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20111ee114_1_hw4_sol

# 20111ee114_1_hw4_sol - EE114 Winter 2011 Problem Set 4...

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Unformatted text preview: EE114, Winter 2011 Problem Set 4 Solution ________________________________________________________________________ Problem Set #4 Solution 1) Let f(x,y) be a function below with value 1 in the shaded region and 0 otherwise. f(x,y) y x a) Give a sketch of f(x,-y) b) Give a sketch of f(-x,-y) flip f(x,y) with respect to the x-axis f(x,-y) f(x,-y) flip f(x,y) with respect to the origin f(-x,-y) y f(-x,-y) y y y x x x x EE114, Winter 2011 Problem Set 4 Solution ________________________________________________________________________ 2) Find the 2D Fourier Transform of the two-dimensional function f(x,y) () ( ) () f x, y = rect ax + b ⋅ sinc cy ⎡⎛ b⎞ ⎤ = rect ⎢ a ⎜ x + ⎟ ⎥ sinc cy a⎠ ⎦ ⎣⎝ = g ( x ) ⋅ h( y ) in otherwords, this is a separable function () so we can find the F.T. of each dimension separately { ( )} = F .T .{ g ( x )} ⋅ F .T .{h ( y )} = G (u ) ⋅ H (v) F .T . f x, y () first find G u F .T . rect ( x ) ←⎯ ⎯ sinc(u ) → F .T . rect ( ax ) ←⎯ ⎯ → ⎛ g x = rect ( a ⎜ x + ⎝ () 1 u sinc( ) a a b j 2π u 1 b⎞ u F .T . ) ←⎯ ⎯ e a → sinc( ) = G u ⎟ a⎠ a a () () now find H v F .T . sinc( y ) ←⎯ ⎯ rect ( v ) → () F .T . h y = sin c( cy ) ←⎯ ⎯ → 1 v rect ( ) = H v c c () finally { ( )} = G (u ) ⋅ H (v) = e F .T . f x, y b j 2π u a b j 2π u 1 u1 v ea u v sinc( ) ⋅ rect ( ) = ⋅ sinc( ) ⋅ rect ( ) ac c a c a a⋅c EE114, Winter 2011 Problem Set 4 Solution ________________________________________________________________________ 3) 2D Convolution a) find non-zero region of f(x,y)*f(x,y) y y y delta delta d -d -d - = d 2 d yd delta y delta d y y y delta d func x tion y 2 -d d func tion x func x tion y = delta d func x tion 2d func tion 2d x x y -2d 2 d 2d d 2 delta d func tion delta 2d y func tion * x delta = d d 2 2 y y y -d d d delta d * x d 2 -d 2 d func x tion d d x d -d * y -d * d func tion x x delta d 2 d -d d func x tion d func tion y func tion x d delta 2d y y = -d 2 d d d d -d d -d y d 2 -d d 2 d d d 2 EE114, Winter 2011 Problem Set 4 Solution ________________________________________________________________________ 3) a) continued y y delta y * d -d d -d delta d 2 d d 2 d Total non-zero region is: y -d 2d 2d -d -2d -2d d delta d func tion 2d func tion 2d d x x 2 d 2 2d 2d func tion x func x tion = d d -d d 2 -d d d delta y y d 2 EE114, Winter 2011 Problem Set 4 Solution ________________________________________________________________________ 3) b) find non zero region of f(x,y)*f(x,y) y y y y x x * x x = y y y y y y * = x x x x x x y y y y x x y y * = x x x x EE114, Winter 2011 Problem Set 4 Solution ________________________________________________________________________ 3) b) continued y y y y y y * = x x x x x x y y y x y * x y = x x y x x So the total non-zero region of f(x,y)*f(x,y) is y y x x EE114, Winter 2011 Problem Set 4 Solution ________________________________________________________________________ 4) Find the total area under the function f(x) rect(u) rect(u) 1 1 u -0.5 0.5 -0.5 0.5 u ...
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