛應用。而另一方面，我們從傅里葉級數、傅里葉變換、FFT、廣義傅里葉級數的實際應用例子中可以明顯地看到它們在解決實際問題時的思路無外乎以下兩點：（1）將復雜的問題分解為若干簡單的問題求解（2）將糾纏的問題通過變換分離開來如果我們拋開傅里葉級數法、傅里葉變換本身，而僅從這兩點出發，我們便可以看到傅里葉級數法、傅里葉變換的更深層次的意義。它為我們提供了一種解決問題的思路，如果我們可以用不同的變換方法將復雜的問題分解化，將糾纏的問題分離化，我們就可以用這種變換來處理問題。六、總結及結論以上綜述了傅里葉級數、傅里葉變換、FFT、廣義傅里葉級數的理論基礎；實現了它們的MatLab可視化；舉例討論了它們在實際各方面的應用。通過對理論基礎的對比我們可以看到它們之間的相互聯系與區別，知道了傅里葉級數、廣義傅里葉級數法的建立基礎，并認識到發展出傅里葉變換、DFT、FFT的理論根基。通過MatLab可視化，我們直觀而形象地看到了它們各自的特征、特點，更好的理解了它們的數學、物理意義。最后通過在實際各方面的應用，我們看到了它們在實際中的應用方式、方法，并對其能夠解決的各類問題有了一個大致的了解。最終，總結出了它們解決實際問題的思路主體在于將復雜的問題分解化，將糾纏的問題分離化。附錄程序：傅里葉系數計算：syms x H k tao Tint(H*exp(1i*2*k*pi/T*x)/T,x,tao/2,tao/2)繪制圖像：H=1。T=2。tao=。for k=1:20 y(k)=abs((H*sin((pi*k*tao)/T))/(pi*k))。 endy=[H*tao/T,y]。x=0:20。stem(x,y,39。bo39。,39。MarkerFaceColor39。,39。b39。)：N=64。H=1。tao=。T=2。y=zeros(1,N+1)。for k=1:N+1。 if(T/2+k*T/N=tao/2)amp。amp。(T/2+k*T/N=tao/2) y(k)=1。 endendY=fft(y)。Y(1)=[ ]。n=fix(length(Y)/2)。x1=50:1/10:50。y1=abs(sin(tao*x1/2)./x1/pi)/pi。plot(x1,y1,39。r39。,39。LineWidth39。,2)hold onplot((1:n).*pi,abs(Y(1:n)/length(Y)/10),39。b.39。,39。LineWidth39。,3)axis([50 50 0 ])hold off：t = 0::。x = sin(2*pi*50*t) + *sin(2*pi*120*t)+2*sin(2*pi*70*t)。y = x + 2*randn(size(t))。Y = fft(y)。freq=(0:length(Y)1)/length(Y)*1000。figure(1)plot(t,y)axis([0 8 8])figure(2)plot(freq,2*abs(Y)/501)axis([0 500 0 2])：a=imread(39。F:\39。)。 x1=rgb2gray(a)。D0=50。D1=65。[M,N]=size(x1)。 filt=zeros(M,N)。 for i=1:M, for j=1:N, x(i,j)=x1(i,j)+100*sin(1*i+1*j)。 if or(abs(iM/2)=D0,abs(iM/2)=D1), filt(i,j)=1。 end。 end。end。freq_im=fft2(x)。 freq_im=fftshift(freq_im)。 y=(1/(M*N))*freq_im。filt_im=freq_im.*filt。new_freq=ifftshift(filt_im)。new_im=ifft2(new_freq,M,N)。y1=abs(filt_im)。figure(1)imshow(x1)。figure(2)imshow(x)。 figure(3)imshow(abs(y))。figure(4)imshow(abs(new_im),[ ])。：⑥[X,Z]=meshgrid((0::3),(0::2))。[Q,R]=cart2pol(X,Z)。R(R==1)=NaN。u=1./sqrt(12.*R.*cos(Q)+R.^2)。figure(1)meshc(X,Z,u)axis([0 3 0 2 0 10])Rin=R。Rin(Rin1)=NaN。Rout=R。Rout(Rout1)=NaN。Uin=1。 Uout=1./Rout。 for l=1:20 Leg=legendre(l,cos(Q))。 Legl=squeeze(Leg(1,:,:))。 uin=Rin.^l.*Legl。 uout=1./Rout.^(l+1).*Legl。 Uin=Uin+uin。 Uout=Uout+uout。endfigure(2)meshc(X,Z,Uin)hold onmeshc(X,Z,Uout)xlabel(39。x39。)axis([0 3 0 2 0 10])：⑥ [X,Z]=meshgrid(5::5)。 [Q,R]=cart2pol (X,Z)。 sqrtR=sqrt(pi/2./R)。 Leg0=legendre(0,cos(Q))。 Bes0=sqrtR.*besselj(0,R)。 qiu=Bes0.*Leg0。⑥：清華大學出版社，、57figure(1)surfc(X,Z,qiu)for k=1:3 Leg=legendre(k,cos(Q))。 Legk=squeeze(Leg(1,:,:))。 Bes=sqrtR.*besselj(k,R)。 qiuk=Bes.*Legk。 figure(k+1) surfc(X,Z,qiuk) end參考文獻[1]， of Mathematical Physics（Volume I），Wiley,1989[2]：北京航天航空大學出版社，2003[3]，：高等教育出版社，1998[4]侯朝煥，閻世尊，：海洋出版社，1990[5]（卷I），：科學出版社，2007[6]：清華大學出版社，2004致謝 本論文是在導師郭懷明老師的悉心指導下完成的。郭老師治學嚴謹，對待科學研究嚴肅認真。從選題到到資料的收集、資料的閱讀、論文內容的研究探討，一直到最終完成論文，郭老師自始至終都對我進行了細心的指導。郭老師對資料的答疑解惑，對論文中涉及到的內容和問題的探討，以及提出來的很多寶貴的建議都是我受益匪淺。在此，對郭懷明老師表示特別的感謝！英文原文167。 Series of the Fundamental follows from the considerations of 167。1 and from the orthogonality of the trigonometric functions that the best approximation in the mean of degree n is obtained by the socalled Fourier polynoialWith （15） The polynomial may also be written in the more concise form （15’） Is virtue of the relation . It is not a priori certain that these polynomials, which yield the best approximation in the mean, also yield a uniform approximation to the function—. it is not certain that the inginite series converges uniformly and represents the function f(x). This question is the central problem of the theory of Fourier series. For the sake of convenience we shall suppose that the function f(x) is initially defined only in the interval ,and the continued periodically beyond this interval by the functional equation . Furthermore, at each jump discontinuity, we requier f(x) to be equal to the arithmetic mean of the “righthand” and “lefthand”limits, and (h0), respectively。 . we set . The following theorem then holds: Every function which is piecewise smooth in the interval and periodic with the period may be expanded in a Fourier series。that is, the Fourier polynomialsConverge to f(x) with increasing n. Moreover, we shall prove: Theconvergence of the Fourier series is uniform in every closed interval in which the function is continuous. We shall forst give the proof for the case of continuous f(x) in which discontinuities occur only in the derivative f’(x). If the expansion coefficients of f’(x) are denoted by and , we have，Since f’(x) is piecewise continuous, we have the pleteness relation。Thus This immediately establishes the absolute and uniform convergence of the infinite serieswhich represents the function f(x), because of the pleteness of the trigonometric functions. To verify that the Fourier series expansion is also valid for functions which are discontinuous but piecewise smooth, we start by considering a particular function of this type, defined by， ， ，This fuction jumps by at the points . Its Fourier coefficients are ， If the Fourier expansion is valid for piecewise continuous functions we have，To justify this equation we first form the function，Which is everywhere continuous and piecewise smooth. By the above reasoning, the Fourier series of this function converges uniformly and is equal to g(x). The coefficients are related to the by the equations， ，If we set and we haveWith increasing n, converges to zero and the sum converges uniformly to g(x). It follows that also coverfes uniformly to g(x) in the interval , and therefore that itself converges uniformly to h（x） in every closed subinterval not including the point x=0. At