MATLAb连续时间傅里叶变换PPT课件

1、1,第八章连续时间傅里叶变换,.,2,连续时间傅里叶变换（CTFT）,将连续时间傅里叶级数(CTFS)推广到既能对周期连续时间信号，又能对非周期连续时间信号进行频谱分析。这是一种重要而强有力的方法，因为有很多信号当从时域来看时呈现出很复杂的结构，但从频域来看却很简单。另外，许多LTI系统的特性行为在频域要比在时域容易理解得多。为了更有效地应用频域方法，重要的是要将信号的时域特性是如何与它的频域特性联系起来的建立直观的认识。,.,3,频谱计算中的问题,连续离散（抽样，抽样间隔如何选取？）无穷积分有限长（截断）,.,4,8.1连续时间傅里叶变换的数值近似,.,5,傅里叶变换的近似表示,.,6,8.

2、2连续时间信号的采样,.,7,8.3理想抽样信号的傅里叶变换（利用卷积定理）,.,8,冲激抽样信号的频谱,.,9,说明,.,10,抽样定理,.,11,8.4DTFT的引出（利用时移性质）,DTFT:Discrete-timeFouriertransform,为研究离散时间系统的频率响应作准备，从抽样信号的傅里叶变换引出：,.,12,离散时间信号的傅里叶变换DTFT就是抽样信号的傅立叶变换。,.,13,比较,.,14,利用快速傅里叶变换计算频谱,.,15,fft函数,FFTDiscreteFouriertransform.FFT(X)isthediscreteFouriertransform(D

3、FT)ofvectorX.Formatrices,theFFToperationisappliedtoeachcolumn.ForN-Darrays,theFFToperationoperatesonthefirstnon-singletondimension.FFT(X,N)istheN-pointFFT,paddedwithzerosifXhaslessthanNpointsandtruncatedifithasmore.FFT实现的是DTFT的一个周期的抽样，实际的频谱近似为,.,16,fft函数的使用说明,.,17,补充说明,.,18,例题,解：,.,19,.,20,画图（利用解析式）

4、,%ss8_2.manddouble_side_exp_spectrum.mTs=0.05;t=-5:Ts:5;x=exp(-2*abs(t);subplot(2,1,1);h=plot(t,x);set(h,linewidth,2);xlabel(t/s);ylabel(exp(-2|t|);N=256;w=-pi/Ts+(0:N-1)/N*(2*pi/Ts);X=4./(w.*w+4);subplot(2,1,2);h=plot(w,X);set(h,linewidth,2);xlabel(omegarad/s);ylabel(X(jomega);,.,21,抽样间隔如何选取？,.,22,

5、(b),%exe4_2_bcde.mclear;T=10;Ts=0.01;t=(-T/2):Ts:(T/2-Ts);N=length(t);x=exp(-2*abs(t);,.,23,X=fft(x,N);X=Ts*fftshift(X);w=-pi/Ts+(0:N-1)/N*(2*pi/Ts);,.,24,SEMILOGYSemi-logscaleplot.SEMILOGY(.)isthesameasPLOT(.),exceptalogarithmic(base10)scaleisusedfortheY-axis.,.,25,abs_X=4./(4+w.*w);subplot(2,1,1);

6、h=semilogy(w,abs(X);set(h,linewidth,2);xlabel(omegarad/s);ylabel(log_1_0(|X(jomega)|);holdonsemilogy(w,abs_X,r:);legend(fft,real);subplot(2,1,2);h=plot(w,unwrap(angle(X);set(h,linewidth,1);xlabel(omegarad/s);ylabel(phi(omega);,.,26,.,27,8.5连续时间傅里叶变换性质,目的：直观、深刻地理解傅里叶变换的性质；主要内容：奇偶虚实性；信号的幅度谱与相位谱尺度变换特性频

7、移性质和调制定理；抽样信号的重建,.,28,方法,.,29,sound函数,SOUNDPlayvectorassound.SOUND(Y,FS)sendsthesignalinvectorY(withsamplefrequencyFS)outtothespeakeronplatformsthatsupportsound.ValuesinYareassumedtobeintherange-1.0=y1);将大于1的部分置为1：y2(position)=1;,.,44,8.6幅度调制和连续时间傅里叶变换,本地载波,解调,.,45,举例,.,46,莫尔斯电报编码,.,47,(a),%exe4_6_a

8、.mclear;loadctftmod.matZ=dashdashdotdot;plot(t,Z,r);,.,48,(b),freqs(bf,af);,.,49,freqs,FREQSLaplace-transform(s-domain)frequencyresponse.H=FREQS(B,A,W)returnsthecomplexfrequencyresponsevectorHofthefilterB/A:giventhenumeratoranddenominatorcoefficientsinvectorsBandA.Thefrequencyresponseisevaluatedatth

9、epointsspecifiedinvectorW(inrad/s).ThemagnitudeandphasecanbegraphedbycallingFREQS(B,A,W)withnooutputarguments.,.,50,传输函数,nb-1nb-2B(s)b(1)s+b(2)s+.+b(nb)H(s)=-=-na-1na-2A(s)a(1)s+a(2)s+.+a(na),.,51,B,A矩阵的写法,.,52,例题,.,53,运行结果,.,54,其他用法,H,W=FREQS(B,A)automaticallypicksasetof200frequenciesWonwhichthefre

10、quencyresponseiscomputed.FREQS(B,A,N)picksNfrequencies.Seealsologspace,polyval,invfreqs,andfreqz（离散系统）.,.,55,(c),.,56,.,57,分析,.,58,(d),.,59,.,60,(e),.,61,相干接收,需要使用本地载波（接收端）同步解调：本地载波与发送端载波同频同相正交调制技术简介,.,62,第一种情况：本地载波与调制载波同频同相,高频信号,恢复出的原始信号,.,63,第二种情况：本地载波与调制载波同频不同相,只有高频信号，经过低通滤波器后被滤除？,.,64,第三种情况：本地载波

11、与调制载波不同频,差拍信号,高频信号,y=x.*cos(2*pi*f1*t);,D-.,y=x.*sin(2*pi*f1*t);,P.-.,y=x.*cos(2*pi*f2*t);,y=x.*sin(2*pi*f2*t);,S.,.,69,总结,本地载波,.,70,8.7由欠采样引起的混叠,.,71,基本题,.,72,MATLAB实现,%exe7_1_a.mT=1/8192;n=0:8191;t=n*T;f0=1000;x=sin(2*pi*f0*t);,.,73,(b),取前50个样本：x(1:50),.,74,(c),%exe7_1_c.m,.,75,fs=8192;T=1/fs;f0=8

12、00;W=2*pi*f0*T;n=0:fs;x=sin(W*n);sound(x,fs);X=fft(x,56);stem(abs(X);,.,76,(d),提示：通过修改exe7_1_c.m中的数据来实现,.,77,.,78,深入题,.,79,.,80,%exe7_1_g.mfs=8192;T=1/fs;n=0:fs*10;t=n*T;%f0=3000/2/pi;%bate=2000;f0=100;bate=5000;x=sin(2*pi*f0*t+bate*t.*t/2);,.,81,sound(x);specgram(x,8192);,.,82,SPECTROGRAM,SPECTROGR

13、AMSpectrogramusingaShort-TimeFourierTransform(STFT,短时傅里叶变换).S=SPECTROGRAM(X)returnsthespectrogramofthesignalspecifiedbyvectorXinthematrixS.Bydefault,Xisdividedintoeightsegmentswith50%overlap,eachsegmentiswindowedwithaHammingwindow.ThenumberoffrequencypointsusedtocalculatethediscreteFouriertransforms

14、isequaltothemaximumof256orthenextpoweroftwogreaterthanthelengthofeachsegmentofX.,.,83,8.7由样本重建信号,零阶保持一阶保持抽样函数,.,84,demo,9sam_inversesam_inverse.m,.,85,Sa函数作为内插函数（理想化）,.,86,Sa函数作为内插函数（理想化）,.,87,Sa函数作为内插函数（理想化）,.,88,sinc函数内插,.,89,sinc函数内插的MATLAB实现,分析：在各抽样值处插入一个sinc函数，大小与抽样值成正比，定义域为全时域（或给定定义域）。时间矩阵：tt=

15、ones(length(n),1)*t-Ts*n*ones(1,length(t)内插函数矩阵：sinc(fs*tt)函数内插：x*sinc(tt)%x为样值函数,.,90,内插函数矩阵,.,91,spline:三次样条内插函数,SPLINECubicsplinedatainterpolation.PP=SPLINE(X,Y)providesthepiecewisepolynomialformofthecubicsplineinterpolanttothedatavaluesYatthedatasitesX,forusewiththeevaluatorPPVALandthesplineutil

16、ityUNMKPP.Xmustbeavector.IfYisavector,thenY(j)istakenasthevaluetobematchedatX(j),henceYmustbeofthesamelengthasX-seebelowforanexceptiontothis.IfYisamatrixorNDarray,thenY(:,.,:,j)istakenasthevaluetobematchedatX(j),hencethelastdimensionofYmustequallength(X)-seebelowforanexceptiontothis.YY=SPLINE(X,Y,XX

17、)isthesameasYY=PVAL(SPLINE(X,Y),XX),thusproviding,inYY,thevaluesoftheinterpolantatXX.ForinformationregardingthesizeofYYseePPVAL.,.,92,举例,clearTs=1;Fs=1/Ts;n=0:10;x=sin(n);t=0:.25:10;x_spline=spline(n,x,t);plot(t,x_spline,b);nTs=0:10;tt=ones(length(n),1)*t-nTs*ones(1,length(t);x_sinc=x*sinc(Fs*tt);ho

18、ldon;plot(t,x_sinc,r);legend(spline,sinc);holdon；stem(n,x,m);hh=findobj(0,type,line);set(hh,linewidth,2);,.,93,结果图形,.,94,8.8连续时间傅里叶变换的符号计算,x1=sym(1/2)*exp(-2*t)*heaviside(t);x2=sym(exp(-4*t)*heaviside(t);,.,95,heaviside：单位阶跃函数,helpheavisideHEAVISIDEUnitStepfunctionf=Heaviside(t)returnsavectorfthesam

19、esizeastheinputvector,whereeachelementoffis1ifthecorrespondingelementoftisgreaterthanzero.举例：symst;y=cos(t)*(heaviside(t+0.5*pi)-heaviside(t-0.5*pi);ezplot(y);,.,96,.,97,解：,.,98,主要代码,%exe4_7_a.mclear;x1=sym(1/2)*exp(-2*t)*heaviside(t);x2=sym(exp(-4*t)*heaviside(t);subplot(2,1,1);ezplot(x1,0,2);legen

20、d(x1);axis(0201);subplot(2,1,2);ezplot(x2,0,2);legend(x2);axis(0201);,.,99,.,100,fourier函数,FOURIERFourierintegraltransform.F=FOURIER(f)istheFouriertransformofthesymscalarfwithdefaultindependentvariablex.F(w)=int(f(x)*exp(-i*w*x),x,-inf,inf)Seealsosym/ifourier,sym/laplace,sym/ztrans.,.,101,主要代码,%exe4

21、_7_a.mx1=sym(1/2)*exp(-2*t)*heaviside(t);x2=sym(exp(-4*t)*heaviside(t);X1=fourier(x1);X2=fourier(x2);subplot(2,1,1);ezplot(abs(X1),-20,20);legend(|X1|)axis(-202000.3);subplot(2,1,2);ezplot(abs(X2),-20,20);legend(|X2|)axis(-202000.3);,.,102,.,103,练习1,closeall;clearall;symstau%exersice1:Fouriertransfo

22、rmofexp(-abs(t)x1=exp(-abs(t);X1=fourier(x1)ezplot(X1,-10,10);axis(-10,1002.1);x11=ifourier(X1,w)figure;x111=simple(x11)ezplot(x111,-10,10);axis(-10,1001.1);,.,104,运行结果1,X1=2/(w2+1)x11=(2*pi*exp(-w)*heaviside(w)+2*pi*heaviside(-w)*exp(w)/(2*pi)x111=exp(-w)*heaviside(w)+heaviside(-w)*exp(w),.,105,练习2

23、,%exersice2:Fouriertransformof%exp(-a*t)*heaviside(t)x2=exp(-a*t)*heaviside(t);X2=fourier(x2);a=2;X22=subs(X2)x22=ifourier(X22)figure;subplot(2,1,1);ezplot(abs(X22),-1010);subplot(2,1,2);ezplot(angle(X22),-1010);,.,106,运行结果2,X22=1/(w*i+2)x22=exp(-2*x)*heaviside(x)H31=2/5-i/5,.,107,练习3,%exersice3:sinusoidalsignalspasstheRClowpassfilterx3=