17 Please answer all. . 6 EEEE3090-E1 3. (a) You have six samples of a signal y which is a function of time t. The samples are taken at times ti, t), tz, ta, ts and to and the signal values at these times are given by Y1. Y2, 73, 74, ys and yo respectively. It is required to fit, using least squares, the following model to the data: y = a +Bt? + y Derive the matrix and vectors X, Y and v defined in the notes for this problem. (5) (b) A variable x2 is plotted as a function of Xi and the following plot is obtained: X2 "K 40 X1 As indicated the direction of maximum variance is at 40 degrees to the x: axis. It is required to define a new set of orthogonal axes so that the new x- axis xi' is in the direction of maximum variance indicated above. 1) Determine the required rotation matrix. [1] ii) Explain how the data in the original co-ordinate frame (x1,x2) can be recovered from the data in the rotated frame (Xi'.x2'). [1] iii) Explain how one could use the variables in the rotated co-ordinate system (x:',x') for lossy data compression. [2] (c) Determine whether the following pairs of vectors are orthonormal, orthogonal, or neither. i) [2-3.5 8] and [1.5 2 0.5) li) [3-2 4] and [1 1 -0.75] [1] 1) Gland (0 0 11 [1] Continued on the next page Will give 10 downvotes for copied or incorrect answer. EEEE3090-E1 - 7 EEEE3090-E1 (d)) [6] ii) Two vectors are given by: Vi = [1 2] and V2 = [34] Using the Gram-Schmidt method, with the vectors in the order shown above, derive two orthonormal vectors, normalising the vectors at each stage. It is required to car y out a QR factorisation on the following mats 6 of 8 Using your answer to part 1), derive the matrices Q and R defin in the notes and verify that: A = QR 171