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Unformatted text preview: EE 261 The Fourier Transform and its Applications Fall 2011 Problem Set Nine Due Friday, December 9 1. (20 points) 2D Fourier Transforms Find the 2D Fourier Transforms of: (a) sin2 πax 1 sin2 πbx 2 (b) e 2 πi ( ax + by ) cos(2 πcx ) (c) cos(2 π ( ax + by )) Hint: Use the addition formula for the cosine. 2. Linear Transformations (20 points) Consider a 2D rectangular function Π( x,y ): Student Version of MATLAB This 3D representation is depicted by the following 2D image, where white corresponds to 1, and black to 0. 1 Student Version of MATLAB This 2D rectangular function is subjected to 3 different linear transformations. The following images (A, B, C) are obtained: A : Student Version of MATLAB B : Student Version of MATLAB C : 2 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. 1 k 1 1 1 k 2 1 1 k 3 1 (b) The 2D Fourier transform of each figure is taken, and the following magnitude plots I, II, and III, below, are obtained. Match figures A, B, and C with their corresponding Fourier transforms. Explain your reasoning. (Hint: Look at the orientation of the crests.) 3 I : Student Version of MATLAB II : Student Version of MATLAB III : 4 Student Version of MATLAB 3. (15 points) 2D Discrete Fourier Transform Let f be a M × N matrix (you can think of f as an M × N image). The 2D DFT of f is given by the following formula: F f [ k,l ] = M 1 X m =0 N 1 X n =0 f [ m,n ] ω ln N ω km M where ω N = e 2 πi/N , ω M = e 2 πi/M . Independent of the problems to follow, we comment that the 2D DFT is ‘separable’ in the following sense. Let f m be the m th row of the matrix f : it’s a vector of length N. Let F f m be the 1D DFT of f m then: F f m [ l ] = N 1 X n =0 f m [ n ] ω ln N = N 1 X n =0 f [ m,n ] ω ln N Then, from the given formula for the 2D DFT, we can easily see that: F f [ k,l ] = M 1 X m =0 F f m [ l ] ω km M In other words, this is a 1D DFT of the vector of length M which consists of the l t h entry in each vector F f m ....
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