HW1_ece220_2007_solution

# HW1_ece220_2007_solution - Homework 1 Solution(ECE220 Fall...

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Unformatted text preview: Homework 1 Solution (ECE220 - Fall 2007) 1. (a) (10 points) I ( x,y ) = cos ( αx )cos( βy ) Prove that: I ( x,y ) = 0 . 5 A ( x,y ) + 0 . 5 B ( x,y ) where A ( x,y ) = cos ( αx + βy ) B ( x,y ) = cos ( αx- βy ) . Solution: Using the cosine identity cos( θ ) = 0 . 5( e jθ + e- jθ ) I ( x,y ) = cos ( αx )cos( βy ) = 0 . 5( e jαx + e- jαx )0 . 5( e jβy + e- jβy ) = 0 . 5 £ . 5( e j ( αx + βy ) + e- j ( αx + βy ) ) + 0 . 5( e j ( αx- βy ) + e- j ( αx- βy ) ) / = 0 . 5[ cos ( αx + βy ) + cos ( αx- βy )] = 0 . 5 A ( x,y ) + 0 . 5 B ( x,y ) (b) (10 points) Say that black is 1 and 0 is white. Draw the following two images: C ( x,y ) = ‰ 1 , | x | < . 5 , | y | < . 5; , else. D ( x,y ) = ‰ 1 , x 2 + y 2 < . 5; , else. Solution: C(x,y) x y-1.5-1-0.5 0.5 1 1.5-1.5-1-0.5 0.5 1 1.5 D(x,y) x y-1.5-1-0.5 0.5 1 1.5-1.5-1-0.5 0.5 1 1.5 2. (20 points) [40] Transformations of the independent variable in analog signals: (a) (10 points) Show that a rectangular pulse can be obtained by subtracting two shifted unit step functions. The unit step function is defined as: u ( t ) = ‰ 1 , t ≥ 0; , else.else....
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HW1_ece220_2007_solution - Homework 1 Solution(ECE220 Fall...

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