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MATLABµÄsymbolic Math Toolbox ÌṩÁËÖ±½ÓÇó½â¸µÀïÒ¶±ä»»¼°Äæ±ä»»µÄº¯Êýfourier()¼°ifourier()Á½Õߵĵ÷ÓøñʽÈçÏ¡£ Fourier ±ä»»µÄµ÷Óøñʽ
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F(v)??????f(x)e?jvxdx
FourierÄæ±ä»»µÄµ÷Óøñʽ
f=ifourier(F):ËüÊÇ·ûºÅº¯ÊýFµÄfourierÄæ±ä»»£¬Ä¬ÈϵĶÀÁ¢±äÁ¿Îªw£¬Ä¬ÈÏ·µ»ØÊÇ
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f=ifourier(f,u):ËüµÄ·µ»Øº¯ÊýfÊÇuµÄº¯Êý£¬¶ø²»ÊÇĬÈϵÄx.
×¢Ò⣺ÔÚµ÷Óú¯Êýfourier()¼°ifourier()֮ǰ£¬ÒªÓÃsymsÃüÁî¶ÔËùÓõ½µÄ±äÁ¿£¨Èçt,u,v,w£©½øÐÐ˵Ã÷,¼´½«ÕâЩ±äÁ¿ËµÃ÷³É·ûºÅ±äÁ¿¡£ Àý4-1 Çóf(t)?e?2tµÄ¸µÁ¢Ò¶±ä»»
½â: ¿ÉÓÃMATLAB½â¾öÉÏÊöÎÊÌ⣺ syms t
Fw=fourier(exp(-2*abs(t)))
Àý4-2 ÇóF(jw)?1µÄÄæ±ä»»f(t) 21??½â£º ¿ÉÓÃMATLAB½â¾öÉÏÊöÎÊÌâ syms t w
ft=ifourier(1/(1+w^2),t)
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Àý4-3 Çóµ÷ÖÆÐźÅf(t)?AG?(t)cos?0tµÄƵÆ×£¬Ê½ÖÐ
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ft=sym('4*cos(2*pi*6*t)*(Heaviside(t+1/4)-Heaviside(t-1/4))'); Fw=simplify(fourier(ft))
17
subplot(121)
ezplot(ft,[-0.5 0.5]),grid on subplot(122)
ezplot(abs(Fw),[-24*pi 24*pi]),grid
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F(j?)??f(t)e????j?tdt?lim?f(n?)e?j?n??
??0n????µ±?×㹻Сʱ£¬½üËÆ¼ÆËã¿ÉÂú×ãÒªÇó¡£ÈôÐźÅÊÇʱÏ޵쬻òµ±Ê±¼ä´óÓÚij¸ö¸ø¶¨ÖµÊ±£¬ÐźÅÒÑË¥¼õµÄºÜÀ÷º¦£¬¿ÉÒÔ½üËÆµØ¿´³ÉʱÏÞÐźÅʱ£¬nµÄȡֵ¾ÍÊÇÓÐÏ޵ģ¬ÉèΪN£¬ÓÐ
F(k)??f(n?)e?j?kn??n?0N?1,0?k?N,?k?2?k ÊÇÆµÂÊÈ¡Ñùµã N?ʱ¼äÐźÅÈ¡Ñù¼ä¸ô?ӦСÓÚÄοüË¹ÌØÈ¡Ñùʱ¼ä¼ä¸ô£¬Èô²»ÊÇ´øÏÞÐźſɸù¾Ý¼ÆË㾫¶ÈÒªÇóÈ·¶¨Ò»¸öƵÂÊ W0ΪÐźŵĴø¿í¡£
Àý4-4 ÓÃÊýÖµ¼ÆËã·¨ÇóÐźÅf(t)?u(t?1)?u(t?1)µÄ¸µÀïÒ¶±ä»»
½â£¬ÐÅºÅÆµÆ×ÊÇF(j?)?2Sa(?)£¬µÚÒ»¸ö¹ýÁãµãÊÇ?£¬Ò»°ã½«´ËƵÂÊÊÓΪÐźŵĴø¿í£¬Èô½«¾«¶ÈÌá¸ßµ½¸ÃÖµµÄ50±¶£¬¼ÈW0=50?£¬¾Ý´ËÈ·¶¨È¡Ñù¼ä¸ô£¬??R=0.02;t=-2:R:2;
f=Heaviside(t+1)-Heaviside(t-1); W1=2*pi*5;
N=500;k=0:N;W=k*W1/N; F=f*exp(-j*t'*W)*R; F=real(F);
W=[-fliplr(W),W(2:501)]; F=[fliplr(F),F(2:501)]; subplot(2,1,1);plot(t,f); xlabel('t');ylabel('f(t)'); title('f(t)=u(t+1)-u(t-1)'); subplot(2,1,2);plot(W,F); xlabel('w');ylabel('F(w)'); title('f(t)µÄ¸¶Êϱ任F(w)');
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1?0.02 2F0jwM)?1?L?b1jw(?)b0B(w)bM(jw)M?bM?1( H(jw)? ?NN?1A(w)aN(jw)?aN?1(jw)?L?a1(jw)?a018
MATLABÐźŴ¦Àí¹¤¾ßÏäÌṩµÄfreqsº¯Êý¿ÉÖ±½Ó¼ÆËãϵͳµÄƵÂÊÏìÓ¦µÄÊýÖµ½â¡£Æäµ÷ÓøñʽÈçÏÂ
H=freqs(b,a,w)
ÆäÖУ¬aºÍb·Ö±ðÊÇH(jw)µÄ·ÖĸºÍ·Ö×Ó¶àÏîʽµÄϵÊýÏòÁ¿£¬wΪÐÎÈçw1:p:w2µÄÏòÁ¿£¬¶¨ÒåϵͳƵÂÊÏìÓ¦µÄƵÂÊ·¶Î§£¬w1ΪƵÂÊÆðʼֵ£¬w2ΪƵÂÊÖÕÖ¹Öµ,pΪƵÂÊÈ¡Ñù¼ä¸ô¡£H·µ»ØwËù¶¨ÒåµÄƵÂʵãÉÏ£¬ÏµÍ³ÆµÂÊÏìÓ¦µÄÑùÖµ¡£
ÀýÈ磬ÔËÐÐÈçÏÂÃüÁ¼ÆËã0~2piƵÂÊ·¶Î§ÄÚÒÔ¼ä¸ô0.5È¡ÑùµÄϵͳƵÂÊÏìÓ¦µÄÑùÖµ a=[1 2 1]; b=[0 1];
h=freqs(b,a,0:0.5:2*pi)
Àý 4-5 Èý½×¹éÒ»»¯µÄbutterworth µÍͨÂ˲¨Æ÷µÄƵÂÊÏìӦΪ
H(jw)?1 32(jw)?2(jw)?2(jw)?1 ÊÔ»³ö¸ÃϵͳµÄ·ù¶ÈÏìÓ¦H(jw)ºÍÏàλÏìÓ¦?(?)¡£
½â ÆäMATLAB³ÌÐò¼°ÏìÓ¦µÄ²¨ÐÎÈçÏ w=0:0.025:5; b=[1];a=[1,2,2,1]; H=freqs(b,a,w); subplot(2,1,1);
plot(w,abs(H));grid; xlabel('\\omega(rad/s)'); ylabel('|H(j\\omega)|'); title('H(jw)µÄ·ùÆµÌØÐÔ'); subplot(2,1,2);
plot(w,angle (H));grid; xlabel('\\omega(rad/s)'); ylabel('\\phi(\\omega)'); title('H(jw)µÄÏàÆµÌØÐÔ');
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Àý 4-6ÒÑÖªÒ»RCµç·ÈçͼËùʾ ϵͳµÄÊäÈëµçѹΪf(t)£¬Êä³öÐźÅΪµç×èÁ½¶ËµÄµçѹy(t).µ±RC=0.04£¬f(t)=cos5t+cos100t, ???t??? ÊÔÇó¸ÃϵͳµÄÏìÓ¦y(t)
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+ f(t) - C R + y(t) -
½â ÓÉͼ¿ÉÖª £¬¸Ãµç·Ϊһ¸ö΢·Öµç·£¬ÆäƵÂÊÏìӦΪ H(jw)?Rjw ?R?1jwCjw?1RC ÓÉ´Ë¿ÉÇó³öÓàÏÒÐźÅcos?0tͨ¹ýLTIϵͳµÄÏìӦΪ
y(t)?H(j0w)co?s0(?t??0( ))¼ÆËã¸ÃϵͳÏìÓ¦µÄMATLAB³ÌÐò¼°ÏìÓ¦²¨ÐÎÈçÏÂ
RC=0.04;
t=linspace(-2,2,1024); w1=5;w2=100;
H1=j*w1/(j*w1+1/RC); H2=j*w2/(j*w2+1/RC); f=cos(5*t)+cos(100*t);
y=abs(H1)*cos(w1*t+angle(H1))+ abs(H2)*cos(w2*t+angle(H2)); subplot(2,1,1); plot(t,f); ylabel('f(t)'); xlabel('Time(s)'); subplot(2,1,2); plot(t,y); ylabel('y(t)'); xlabel('Time(s)'); Èý¡¢ ÉÏ»úʵÑéÄÚÈÝ
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