信號(hào)與系統(tǒng)實(shí)驗(yàn)二傅里葉分析及應(yīng)用實(shí)驗(yàn)報(bào)告

1、實(shí)驗(yàn)二傅里葉分析及應(yīng)用一、實(shí)驗(yàn)?zāi)康模ㄒ唬┱莆帐褂胢atlab進(jìn)行周期信號(hào)傅里葉級(jí)數(shù)展開和頻譜分析1、學(xué)會(huì)使用matlab分析傅里葉級(jí)數(shù)展幵，深入理解傅里葉級(jí)數(shù)的物理含義2、學(xué)會(huì)使用matlab分析周期信號(hào)的頻譜特性（二）掌握使用matlab求解信號(hào)的傅里葉變換并分析傅里葉變換的性質(zhì)1、學(xué)會(huì)運(yùn)用matlab求連續(xù)時(shí)間信號(hào)的傅里葉變換2、學(xué)會(huì)運(yùn)用matlab求連續(xù)時(shí)間信號(hào)的頻譜圖3、學(xué)會(huì)運(yùn)用matlab分析連續(xù)時(shí)間信號(hào)的傅里葉變換的性質(zhì)（三）掌握使用matlab完成信號(hào)抽樣并驗(yàn)證抽樣定理1、學(xué)會(huì)運(yùn)用matlab完成信號(hào)抽樣以及對(duì)抽樣信號(hào)的頻譜進(jìn)行分析2、學(xué)會(huì)運(yùn)用matlab改變抽樣吋間間隔，觀察抽

2、樣后信號(hào)的頻譜變化3、學(xué)會(huì)運(yùn)用matlab對(duì)抽樣后的信號(hào)進(jìn)行重建二、實(shí)驗(yàn)條件裝用matlab r2015a的電腦。三、實(shí)驗(yàn)過程（1）試求出該信號(hào)的傅里葉級(jí)數(shù)自己求或參見課本p112或p394,利用matlab編 程實(shí)現(xiàn)其各次諧波如1、3、5、13、49的疊加，并驗(yàn)證其收斂性；程序：t=-10:0.001:10;y= (sawtooth(pi*(t + 1),0.5)+l)/2;plot (t, y) , xlabel ( ' t1 ) , ylabel ('三角波信號(hào) 1); axis(-2,2,0,1.1); n_max=1,3,5,11,47;n=length(n_max)

3、; for k=l:nn=l:2:n_max(k); b=4./(pi.a2)*(n.2); x=0.5+b*cos(pi*n1*t); figure,plot(tz x);hold on; plot (t,y);hold off;xlabel ( t * ) , ylabel ( * 部分和的波形 1 ); axis ( -3,3,0,1);title ( 最大諧波數(shù)=1, num2str (n_max (k);end結(jié)果:figure 1. |figure 2-figure 3 a file edit viewimert tools desktopwindow help »ied

4、it view imert tools desktopwindow hdp« edit view insert tools desktop window helpu jq 0 a nj id i k f； © e ed i k 、'打 ® w k q | s 0嵌大術(shù)波鍬人諮波8 6 4 o.&o.86228 6 o.0.4.2«人浙波tt=58 6 4 2 o.o.o.o.xbsv-.v r 2 l rz i極大淅波教銀大偕波數(shù)=47（2）用matlab分析該周期三角信號(hào)的頻譜三角形式或指數(shù)形式均可。當(dāng)周期三角信號(hào)的周期（如由2ms+l

5、ms或由2ms+4ms）和寬度（如2ms今lms）分別變化時(shí)，試觀察分析其頻譜的變化。程序：n=-20 : 20; t=10; wl=2*pi/t;fn=2./(n.*n.*wl.*wl);subplot(311),stem(n*wl,fn),title(t=10)；%axis ( -10,10,-0.1,20); t=5;wl=2*pi/t;fn=2/(n*n*wl*wl);subplot(312),stem(n*wl,fn),title(1t=51);axis(-20,20,-0.1,3);t=1;wl=2*pi/t;fn=2./(n.*n.*wl.*wl);subplot(313),st

6、em(n*wl,fn),title(*t=1*);axis ( -100, 100,-0.01, 0.1);結(jié)果：t=101050ii1 1 1115-10-5051015t=5b <)<> i-20-15-10-505101520t=12、分別利用matlab符號(hào)運(yùn)算求解法和數(shù)值計(jì)算法求下圖所示信號(hào)的ft,并畫出其 頻譜圖(包括幅度譜和相位譜)f注：圖中時(shí)間單位為：毫秒(ms)l。程序：1.ft=sym(1 (t + 2)* (heaviside(t+2)-heaviside(t + 1) + (heaviside(t + 1)-heaviside(t-1) + (-t+2

7、)*(heaviside(t-1)-heaviside (t-2) )；fw=simplify(fourier(ft);subplot(311);ezplot(ft, -3,3);subplot(312);ezplot(abs(fw),-10,10,0,4);title('-')；phase=atan(imag(fw)/real(fw);subplot(313);ezplot(phase)/title(1ial»x1)；ft=s:，t»r <t-*-2) < (heavisxde <r+2>-hes 1 de(t+1 + (heavi

8、s 1 de)-hea*/i-1 de 11-1 14-(-1+2 «hea'/islde (t-1 -heaviside t-2 i 回托 fv=sxaplify(fourier(ft): subplot(221).ezplot(ft.-3, 3:>:subplot <222):esplot (abs(fv), -10,10.0, 4) :titlec 播度is. > edit view insert tools desktop window helpphase=atan(iaag(f<)/real(fw): subplot (223) ezplot

9、 (phase) :titlec 相位謂'):filefigure 1desktop window <j ji ； k % -、c ® w x a 相位譜1 and windowntitieahtitledntxtledntixledntxtledntxtled0.50 -0.5-1 -505w程序2:dt=0.01;t=-4:dt:4;ft= (t+2) .*(heaviside(t+2)-heaviside(t + 1) + (heaviside(t + 1)-heaviside(t一1) + (-t+2) .*(heaviside(t-1)-heaviside (

10、t-2);n=2000;k=-n:n;w=pi*k/(n*dt);f=dt*ft*exp(一1j*t *w);f=abs(f);plot (w,f); axis(-10 10 0 3.2);*figure 1-n |file edit view insert tools desktop window help bi：n;)i*k/ (n*dt):/jt= (t+2). * (heaviside (t+2)-heaviside (t+1) )+(heaviside (t+1 )-heaviside(t-1 >)+<-1+2). t(heaviside(t-l)-heaviside(t

11、-2); =2000;f=dt*ft*exp(-lj»t, <v): :=abs(f):clot (v, f); axis(:-l0 10 0 3.2);jnd window ltitiea ltitled :using 拿 natrix dimensions must agree.3、試用matlab命令求f（j幼的傅里葉反變換，并繪出其吋域信號(hào)圖3 +j69 5 +j69注意：（1）寫代碼時(shí)j今i程序:syms t;fw=syin ( ' 10/ (3 + iw) -4/ (5 + iw)'); ft=simplify(ifourier(fw,t); ezp

12、lot(abs (ft);sym(. 10/(3+i曹)-4/ (5+iv).) -simplify(ifourler (fw, t): plot(abs(ft).figure 1file edit view insert tools desktop window help a s nnnd windowt»5i)*sign(t)<(exp(t*2i)<5ititled4、己知門函數(shù)自身卷積為三角波信號(hào)，試用matlab命令驗(yàn)證ft的時(shí)域卷積定理。注：即驗(yàn)證門函數(shù)ft的平方與和應(yīng)三角波信號(hào)的ft后結(jié)果是否一致，可結(jié)合頻譜閣觀察分析程序：dt=0.01;t=-5:dt:5;

13、ft = (t=-05)-(t=05);f=conv(ft,ft)*dt;n=length(f);tt=(0:n-1)*dt-10;subplot (221) , plot (t, ft) ; axis ( -1,1,0, 1.1) / title ( ' f (t)');subplot(222),plot(tt,f);axis(-1.5,1.5,0,1.1);title('f(t)*f(t)');n=2000;k=-n:n;w=pi*k/(n*dt);fwl=dt*f*exp(-1j*tt1*w);subplot (223) ;plot (w, abs (fw

14、l) ) ; xlabel ( 1 w' ) , ylabel ( ' f (f (t)卷積f (t)'); axis ( -20 20 -0.3 1.2) ; title ( *f (f (t)(t)');fw2=dt*ft*exp(一1j*t *w);subplot(224);plot(w,abs(fw2.*fw2);xlabel('w'), ylabel ('f (w)*f (w)'); axis ( -20 20 -0.3 12);title(f(w)(w) );結(jié)果:editjbreakpantsrun bin r -

15、d:programfilesdmatlabbinuntitled.med.m +f"conv(ft. ft)*dt;n-length(f):tt-(0:n-1)*dt-10; subplot(221).plot(t,ft):axis<-l,1,0,1.1:title subplot (222), plot (tt, f) :axis(-l. 5, 1. 5, 0,1.1) ;t n=2000:k=-n:n:w=pi*k/(n*dt> :fwl-dt*f*exp(-lj*tt* *w):subplot (223) plot(w. abs(frl):xlabel( a ) y

16、labe axis(-20 20 -0.3 1.2) :titlecf<f <t)卷稅f <t)r> fw2=dt*ft*esp(-lj*t，*v):subplot(224):plot(w, abs(fw2. »fw2);xlabelc v'), axis(：-20 20 -0.3 1. 2) :title(, f (w) *f (w) *);file edit view insert tools desktop window helpn|ao0.5nd windowziatlab? see resources for getting started.

17、titledtitled01匕1-0.500.5f(f(t)卷積ft)(gdisgd(mir.(m)ll-20-1o2010o-1o-2o20105、設(shè)有兩個(gè)不同頻率的余弦信號(hào)，頻率分別為./;=100hz, /2= 3800hz；現(xiàn)在使用抽樣頻率乂 =4000/z對(duì)這三個(gè)信號(hào)進(jìn)行抽樣，使用matlab命令畫出各抽樣信號(hào)的波形和頻譜，并分析其頻率混疊現(xiàn)象建議：抽樣信弓的頻譜圖橫坐標(biāo)范圍不小于 -10000hz10000hz 或-20000*pi20000*pi rad/s程序：f s=4 000; ts=l/f s; %抽樣信號(hào)dt=o.00001;tl=-0.1:dt:0.1;ftl=cos

18、(2*pi*100*tl);ft2=cos(2*pi*3800*tl);t2=-0.1:t s:0.1;fstl=cos(2*pi*100*t2);subplot(221);stem(t2,fstl);xlabel('t');ylabel(f(t); axis (0,0.01,-1.5,1.5) ; title (，f=100hz 抽樣后的信號(hào) 1);n=2000;k=-n:n;w=pi*k/(n*dt);fswl=ts*fstl*exp(-1j*t2'*w);subplot(222);plot(w,abs(fswl);xlabel('w');ylabe

19、l('fi(w)'); axis (-60000, 60000,-0.01, 0.11) ; title (，f=100hz抽樣信號(hào)的頻譜);fst2=cos(2*pi*3800*t2);subplot(223);stem(t2,fst2);xlabel(1t1);ylabel(1f(t) )； axis ( 0, 0.01, -1.1,1.1 ) ; title ( ' f=3800hz抽樣后的信號(hào)：);fsw2=ts*fst2*exp(-lj*t2*w；subplot(224);plot(w,abs(fsw2);xlabel ( 1w1);ylabel ( 1f2

20、(w)'); axis ( -60000, 60000,-0.01, 0.11) / title ( ' f=3800hz抽樣信號(hào)的頻譜 1 );結(jié)果：rs,"誦抽稗后的信兮1.500.0020.0040.0060.0080.01f=100hz抽樣信號(hào)的頻譜f=3800hz抽樣后的信號(hào)f=3800hz抽樣信號(hào)的頻譜w6、結(jié)合抽樣定理，利用matlab編程實(shí)現(xiàn)sz(z)信號(hào)經(jīng)過沖激脈沖抽樣后得到的抽 樣信號(hào)/y(z)及其頻譜建議：沖激脈沖的周期分別取4*pi/3 s、pi s、2*pi/3 s三種情況對(duì)比， 并利用乂構(gòu)建心信號(hào)。程序：ts=2*pi/3;%抽樣信號(hào)，ts分別等于4*pi/3 s、pi s、2*pi/3 sdt=o.1; t=-10*pi:dt:10*pi;tsl=-10*pi:ts:10*pi;ft=sinc(t/pi);fstl=sinc(tsl/pi);subplot(311);plot(t,ft,1:);hold on;stem(tsi,fstl);grid on;xlabel(*t1);ylabel(*f(t)1);axis(-20,20,-0.2,1.1);title ( ' t=2*pi/3 s的抽樣信號(hào))；n=2000;k=-n:n;w=pi*k/(n*dt)