傅里叶变换f(t)=(e^-2t)ε(t+1) 请教一道信号与系统习题 求E∞ x(t)=[e∧-2t]*ε...
\u6c42f(t)=e^-2t*u(t)\u7684\u5085\u7acb\u53f6\u53d8\u6362\u7528matlab\u8fd9\u91cct\u5df2\u7ecf\u662f\u7b26\u53f7\u53d8\u91cf\uff0c\u5c31\u4e0d\u7528\u5bf9\u5b83\u8d4b\u503c\uff08\u7b2c\u4e8c\u884c\uff09\u3002\u6211\u5199\u4e86\u4e2a\u7b80\u5355\u7684\u4ee3\u7801\uff1aclear all;
close all;
clc;
%\u5e26\u6709\u9636\u8dc3\u51fd\u6570\u7684Laplace\u53d8\u6362
syms t s
f=exp(-t)*heaviside(t-2) %heaviside \u9636\u8dc3\u51fd\u6570\u7684\u8868\u793a
F=laplace(f,t,s)\u8fd0\u884c\u7684\u7a0b\u5e8f\u622a\u56fe\u5982\u4e0b\uff1af =heaviside(t - 2)/exp(t) F =1/(exp(2)*exp(2*s)*(s + 1))
\u5e0c\u671b\u80fd\u6709\u5e2e\u52a9\u3002
\u8bbeP=\uff08e^x+f(x)\uff09y\uff0c\u8bbeQ=-f(x)\uff0c\u6b32\u4e0e\u8def\u5f84\u65e0\u5173\uff0c\u987b\u6210\u7acbP'y=Q'x\uff0c\u5373e^x+f(x)=-f'(x)\u2605\u5219\u95ee\u9898\u6210\u4e3a\uff0c\u6c42\u5fae\u5206\u65b9\u7a0b\u2605\u7684\u6ee1\u8db3\u521d\u59cb\u6761\u4ef6f(0)=0\u7684\u89e3\u3002\u89e3\u5f97f(x)=(1/2)*\u3010e^x-e^(-x)\u3011\u3002
答案如图所示
绛旓細5銆佹寮﹀嚱鏁癋(ejw0t)=2(w-w0)锛岀浉褰撲簬鏄洿娴佷俊鍙风殑绉讳綅銆侳(sinw0t)=F((ejw0t-e-jw0t)/2)=((w-w0)-(w+w0))F(sinw0t)=F((e銆6銆佸崟浣嶅啿鍑诲簭鍒梛w0t-e-jw0t)/2j)=j((w-w0)-(w+w0)) T(t)=(t-Tn) -杩欐槸涓涓懆鏈熷嚱鏁帮紝姣忛殧T鍑虹幇涓涓啿鍑伙紝鍛ㄦ湡鍑芥暟鐨鍌呴噷鍙跺彉鎹鏄...
绛旓細= (1/2蟺) 鈭 (1/2鈭歛) 鈭毾 e^(-蠅^2/4a) e^(j蠅t) d蠅 = (1/4鈭毾a) 鈭 e^(-(蠅-2at)^2/4a) d(蠅-2at)= (1/2鈭歛) e^(-a^2t^2) 锛屽叾涓琛ㄧず鍦嗗懆鐜囥傚洜姝わ紝鏍规嵁瀵瑰伓鎬у師鐞嗭紝鍑芥暟f(t) = e^(-at^2) 鐨鍌呴噷鍙跺彉鎹涓篎(蠅) = (1/2鈭歛) e^(-蠅^2...
绛旓細搴旂敤锛1銆佹媺鏅媺鏂彉鎹富瑕佺敤浜庣數璺垎鏋愶紝浣滀负瑙ｅ井鍒嗘柟绋嬬殑寮烘湁鍔涘伐鍏凤紙灏嗗井绉垎杩愮畻杞寲涓轰箻闄よ繍绠楋級銆2銆鍌呴噷鍙跺彉鎹鍦ㄧ墿鐞嗗銆佺數瀛愮被瀛︾銆佹暟璁恒佺粍鍚堟暟瀛︺佷俊鍙峰鐞嗐佹鐜囪銆佺粺璁″銆佸瘑鐮佸銆佸０瀛︺佸厜瀛︺佹捣娲嬪銆佺粨鏋勫姩鍔涘绛夐鍩熼兘鏈夌潃骞挎硾鐨勫簲鐢紙渚嬪鍦ㄤ俊鍙峰鐞嗕腑锛屽倕閲屽彾鍙樻崲鐨勫吀鍨嬬敤閫旀槸灏嗕俊鍙...
绛旓細1鐨鍌呴噷鍙跺彉鎹鏄2蟺未锛坱锛夈鍌呯珛鍙跺彉鎹瀵规湁澶氱瀹氫箟褰㈠紡锛屽鏋滈噰鐢ㄤ笅鍒楀彉鎹㈠銆傚嵆锛欶锛埾夛級=鈭紙鈭烇紝-鈭烇級f(t)e^(-i蠅t)dtf(t)=(1/2蟺)鈭紙鈭烇紝-鈭烇級F锛埾夛級e^(i蠅t)d蠅銆備护锛f(t)=未锛坱锛夛紝閭ｄ箞锛氣埆锛堚垶锛-鈭烇級未锛坱锛塭^(-i蠅t)dt=1銆傝屼笂寮忕殑鍙嶅彉鎹細锛1/2蟺...
绛旓細鏃犺В銆俛0绉笉鍑烘潵锛e^2x鍦-蟺锝炏鍖洪棿鍐呯Н鍒 an鏄彲浠ョН鐨勫嚭鏉ワ紝鐢ㄥ垎閮ㄧН鍒嗘硶銆傝f(x)=sinax, -蟺鈮鈮は, a>0锛屽皢鍏跺睍寮鎴愪互2蟺涓哄懆鏈熺殑鍌呴噷鍙绾ф暟 鍒ゆ柇鍌呴噷鍙剁骇鏁 鐨勬敹鏁涙у厖鍒嗗繀瑕佹潯浠讹紝浣嗘槸瀵逛簬瀹為檯闂涓嚭鐜扮殑鍑芥暟锛屾湁寰堝绉嶅垽鍒潯浠跺彲鐢ㄤ簬鍒ゆ柇鏀舵暃鎬с傛瘮濡倄(t)鐨勫彲寰ф垨绾ф暟鐨勪竴鑷...
绛旓細sintcost=1/2sin2t F(1/2sin2t)=鈭(-鈭烇紝+鈭) 1/2sin2t 路 e^-jwt dt 鐢ㄦ鎷夊叕寮忓彲寰楀師寮= 1/2鈭(-鈭烇紝+鈭) j/2( e^-2jt - e^2jt )e^-jwt dt =j/4鈭(-鈭烇紝+鈭) e^-j(w+2)t - e^-j(w-2)t dt 鐢ㄎ村嚱鏁扮殑鍌呮皬鍙樻崲 寰楀師寮= j/2 蟺[未(w+2)-未(w...
绛旓細F(s)=[e^-(s-2)]/(s+2)=e^2 * e^(-s) /(s+2)鎵浠ョ涓琛屽搴旂殑a=1 鏌ヨ〃 1/(s+2) 瀵瑰簲 e^(-2t)鎵浠 f(t-1)=e^2 * e^(-2t)灏嗗乏杈瑰紡瀛愮殑t-1鐢╰0鏉ユ浛鎹,鍒 f(t0)=e^2 * e^(-2)*(t0+1)=e^(-2 * t0)鍗 f(t-1)=e^[-2*(t-1)]搴旇鏄繖鏍,
绛旓細鏍规嵁娆ф媺鍏紡寰梥inw0t=(e^jw0t-e^(-jw0t)/(2j)銆傚洜涓虹洿娴佷俊鍙1鐨鍌呴噷鍙跺彉鎹涓2蟺未(w)銆傝宔^jw0t鏄洿娴佷俊鍙峰倕閲屽彾鍙樻崲鐨勯绉汇傛墍浠^jw0t鐨勫倕閲屽彾鍙樻崲涓2蟺未(w-w0),鍚岀悊e^(-jw0)鐨勫倕閲屽彾鍙樻崲涓2蟺未(w+w0)銆傛墍浠F(jw)=[蟺未(w-w0)-蟺未(w+w0)]/j銆傚倕閲屽彾鍙樻崲锛...
绛旓細(8*E*Sin[蟺*w])/w
绛旓細t=[0:0.01:10];x=(2-exp(-2*t));plot(t,x);xlabel('t');ylabel('x(t)');title('x(t)=(2-exp(-2*t)');grid on
