fourier transform formula list - EE 261 The Fourier...

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EE 261 The Fourier Transform and its Applications This Being an Ancient Formula Sheet Handed Down To All EE 261 Students Integration by parts: b a u ( t ) v ( t ) dt = u ( t ) v ( t ) t = b t = a - b a u ( t ) v ( t ) dt Even and odd parts of a function: Any function f ( x ) can be written as f ( x ) = f ( x ) + f ( - x ) 2 + f ( x ) - f ( - x ) 2 (even part) (odd part) Geometric series: N n =0 r n = 1 - r N +1 1 - r N n = M r n = r M (1 - r N - M +1 ) (1 - r ) Complex numbers: z = x + iy , ¯ z = x - iy , | z | 2 = z ¯ z = x 2 + y 2 1 i = - i x = Re z = z + ¯ z 2 , y = Im z = z - ¯ z 2 i Complex exponentials: e 2 πit = cos 2 πt + i sin 2 πt cos 2 πt = e 2 πit + e - 2 πit 2 , sin 2 πt = e 2 πit - e - 2 πit 2 i Polar form: z = x + iy z = re , r = x 2 + y 2 , θ = tan - 1 ( y/x ) Symmetric sum of complex exponentials (special case of geometric series): N n = - N e 2 πint = sin(2 N + 1) πt sin πt Fourier series If f ( t ) is periodic with period T its Fourier series is f ( t ) = n = -∞ c n e 2 πint/T c n = 1 T T 0 e - 2 πint/T f ( t ) dt = 1 T T/ 2 - T/ 2 e - 2 πint/T f ( t ) dt Orthogonality of the complex exponentials: T 0 e 2 πint/T e - 2 πimt/T dt = 0 , n = m T , n = m The normalized exponentials (1 / T ) e 2 πint/T , n = 0 , ± 1 , ± 2 , . . . form an orthonormal basis for L 2 ([0 , T ]) Rayleigh (Parseval): If f ( t ) is periodic of period T then 1 T T 0 | f ( t ) | 2 dt = k = -∞ | c k | 2 The Fourier Transform: F f ( s ) = -∞ f ( x ) e - 2 πisx dx The Inverse Fourier Transform: F - 1 f ( x ) = -∞ f ( s ) e 2 πisx ds Symmetry & Duality Properties : Let f - ( x ) = f ( - x ). FF f = f - F - 1 f = F f - F f - = ( F f ) - If f is even (odd) then F f is even (odd) If f is real valued, then F f = ( F f ) - Convolution : ( f * g )( x ) = -∞ f ( x - y ) g ( y ) dy f * g = g * f ( f * g ) * h = ( f * g ) * h f * ( g + h ) = f * g + f * h Smoothing: If f (or g ) is p -times continuously dif- ferentiable, p 0, then so is f * g and d k dx k ( f * g ) = ( d k dx k f ) * g Convolution Theorem: F ( f * g ) = ( F f )( F g ) F ( fg ) = F f * F g 1
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Autocorrelation : Let g ( x ) be a function satisfying -∞ | g ( x ) | 2 dx < (finite energy) then ( g g )( x ) = -∞ g ( y ) g ( y - x ) dy = g ( x ) * g ( - x ) Cross correlation : Let g ( x ) and h ( x ) be functions with finite energy. Then ( g h )( x ) = -∞ g ( y ) h ( y + x ) dy = -∞ g ( y - x ) h ( y ) dy = ( h g )( - x ) Rectangle and triangle functions Π( x ) = 1 , | x | < 1 2 0 , | x | ≥ 1 2 Λ( x ) = 1 - | x | , | x | ≤ 1 0 , | x | ≥ 1 F Π( s ) = sinc s = sin πs πs , F Λ( s ) = sinc 2 s Scaled rectangle function Π p ( x ) = Π( x/p ) = 1 , | x | < p 2 0 , | x | ≥ p 2 , F Π
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