HW Problem 14 All vectors have length N = 9. (i) (4 pts.) The entries of the time-domain vector x ) = [2 -1 -1 2 -1 -1 2 -1 -1 7 are given by 2 coswn, where n = 0 : 8. What is the value of w? Express x() 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 x2) = [0 1 -1 0 1 -1 0 1 -1 7 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 3) [n] = cos(2un/9) + 3 cos(4an/9) , n = 0, ....8 (iv) (4 pts.) Determine and display the DFT X(4) of x 4)[n] = sin(8xn/9) , n = 0, . ..,8 (v) (4 pts.) If the time-domain vector x() has DFT X(5) = [ 18 0 97 27 0 0 27 -97 0 7 write an equation for r)[n], where n = 0 : 8. Your equation should be in terms of cosines and sines.HW Problem 15 A real-valued signal vector s of length N = 8 has DFT S = 16 21 Z2 23 -4 7+j 2j -4+357 (i) (2 pts.) What are the values of 21, 22 and z3? (ii) (3 pts.) Without inverting the DFT S, evaluate the sum of the entries of the time domain vector s. (iii) (3 pts.) Without inverting the DFT S, evaluate the sum $[0] + $[2] + $[4] + $ [6] (iv) (4 pts.) Display the amplitude (IS[) and phase (ZS[k]) spectra as vectors. (v) (6 pts.) Write an equation for s (where n = 0 : 7) in the form of a constant plus four real-valued sinusoids with frequencies in the range (0, a]. (vi) (2 pts.) In MATLAB, generate the vector s given by the equation found in part (iv) and verify that fit (s) agrees with S.HW Problem 16 The vector s is given by s= [abcdefgh]" (i) (8 pts.) Display the following vectors: . s() = s+ Rs . s(2) = PARs - s . (3) = F's + F 's . s(4) = Ps + P's (ii) (12 pts.) Express the following vectors in terms of P, R, F and s: . s5) = [h-b a-c b-d c-e d-f e-g f-h g-all . 80) = [ate b -f ctg d-h eta f -b g+c h-d]. . s(7) = [0 v2b 2c V/2d 0 -V2f -2g -V2h ]