# hw2 - 16:04 Homework#2 EECS 556 W11 Due Fri Jan 21 by...

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January 11, 2011 16:04 1 Homework #2, EECS 556, W11. Due Fri. Jan. 21 , by 1:00PM Skills and Concepts 2D FT, sampling, filtering via 2D FFT Problems 1. [15] Find the just-resolved distances (according to Rayleigh’s resolution criterion given in the notes) for a system having a “square” frequency response H ( ν X , ν Y ) = rect 2 parenleftBig ν X 2 ν max , ν Y 2 ν max parenrightBig and for one having a “disk” frequency response H ( ρ ) = rect parenleftBig ρ 2 ν max parenrightBig , where ν max = 5 mm - 1 . Which system would you rather use for imaging and why? Hint: M ATLAB ’s fzero function may be useful here. 2. [5] Prove Parseval’s theorem for the 2D FT. 3. [5] Prove the rotation property of the 2D FT. 4. [5] Find a version of Parseval’s theorem that is appropriate for circularly symmetric functions. 5. [15] Generalize the three main results of Shannon’s ideal “impulse” sampling theory to the more realistic case where we account for finite detector size in the sampling relationship: g d [ n, m ] = 1 Δ X Δ Y integraldisplay ( m +1 / 2)Δ Y ( m - 1 / 2)Δ Y integraldisplay ( n +1 / 2)Δ X ( n - 1 / 2)Δ X g a ( x, y ) d x d y .

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