第五章 非周期信号的频域分析 - 杭州师范学院

1、 0TdtetfTCTTtjnTn220)(1 dtetfTCTTtjnTTnT220)(1limlim T0 dtetfdtetfTCjFtjtjnTTnTTT 220)(limlim dejFtftftjTT)(21lim dtetftfFjFtjT)( tfjF dejFjFFtftj)(211频谱演变的定性观察频谱演变的定性观察112T122 0lim nnnTTjFCTCjF dttf 202 ttAtf 22 dtAedtetfjFtjtj 2 SaAjF 20 nSaTACnnC jF 0 atuetfat jadteedtetfjFtjattj 10 221 ajF aarct

2、g 22002 aadteedteedteejFtjattjattjta 222 aajF 0 jF dettFtj 211 dtettj 21 21 dteFtj 010001sgntttt teFjFtasgn1 2200121 ajdteedteejFtjattjat0a jajtFa22limsgn220 jF 2 jF 0202 j tetj0 21 dtetj 0200 dteeFjFtjtj jF 020 tjeFjF 0000cos21cos00 tFeettjtj jF 0020TeCtftjnnn 0200 nCeFCeCFjFnntjnnntjnnn nTnTtt00 n

3、tjnnTeTnTtt00001 nnTnnTtF0000120 jFtfjFtf2211 jbFjaFtfbtaf2121 jFtf jFtfjFtf* 求：求：)(tf的傅立叶变换的傅立叶变换)(tf2212t)()()()()(22tututututf)(2)2/()(SaSajF2 00tjejFttf jFtf 2 SaAjF Ttftf 1 TjeSaAjF 21 jFtf ajFaatf 1)2(21F f(t/2)0t1)2( tf04/4/t压缩压缩1)2(2F20244扩展扩展0扩展扩展压缩压缩abjeajFabatfF)(1)()(1)(1/)()(10)()()(/)(

4、ajFeadefeaabtdefabatadtebatfbatfFabjjabjabjtja若若a 0,则有绝对值则有绝对值带有尺度变换的时移特性带有尺度变换的时移特性 jFtf ftF2 tfjF tfjF t 2 tjt 2j0tt 0tjetje0020, atueatja 10,1bjtb ueb2 tp 2 Sa2tSa p2 tsgn j2t 2 sgnj tSa 2 Satp 2 Satp22 Satp221 fjtF2 2212ptSa 2ptSa)()(jFtf)()(jFjtfdtd)()()(jFjtfdtdkkk)()0()(1)(FjFjdft若若6. 时域微分与积分

5、性质时域微分与积分性质这一性质可将时域微分转变为频域的代数运算。这一性质可将时域微分转变为频域的代数运算。积分性质积分性质)(0)()1 ()(222tttEtf22)(dttfd)(FE2440dttdf)(E2E222t0E2E2E422t0 三角脉冲三角脉冲)(tf220tE)(2)()(2)(2222tttEdttfd)4(24sin8)2(2)()(222222 SaEEeeEjFjjj)4(2)(2SaEjFtdtuty)()()()(1)()(jtuFjF1)( F)()(jFtf00)(jFetftjttf0cos)()()(jFtf)(21cos000tjtjeet)()(2

6、10000jFjFcos)(0ttfF)()(0jFtfF)(2100tjtjeetf0)(0F)(021F)(F00)(021F频移特性频移特性)()(11jFtfF)()(22jFtfF)()()(*)(2121jFjFtftfF)()(11jFtfF)()(22jFtfF)(*)(21)()(2121jFjFtftfF tttttf 2sin2sin2 tSatSatf 242 2ptSa 4212ptSa 42*4pptfF tfF ddtetfjFddtetfjFdtdejFtfdttftfdttfEtjtjtjf*2212121 221FGdjFEf221非周期信号的表示：非周期信

7、号的表示： 离散时间傅立叶变换离散时间傅立叶变换 一、从傅立叶级数到傅立叶变换一、从傅立叶级数到傅立叶变换1. 傅立叶变换傅立叶变换 NmmNjkNemFNkf 21)(周期信号周期信号 NkmNjkNekfmF 2 kfkfNNlim当N 时， kfkfN,20dN ,211dNNm2 00mkjmkekfmF信号的频谱信号的频谱 deeFdeeFkfkjjNmkjmjm 22121 kjkjekfeF)(二、常用信号的离散时间傅立叶变换二、常用信号的离散时间傅立叶变换 1| ,akuakfk1. 0)(kkjkkkjjeaekfeF0)(kkjae)(| )(|11jejjjeeFaeco

8、s211|(|2)aaeFjcos1sin)(1aatgej-2 - 0 2 | )(|jeF 1| ,|aakfk10|)(kkjkkkjkkkjkjeaeaeaeF10)()(kkjkkjaeae22cos211111aaaaeaeaejjj 2. 3 矩形脉冲序列矩形脉冲序列 01kf11|NkNk 11NNkkjkjkjeekfeF11112020NnnjNjNnNnjeee1NknjNjNjjNjNjjeeeeeeeF11111211112sin21sin1N21N kkf 1)(kjkjekeF1.1.线性特性线性特性若若则则a，b为任意常数为任意常数 jjeFkfeFkf1111 jjebFeaFkbfkaf1111 jeFkf jjeFk