PS-9-2011-Solutions

# PS-9-2011-Solutions - EE 261 The Fourier Transform and its...

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Unformatted text preview: EE 261 The Fourier Transform and its Applications Fall 2011 Problem Set Nine Solutions 1. (20 points) 2D Fourier Transforms Find the 2D Fourier Transforms of: (a) sin2 πax 1 sin2 πbx 2 Solution: Because the function is separable we have F (sin2 πax 1 sin2 πbx 2 ) = F sin2 πax 1 F sin2 πbx 2 = 1 2 i ( δ ( ξ 1- a )- δ ( ξ 1 + a )) 1 2 i ( δ ( ξ 2- b )- δ ( ξ 2 + b )) =- 1 4 ( δ ( ξ 1- a ) δ ( ξ 2- b )- δ ( ξ 1- a ) δ ( ξ 2 + b )- δ ( ξ 1 + a ) δ ( ξ 2- b ) + δ ( ξ (1 + a ) δ ( ξ 2 + b )) =- 1 4 ( δ ( ξ 1- a,ξ 2- b )- δ ( ξ 1- a,ξ 2 + b )- δ ( ξ 1 + a,ξ 2- b ) + δ ( ξ 1 + a ) δ ( ξ 2 + b ) The plot of this would be four spikes, two up, two down, at the four vertices of the rectangle ( a,b ), (- a,b ), ( a,- b ), (- a,- b ). (b) e- 2 πi ( ax + by ) cos(2 πcx ) Solution: Again, this is separable F n e- 2 πi ( ax + by ) cos(2 πcx ) o = F n e- 2 πiax cos(2 πcx ) e- 2 πiby o = F e- 2 πiax cos(2 πcx ) F n e- 2 πiby o = δ ( ξ + a ) * 1 2 [ δ ( ξ + c ) + δ ( ξ- c )] δ ( η + b ) = 1 2 [ δ ( ξ + a + c ) + δ ( ξ + a- c )] δ ( η + b ) This can also be expressed using the multi-dimensional notation for the product of delta functions of different variables F f ( ξ,η ) = 1 2 [ δ ( ξ + a + c,η + b ) + δ ( ξ + a- c,η + b )] 1 (c) cos(2 π ( ax + by )) Hint: Use the addition formula for the cosine. Solution: One way to do this is to write cos(2 π ( ax + by )) = cos2 πax cos2 πby- sin2 πax sin2 πby, i.e. , as a separable function. Then (much like part (a)) F (cos(2 π ( ax + by ))) = F (cos2 πax cos2 πby )- F (sin2 πax sin2 πby ) = F (cos2 πax ) F (cos2 πby )- F (sin2 πax ) F (sin2 πby ) = 1 4 ( δ ( ξ- a ) + δ ( ξ + a ))( δ ( η- b ) + δ ( η + b )) + 1 4 ( δ ( ξ- a )- δ ( ξ + a ))( δ ( η- b )- δ ( η + b )) = 1 4 ( δ ( ξ- a ) δ ( η- b ) + δ ( ξ- a ) δ ( η + b ) + δ ( ξ + a ) δ ( η- b ) + δ ( ξ + a ) δ ( η + b )) + 1 4 ( δ ( ξ- a ) δ ( η- b )- δ ( ξ- a ) δ ( η + b )- δ ( ξ + a ) δ ( η- b ) + δ ( ξ + a ) δ ( η + b )) These are some combinations and cancellations here, leading to F (cos(2 π ( ax + by ))) = 1 2 δ ( ξ- a ) δ ( η- b ) + 1 2 δ ( ξ + a ) δ ( η + b ) = 1 2 ( δ ( ξ- a,η- b ) + δ ( ξ + a,η + b )) The plot is two spikes at opposite corners of the rectangle, ( a,b ) and (- a,- b ). 2. Linear Transformations (20 points) Consider a 2D rectangular function Π( x,y ): 2 Student Version of MATLAB This 3D representation is depicted by the following 2D image, where white corresponds to 1, and black to 0. Student Version of MATLAB This 2D rectangular function is subjected to 3 different linear transformations. The following images (A, B, C) are obtained: A : 3 Student Version of MATLAB B : Student Version of MATLAB C : Student Version of MATLAB (a) Each of the figures is a result of a horizontal shear. If | k 1 | > | k 2 | > | k 3 | , match the following linear transformations with figures A, B and C....
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PS-9-2011-Solutions - EE 261 The Fourier Transform and its...

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