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ENEE222_Problem_Set_5_&_Solutions_HW_13,14&15

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HW Problem 13 Let V = v (0) v (1) v (2) v (3) v (4) v (5) v (6) v (7) be the matrix of Fourier sinusoids of length N = 8. (i) (6 pts.) If x = 3 1 −5 3 3 1 −5 3 T , use project ... ions to represent x in the form x = Vc. Verify that x is a linear combination of four columns of V (only). (ii) (6 pts.) Repeat for y = 0 0 2 0 0 0 −2 0 T , expressing it as y = Vd. Verify that y is a linear combination of four columns of V. (iii) (2 pts.) Verify your results in (i) and (ii) using the FFT command in MATLAB (which will generate the vectors 8c and 8d). (iv) (2 pts.) Verify that the vectors x and y are orthogonal by computing their inner product. Do your answers to (i) and (ii) above also support this conclusion? (v) (4 pts.) If s = x + y, use your results from (i) and (ii) above to obtain the least squares approximation ˆs of s in terms of v (0) , v (1) and v (7). Display the entries of ˆs. Also, compute the squared error norm ks − ˆsk 2 . HW Problem 14 All vectors have length N = 9. (i) (4 pts.) The entries of the time-domain vector x (1) = 2 −1 −1 2 −1 −1 2 −1 −1 T are given by 2 cos ωn, where n = 0 : 8. What is the value of ω? Express x (1) as the sum of two Fourier sinusoids. By considering the appropriate column of the Fourier matrix V, determine and display the DFT X(1) . (ii) (4 pts.) Similarly, express the time-domain vector x (2) = 0 1 −1 0 1 −1 0 1 −1 T as a linear combination of the same two Fourier sinusoids as in part (i). Hence determine and display the DFT X(2) . (iii) (4 pts.) Determine and display the DFT X(3) of x (3)[n] = cos(2πn/9) + 3 cos(4πn/9) , n = 0, . . . , 8 (iv) (4 pts.) Determine and display the DFT X(4) of [Show More]

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