20105ee211A_1_HW3Solutions

# 20105ee211A_1_HW3Solutions - 1 EE 211A Digital Image...

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Unformatted text preview: 1 EE 211A Digital Image Processing I Fall Quarter, 2010 Handout 12 Instructor: John Villasenor Homework 3 Solutions Textbook problems: 4.3 The spectrum of x 2 2 cos 4 π has delta functions of strength 2 at u = ± 2 and is described by ( , ) 2 ( 2, ) : F u v u v δ = ± v u The spectrum of y 3 2 cos π is described by : ) 3 , ( 2 1 ) , ( ± = v u v u F δ v u The spectrum of the product y x 3 2 cos 2 2 cos 4 π π is the convolution of the spectra above and is described by ( , ) ( 2, 3) : F u v u v δ = ± ± v u 2 Reconstruction cases: (1) Sampling at 5 . = ∆ = ∆ y x is a sampling frequency of 2, so the spectrum ) , ( v u F will be replicated at intervals of 2, 2 in u , v . The reconstruction filter will only let through those frequencies with | u |, | v | ≤ 1. This gives a spectrum for the reconstructed signal: v u Thus, . 2 cos 8 ) , ( ~ y y x f π = (2) Sampling at , 5 2 . = ⇒ = ∆ = ∆ s F y x cutoff for reconstruction is 2.5. c F = ± This gives a reconstructed spectrum: v u Stated in another way, ) 2 , 2 ( ) 2 , 2 ( ) 2 , 2 ( ) 2 , 2 ( ) , ( + + + +- +- + +-- = v u v u v u v u v u F δ δ δ δ ) 2 2 ( 2 cos 2 ) 2 2 ( 2 cos 2 ) , ( ~ y x y x y x f- + + = π π y x π π 4 cos 4 cos 4 = 4.9 We know, from p.85, that comb ) , , , ( y x...
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20105ee211A_1_HW3Solutions - 1 EE 211A Digital Image...

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