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
esult 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 Matla
®
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 fo
x =
(
0 ,
√
3
2
,
√
3
2
, 0 ,−
√
3
2
,−
√
3
2
, 0 ,
√
3
2
,
√
3
2
)T
on [c , d] = [0 , 3] .
2