soln6 - R. Balan Homework #6 Solutions MATH 464 1. Note the...

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Unformatted text preview: R. Balan Homework #6 Solutions MATH 464 1. Note the function f is the convolution between the box function and the unit Gaussian function , that is: f ( x ) = Z - ( u ) ( x- u ) du , ( u ) = 1 , | u | < 1 2 , | u | > 1 2 , ( u ) = e- | u | 2 Thus the Fourier transform is the product of the Fourier transforms of and , respectively: F ( s ) = sinc ( s ) e- | s | 2 = sinc ( s ) ( s ) 2. Note f 1 =- 1 2 df dx The Fourier transforms of f and f 1 are: F ( s ) = e- 2 s 2 respectively F 1 ( s ) =- isF ( s ) =- is e- 2 s 2 . Let g = f * f and g 1 = f * f 1 . Then their Fourier transforms are: G ( s ) = F ( s ) 2 = e- 2 2 s 2 G 1 ( s ) = F ( s ) F 1 ( s ) =- is 2 e- 2 2 s 2 Next we need to compute the inverse Fourier transforms. Again apply the dilation and the derivative rules, and get: g ( x ) = r 2 e- x 2 / 2 g 1 ( x ) =- 1 2 dg dx = x 2 r 2 e- x 2 / 2 3....
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This note was uploaded on 05/05/2010 for the course MATH 464 taught by Professor Staff during the Spring '08 term at Maryland.

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soln6 - R. Balan Homework #6 Solutions MATH 464 1. Note the...

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