Please complete both problems, each having subproblems. Problem 1 requires hand calculations whereas problem 2 requires hand calculations and MatLab usage.
AERO 300: AEROSPACE ENGINEERING ANALYSIS Spring 2021 Cal Poly San Luis Obispo Homework 7 Assigned 05/28 — Due by noon on 06/05 Problem 1 (50 points) Note: This problem should be done by hand. 1. (20 points) (Primitive) roots of unity. (a) (12 points) Find all 6th roots of unity. (b) (8 points) Find all primitive 6th roots of unity. 2. (30 points) Consider the vector x = (1 , 1 , 1 , 1)T . (a) (5 points) Compute the 4× 4 Fourier matrix F4 . (b) (2 points) Compute the conjugate, F 4 , of the above Fourier matrix. (c) (7 points) Verify that F 4 = F −1 4 by showing that F 4F4 = F4F 4 = I4 , where I4 is the 4× 4 identity matrix. (d) (3 points) Compute the Discrete Fourier Transform (DFT) of the vector x . (e) (7 points) Compute the Fast Fourier Transform (FFT) of x and compare your answer to the result of part (d). Does this match your expectation? (f) (6 points) Let y = (2 , 0 , 0 , 0)T . Compute the inverse FFT of y . Does the answer match your expectation? Problem 2 (50 points) Given an interval [c , d] and a positive integer n , let tj = c + j∆t for j = 0 , 1 , , . . . , n − 1 , where ∆t = (d− c)/n . Let x = (x0 , x1 , . . . , xn−1)T denote a vector in Rn . Define y = a + ib = Fnx as the Discrete Fourier Transform of x , where, for k = 0 , 1 , . . . , n− 1 , ak = Re(yk) and bk = Im(yk) are respectively the real and imaginary parts of the component yk of y . 1 Then, we know from class that the real function Pn(t) = 1√ n n−1∑ k=0 [ ak cos ( 2πk n · t− c ∆t ) − bk sin ( 2πk n · t− c ∆t )] satisfies Pn(tj) = xj for j = 0 , 1 , . . . , n− 1 . Furthermore, since xj ∈ R for j = 0 , 1 , . . . , n − 1 , we also know from lecture that y0 ∈ R and yn−k = yk for k = 1 , . . . , n− 1 . As a result, for n even, the Fourier series Pn(t) = a0√ n + 2√ n n/2−1∑ k=1 [ ak cos ( 2πk n · t− c ∆t ) −bk sin ( 2πk n · t− c ∆t )] + an/2√ n cos ( π · t− c ∆t ) . (1) satisfies Pn(tj) = xj for j = 0 , 1 , . . . , n− 1 . 1. (20 points) For n odd, derive a similar expression to Equation (1) for the interpolating Fourier series through (tj , xj) , j = 0 , 1 , . . . , n− 1 . 2. (20 points) Implement your expression from part 1 in a Matlab ® function. Hint: You are welcome to use the function trigInterp.m on Canvas as a template. 3. (10 points) Use the function from part 2 to plot the interpolating Fourier series for x = ( 0 , √ 3 2 , √ 3 2 , 0 ,− √ 3 2 ,− √ 3 2 , 0 , √ 3 2 , √ 3 2 )T on [c , d] = [0 , 3] . 2