MATLAb连续时间傅里叶变换PPT课件[通用]

1、MATLAb连续时间傅里叶变换 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。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）8.1 连续时间傅里叶变换的数值近似傅里叶变换的近似表示8.2 连续时间信号的采样8.3 理想抽样信号的傅里叶变换（利用卷积定理）冲激抽样

2、信号的频谱说明抽样定理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率响应作准备，从抽样信号的傅里叶变换引出：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。比较利用快速傅里叶变换计算频谱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. Fo

3、r N-D arrays, the FFT operation operates on the first non-singleton dimension. FFT(X,N) is the N-point FFT, padded with zeros if X has less than N points and truncated if it has more.FFT实现的是DTFT的一个周期的抽样，实际的频谱近似为fft函数的使用说明补充说明例题解：画图（利用解析式）% ss8_2.m and double_side_exp_spectrum.mTs=0.05; t=-5:Ts:5;x=e

4、xp(-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(omega rad/s);ylabel(X(jomega);抽样间隔如何选取？(b)% exe4_2_bcde.mclear;T=10;Ts=0.01; t=(-T/2):Ts:(T/2-Ts);N=len

5、gth(t);x=exp(-2*abs(t);X=fft(x,N);X=Ts*fftshift(X);w=-pi/Ts+(0:N-1)/N*(2*pi/Ts);SEMILOGY Semi-log scale plot. SEMILOGY(.) is the same as PLOT(.), except a logarithmic (base 10) scale is used for the Y-axis.abs_X=4./(4+w.*w);subplot(2,1,1);h=semilogy(w,abs(X);set(h,linewidth,2);xlabel(omega rad/s);yl

6、abel(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);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移性质和调制定理；抽样信号的重建方法sound函数SOUND Play vector as sound. SOUND(Y,FS) send

7、s 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);将大于1的部分置为1：y2(position)=1;8.6 幅度调制和连续时间傅里叶变换本地载波解调举例莫尔斯电报编码A .-H .O -V -B -I .P .-.W .-C -.-.J .-Q -.-X -.-D -.K -.-R .-.Y -.-E .L .-.S Z -.F

8、 .-.M -T -G -.N -.U .- (a)% exe4_6_a.mclear;load ctftmod.matZ=dash dash dot dot;plot(t,Z,r);(b)freqs(bf,af);freqsFREQS Laplace-transform (s-domain) frequency response. H = FREQS(B,A,W) returns the complex frequency response vector H of the filter B/A:given the numerator and denominator coefficients

9、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 arguments. 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)B,A 矩阵的写

10、法例题运行结果其他用法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（离散系统）.(c)分析(d)(e)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第二种情况：

11、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？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 . . .总结本地载波8.7 由欠采样引起的混叠基本题MATLAB实现% exe7_1_a.mT=1/8192;n=0:8191;t=n*T;f0=1000;x=sin(2*pi*f0*t);(b)取前50个样本：x(1:50)(c)% exe7_1_c.mfs=8192;T=1/fs;f0=

12、800;W=2*pi*f0*T;n=0:fs;x=sin(W*n);sound(x,fs);X=fft(x,56);stem(abs(X);(d)提示：通过修改exe7_1_c.m中的数据来实现深入题% 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);sound(x);specgram(x,8192);SPECTROGRAMSPECTROGRAM Spectrogram using a Short-Time Fourie

13、r Transform (STFT, 短时傅里叶变换). S = SPECTROGRAM(X) returns the spectrogram of the signal 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 Fo

14、urier transforms is equal to the maximum of 256 or the next power of two greater than the length of each segment of X.8.7 由样本重建信号零阶保持一阶保持抽样函数demo9sam_inversesam_inverse.mSa函数作为内插函数（理想化）Sa函数作为内插函数（理想化）Sa函数作为内插函数（理想化）sinc函数内插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=ones(l

15、ength(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为样值函数内插函数矩阵spline: 三次样条内插函数SPLINE Cubic spline data 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 PPVA

16、L 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(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

17、 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 = 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.举例clearTs=1;Fs=1/Ts;n = 0:10; x = sin(n);t

18、= 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);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);结果图形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=s

19、ym(exp(-4*t)*heaviside(t);heaviside：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 corresponding element of t is greater than zero.举例：syms t;y=cos(t)*(heaviside(t+0.5*pi)-heaviside(t-0.5*pi);e

20、zplot(y);解： 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);legend(x1);axis(0 2 0 1);subplot(2,1,2);ezplot(x2,0,2);legend(x2);axis(0 2 0 1);fourier函数FOURIER Fourier integral transform. F = FOURIER(f) is the Fourier transform of the

21、 sym scalar f with default independent variable x.F(w) = int(f(x)*exp(-i*w*x),x,-inf,inf)See also sym/ifourier, sym/laplace, sym/ztrans.主要代码% exe4_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|)axi

22、s(-20 20 0 0.3);subplot(2,1,2);ezplot(abs(X2),-20,20);legend(|X2|)axis(-20 20 0 0.3);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,

23、10 0 1.1);运行结果1X1 = 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)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=subs(X2)x22=ifourier(X22)figure;subplot(2,1,1);ezplot(abs(X22),-10 10);subplot(2,1,2);ezplot(angle(X22),-10 10);运行结果2X22 = 1