Signals & Systems - Reference Tables 1 Table of Fourier Transform Pairs Function, f(t) Fourier Transform, F(w) Definition of Inverse Fourier Transform ò ¥ ¥- = ww p w deFtf tj )( 2 1 )( Definition of Fourier Transform ò ¥ ¥- - = dtetfF tjw w )()( )( 0 ttf - 0 )( tj eF w w - tj etf 0 )( w )( 0 ww - F )( tf a )( 1 a w a F )(tF )(2 wp - f n n dt tfd )( )()( ww Fj n )()( tfjt n - n n d Fd w w)( ò ¥- t df tt )( )()0( )( wdp w w F j F + )(t d 1 tj e 0 w )(2 0 wwpd - (t) sgn w j 2 Signals & Systems - Reference Tables 2 t j p 1 )sgn( w )( tu w wpd j 1 )( + å ¥ -¥=n tjn n eF 0 w å ¥ -¥= - n n nF )(2 0 wwdp )( t t rect ) 2 ( w t tSa ) 2 ( 2 Bt Sa B p )( B rect w )(ttri ) 2 ( 2 w Sa ) 2 () 2 cos( tt p t rect t A 22 ) 2 ( )cos( w t p w t t p - A )cos( 0 t w [] )()( 00 wwdwwdp ++- )sin( 0 t w [] )()( 00 wwdwwd p +-- j )cos()( 0 ttu w [] 22 0 00 )()( 2 ww w wwdwwd p - +++- j )sin()( 0 ttu w [] 22 0 2 00 )()( 2 ww w wwdwwd p - ++-- j )cos()( 0 tetu t w a- 22 0 )( )( waw wa j j ++ + Signals & Systems - Reference Tables 3 )sin()( 0 tetu t w a- 22 0 0 )( waw w j++ t e a- 22 2 wa a + )2/( 22 s t e - 2/ 22 2 ws ps - e t etu a- )( wa j + 1 t tetu a- )( 2 )( 1 wa j + Ø Trigonometric Fourier Series () å ¥ = ++= 1 000 )sin()cos()( n nn ntbntaatf ww where ò òò = == T n T T n dtnttf T b dtnttf T adttf T a 0 0 0 0 0 0 )sin()( 2 and, )cos()( 2 , )( 1 w w Ø Complex Exponential Fourier Series ò å - ¥ -¥= == T ntj n n ntj n dtetf T FeFtf 0 0 )( 1 where, )( w w Signals & Systems - Reference Tables 4 Some Useful Mathematical Relationships 2 )cos( jxjx ee x - + = j ee x jxjx 2 )sin( - - = )sin()sin()cos()cos()cos( yxyxyx m=± )sin()cos()cos()sin()sin( yxyxyx ±=± )(sin)(cos)2cos( 22 xxx -= )cos()sin(2)2sin( xxx = )2cos(1)(cos2 2 xx += )2cos(1)(sin2 2 xx -= 1)(sin)(cos 22 =+ xx )cos()cos()cos()cos(2 yxyxyx ++-= )cos()cos()sin()sin(2 yxyxyx +--= )sin()sin()cos()sin(2 yxyxyx ++-= Signals & Systems - Reference Tables 5 Useful Integrals ò dxx)cos( )sin(x ò dxx)sin( )cos(x - ò dxxx )cos( )sin()cos( xxx + ò dxxx )sin( )cos()sin( xxx - ò dxxx )cos( 2 )sin()2()cos(2 2 xxxx -+ ò dxxx )sin( 2 )cos()2()sin(2 2 xxxx -- ò dxe xa a e x a ò dxxe xa ú û ù ê ë é - 2 1 a a x e x a ò dxex xa2 ú û ù ê ë é -- 32 2 22 aa x a x e x a ò + x dx ba x ba b + ln 1 ò + 222 x dx ba )(tan 1 1 a b ab x - . Systems - Reference Tables 1 Table of Fourier Transform Pairs Function, f(t) Fourier Transform, F(w) Definition of Inverse Fourier Transform ò ¥ ¥- = ww. Inverse Fourier Transform ò ¥ ¥- = ww p w deFtf tj )( 2 1 )( Definition of Fourier Transform ò ¥ ¥- - = dtetfF tjw w )()( )( 0 ttf - 0 )( tj eF w w - tj