Table of Fourier Transform Pairs

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Table of Fourier Transform Pairs

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  1. Table of Fourier Transform Pairs Function, f(t) Fourier Transform, F(w) Definition of Inverse Fourier Transform Definition of Fourier Transform ¥ ¥ 1 jwt - jwt f (t ) = 2p ò F (w )e dw F (w ) = ò f (t )e dt -¥ -¥ f (t - t 0 ) F (w )e - jwt0 f (t )e jw 0t F (w - w 0 ) f (at ) 1 w F( ) a a F (t ) 2pf (-w ) d n f (t ) ( jw ) n F (w ) dt n (- jt ) n f (t ) d n F (w) dw n t F (w ) + pF (0)d (w ) ò f (t )dt jw -¥ d (t ) 1 e jw 0 t 2pd (w - w 0 ) sgn (t) 2 jw Signals & Systems - Reference Tables 1
  2. 1 sgn(w ) j pt u (t ) 1 pd (w ) + jw ¥ ¥ å Fn e jnw 0t 2p å Fnd (w - nw 0 ) n = -¥ n = -¥ t wt rect ( ) tSa( ) t 2 B Bt w Sa( ) rect ( ) 2p 2 B tri (t ) w Sa 2 ( ) 2 pt t Ap cos(wt ) A cos( )rect ( ) 2t 2t t (p ) 2 - w 2 2t cos(w 0 t ) p [d (w - w 0 ) + d (w + w 0 )] sin(w 0 t ) p [d (w - w 0 ) - d (w + w 0 )] j u (t ) cos(w 0 t ) p [d (w - w 0 ) + d (w + w 0 )] + 2 jw 2 2 w0 - w u (t ) sin(w 0 t ) p 2 [d (w - w 0 ) - d (w + w 0 )] + 2w 2 2j w0 - w u (t )e -at cos(w 0 t ) (a + jw ) w 0 + (a + jw ) 2 2 Signals & Systems - Reference Tables 2
  3. u (t )e -at sin(w 0 t ) w0 w 0 + (a + jw ) 2 2 e -a t 2a a2 +w2 2 /( 2s 2 ) 2 w2 / 2 e -t s 2p e -s u (t )e -at 1 a + jw u (t )te -at 1 (a + jw ) 2 Ø Trigonometric Fourier Series ¥ f (t ) = a 0 + å (a n cos(w 0 nt ) + bn sin(w 0 nt ) ) n =1 where 1 T 2T a0 = T ò0 f (t )dt , a n = ò f (t ) cos(w 0 nt )dt , and T0 2T bn = ò f (t ) sin(w 0 nt )dt T 0 Ø Complex Exponential Fourier Series ¥ 1T f (t ) = å Fn e jwnt , where Fn = ò f (t )e - jw 0 nt dt T 0 n = -¥ Signals & Systems - Reference Tables 3
  4. Some Useful Mathematical Relationships e jx + e - jx cos( x) = 2 e jx - e - jx sin( x) = 2j cos( x ± y ) = cos( x) cos( y ) m sin( x) sin( y ) sin( x ± y ) = sin( x) cos( y ) ± cos( x) sin( y ) cos(2 x) = cos 2 ( x) - sin 2 ( x) sin( 2 x) = 2 sin( x) cos( x) 2 cos2 ( x) = 1 + cos(2 x) 2 sin 2 ( x) = 1 - cos(2 x) cos 2 ( x) + sin 2 ( x) = 1 2 cos( x) cos( y ) = cos( x - y ) + cos( x + y ) 2 sin( x) sin( y ) = cos( x - y ) - cos( x + y ) 2 sin( x) cos( y ) = sin( x - y ) + sin( x + y ) Signals & Systems - Reference Tables 4
  5. Useful Integrals sin(x) ò cos( x)dx - cos(x) ò sin( x)dx cos( x) + x sin( x) ò x cos( x)dx sin( x) - x cos( x) ò x sin( x)dx òx 2 cos( x)dx 2 x cos( x) + ( x 2 - 2) sin( x) òx 2 sin( x)dx 2 x sin( x) - ( x 2 - 2) cos( x) ax òe dx e ax a ax ò xe dx éx 1 ù e ax ê - 2 ú ëa a û 2 ax òx e dx é x 2 2x 2 ù e ax ê - 2 - 3 ú ëa a a û dx 1 ò a + bx b ln a + bx dx 1 bx ò a 2 + b 2x2 tan -1 ( ) ab a Signals & Systems - Reference Tables 5
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