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1、尹霄丽Electronic Engineer School of BUPT第八章 连续时间傅里叶变换Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用连续时间傅里叶变换（CTFT） 将连续时间将连续时间傅里叶级数傅里叶级数(CTFS)推广到既能对推广到既能对周期周期连连续时间信号，又能对续时间信号，又能对非周期非周期连续时间信号进行频谱分析。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，但域来看时呈现出很复杂的结构，但从频域来看却很简
2、单从频域来看却很简单。另外，许多另外，许多LTI系统的特性行为在频域要比在时域容易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信号的时域特性是如何与它的频域特性联系联系起来的建立起来的建立直观的认识直观的认识。Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用频谱计算中的问题n连续连续离散（抽样，抽样间隔如何选取？）离散（抽样，抽样间隔如何选取？）n无穷积分无穷积分有限长（截断）有限长（截断）Signals and S
3、ystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用8.1 连续时间傅里叶变换的数值近似Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用傅里叶变换的近似表示Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用8.2 连续时间信号的采样Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用8.3 理想抽样信号的傅里叶变换（利用卷积定理）Signals and SystemsEE BUPTEE of
4、BUPTMATLAB在信号与系统课程中的应用冲激抽样信号的频谱Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用说明Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用抽样定理Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用8.4 DTFT的引出（利用时移性质）DTFT:Discrete-time Fourier transform为研究离散时间系统的频为研究离散时间系统的频率响应作准备，从抽样信率响应作准备，从抽样信号的
5、傅里叶变换引出：号的傅里叶变换引出：Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用离散时间信号离散时间信号的傅里叶变换的傅里叶变换DTFTDTFT就是抽样就是抽样信号的傅立叶信号的傅立叶变换。变换。Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用比较Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用利用快速傅里叶变换计算频谱利用快速傅里叶变换计算频谱Signals and SystemsEE BUPTEE of BU
6、PTMATLAB在信号与系统课程中的应用fft函数FFT Discrete Fourier transform. FFT(X) is the discrete Fourier transform (DFT) of vector X. For matrices, the FFT operation is applied to each column. For N-D arrays, the FFT operation operates on the first non-singleton dimension. FFT(X,N) is the N-point FFT, padded with ze
7、ros if X has less than N points and truncated if it has more.FFT实现的是实现的是DTFT的一个周期的抽样，实际的频谱的一个周期的抽样，实际的频谱近似为近似为Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用fft函数的使用说明Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用补充说明Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用例题解：Signals an
8、d SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用画图（利用解析式）Ts=; 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(
9、h,linewidth,2);xlabel(omega rad/s);ylabel(X(jomega);Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用抽样间隔如何选取？Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用(b)clear;T=10;Ts=0.01; t=(-T/2):Ts:(T/2-Ts);N=length(t);x=exp(-2*abs(t);Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用X=fft
10、(x,N);X=Ts*fftshift(X);w=-pi/Ts+(0:N-1)/N*(2*pi/Ts);Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用SEMILOGY Semi-log scale plot. SEMILOGY(.) is the same as PLOT(.), except a logarithmic (base 10) scale is used for the Y-axis.Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用abs_X=4./(4+w.
11、*w);subplot(2,1,1);h=semilogy(w,abs(X);set(h,linewidth,2);xlabel(omega rad/s);ylabel(log_1_0(|X(jomega)|);hold onsemilogy(w,abs_X,r:);legend(fft,real);subplot(2,1,2);h=plot(w,unwrap(angle(X);set(h,linewidth,1);xlabel(omega rad/s);ylabel(phi(omega);Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的
12、应用Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用8.5 连续时间傅里叶变换性质n目的：直观、深刻地理解傅里叶变换的性质；目的：直观、深刻地理解傅里叶变换的性质；n主要内容：主要内容：n奇偶虚实性；信号的幅度谱与相位谱奇偶虚实性；信号的幅度谱与相位谱n尺度变换特性尺度变换特性n频移性质和调制定理；频移性质和调制定理；n抽样信号的重建抽样信号的重建Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用方法方法Signals and SystemsEE BUPTEE of BUPTM
13、ATLAB在信号与系统课程中的应用sound函数SOUND Play vector as sound. SOUND(Y,FS) sends the signal in vector Y (with sample frequency FS) out to the speaker on platforms that support sound. Values in Y are assumed to be in the range -1.0 = y 1);position=find(y21);2.2.将大于将大于1 1的部分置为的部分置为1 1：y2(position)=1;y2(position)
14、=1;Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用8.6 幅度调制和连续时间傅里叶变换本地载波本地载波解调Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用举例Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用莫尔斯电报编码A .-H .O -V -B -I .P .-.W .-C -.-.J .-Q -.-X -.-D -.K -.-R .-.Y -.-E .L .-.S Z -.F .-.M -T -G -.N
15、-.U .-Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用 (a)clear;Z=dash dash dot dot;plot(t,Z,r);Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用(b)freqs(bf,af);Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用freqsFREQS Laplace-transform (s-domain) frequency response. H = FREQS(B,A,
16、W) returns the complex frequency response vector H of the filter B/A:given the numerator and denominator coefficients in vectors B and A. The frequency response is evaluated at the points specified in vector W (in rad/s). The magnitude and phase can be graphed by calling FREQS(B,A,W) with no output
17、arguments. Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用传输函数 nb-1 nb-2 B(s) b(1)s + b(2)s + . + b(nb) H(s) = - = - na-1 na-2 A(s) a(1)s + a(2)s + . + a(na)Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用B,A 矩阵的写法Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用例题Signals and Syste
18、msEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用运行结果Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用其他用法H,W = FREQS(B,A) automatically picks a set of 200 frequencies W on which the frequency response is computed. FREQS(B,A,N) picks N frequencies. See also logspace, polyval, invfreqs, and freqz（离散（离散系统）系统
19、）.Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用(c)Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用分析Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用(d)Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用Signals and SystemsE
20、E BUPTEE of BUPTMATLAB在信号与系统课程中的应用(e)Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用相干接收n需要使用本地载波（接收端）需要使用本地载波（接收端）n同步解调：本地载波与发送端载波同频同相同步解调：本地载波与发送端载波同频同相n正交调制技术简介正交调制技术简介Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用第一种情况：本地载波与调制载波同频同相高频信号高频信号恢复出的原始信号Signals and SystemsEE BUPTEE of B
21、UPTMATLAB在信号与系统课程中的应用第二种情况：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？Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用第三种情况：本地载波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 . . .Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用总结本地载波本
22、地载波Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用8.7 由欠采样引起的混叠Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用基本题Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用MATLAB实现nT=1/8192;nn=0:8191;nt=n*T;nf0=1000;nx=sin(2*pi*f0*t);Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用(b)
23、取前50个样本：x(1:50)Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用(c)Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用nfs=8192;nT=1/fs;nf0=800;nW=2*pi*f0*T;nn=0:fs;nx=sin(W*n);nsound(x,fs);nX=fft(x,56);nstem(abs(X);Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用(d)提示：通过修改中的数据来实现Signal
24、s and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用深入题Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用fs=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*
25、t+bate*t.*t/2);Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用sound(x);specgram(x,8192);Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用SPECTROGRAMSPECTROGRAM Spectrogram using a Short-Time Fourier Transform (STFT, 短时傅里叶变换短时傅里叶变换). S = SPECTROGRAM(X) returns the spectrogram of the signa
26、l specified by vector X in the matrix S. By default, X is divided into eight segments with 50% overlap, each segment is windowed with a Hamming window. The number of frequency points used to calculate the discrete Fourier transforms is equal to the maximum of 256 or the next power of two greater tha
27、n the length of each segment of X.Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用8.7 由样本重建信号零阶保持零阶保持一阶保持一阶保持抽样函数抽样函数Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用demoSignals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用Sa函数作为内插函数（理想化）函数作为内插函数（理想化）Signals and SystemsEE BUPTEE of BUPTM
28、ATLAB在信号与系统课程中的应用Sa函数作为内插函数（理想化）Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用Sa函数作为内插函数（理想化）Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用sinc函数内插Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用sinc函数内插的MATLAB实现n分析：在各抽样值处插入一个分析：在各抽样值处插入一个sinc函数，大小与抽样值函数，大小与抽样值成正比，定义域为全时域（或给定定义
29、域）。成正比，定义域为全时域（或给定定义域）。n时间矩阵：时间矩阵：tt=ones(length(n),1)*t-Ts*n*ones(1,length(t)n内插函数矩阵：内插函数矩阵：sinc(fs*tt) n函数内插：函数内插：x*sinc(tt) %x为样值函数为样值函数Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用内插函数矩阵Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用spline: 三次样条内插函数三次样条内插函数SPLINE Cubic spline dat
30、a interpolation. PP = SPLINE(X,Y) provides the piecewise polynomial form of the cubic spline interpolant to the data values Y at the data sites X, for use with the evaluator PPVAL and the spline utility UNMKPP. X must be a vector. If Y is a vector, then Y(j) is taken as the value to be matched at X(
31、j), hence Y must be of the same length as X - see below for an exception to this. If Y is a matrix or ND array, then Y(:,.,:,j) is taken as the value to be matched at X(j), hence the last dimension of Y must equal length(X) - see below for an exception to this. YY = SPLINE(X,Y,XX) is the same as YY
32、= PVAL(SPLINE(X,Y),XX), thus providing, in YY, the values of the interpolant at XX. For information regarding the size of YY see PPVAL.Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用举例举例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
33、=ones(length(n),1)*t-nTs*ones(1,length(t);x_sinc=x*sinc(Fs*tt);hold on; plot(t,x_sinc,r);legend(spline,sinc);hold on；stem(n,x,m);hh=findobj(0,type,line);set(hh,linewidth,2);Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用结果图形结果图形Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用8.8 连续时间傅里叶
34、变换的符号计算x1=sym(1/2)*exp(-2*t)*heaviside(t);x2=sym(exp(-4*t)*heaviside(t);Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用heaviside：单位阶跃函数：单位阶跃函数help heaviside HEAVISIDE Unit Step function f=Heaviside(t) returns a vector f the same size as the input vector, where each element of f is 1 if the co
35、rresponding element of t is greater than zero.举例：举例：syms t;y=cos(t)*(heaviside(t+0.5*pi)-heaviside(t-0.5*pi);ezplot(y);Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用解：解：Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用 主要代码clear;x1=s
36、ym(1/2)*exp(-2*t)*heaviside(t);x2=sym(exp(-4*t)*heaviside(t);subplot(2,1,1);ezplot(x1,0,2);legend(x1);axis(0 2 0 1);subplot(2,1,2);ezplot(x2,0,2);legend(x2);axis(0 2 0 1);Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用fourier函数nFOURIER Fou
37、rier integral transform.n F = FOURIER(f) is the Fourier transform of the sym scalar f with default independent variable x.nF(w) = int(f(x)*exp(-i*w*x),x,-inf,inf)nSee also sym/ifourier, sym/laplace, sym/ztrans.Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用主要代码x1=sym(1/2)*exp(-2*t)*heaviside(
38、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(-20 20 0 0.3);subplot(2,1,2);ezplot(abs(X2),-20,20);legend(|X2|)axis(-20 20 0 0.3);Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用Signals and SystemsEE BUPTEE of BUPTMATLAB在信号
39、与系统课程中的应用练习练习1close all;clear all;syms t a u% exersice 1: Fourier transform of exp(-abs(t)x1=exp(-abs(t);X1=fourier(x1)ezplot(X1,-10,10);axis(-10,10 0 2.1);x11=ifourier(X1,w)figure;x111=simple(x11)ezplot(x111,-10,10);axis(-10,10 0 1.1);Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用运行结果运行结果1X
40、1 = 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)Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用练习练习2% exersice 2: Fourier transform of % exp(-a*t)*heaviside(t)x2=exp(-a*t)*heaviside(t);X2=fourier(x2);a=2;X22=
41、subs(X2)x22=ifourier(X22)figure;subplot(2,1,1);ezplot(abs(X22),-10 10);subplot(2,1,2);ezplot(angle(X22),-10 10);Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用运行结果运行结果2X22 = 1/(w*i + 2) x22 = exp(-2*x)*heaviside(x) H31 = 2/5 - i/5Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用练习练习3% ex
42、ersice 3: sinusoidal signals pass the RC low pass filterx3=sin(t)+sin(3*t);H31=subs(X22,w,1)H33=subs(X22,w,3)x33=abs(H31)*sin(t+angle(H31)+abs(H33)*sin(3*t+angle(H33);figure;subplot(2,1,1);ezplot(x3,-10 10);subplot(2,1,2);ezplot(x33,-10 10);Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用运行结果运
43、行结果3H31 = 2/5 - i/5 H33 = 2/13 - (3*i)/13Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用练习练习4% exersice 4: Ideal low pass filterx4=x3;H4=heaviside(u+2)-heaviside(u-2);H41=subs(H4,u,1);H43=subs(H4,u,3);x44=abs(H41)*sin(t+angle(H41)+abs(H43)*sin(3*t+angle(H43);figuresubplot(2,1,1);ezplot(x4,-10
44、 10);subplot(2,1,2);ezplot(x44,-10 10);Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用运行结果运行结果4Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用练习练习5% exersice 5: Hilbert transformx5=sin(t);H5=-i*sign(u);H51=subs(H5,u,1);x55=abs(H51)*sin(t+angle(H51);figuresubplot(2,1,1);ezplot(x5,-10 10);subplot(2,1,2);ezplot(x55,-10 10);Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用运行结果运行结果5Signals and SystemsEE BUPTEE of BUPTMATLAB在信号与系统课程中的应用函数总结nsoundnloadnsincnsplinenfft/ifftnfftshiftnfreqsnsemilogynconjnabsnanglefourier/ifourierdiracheaviside
