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Table of Fourier Transform Pairs Table of Fourier Transform Pairs Function, f(t) Fourier Transform, F() Definition of Inverse Fourier Transform Definition of Fourier Transform A A 1 f (t) * F()e j t d F() * f (t)e j t dt 2 A A jt f (t t0 ) F()e 0 j t f (t)e 0 F( 0 ) f (t) 1 F( ) F(t) 2 f () d n f (t) ( j) n F() dt n ( jt) n f (t) d n F() d n t F() F(0) () * f ( )d A j (t) 1 j t e 0 2 ( 0 ) sgn (t) 2 j Signals & Systems - Reference Tables 1 Fourier Transform Table UBC M267 Resources for 2005 F (t) F (ω) Notes (0) ∞ b iωt f(t) f(t)e− dt Definition. (1) Z−∞ 1 ∞ f(ω)eiωt dω f(ω) Inversion formula. (2) 2π Z−∞ b b f( t) 2πf(ω) Duality property. (3) − at 1 e−b u(t) a constant, e(a) > 0 (4) a + iω < a t 2a e− | | a constant, e(a) > 0 (5) a2 + ω2 < 1, if t < 1, sin(ω) β(t)= | | 2sinc(ω)=2 Boxcar in time. (6) 0, if t > 1 ω | | 1 sinc(t) β(ω) Boxcar in frequency. (7) π f 0(t) iωf(ω) Derivative in time. (8) 2 f 00(t) (iω)bf(ω) Higher derivatives similar. (9) d tf(t) i f(ω) Derivative in frequency. (10) dω b d2 t2f(t) i2 bf(ω) Higher derivatives similar. (11) dω2 iω0t e f(t) f(ω ω0) Modulation property. (12) −b t t 0 b iωt0 f − ke− f(kω) Time shift and squeeze. (13) k b (f g)(t) f(ω)g(ω) Convolution in time. (14) ∗ 0, if t<0 1 u(t)= b + πδb (ω) Heaviside step function. (15) 1, if t>0 iω iωt0 δ(t t0)f(t) e− f(t0) Assumes f continuous at t0. (16) − iω0t e 2πδ(ω ω0) Useful for sin(ω0t), cos(ω0t). (17) − ∞ ∞ Convolution: (f g)(t)= f(t u)g(u) du = f(u)g(t u) du. ∗ − − Z−∞ Z−∞ ∞ 1 ∞ 2 Parseval: f(t) 2 dt = f(ω) dω. | | 2π Z−∞ Z−∞ b 1 sgn() j t u(t) 1 () j A A jn 0t Fn e 2 Fn ( n 0 ) nA nA t rect( ) Sa( ) 2 B Bt Sa( ) rect( ) 2 2 B tri(t) Sa 2 ( ) 2 t t A cos( ) Acos( )rect( ) 2 2 ( ) 2 2 2 cos( 0t) ( 0 ) ( 0 ) sin( t) 0 ( ) ( ) j 0 0 u(t) cos( t) j 0 ( ) ( ) 0 0 2 2 2 0 2 u(t)sin( 0t) ( ) ( ) 0 0 2 2 2 j 0 t ( j) u(t)e cos( 0t) 2 2 0 ( j ) Signals & Systems - Reference Tables 2 t u(t)e sin( 0t) 0 2 2 0 ( j ) e t 2 2 2 2 2 2 2 e t /(2 ) 2 e / 2 u(t)e t 1 j u(t)te t 1 ( j) 2 Trigonometric Fourier Series A f (t) a0 an cos( 0 nt) bn sin( 0 nt) n1 where T 1 T 2 a * f (t)dt , a * f (t) cos( nt)dt , and 0 0 n 0 T T 0 T 2 bn * f (t)sin( 0 nt)dt T 0 Complex Exponential Fourier Series A T 1 j nt j 0nt f (t) Fn e , where Fn * f (t)e dt nA T 0 Signals & Systems - Reference Tables 3 Some Useful Mathematical Relationships e jx e jx cos(x) 2 e jx e jx sin(x) 2 j cos(x M y) cos(x) cos(y) sin(x)sin(y) sin(x M y) sin(x) cos(y) M cos(x)sin(y) cos(2x) cos 2 (x) sin 2 (x) sin(2x) 2sin(x) cos(x) 2cos2 (x) 1 cos(2x) 2sin 2 (x) 1 cos(2x) cos 2 (x) sin 2 (x) 1 2 cos(x) cos(y) cos(x y) cos(x y) 2sin(x)sin(y) cos(x y) cos(x y) 2sin(x) cos(y) sin(x y) sin(x y) Signals & Systems - Reference Tables 4 Useful Integrals * cos(x)dx sin(x) * sin(x)dx cos(x) * x cos(x)dx cos(x) x sin(x) * x sin(x)dx sin(x) x cos(x) * x 2 cos(x)dx 2x cos(x) (x 2 2)sin(x) * x 2 sin(x)dx 2x sin(x) (x 2 2) cos(x) * e x dx ex a x * xe dx x ! x 1 1 e " 2 #a a 2 3 2 x 2 * x e dx ! x 2x 2 1 e x " 2 # a a 2 a 3 3 dx 1 * ln x x dx 1 x * tan 1 ( ) 2 2 x 2 Signals & Systems - Reference Tables 5 Your continued donations keep Wikibooks running! Engineering Tables/Fourier Transform Table 2 From Wikibooks, the open-content textbooks collection < Engineering Tables Jump to: navigation, search Fourier transform Fourier transform Signal Remarks unitary, angular frequency unitary, ordinary frequency 10 The rectangular pulse and the normalized sinc function Dual of rule 10. The rectangular function is an idealized 11 low-pass filter, and the sinc function is the non-causal impulse response of such a filter. 12 tri is the triangular function 13 Dual of rule 12. Shows that the Gaussian function exp( - at2) is its own 14 Fourier transform. For this to be integrable we must have Re(a) > 0. common in optics a>0 the transform is the function itself J (t) is the Bessel function of first kind of order 0, rect is 0 the rectangular function it's the generalization of the previous transform; T (t) is the n Chebyshev polynomial of the first kind. Un (t) is the Chebyshev polynomial of the second kind Retrieved from "http://en.wikibooks.org/wiki/Engineering_Tables/Fourier_Transform_Table_2" Category: Engineering Tables Views .
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