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**Unformatted text preview: **E R IAIN TIT TE FTE HN L Y
SCHOOL of ELECTRICAL and COMPUTER ENGINEERING E E2 2 mm r2 1
Problem Set #3 Assigned: Week of 07—Jun—10
Due Date: 14[15—Jun—10 Warning: This is one of the more challenging homeworks of the semester. Do not leave this one
until the last minute! Start early! First quiz is in class on Friday 11—June—10. Remember there are no makeup exams; if you have
a valid, documented excuse for missing the exam, your average will be computed using the other
quizzes. Ooersleepz'ng 07“ lack of studying is not a valid excuse. The Fourier series material on this homework will not explicitly be on Quiz #1, but you will ﬁnd
that many of the basic manipulations needed to do the homework will give you some additional
experience that might help on Quiz #1, so it would not hurt to start this homework early. Please check T—square chat and announcements daily. All ofﬁcial course announcements will be
posted there. ALL of the STARRED problems should be turned in for grading. After this assignment is handed in by everyone, a solution to all the starred problems will be posted
to the web. PROBLEM 3.1*: Suppose we wanted to ﬁnd a signal a:(t) of the form i cos(*) for *2 < t < 2
$05) T { 0 otherwise that has an instantaneous frequency (in Hertz) given by : sm2(—t) for72<t<2. IaI Find such an mltl.
(b) Now let : 467(t+2)/2 for72<t<2. Find such an PRQBLEM 32*: Which of the following signals are periodic? For those that are periodic, ﬁnd the fundamental
frequency in radians, (do, and the non—zero F.S. (Fourier Series) coefﬁcients ak for all k. (Important:
you should be able to do this problem without actually having to compute any integrals! While
working the problem, stop yourself the moment you write an integral sign and spend some time
thinking. You can generally rewrite the expressions using Euler’s formula and expand things out,
and/ or use trignometric identities.) OO 1 $(t) = 13 + E — sin(187r]€t + 77/3)
H \/k2 + 1 PROBLEM 3.3*: Unlike the previous problem, you will have to do some integration here. Consider a periodic
function a;(t), with fundamental period To, where a single period is speciﬁed by $(t) = cos for 7T0/2 g t g T0/2
0 (a) Draw a labeled sketch of $(t) for at least 3 periods. (The function should never go negative.) (b) Use the fact that the “DC.” coefﬁcient, a0, is simply given by the integral of $(t) over a
single period, divided by its period, to compute a0. (c) Use the Fourier series analysis integral to compute the F.S. coefﬁcients ak for the periodic
function x(t) for all k. Simplify your answer as much as possible. (There is no easy way to do
this problem given your current knowledge, but look for symmetries where you can.) If you
get something with an indetermine form (i.e., 0/0), evaluate it using L’Hopital’s rule or ﬁnd
an alternative method. Make sure that the values you get for a0 in parts (b) and (0) match. Hint: Your ﬁnal answer should look like: (71);“ t t X —
[some cons an ] 77(1 7 [some integerlkZ) 77 We’re not telling what “some constan and “some integer” are, you’ll need to find that out. If your answer isn’t as simple as this keep working at it. (d) Suppose y(t) : a:(t) — a0. Sketch y(t) Sanity cheek: Since $(t) is symmetric, i.e., $(t) = 30(7t), we know that the F.S. coefﬁcients must
consist solely of real values. If you’re thinking in terms of polar coordinates, the phases must all
be either 0 or 71'. That gives you a quick way to check to see if your answer makes sense. PRQBLEM 3.4*: A signal 30(75) is periodic With period To = 4. Therefore it can be represented as a Fourier series of the form $(t): i ak€j(27T/4)kt. kznm It is known that the Fourier series coefﬁcients for this representation of a particular signal $(t) are
given by the integral 10
ak : i/e’tﬂeijQW/letdt. (1)
(i (a) In the expression for ak in Equation (1) above, the integral and its limits deﬁne the signal Determine an equation for $(t) that is valid over one period. (b) Using your result from part (a)7 draw a plot of a:(t) over the range —4 S t S 4 seconds. Label
it carefully. (c) Determine a0, the DC value of PRQBLEM 3.5: This is not starred, but the results are required for further problems. We have seen that a periodic signal 30(t) can be represented by the Fourier series ax; m) = Z akejkwot, (2)
him where mo 2 27T/T0 = 271' f0. It turns out that we can transform many operations on the signal
into corresponding operations on the Fourier coefficients ak. For example, suppose that we want
to consider a new periodic signal y(t) = Chg—gt). What would the Fourier coefﬁcients be for y(t)? To see this, we simply need to differentiate the Fourier series representation; i.e., W): d:c(t) :1 i akejkwot : i aki [ejkwot] : i ak[(jkw0)€jkw0t]_ (3) k:—oo h:—oo h:—oo Thus, we see that y(t) is also in the Fourier series form w .
ya): 2 516631690015, where bk: (jkw0)ak k:—oo but in this case the Fourier series coefficients are related to the Fourier series coefﬁcients of $(t)
by bk 2 (jhwomk. This is a nice result because it allows us to ﬁnd the Fourier coefﬁcients without
actually doing the differentiation of $(t) and without doing any tedious evaluation of integrals to
obtain the Fourier coefﬁcients bk. It is a general result that holds for every periodic signal and its
derinatine We can use this style of manipulation to obtain some other useful results for Fourier series. In
each case below, use Equation (2) as the starting point and the given deﬁnition for y(t) to express
y(t) as a Fourier series and then manipulate the equation so that you can pick off an expression for
the Fourier coefﬁcients bk, as a function of the original coefﬁcients ak. (a) Suppose that y(t) : Ax(—t) where A is a real number; i.e., y(t) is just a scaled, time—reversed
version of Show that the Fourier coefﬁcients for y(t) are bk 2 Aa_k, Given this knowlege,
show that if $(t) : a:(—t), all the coefﬁcients must be purely real. . (b) Similarly show that if x(t) = 730(7t), all the coefﬁcients must be purely imagianry. (c) Suppose that y(t) = $(t + td) where td is a real number; i.e., y(t) is just an advanced version
of Show that the Fourier coefﬁcients for y(t) in this case are bk : akejkwotd. (d) Referring to part (b), what happens to the FS coefﬁcients when we delay the signal by 1/2
of a period? Explain. (e) Referring to part (b), what happens to the FS coefﬁcients when we delay the signal by 1/4
of a period? Explain. (f) Referring to part (b), what happens to the FS coefficients when we delay the signal by an
integer number of periods? Explain. (g) Suppose y(t) 2 ﬂoat) where oz is a positive number. For oz > 1, the signal is compressed; for
CM < 1, the signal is expanded. Make a convincing argument that the FS coefficients remain
unchanged. PRQBLEM 35*:
Consider the periodic function 30(75) plotted below. —2T0 —T0 _no A T0 16 This signal can be represented by the Fourier series with (do = 27T[T0. The other Fourier coefﬁcients ak for k 75 0 in the Fourier series representation of $(t) are “ E
blIll4 ak=—. —sz_ down the Fourier series coefﬁcients b0 and bk. for k 7E 0 for the Fourier series representation
of the periodic signal without evaluating any integrals. (e) If we add a constant value of minus one—quarter to Mt), we obtain the signal w(t) : $(t)—0.25.
Make a plot of the signal w(t). This signal can also be represented by a Fourier series, EX) 112(75): E ckejkwot. k:—oo because it is periodic with period To. Explain how 00 and Ch. are related to (L0 and ak. (Note:
You should not have to evaluate any new integrals explicitly to answer this question.) (f) Let 2(t) also be periodic with period To. Its FS coefﬁcients dk : ak(—1)k Sketch ...

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