AAE 301 Problem Set 4
School of Aeronautics and Astronautics, Purdue University Fall 2023
Issued 9/15; Due 9/22 11:59pm
All work should be done by hand (without the use of a calculator/computer) unless specified otherwise.
Show all work for full credit. If you have code, publish as PDF and attach to the end of your homework.
Exercise 2.4.1 (modified version, use information given in this homework)
Problem 1. Consider the function f in L2 (0, 2⇡) given by
f (t) = 2
= 1
0t<⇡
⇡  t < 2⇡.
if
if
Find the sine and cosine Fourier series expansion (3.1) for f , that is, find the Fourier series for f of the form:
f (t) = a0 +
1
X
↵k cos(kt) +
k
sin(kt).
k=1
Choose a partial Fourier series approximation p100 (t) for f (t). Then plot p100 (t) and f (t) on the same graph.
Compute the error
s
Z 2⇡
1
2
kf pn k =
|f (t) pn (t)| dt
2⇡ 0
in Matlab. Does this Fourier series converge for t = j⇡ where j is an integer, and if so what does it converge
to; see the Dirichlet convergence Theorem 2.3.4.
Problem 4. Consider the function f in L2 (0, 2⇡) given by
f (t) = 4 + 2 cos(2t)
4 sin(2t) + 4 cos(3t) + 8 sin(3t) + 6 cos(5t)
4 sin(8t).
Compute the norm kf kL2 (0,2⇡) for this function. Express f as a Fourier series of the form f =
Plot the power spectrum for f .
P1
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.
2⇡ikt
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Problem 5. Consider the function f in L2 (0, 1) given by
f (t) = 2
4 cos2 (⇡t) + 4 cos(2⇡t) + 6 cos(6⇡t)
10 sin(6⇡t) + 12 cos(8⇡t)
Compute the norm kf kL2 (0,1) for this function. Express f as a Fourier series of the form f =
Plot the power spectrum for f .
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%--------------------------------------------------% Calculations & Output
%--------------------------------------------------% Initialize arrays for t and p
t = linspace(0, 2 * pi, 10000);
p = zeros(size(t));
%---------------------------
Problem 1
%--------------------------% Create piecewise function
syms t1;
f1 = piecewise(0 < t1 <= pi, 2, pi < t1 <= 2*pi, -1);
% Calculate fourier approximation of function
p1 = p;
for k = 1:1:100
p1 = p1 + (sin(k * t) / (k * pi)) * (3 - 3 * cos(pi * k));
end
p1 = p1 + (1 / 2);
% Calculate error of fourier series
error1 = 0;
for k = 101:1:10000
error1 = error1 + ((3 - 3 * cos(pi * k)) / (pi * k)) ^ 2;
end
error1 = sqrt(error1) / sqrt(2);
%---------------------------
Problem 1 Plots
%--------------------------% Function and Fourier Approximation plots
figure(1)
hold on
fplot(f1)
plot(t,p1)
axis([0,2 * pi,-1.5,2.5])
xlabel('t')
legend('Function','Fourier Series')
title('The graph for f(t) and p100(t) [Problem 1]')
%---------------------------
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Published with MATLAB® R2022b
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