MATLAB实验二傅里叶分析报告及应用

1、实用文档实验二傅里叶分析及应用一、实验目的（一）掌握使用Matlab进行周期信号傅里叶级数展开和频谱分析1、学会使用Matlab分析傅里叶级数展开，深入理解傅里叶级数的物理含义2、学会使用Matlab分析周期信号的频谱特性（二）掌握使用Matlab求解信号的傅里叶变换并分析傅里叶变换的性质1、学会运用Matlab求连续时间信号的傅里叶变换2、学会运用Matlab求连续时间信号的频谱图3、学会运用Matlab分析连续时间信号的傅里叶变换的性质（三）掌握使用Matlab完成信号抽样并验证抽样定理1 、学会运用MATLA完成信号抽样以及对抽样信号的频谱进行分析2 、学会运用MATLA改变抽样时间间隔

2、，观察抽样后信号的频谱变化3 、学会运用MATLA对抽样后的信号进行重建二、实验条件Win7 系统，MATLAB R2015a三、实验内容1、分别利用Matlab符号运算求解法和数值计算法求下图所示信号的FT,并画出其频谱图（包括幅度谱和相位谱）注：图中时间单位为：毫秒（ms）。wxphase speetrunnCode:ft = sym( (t+2)*(heaviside(t+2)-heavisi de(t+1)+(heaviside(t+1)-heav iside(t-1)+(2-t)*(heaviside( t-1)-heaviside(t-2);fw = simplify(fourie

3、r(ft);subplot(2, 1, 1); ezplot(abs(fw); gridon ;title( amp spectrum );phi = atan(imag(fw)/ real(fw);subplot(2, 1,2);ezplot(phi); gridon ;title( phase spectrum);符号运算法Code:dt = 0.01;t = -2: dt: 2;ft(t+2).*(uCT(t+2)-uCT(t+1)+(uCT(t+1)-uCT(t-1)+(2-t).*(uCT (t-1)-uCT(t-2);N = 2000;k = -N: N;w = pi * k /

4、(N*dt);fw = dt*ft*exp(-i*t*w);fw = abs(fw); plot(w, fw), grid on; axis(-2*pi 2*pi -1 3.5);数值运算法2、试用Matlab命令求Fj )1 -的傅里叶反变换，并绘出其时域信号图3 + 他 5 + jcoCode：syms t ;fw =sym( 10/(3+i*w)-4/(5+i*w);ft = ifourier(fw, t);ezplot(ft), gridon;(20 - exp(-3 t) heaviside(t) -8 - exp(-5 t) heaviside(t)/( 2 -)两个单边指数脉冲的

5、叠加3、已知门函数自身卷积为三角波信号，试用Matlab命令验证FT的时域卷积定理Codef = sym( heaviside(t+1) - heaviside(t-1);fw = simplify(fourier(f);F = fw.*fw;subplot(211);ezplot(abs(F), -9, 9), gridontitle(FWA2) tri =sym( (t+2)*heaviside(t+2)-2*t*heaviside(t)+(t-2)*heaviside(t-2);Ftri = fourier(tri);F = simplify(Ftri);subplot(212);ezp

6、lot(abs(F), -9, 9), gridon ;title( tri FT )FW1-8-6-202468wtri FT-吕024684、设有两个不同频率的余弦信号，频率分别为匸=100Hz , f 3800Hz ；现在使用抽样频率fs =4000Hz对这三个信号进行抽样,使用MATLAB令画出各抽样信号的波形和频谱，并分析其频率混叠现象Code :t2 = -0.007:ts:0.007; fst = cos(2*f1*pi*t2);fl = 100;% fl = 100 hzsubplot(223);plot(t1, ft,:),ts = 1/4000;% sample = 400

7、0hzhold ondt = 0.0001;stem(t2, fst), gridon ;t1 = -0.007:dt:0.007;axis(-0.006 0.006 -1.5 1.5)ft = cos(2*f1*pi*t1);xlabel( Time/s),ylabel(fs(t)subplot(221);plot(t1,ft),grid on ;title(Sample sig nal); holdoffaxis(-0.006 0.006 -1.5 1.5)xlabel( Time/s),ylabel(f(t)fsw=ts*fst*exp(-1i*t2*w);title(Cos ine c

8、urve);subplot(224); plot(w, abs(fsw),grid onN = 5000; k = -N:N;axis(-20000 20000 0 0.006)w = 2*pi*k/(2*N+1)*dt);xlabel( omega ),ylabel(fsw )fw = ft*dt*exp(-1i*t1*w);title( Sample freq spectrum);subplot(222);plot(w, abs(fw); gridon ;axis(-20000 20000 0 0.005);xlabel( omega ), ylabel(f(w)title(Cos fre

9、q spectrum);Cos ine curve-3x 10 Cos freq spectrum-3x 10 Sample freq spectrumSample sig nalw s4Time/sx 10-3x 10f1 = 100Hz将代码中f1设为3800即可JCos ine curve0-3-1-5!32154-3x 10 Cos freq spectrum0Time/s5X 100-2-10 1x210Sample sig nal.I42w s4x 10 Sample freq spectrum-50Time/s5-3X 10x 100 -2f2 = 3800Hz5、结合抽样定理，

10、利用MATLA编程实现Sa(t)信号经过冲激脉冲抽样后得到的抽样信号fs t及其频谱建议：冲激脉冲的周期分别取4*pi/3 s、pi s、2*pi/3 s三种情况对比，并利用fst构建Sa(t)信号(*改动第一行代码即可)Ts = 4/3;% impulse period = 4*pi/3t1 = -5:0.01:5;ft = si nc(t1);subplot(2, 2, 1)plot(t1, ft), gridonaxis(-6 6 -0.5 1.2) title( Sa(t)N = 500; k = -N: N;W = pi*k / (N*0.01);Fw = 0.01*ft*exp(-

11、1i*t1*W);subplot(2, 2, 2)plot(W, abs(Fw), gridonaxis(-30 30 -0.05 1.5)title( Sa(t) freq spectrum)t2 = -5: Ts: 5;fst = sin c(t2);subplot(2, 2, 3)plot(t1, ft,:), hold onstem(t2, fst), gridonaxis(-6 6 -0.5 1.2)title( Sampli ng sig nal)Fsw = Ts*fst*exp(-1i*t2*W);subplot(2, 2, 4)plot(W, abs(Fsw), gridona

12、xis(-50 50 -0.05 1.5)title( spectrum of Samplingsignal)Sa(t)Sa(t) freq spectrumSampli ng sig nalspectrum of Sampli ng sig nal冲激脉冲的周期=4*pi/3 sSa(t)Sa(t) freq spectrumSampli ng sig nalspectrum of Sampli ng sig nal冲激脉冲的周期=pi sSa(t)Sa(t) freq spectrumSampli ng sig nalspectrum of Sampli ng sig nal冲激脉冲的周期

13、=2*pi/3 s6、已知周期三角信号如下图所示注：图中时间单位为：毫秒(ms):(1)试求出该信号程实现其各次谐波如1、3、5、13、49的叠加，并验证其收敛性;a。=a谐波幅度收敛速度：；bn = 0原始波形:Original wave1 class H-wave3 class H-wave11I1pLL0.8-0.8-0.6-0.6-Xja-0.4 /0.47 Z k/W11r0.2p 10.20irr0rr,r-4-2024113 class H-wav e10.80.80.6- -0.60.4 0.40.2irr0.2-4-20-4-249 class H-wav e-4-2第k阶谐

14、波波形The Itimes superpose-4-2024The 13times superpose10.80.60.40.2010.80.60.40.20The 49times superpose-4-2024-4-2024前K次谐波的叠加The 3times superpose10.80.60.40.20Code：figure(1);t = -2*pi: 0.001: 2*pi;f = abs(sawtooth(0.5*pi*t, 0.5); plot(t, f), gridon ;axis(-4, 4, -1,2) title( Origi nal wave);nclass = 1,3

15、, 13, 49;figure(2);N = 4;a0 = 1/2;for k = 1: Nn = n class(k);an = 4./( n*pi).A2);ft = an *cos(pi* n*t);ft = ft + a0;subplot(2, 2, k); plot(t, ft);axis(-4, 4, 0, 1)title( num2str( nclass(k), class H-wave );endfigure(3);N = 4;a0 = 1/2;for k = 1: Nn = 1: 2: n class(k);an = 4./( n*pi).A2);ft = an *cos(p

16、i* n*t);ft = ft + a0;subplot(2, 2, k); plot(t, ft);axis(-4, 4, 0, 1)title(The,num2str(nclass(k),timessuperpose );end周期为2ms周期为1ms(2)用Matlab分析该周期三角信号的频谱三角形式或指数形式均可。当周期三角 信号的周期(如由2ms 1ms或由2ms 4mS和宽度(如2ms 1mS分别变化时，试观察 分析其频谱的变化。dt=0.01;t=-4:dt:4;ft=(t=-1 &t0&t v=1).*(1-t)；%subplot(2,1,1) %plot(t,ft);grid on n=2000;k=-n:n;w=pi*k/( n*dt); f=dt*ft*exp(-i*t*w); f=abs(f);%subplot(2,1,2)plot(w,f);axis(-20 20 0 1.1),grid on;dt=0.01;t=-4:dt:4;ft=(t=-0.5&t0 &t2fmax)，采样之后的 数字信号才能完整保留原始信号中的信息。第四题中， f=3800hz时，采样频 率4000hz明显小于f,故发生了混叠。2、谐波叠加实验：可以看出随着波次的