HW2_solutions - ECE 380 Biomedical Imaging Spring 2010...

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1 ECE 380 Biomedical Imaging Spring 2010 Homework 2 Solutions ________________________________________________________ 1. One-Dimensional Fourier Transforms (25 points) a) ( ) ( 8) f x x  2 16 ( ) ( 8) =e j kx jk F k x e dx  b) ( ) cos(3 ) f x x 2 33 2 2 3 2 3 ( ) cos(3 ) = ( )e 2 1 = (e e ) 2 j kx j x j x j kx j kx j x j kx j x F k x e dx ee dx e e dx     By using the properties of the delta function 1 ( ) [ ( ) ( )] 22 2 F k k k      c) 6 ( ) ( ) x f x e u x 62 0 (6 2 ) (6 2 ) 0 0 ( ) ( ). = e 1 (6 2 ) 6 2 x j kx x j kx x j k x j k F k e u x e dx e dx e e dx j k j k            
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2 2. Two-Dimensional Fourier Transforms (30 points) a) A 2D Fourier transform ( , ) F u v of an object ( , ) f x y is equivalent to 2 ( ) ( , ) ( , ) j ux vy F u v f x y e dxdy      We have to show that multi-dimensional Fourier transforms can be computed by performing one-dimensional Fourier transforms in each dimension consecutively.
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This note was uploaded on 04/12/2010 for the course ECE 380 taught by Professor Prof.a during the Spring '10 term at University of Illinois at Urbana–Champaign.

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HW2_solutions - ECE 380 Biomedical Imaging Spring 2010...

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