formula sheet

# formula sheet - EE 261 The Fourier Transform and its...

This preview shows pages 1–3. Sign up to view the full content.

EE 261 The Fourier Transform and its Applications This Being an Ancient Formula Sheet Handed Down To All EE 261 Students Integration by parts: Z b a u ( t ) v 0 ( t ) dt = ± u ( t ) v ( t ) ² t = b t = a - Z b a u 0 ( 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 X n =0 r n = 1 - r N +1 1 - r N X 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 = cos2 πt + i sin2 πt cos2 πt = e 2 πit + e - 2 πit 2 , sin2 πt = e 2 πit - e - 2 πit 2 i Polar form: z = x + iy z = re , r = p x 2 + y 2 = tan - 1 ( y/x ) Symmetric sum of complex exponentials (special case of geometric series): N X 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 ) = X n = -∞ c n e 2 πint/T c n = 1 T Z T 0 e - 2 πint/T f ( t ) dt = 1 T Z T/ 2 - T/ 2 e - 2 πint/T f ( t ) dt Orthogonality of the complex exponentials: Z T 0 e 2 πint/T e - 2 πimt/T dt = ( 0 , n 6 = 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 Z T 0 | f ( t ) | 2 dt = X k = -∞ | c k | 2 The Fourier Transform: F f ( s ) = Z -∞ f ( x ) e - 2 πisx dx The Inverse Fourier Transform: F - 1 f ( x ) = Z -∞ f ( s ) e 2 πisx ds : 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 ) = Z -∞ 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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Autocorrelation : Let g ( x ) be a function satisfying R -∞ | g ( x ) | 2 dx < (ﬁnite energy) then ( g ? g )( x ) = Z -∞ 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 ) = Z
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 12/04/2011 for the course EE 263 at Stanford.

### Page1 / 5

formula sheet - EE 261 The Fourier Transform and its...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online