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Unformatted text preview: EE 261 The Fourier Transform and its Applications Fall 2011 Problem Set Eight Due Friday, December 2 1. (20 points) A True Story : Professor Osgood and a graduate student were working on a discrete form of the sampling theorem. This included looking at the DFT of the discrete rect function f [ n ] = ( 1 , | n | N 6 ,- N 2 + 1 n <- N 6 , N 6 < n N 2 The grad student, ever eager, said Let me work this out. A short time later the student came back saying I took a particular value of N and I plotted the DFT using MATLAB (their FFT routine). Here are plots of the real part and the imaginary part. 10 8 6 4 2 2 4 6 8 10 2 2 4 6 8 k real part of F(k) 10 8 6 4 2 2 4 6 8 10 2 1 1 2 k imaginary part of F(k) 1 (a) Produce these figures. Professor Osgood said, That cant be correct. (b) Is Professor Osgood right to object? If so, what is the basis of his objection, and produce the correct plot. If not, explain why the student is correct. 2. (15 points) DFTs frequency response and spectral leakage Compute the DFT of a discrete signal of frequency x/N : f [ k ] = e 2 ixk/N i.e., find a closed form expression for F f . This is the frequency response of the DFT. Qual- itatively, how is the spectrum different when x is an integer and when x is not an integer? Plot the magnitude of the frequency response for x = 2 . 5 and N = 8. Does the plot agree with what you expect? 3. (20 points) The DFT diagonalizes circulant matrices: A quals problem A discrete linear system with input v R n and output w R n is given by matrix multipli- cation, w = Av for an n n matrix A . The following three statements are equivalent (and are taken as known): (a) The system is time-invariant (or shift-invariant). (b) A is a circulant matrix. (c) A acts by convolution, meaning there is an h R n such that Av = h * v for all v R n . Show the following: Theorem 1 A matrix A is circulant if and only if it is diagonalized by the discrete Fourier transform F , i.e., F A F- 1 = or equivalently F A = F , where is a diagonal matrix....
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