320 HW 7 - ECE 320 Networks and Systems Spring 2007–2008...

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Unformatted text preview: ECE 320 Networks and Systems Spring 2007–2008 Problem Set 7 Due March April 7, 2008 1. Let σ ( x ) denote a two dimensional signal where x = ( x 1 , x 2 ) T in rectangular coordinates. Let Σ( k ) denote the two dimensional Fourier transform of σ ( x ) where k = ( k 1 , k 2 ) T in rectangular coordinates. The 2-D Fourier transform pair is Σ( k ) = Z + ∞-∞ Z + ∞-∞ σ ( x ) exp(- i 2 π k T x )d 2 x σ ( x ) = Z + ∞-∞ Z + ∞-∞ Σ( k ) exp( i 2 π k T x )d 2 k . (a) Suppose σ ( x ) has the special structure that σ ( x ) = σ 1 ( x 1 ) σ 2 ( x 2 ). Give a formula for the 2-D Fourier transform of σ ( x ) in terms of the 1-D Fourier transforms of σ 1 ( x 1 ) and σ 2 ( x 2 ) which are denoted by Σ 1 ( k 1 ) and Σ 2 ( k 2 ), respectively. (b) Define r d ( x ) = ‰ 1 , | x | ≤ d/ 2 , otherwise . Suppose σ ( x ) = r d 1 ( x 1 ) r d 2 ( x 2 ). Give a formula containing no integral signs for the 2-D Fourier transform of σ ( x ) which is denoted by Σ( k ). (c) Draw a plot of the 2-D signal and 2-D Fourier transform for Problem 1b for the case of d 1 = 1 and d 2 = 2. I suggest using the Matlab mesh command. (d) Define t d ( x ) = ‰ 1- 2 | x | /d, | x | ≤ d/ 2 , otherwise which is a triangle centered at the origin. Suppose σ ( x ) = t d 1 ( x 1 ) t d 2 ( x 2 ). Give a formula con- taining no integral signs for the 2-D Fourier transform of...
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This note was uploaded on 05/22/2008 for the course ECE 3200 taught by Professor Fine during the Spring '06 term at Cornell University (Engineering School).

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320 HW 7 - ECE 320 Networks and Systems Spring 2007–2008...

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