3.7 傅里葉變換的基本性質(zhì)（經(jīng)典實(shí)用）

1、3.7 傅里葉變換的基本性質(zhì)1 為常數(shù) i n i ii n i ii ii a FatfaFT ni FtfFT 11 )( )( ), 2 , 1( )()( )(tf 2 2 1 2 t 求：求：)(tf的傅里葉變換的傅里葉變換 3.7 傅里葉變換的基本性質(zhì)2 )()() 2 () 2 ()( tututututf ) 2 ()( SaEtEGFT )(tf 2 2 1 2 t )(2) 2 ()( SaSaF 2 )(F 2 3 3.7 傅里葉變換的基本性質(zhì)3 )( )( arctan)()()()( 22 R X XRF )()()( )()()( )( jXReF dtetftfF

2、TF j tj 3.7 傅里葉變換的基本性質(zhì)4 tdttfXtdttfRsin)()(cos)()( )()( sin)(cos)( )()()( jXR tdttfjtdttf dtetftfFTF tj )()()()( )()()()( FF XXRR 3.7 傅里葉變換的基本性質(zhì)5 arctan)( 1 )( 22 F 0)()( tuetf t j F 1 )( 22 )( j F 2222 )()( XR 3.7 傅里葉變換的基本性質(zhì)6 也是實(shí)偶函數(shù)是實(shí)偶函數(shù)若)(,)(Ftf 0 cos)(2)( sin)(cos)()( tdttfR tdttfjtdttfF ) 2 ()(

3、SaEtGFT 則為虛奇函數(shù)是實(shí)奇函數(shù)若)(,)(Ftf 0 sin)(2)( sin)(cos)()( tdttfjjX tdttfjtdttfF j tFT 2 )sgn( 3.7 傅里葉變換的基本性質(zhì)7 為實(shí)函數(shù)令)()()(:tgtjgtf dtetftfFTF tj )()()( )()()cos()()sin()( )sin()cos()()( jXRdtttgjdtttg dttjttjgdtetjgF tj )()()()( )()()()( FF XXRR 3.7 傅里葉變換的基本性質(zhì)8 為非零實(shí)常數(shù)則設(shè)a a F a atfFTFtfFT)( 1 )(),()( )( 1

4、)( 1 1 )( )()( a F a def a d a ef dteatfatfFT a j a j tj a tata , 0設(shè) )( 1 )(,0 a F a atfFTa 時(shí) )()(FtfFT 3.7 傅里葉變換的基本性質(zhì)9 )( 1 )( a F a atfFT ) 2 1 ( tf )(tf )2( tf )2(2F )(F ) 2 ( 2 1 F 1a 2a 2 1 a 3.7 傅里葉變換的基本性質(zhì)10 3.7 傅里葉變換的基本性質(zhì)11 2 )0(2)0()0()0(BfBFFf dttfF dtetfF tj )()0( )()( dFf deFtf tj )( 2 1

5、)0( )( 2 1 )( 3.7 傅里葉變換的基本性質(zhì)12 dFF F Ftf )()2(;)0() 1 ( :,)( ),(,)( 試計(jì)算下列值也不必做積分運(yùn)算的表達(dá)式 不必求出其頻譜密度為如圖所示信號(hào) 0 0 11 1 t )(tf deFtf dtetfF tj tj )( 2 1 )( )()( dFf dttfF )( 2 1 )0( )()0( 2)0(2)( 5 . 1)0( fdF F 3.7 傅里葉變換的基本性質(zhì)13 )(2)(),()(ftFFTFtfFT則設(shè) deFtf tj )( 2 1 )( deFtf tj )( 2 1 )( )( 2 1 )( 2 1 )(tF

6、FTdtetFf tj )(2)(ftFFT )(2)( )( ftFFT tf 若為偶函數(shù) 3.7 傅里葉變換的基本性質(zhì)14 )(2 1 FT 1)(tFT )(2)(),()(ftFFTFtfFT則設(shè) 3.7 傅里葉變換的基本性質(zhì)15 )1() 1()(uutSaFT 的頻譜函數(shù)試求抽樣函數(shù)例 t t tSa sin )( EE ) 2 ()( SaEtfFT )()( 2 1 , 2SatfFTE )(2)(2)(fftSaFT 10 1 )( tSaFT )(2)(ftFFT 1 11 )()(tSatf 3.7 傅里葉變換的基本性質(zhì)16 0)()( 1 atuetf at ja F

7、1 )( 1 )( 1 )(:F jta tf的傅里葉變換求例 )(2)(2)()( 11 ueftFFTF a )(2)(ftFFT jta tFtf 1 )()( 1 )(2)( ueF a 3.7 傅里葉變換的基本性質(zhì)17 ).( )(0 )( )( ,)3(223 0 0 177 tfF P 所對(duì)應(yīng)的時(shí)間函數(shù) 其他 求性利用時(shí)域與頻域的對(duì)稱(chēng)習(xí)題 )()()( 00 0 uuF)(2)(ftFFT )()()( 00 0 tututF )(2)( 2 ) 2 2 (2)( 0 2 00 0 0 fSaSatFFT ) 2 ()( SaEtEGFT )()()( 0 2 2 0 fSaf

8、)()( 0 2 2 0 tSatf 3.7 傅里葉變換的基本性質(zhì)18 )()(: )()(: 0 0 FettfFT FtfFT tj 則 設(shè) 0, )( 1 )( aba a Fe a batfFT a b j 為實(shí)常數(shù) 3.7 傅里葉變換的基本性質(zhì)19 ).52(),()()7(293 179 tfFTFtfFTP求設(shè) 5 )()5( j eFtfFT 2 5 ) 2 ( 2 1 )52( j eFtfFT ) 2 ( 2 1 )2( FtfFT 2 5 ) 2 ( 2 1 ) 2 5 (2)52( j eFtfFTtfFT )( 1 )( a Fe a batfFT a b j 2 5

9、 ) 2 ( 2 1 )52( j eFtfFT 3.7 傅里葉變換的基本性質(zhì)20 . 23 號(hào)的頻譜 求三脈沖信例 )()()()( 000 TtfTtftftf )( 0 tf ) 2 ()()( 00 SaEtfFTF )cos(21) 2 ()1)( )()()()( 0 000 TSaEeeF eFeFFF TjTj TjTj 3.7 傅里葉變換的基本性質(zhì)21 )cos(21) 2 ()(TSaEF 3.7 傅里葉變換的基本性質(zhì)22 . )2()()(33 試求其頻譜 信號(hào)已知雙例 tSatSatfSa cc c )1() 1()(uutSaFT )()()1() 1()( 0cc

10、ccc c uuuuF )2()()2()()( 00 tftftSatSatf c c c c )1)()( )()()( 2 2 00 j cc j euu eFFF 3.7 傅里葉變換的基本性質(zhì)23 )1)()()( 2 j cc euuF )sin(2)2cos(22 )2sin()2cos(1 )2sin()2cos(11 22 2 je j )()( )sin(2)( cc uuF )2()()(: tSatSatfSa c c c c 信號(hào)雙 3.7 傅里葉變換的基本性質(zhì)24 )2()()(: tSatSatfSa c c c c 信號(hào)雙 c 3.7 傅里葉變換的基本性質(zhì)25 )

11、()( )sin(2)( cc uuF )2()()(: tSatSatfSa c c c c 信號(hào)雙 c 3.7 傅里葉變換的基本性質(zhì)26 為常數(shù)則設(shè) 00), ()(),()( 0 FetfFTFtfFT tj )()()()( 0 )( 000 FdtetfdteetfetfFT tjtjtjtj 2 cos 2 sin 0000 00 tjtjtjtj ee t j ee t )()( 2 1 cos)( 000 FFttfFT )()( 2 sin)( 000 FF j ttfFT 3.7 傅里葉變換的基本性質(zhì)27 ., , )(),cos()()( 43 0 試求其頻譜函數(shù)如圖所示

12、 脈寬為脈幅為為矩形脈沖 其中 已知矩形調(diào)幅信號(hào)例 E tGttGtf ) 2 ()()( 1 SaEGtGFT )()( 2 1 cos)( 000 FFttfFT 2 )( 2 2 )( 2 )()( 2 1 )( 00 0101 Sa E Sa E GGF 3.7 傅里葉變換的基本性質(zhì)28 )cos()()( 0t tGtf 2 )( 2 2 )( 2 )( 00 Sa E Sa E F 3.7 傅里葉變換的基本性質(zhì)29 .),cos()(53 0 號(hào)的頻譜利用頻移定理求余弦信已知例ttf )cos(1)( 0t tf )()( 2 1 cos)( 000 FFttfFT )(2 1 F

13、T )()( )(2)(2 2 1 )()( 00 00 tfFTF )()()cos( 000 tFT )()()sin( 000 jjtFT 3.7 傅里葉變換的基本性質(zhì)30 0 0 0 )( )( )cos( 0t FT 0 0 0 )( )( )sin( 0t jFT )()()cos( 000 tFT )()()sin( 000 jjtFT 3.7 傅里葉變換的基本性質(zhì)31 )()( )( ),()(Fj dt tfd FTFtfFT n n n 則設(shè) )( 1 )( j tuFT 1)(1)( 1 )( j j jtFT jjtFT1)( jjj j jtFT)()()( 1 )(

14、)( 22 3.7 傅里葉變換的基本性質(zhì)32 )()( )( ),()( 1 tfjt d Fd FTFtfFT n n n 則設(shè) dtetfF tj )()( dtejttfdt d de tf d dF tj tj )()( )( d dF jttfFT )( )( dtejttfdt d ed tf d Fd tjn n tjn n n )()( )( )()( F d d jtftFT n n nn 3.7 傅里葉變換的基本性質(zhì)33 ).(, 2 0 2 ) 2 1 ( )( 63 F t ttE tf 求其頻譜如圖所示 已知三角脈沖信號(hào)例 ) 4 ( 2 )( 2 Sa E F 3.

15、7 傅里葉變換的基本性質(zhì)34 )()( F d d jtftFT n n nn d dF jttfFT )( )() 2 ( 2 1 )2( FtfFT ) 2 ( 2 ) 2 ( 2 1 )2( F d dj F d d jttfFT ) 2 ( 4 ) 2 ( 2 )2( F j F d dj ttfFT )2(),()() 1 (293 179 ttfFTFtfFTP求設(shè) 3.7 傅里葉變換的基本性質(zhì)35 d dF jttfFT )( )( ) 2 ( 4 ) 2 ( 2 2 1 2 1 )2(2 2 1 )2( F j F d d jttfFTttfFT ) 2 ( 2 ) 2 ( 4

16、 )2( F d dj F j ttfFT ).2(),()() 1 (293 179 ttfFTFtfFTP求設(shè) )( 1 )( a F a atfFT 3.7 傅里葉變換的基本性質(zhì)36 )( 1 )()0()(),()( F j FdfFTFtfFT t 則設(shè) )( 1 )()( 1 )()( )()( )()( )()( F j Fde j f ddtetuf dtedtuf dtedfdfFT j tj tj tj tt )( 1 )0()()( F j FdfFT t 3.7 傅里葉變換的基本性質(zhì)37 )( 1 )0()()( F j FdfFT t j tuFT 1 )()( .:

17、號(hào)的傅里葉變換利用積分特性求階躍信例 t dtu)()()()( t dFTtuFT )( 1 )0()()( F j FtuFT 1)(F 3.7 傅里葉變換的基本性質(zhì)38 )(1 )0( )0(0 )( 73 0 0 0 tt tt t t t ty 的頻譜求圖示的截平斜變信號(hào)例 0 t 0 1t )(tf t t dfty)()( )( 1 )0()()( F j FdfFT t )() 2 ()( 2 0 0 Fe t SatfFT t j 3.7 傅里葉變換的基本性質(zhì)39 t dfty)()( )( 1 )0()()( F j FdfFT t 2 0 0 ) 2 ()( t j e

18、t SaF 2 0 0 ) 2 ( 1 )()()( t jt e t Sa j dfFTtyFT )(1 )0( )0(0 )( 73 0 0 0 tt tt t t t ty 的頻譜求圖示的截平斜變信號(hào)例 3.7 傅里葉變換的基本性質(zhì)40 0 )(F )( 2 1 0 F )cos()( 0t tfFT 0 0 )( 2 1 0 F )()( 2 1 cos)( 000 FFttfFT 3.7 傅里葉變換的基本性質(zhì)41 )(tg )cos( 0t )cos()( 0t tg m 0m 0 0 0 )( 0m )( 0m )cos()( 0t tgFT 3.7 傅里葉變換的基本性質(zhì)42 )cos()( 0t tg )cos( 0t )( 0 tg )(tg m 0m 0 0 0 )( 0m )( 0m )cos()( 0t tgFT 3.7