Formulas

# Formulas - EE 261 The Fourier Transform and its...

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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 = ² u ( t ) v ( t ) ³ 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 cos 2 = e 2 πit + e - 2 πit 2 , sin 2 = e 2 πit - e - 2 πit 2 i Polar form: z = x + iy z = re ,r = µ x 2 + y 2 = tan - 1 ( y/x ) Symmetric sum of complex exponentials (special case of geometric series): N ´ n = - N e 2 πint = sin(2 N +1) sin 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 - 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 , ² 1 , ² 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 < (Fnite energy) then ( g±g )( x )= ² -∞ g ( y ) g ( y - x ) dy = g ( x ) * g ( - x ) Cross correlation : Let g ( x ) and h ( x ) be functions with Fnite energy. Then ( g±h )( x ² -∞ g ( y ) h ( y + x ) dy = ² -∞ g ( y - x ) h ( y ) dy = ( h±g )( - x ) Rectangle and triangle functions Π( x ³ 1 , | x | < 1 2 0 , | x |≥ 1 2 Λ( x ³ 1 -| x | , | x |≤ 1 0 , | x 1 F Π( s ) = sinc s = sin πs , F Λ( s ) = sinc 2 s Scaled rectangle function Π p ( x )=Π( x/p ³ 1 , | x | < p 2 0 , | x p 2 , F Π p ( s p sinc
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Formulas - EE 261 The Fourier Transform and its...

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