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# bracewell - The Fourier Transform and its Applications The...

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Unformatted text preview: The Fourier Transform and its Applications The Fourier Transform: F ( s ) = Z ∞-∞ f ( x ) e- i 2 πsx dx The Inverse Fourier Transform: f ( x ) = Z ∞-∞ F ( s ) e i 2 πsx ds Symmetry Properties : If g ( x ) is real valued, then G ( s ) is Hermitian: G (- s ) = G * ( s ) If g ( x ) is imaginary valued, then G ( s ) is Anti-Hermitian: G (- s ) =- G * ( s ) In general: g ( x ) = e ( x ) + o ( x ) = e R ( x ) + ie I ( x ) + o R ( x ) + io I ( x ) G ( s ) = E ( s ) + O ( s ) = E R ( s ) + iE I ( s ) + iO I ( s ) + O R ( s ) Convolution : ( g * h )( x ) 4 = Z ∞-∞ g ( ξ ) h ( x- ξ ) dξ Autocorrelation : Let g ( x ) be a function satisfying R ∞-∞ | g ( x ) | 2 dx < ∞ (finite energy) then Γ g ( x ) 4 = ( g * ? g )( x ) 4 = Z ∞-∞ g ( ξ ) g * ( ξ- x ) dξ = g ( x ) * g * (- x ) Cross correlation : Let g ( x ) and h ( x ) be functions with finite energy. Then ( g * ? h )( x ) 4 = Z ∞-∞ g * ( ξ ) h ( ξ + x ) dξ = Z ∞-∞ g * ( ξ- x ) h ( ξ ) dξ = ( h * ? g ) * (- x ) The Delta Function: δ ( x ) • Scaling: δ ( ax ) = 1 | a | δ ( x ) • Sifting: R ∞-∞ δ ( x- a ) f ( x ) dx = f ( a ) R ∞-∞ δ ( x ) f ( x + a ) dx = f ( a ) • Convolution: δ ( x ) * f ( x ) = f ( x ) • Product: h ( x ) δ ( x ) = h (0) δ ( x ) • δ 2 ( x ) - no meaning • δ ( x ) * δ ( x ) = δ ( x ) • Fourier Transform of δ ( x ): F{ δ ( x ) } = 1 • Derivatives: – R ∞-∞ δ ( n ) ( x ) f ( x ) dx = (- 1) n f ( n ) (0) – δ ( x ) * f ( x ) = f ( x ) – xδ ( x ) = 0 – xδ ( x ) =- δ ( x ) • Meaning of δ [ h ( x )]: δ [ h ( x )] = X i δ ( x- x i ) | h ( x i ) | The Shah Function: III( x ) • Sampling: III( x ) g ( x ) = ∑ ∞ n =-∞ g ( n ) δ ( x- n ) • Replication: III( x ) * g ( x ) = ∑ ∞ n =-∞ g ( x- n ) • Fourier Transform: F{ III( x ) } = III(...
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bracewell - The Fourier Transform and its Applications The...

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