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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 sx dx The Inverse Fourier Transform: f ( x ) = Z - F ( s ) e i 2 sx 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 ( ) h ( x- ) d Autocorrelation : Let g ( x ) be a function satisfying R - | g ( x ) | 2 dx < (finite energy) then g ( x ) 4 = ( g * ? g )( x ) 4 = Z - g ( ) g * ( - x ) d = g ( x ) * g * (- x ) Cross correlation : Let g ( x ) and h ( x ) be functions with finite energy. Then ( g * ? h )( x ) 4 = Z - g * ( ) h ( + x ) d = Z - g * ( - x ) h ( ) d = ( h * ? g ) * (- x ) The Delta Function: ( x ) Scaling: ( ax ) = 1 | a | ( x ) Sifting: R - ( x- a ) f ( x ) dx = f ( a ) R - ( x ) f ( x + a ) dx = f ( a ) Convolution: ( x ) * f ( x ) = f ( x ) Product: h ( x ) ( x ) = h (0) ( x ) 2 ( x ) - no meaning ( x ) * ( x ) = ( x ) Fourier Transform of ( x ): F{ ( x ) } = 1 Derivatives: R - ( n ) ( x ) f ( x ) dx = (- 1) n f ( n ) (0) ( x ) * f ( x ) = f ( x ) x ( x ) = 0 x ( x ) =- ( x ) Meaning of [ h ( x )]: [ h ( x )] = X i ( x- x i ) | h ( x i ) | The Shah Function: III( x ) Sampling: III( x ) g ( x ) = n =- g ( n ) ( 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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