EE  The Fourier Transform and its Applications This Being an Ancient Formula Sheet Handed Down To All EE  Students Integration by parts dt dt Even and odd parts of a function Any function can be writ
273K - views

EE The Fourier Transform and its Applications This Being an Ancient Formula Sheet Handed Down To All EE Students Integration by parts dt dt Even and odd parts of a function Any function can be writ

Similar presentations


Download Pdf

EE The Fourier Transform and its Applications This Being an Ancient Formula Sheet Handed Down To All EE Students Integration by parts dt dt Even and odd parts of a function Any function can be writ




Download Pdf - The PPT/PDF document "EE The Fourier Transform and its Applic..." is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.



Presentation on theme: "EE The Fourier Transform and its Applications This Being an Ancient Formula Sheet Handed Down To All EE Students Integration by parts dt dt Even and odd parts of a function Any function can be writ"— Presentation transcript:


Page 1
EE 261 The Fourier Transform and its Applications This Being an Ancient Formula Sheet Handed Down To All EE 261 Students Integration by parts: dt dt Even and odd parts of a function: Any function can be written as )= )+ (even part) (odd part) Geometric series: =0 +1 (1 +1 (1 Complex numbers: iy , iy =Re + ,y =Im Complex exponentials: πit = cos 2 πt sin 2 πt cos 2 πt πit πit sin 2 πt πit πit Polar form: iy z re i ,r , = tan y/x Symmetric sum of complex exponentials (special case of geometric series): πint sin(2 +1) πt sin πt Fourier series If ) is periodic with period its Fourier series is )= πint/T πint/T dt T/ T/ πint/T dt Orthogonality of the complex exponentials: πint/T πimt/T dt ,n T, n The normalized exponentials (1 πint/T ,... form an orthonormal basis for ([0 ,T ]) Rayleigh (Parseval): If ) is periodic of period T then dt The Fourier Transform: )= πisx dx The Inverse Fourier Transform: )= πisx ds Symmetry & Duality Properties Let )= ). FF =( If is even (odd) then is even (odd) If is real valued, then =( Convolution )( )= dy ( =( )= Smoothing: If (or )is -times continuously dif- ferentiable, 0, then so is and dx )=( dx Convolution Theorem: )=( )( fg )= ∗F
Page 2
Autocorrelation : Let ) be a function satisfying dx < (finite energy) then g?g )( )= dy Cross correlation : Let ) and ) be functions with finite energy. Then g?h )( )= dy dy h?g )( Rectangle and triangle functions Π( )= | Λ( )= −| | | Π( ) = sinc sin πs πs Λ( ) = sinc Scaled rectangle function )=Π( x/p )= | )= sinc ps Scaled triangle function )=Λ( x/p )= −| x/p | | )= sinc ps Gaussian πt )= πs Gaussian with variance )= )= One-sided exponential decay )= ,t< at ,t )= +2 πis Two-sided exponential decay )= +4 Fourier Transform Theorems Linearity: F{ αf )+ βg αF )+ βG Stretch: F{ ax Shift: F{ πas Shift & stretch: F{ ax πsb/a Rayleigh (Parseval): dx ds dx ds Modulation: F{ ) cos(2 πs )+ )] Autocorrelation: F{ g?g Cross Correlation: F{ g?f Derivative: F{ =2 πisG F{ =(2 πis F{ =( Moments: dx (0) xf dx (0) dx =( (0) Miscellaneous: d (0) )+ πs The Delta Function: Scaling: ax )= Sifting: dx dx Convolution: )= )= )= )) Product: )= (0)
Page 3
Fourier Transform: =1 )) = πisa Derivatives: dx =( 1) (0) )= x )=0 x )= Fourier transform of cosine and sine cos 2 πat )+ )) sin 2 πat )) Unit step and sgn )= ,t ,t> )= )+ πis sgn ,t< ,t> sgn ( )= πis The Shah Function: III III )= ,III )= np Sampling: III )= Periodization: III )= Scaling: III ax )= III /a ), a> Fourier Transform: III III III III /p Sampling Theory For a bandlimited function ) with ) = 0 for | p/ = III )= ) sinc( )) k/p Fourier Transforms for Periodic Functions For a function ) with period , let )= )Π( ). Then )= nL )= The complex Fourier series representation: )= πi where L/ L/ πi dx Linear Systems Let be a linear system, )= Lv ), with impulse response t, )= L ). Superposition integral: )= t, d A system is time-invariant if: )= )] In this case )= ) and acts by convolution: )= Lv )= d =( )( The transfer function is the Fourier transform of the impulse response, The eigenfunctions of any linear time-invariant system are πiνt , with eigen- value ): Le πiνt πiνt The Discrete Fourier Transform th root of unity: Let πi/N . Then = 1 and the powers ,... are distinct and evenly spaced along the unit circle. Vector complex exponentials: =(1 ,..., 1) =(1 ,ω, ,..., =(1 , , ,..., 1) Cyclic property =1 and 1 , , ,... are distinct
Page 4
The vector complex exponentials are orthogonal: ,k 6 mod N, k mod The DFT of order accepts an -tuple as input and returns an -tuple as output. Write an -tuple as f =(f [0] [1] ,..., 1]). =0 Inverse DFT: =0 Periodicity of inputs and outputs: If F then both f and F are periodic of period Convolution (f )[ ]= =0 ]g Discrete ]= ,m mod ,m 6 mod DFT of the discrete DFT of vector complex exponential N Reversed signal: f ]=f =( DFT Theorems Linearity: )= Parseval: F (f Shift: Let ]=f ]. Then )= Modulation: )= Convolution: (f )=( )( (f )= ∗F The Hilbert Transform The Hilbert Transform of ): )= πx )= d (Cauchy principal value) Inverse Hilbert Transform −H Impulse response: πx Transfer function: sgn ( Causal functions: ) is causal if ) = 0 for x< 0. A casual signal Fourier Transform )= )+ iI ), where )= H{ Analytic signals: The analytic signal representation of a real-valued function ) is given by: )= Narrow Band Signals: )= ) cos[2 πf )] Analytic approx: [2 πf )] Envelope: Phase: arg[ )] = 2 πf Instantaneous freq: dt Higher Dimensional Fourier Transform In dimensions: )= πi (x Inverse Fourier Transform: (x )= πi d In 2-dimensions (in coordinates): , )= πi ,x dx dx The Hankel Transform (zero order): )=2 (2 πr rdr The Inverse Hankel Transform (zero order): )=2 (2 πr ρd Separable functions: If ,x )= ) then , )= Two-dimensional rect: Π( ,x )=Π( )Π( Π( , ) = sinc sinc Two dimensional Gaussian: ,x )= Fourier transform theorems
Page 5
Shift: Let ( )(x )= (x ). Then )( )= πi Stretch theorem (special): ,a ,x )) = || Stretch theorem (general): If is an invertible matrix then )) = det Stretch and shift: +b )) = exp(2 πi det III s and lattices III for integer lattice III (x )= (x ,n ,x III III A general lattice can be obtained from the integer lat- tice by ) where is an invertible matrix. III (x )= ∈L (x )= det III If ) then the reciprocal lattice is Fourier transform of III III det III Radon transform and Projection-Slice Theorem Let ,x ) be the density of a two-dimensional region. A line through the region is specified by the angle of its normal vector to the -axis, and its directed distance from the origin. The integral along a line through the region is given by the Radon transform of ρ, )= ,x cos sin dx dx The one-dimensional Fourier transform of with re- spect to is the two-dimensional Fourier transform of )( r, )= , , cos φ, sin The list being compiled originally by John Jackson a person not known to me, and then revised here by your humble instructor