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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: 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

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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)

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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

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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

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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

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