ME 365
HOMEWORK SET 9
Spring 2010
Out: Thursday, April 1, 2010
Due: Thursday, April 8, 2010
1
Solve each problem on separate sets of sheets.
Fourier Series
Problem 1 (Fourier Series)
(a)
Find the Fourier coefficients and write the Fourier series representation of
the function
0
5
0
f
3
0
5
x
x
x
The function has a period of 10.
(b)
How should f(x) be defined at x=5, x=0 and x=5 in order that the Fourier series will converge to
f(x) for
5
5
x
?
(Hint: plot the Fourier series and see where it goes to at the points of discontinuity)
(c)
Find the Fourier series representation of the following functions defined over their fundamental
intervals (0 to T or
–
T/2 to T/2) as follows:
i.
x(t)=t over [1,1]
ii.
x(t)=t+1 over [1,1]
iii. x(t)=t
3
over [1,1]
Problem 2 (Magnitude and Phase Spectra)
For the periodic signals in problem 1(a), and 1(c.i) through 1(c.iii), express the Fourier series
representations you obtain as
a.
sine and cosine series
b.
plot their magnitude and phase spectra using the first 10 terms in each series (for the phase
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 Spring '08
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 Fourier Series, Fourier series representation, 10m/s, 0.01 61 j

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