求u(t)的傅里叶变换 信号与系统:u(1-t)的傅里叶变换，谢啦！
u(t)\u7684\u5085\u91cc\u53f6\u53d8\u6362\u600e\u4e48\u8bc1\u660e\uff08\u8fc7\u7a0b \u8c22\u8c22\uff09\u6211\u5f00\u59cb\u4e5f\u662f\u697c\u4e0a\u5927\u54e5\u90a3\u6837\u7b97\u7684\uff0c\u5176\u5b9e\u662f\u9519\u7684
u(t)\u4e0d\u6ee1\u8db3\u5085\u91cc\u53f6\u53d8\u6362\u7edd\u5bf9\u53ef\u79ef\u7684\u6761\u4ef6\u6240\u4ee5\u4e0d\u80fd\u76f4\u63a5\u7528\u5b9a\u4e49\u6c42\u3002
\u6240\u4ee5\u5c06u(t)\u8f6c\u6362\u4e3a1/2\uff081+sgn\uff08t\uff09\uff09
sgn\uff08t\uff09\u4e3a\u7b26\u53f7\u51fd\u6570
F[u(t)]\uff1dF[1/2]+F[1/2sgn(t)]
=\u03c0\u03b4(\u03a9)+1/j\u03a9
u(1-t)\u7684\u5085\u91cc\u53f6\u53d8\u6362\u7b49\u4e8e\uff1ae^\uff08-jw\uff09*(-1/jw+\u03c0*Delta(w))
f(t)\u662ft\u7684\u5468\u671f\u51fd\u6570\uff0c\u5982\u679ct\u6ee1\u8db3\u72c4\u91cc\u8d6b\u83b1\u6761\u4ef6\uff1a\u5728\u4e00\u4e2a\u4ee52T\u4e3a\u5468\u671f\u5185f(X)\u8fde\u7eed\u6216\u53ea\u6709\u6709\u9650\u4e2a\u7b2c\u4e00\u7c7b\u95f4\u65ad\u70b9\uff0c\u9644f\uff08x\uff09\u5355\u8c03\u6216\u53ef\u5212\u5206\u6210\u6709\u9650\u4e2a\u5355\u8c03\u533a\u95f4\uff0c\u5219F\uff08x\uff09\u4ee52T\u4e3a\u5468\u671f\u7684\u5085\u91cc\u53f6\u7ea7\u6570\u6536\u655b\uff0c\u548c\u51fd\u6570S\uff08x\uff09\u4e5f\u662f\u4ee52T\u4e3a\u5468\u671f\u7684\u5468\u671f\u51fd\u6570\uff0c\u4e14\u5728\u8fd9\u4e9b\u95f4\u65ad\u70b9\u4e0a\uff0c\u51fd\u6570\u662f\u6709\u9650\u503c\u3002
\u5728\u4e00\u4e2a\u5468\u671f\u5185\u5177\u6709\u6709\u9650\u4e2a\u6781\u503c\u70b9\uff1b\u7edd\u5bf9\u53ef\u79ef\u3002\u5219\u6709\u4e0b\u56fe\u2460\u5f0f\u6210\u7acb\u3002\u79f0\u4e3a\u79ef\u5206\u8fd0\u7b97f(t)\u7684\u5085\u7acb\u53f6\u53d8\u6362\uff0c\u2461\u5f0f\u7684\u79ef\u5206\u8fd0\u7b97\u53eb\u505aF(\u03c9)\u7684\u5085\u7acb\u53f6\u9006\u53d8\u6362\u3002F(\u03c9)\u53eb\u505af(t)\u7684\u50cf\u51fd\u6570\uff0cf(t)\u53eb\u505aF(\u03c9)\u7684\u50cf\u539f\u51fd\u6570\u3002F(\u03c9)\u662ff(t)\u7684\u50cf\u3002f(t)\u662fF(\u03c9)\u539f\u50cf\u3002
\u6269\u5c55\u8d44\u6599
\u82e5\u51fd\u6570f\uff08x\uff09 \u7684\u5085\u91cc\u53f6\u53d8\u6362\u4e3af\uff08w\uff09\uff0c\u5219\u5bf9\u4efb\u610f\u5b9e\u6570w0 \uff0c\u51fd\u6570\u4e5f\u5b58\u5728\u5085\u91cc\u53f6\u53d8\u6362\uff0c\u4e14\u5176\u5085\u91cc\u53f6\u53d8\u6362f\uff08w0\uff09\uff0c\u4e5f\u5c31\u662f\u8bf4\uff0cf\uff08w0\uff09 \u53ef\u7531f\uff08w0\uff09\u5411\u53f3\u5e73\u79fbw0\u5f97\u5230\u3002
\u5085\u91cc\u53f6\u53d8\u6362\u5728\u7269\u7406\u5b66\u3001\u7535\u5b50\u7c7b\u5b66\u79d1\u3001\u6570\u8bba\u3001\u7ec4\u5408\u6570\u5b66\u3001\u4fe1\u53f7\u5904\u7406\u3001\u6982\u7387\u8bba\u3001\u7edf\u8ba1\u5b66\u3001\u5bc6\u7801\u5b66\u3001\u58f0\u5b66\u3001\u5149\u5b66\u3001\u6d77\u6d0b\u5b66\u3001\u7ed3\u6784\u52a8\u529b\u5b66\u7b49\u9886\u57df\u90fd\u6709\u7740\u5e7f\u6cdb\u7684\u5e94\u7528\uff08\u4f8b\u5982\u5728\u4fe1\u53f7\u5904\u7406\u4e2d\uff0c\u5085\u91cc\u53f6\u53d8\u6362\u7684\u5178\u578b\u7528\u9014\u662f\u5c06\u4fe1\u53f7\u5206\u89e3\u6210\u9891\u7387\u8c31\u2014\u2014\u663e\u793a\u4e0e\u9891\u7387\u5bf9\u5e94\u7684\u5e45\u503c\u5927\u5c0f\uff09\u3002
\u53c2\u8003\u8d44\u6599\u6765\u6e90\uff1a\u767e\u5ea6\u767e\u79d1-\u5085\u91cc\u53f6\u53d8\u6362
单位阶跃函数 u(t) 可以写成常数1和符号函数的和除以2。 （见图。）
u(t)={1+ sgn(t)}/2
常数1的傅里叶变换是纯实的， 等于2πδ（w）。
符号函数的定义是：sgn(t)=1, 当 t>=0; =-1 当 t<0.
它是一奇函数。奇函数的傅里叶变换是纯虚的， 等于2(1/jw) 。
所以： u(t)={1+ sgn(t)}/2 的傅里叶变换 = (2πδ(w)+ 2(1/jw))/2 = πδ(w)+ (1/jw)
绛旓細u锛坱锛=1/jw+pai*鍐叉縺鍑芥暟锛坵锛夛紝浠旂棰戝煙寰锛屾椂鍩*-jt锛屾渶鍚庣瓑寮忎袱娈*j灏卞彲浠ヤ簡銆傚湪涓嶅悓鐨勭爺绌堕鍩燂紝鍌呯珛鍙跺彉鎹㈠叿鏈夊绉嶄笉鍚岀殑鍙樹綋褰㈠紡锛屽杩炵画鍌呯珛鍙跺彉鎹㈠拰绂绘暎鍌呯珛鍙跺彉鎹傛渶鍒濆倕绔嬪彾鍒嗘瀽鏄綔涓虹儹杩囩▼鐨勮В鏋愬垎鏋愮殑宸ュ叿琚彁鍑虹殑銆傚倕绔嬪彾鍙樻崲鍙垎鏋愪俊鍙风殑鎴愬垎锛屼篃鍙敤杩欎簺鎴愬垎鍚堟垚淇″彿銆傝澶氭尝...
绛旓細u(t)={1+ sgn(t)}/2 甯告暟1鐨勫倕閲屽彾鍙樻崲鏄函瀹炵殑锛 绛変簬2蟺未锛坵锛銆傜鍙峰嚱鏁扮殑瀹氫箟鏄細sgn(t)=1, 褰 t>=0; =-1 褰 t<0.瀹冩槸涓濂囧嚱鏁般傚鍑芥暟鐨勫倕閲屽彾鍙樻崲鏄函铏氱殑锛 绛変簬2(1/jw) 銆傛墍浠ワ細 u(t)={1+ sgn(t)}/2 鐨勫倕閲屽彾鍙樻崲 = (2蟺未(w)+ 2(1/jw))/2 = ...
绛旓細u(t)涓嶆弧瓒鍌呴噷鍙跺彉鎹缁濆鍙Н鐨勬潯浠舵墍浠ヤ笉鑳界洿鎺ョ敤瀹氫箟姹傘傛墍浠ュ皢u(t)杞崲涓1/2锛1+sgn锛坱锛夛級sgn锛坱锛変负绗﹀彿鍑芥暟 F[u(t)]锛滷[1/2]+F[1/2sgn(t)]=蟺未(惟)+1/j惟
绛旓細鏍规嵁鍌呴噷鍙跺彉鎹㈢殑瀹氫箟锛屽浜庨樁璺冨嚱鏁皍(t)锛屽叾鍌呴噷鍙跺彉鎹(蠅)瀹氫箟涓猴細U(蠅) = 鈭玔0,鈭) u(t) * exp(-j蠅t) dt 鏍规嵁闃惰穬鍑芥暟鐨勫畾涔夛紝鍦╰=0涔嬪墠u(t)绛変簬0锛屽湪t=0涔嬪悗u(t)绛変簬1銆傚洜姝わ紝鍙互灏嗕笂杩扮Н鍒嗗垎涓轰袱涓儴鍒嗚绠楋細U(蠅) = 鈭玔0,鈭) 0 * exp(-j蠅t) dt + 鈭玔0,...
绛旓細e^(jw))锛岄偅涔坲(t-2)鐨勫倕閲屽彾鍙樻崲涓篹^(-j2w)*U(e^(jw))锛鏁卽(t)-u(t-2)鐨勫倕閲屽彾鍙樻崲涓篬1-e^(-j2w)]*U(e^(jw))锛涙牴鎹畉*u(t)鐨勫倕閲屽彾鍙樻崲涓簀*[U(e^(jw))鐨勫鏁癩锛屾墍浠*[u(t)-u(t-2)]鐨勫倕閲屽彾鍙樻崲涓簀*{[1-e^(-j2w)]*U(e^(jw))鐨勫鏁皚銆
绛旓細1銆丗(w)=鈭 f(t)e* dt 锛 绉垎鑼冨洿鏄粠-鈭炲埌+鈭烇紝e鐨勬寚鏁版槸-jwt銆傚氨鏄鍌呴噷鍙跺彉鎹鐨勮〃杈惧紡銆傛琛ㄨ揪寮忓氨鏄竴涓嚜鍙橀噺涓簑鐨勫嚱鏁帮紝鐒跺悗鎶奧=0甯﹀叆涓婂紡锛屽彉鎴怓(0)=鈭 f(t) dt锛屽氨鏄f(t)浠-鈭炲埌+鈭炵殑绉垎锛岀敱浜巉(t)鐨t鐨勮寖鍥翠负-1鍒1,鍒欑Н鍒嗚寖鍥村彉鎴-1鍒1,绉垎鐨勭墿鐞嗘剰涔夊氨鏄:...
绛旓細绛旀濡備笅鍥撅細绗﹀彿鍑芥暟涓嶆槸缁濆鍙Н鐨勫嚱鏁帮紝涓嶅瓨鍦ㄥ父涔変笅鐨勫倕閲屽彾鍙樻崲銆傚湪鑰冭檻骞夸箟鍑芥暟鐨勬潯浠朵笅鏄彲姹鐨勶紝浣嗕笉鑳界敤瀹氫箟寮廎(jw)=鈭玣(t)e^{-jwt}dt鏉ユ眰銆傚彲浠ュ湪宸茬煡u锛坱锛夌殑鎯呭喌涓嬶紝閫氳繃鍏辫江瀵圭О鎬ф眰寰椼傚湪涓嶅悓鐨勭爺绌堕鍩燂紝鍌呯珛鍙跺彉鎹㈠叿鏈夊绉嶄笉鍚岀殑鍙樹綋褰㈠紡锛屽杩炵画鍌呯珛鍙跺彉鎹㈠拰绂绘暎鍌呯珛鍙...
绛旓細绉垎 S exp(-at-jwt)u(t)dt= S exp(-(jw+a锛塼)u(t) dt =exp(-(jw+a)t)/-(a+jw)|0-->鏃犵┓锛=锛0-1锛/-锛坅+jw锛=1/(a+jw);鏈鍚庣殑缁撴灉璐熻礋寰楁锛屾病鏈夎礋鍙峰暒銆
绛旓細f(t) = (e^-at)u(t)F(j蠅) = 鈭玔0, 鈭瀅 (e^-at)u(t)e^(-j蠅t)dt = 1/(j蠅+a)鍌呴噷鍙跺彉鎹鏄竴涓瀵瑰簲鐨勩傛墍鏈夊弽鍙樻崲閮藉彲鐢ㄦ鍙樻崲寰楀埌銆
绛旓細3u(t)鐨勫倕閲屽彾鍙樻崲鏄2(1/jw)銆傚倕閲屽彾鍙樻崲鏄妸鐪嬩技鏉備贡鏃犵珷鐨勪俊鍙疯冭檻鎴愮敱涓瀹氭尟骞呫佺浉浣嶃侀鐜囩殑鍩烘湰姝ｅ鸡淇″彿缁勫悎鑰屾垚锛屾槸灏嗗嚱鏁板悜涓缁勬浜ょ殑姝ｅ鸡銆佷綑寮﹀嚱鏁板睍寮锛屽倕閲屽彾鍙樻崲鐨勭洰鐨勫氨鏄壘鍑鸿繖浜涘熀鏈寮︿俊鍙蜂腑鎸箙杈冨ぇ淇″彿瀵瑰簲鐨勯鐜囷紝浠庤屾壘鍑烘潅涔辨棤绔犵殑淇″彿涓殑涓昏鎸姩棰戠巼鐗圭偣銆
