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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
Problem Set #1 ASSIgned: Week of 18 May 09
Due Date: 22 May 09 The web site for the course uses Web—CT:
weBct.gatecE.eHu
=> Please check the “Bulletin Board” daily. All ofﬁcial course announcements will be posted there. ALL of the STARRED problems should be turned in for grading. Some of the problems have solutions that can be found on the CD—ROM or under ”Word” on
WebCT. After this assignment is handed in by everyone, a solution to all the starred problems will
be posted to the web. Special case: This homework will be due at the start of lecture on Wednesday. There
will be no credit given for late homeworks. A complex number is just an ordered pair of real numbers. Several different mathematical notations can be used to represent complex numbers. In rectangular form we will use all of the following notations: Z : (any)
2 m+jy wherejzx/il
= Rejszermjzf
Note that 2' : \/—1 in most math courses. The pair (133/) can be drawn as a vector, such that a: is the horizontal coordinate and y the vertical coordinate in a two—dimensional space. Addition of
complex numbers is the same as vector addition; i.e., add the real parts and add the imaginary parts.
In polar form we will use these notations:
z : IzIej argz
: [€19
: r16 where M : r : x/x2 —I— y2 and argz : 6 : arctan(y/a:). In a vector drawing, r is the length and 6
the direction of the vector.
EIllIer’S Ebrmllla' raj” :rcos6+jrsin6 can be used to convert between Cartesian and polar forms.
Before starting this homework we strongly recommend reading through Chapter A of Signal Processing First. PRQBLEM1.1*: Consider the sinusoidal signal plotted below. The Sinusoid xgt)
1 I
0.5 —
CD
'0
E
a 0 
E
(S
—0.5 
_1  l l    
£0 —10 0 10 20 39 40 SD 69 time (t) in seconds
This signal may be written in the form Acos(wt + (15). Find A, w, and gb. PRQBLEM 1.2: (This is an unstarred problem. Unstarred problems are optional.) I want you to get practice
looking for example problems on the CD—ROM, or equivalently, the users.ece.gatech.edu/ spﬁrst
mirror. Go to “A. Complex Numbers,” and then click on “Homework.” Try some problems that
you ﬁnd interesting to get practice with complex numbers. PROBLEM 1.3*: You do not need to, and should not, use your calculator on this problem. Just focus
in on what you need to answer the questions. Most of what you see doesn’t actually play a role
in the answer. Show your work and explain your reasoning to convince the grader that
you didn’t use a calculator. {a} Suppose A = 7(114 + 45)[21 + 3sin(0.17T)H3 + 2cos(0.27r)l exp(ij7T/5)
B = (93 + 54)[9 + 4cos(0.27r)H39 + 4sin(0.27r)l(1 + j) Compute the angle of the polar representation of the complex number A X B. {bl Suppose C = 3exp(j0.47r) exp(j0.1257r) exp(j[0.9 + arctan(0.13)l)
D 4exp(j0.27r)(3 i 43') exp(j0.2337r) exp(j[0.8 + arctan(0.09)l) Compute the magnitude of the polar representation of the complex number C X D. PRQBLEM 1.4*: You can do parts (b) and (c) of this problem without a calculator if you remember your 30—60—90 triangle:
«E
300 90°
1
2 60'
(a) For V = 71 + j, express the inverse V‘1 in rectangular form. In addition, plot V‘1 as a (b) vector. (Hint: see Section A—5.5 on p. 437 of the text.)
Simplify U = —.
—1 — N3
Express your answer in polar form. Assume that c is a positive real number. In addition,
plot the vector U. Simplify W : j3(—1 — j\/§) Express your answer in polar form. In addition, plot 3'3 and
W as vectors. For this one, you’ll deﬁnitely need your calculator. Compute 46—3067r i 563037r and express
your answer in polar form. Do this ﬁrst by converting each term to rectangular form, sub—
tracting in rectangular form, and the converting the rectangular form of the result back into
polar form. At this point, don’t use any fancy complex number facilities of your calculator
(if it has them); just use the add, subtract, multiply, cos, sin, sqrt, and atan keys. Show your
grader the results of intermediate steps. (Watch out for the “wrong quadran ” issue discussed
in Section A—2.4 on p. 431 of the text.) Then, check your work by typing the commands into MATLAB: a = 4 * exp(j * 0.6 * pi)
b = 5 * exp(j * 0.3 * pi)
m abch) ang1e(c) Make sure MATLAB tells you what you are expecting to see. Finally, if you have a calculator with fancy complex number facilities, you should double—check
your answers using them. If you calculator doesn’t have fancy complex number facilities,
that’s OK, you can get by without having them. PRQBLEM 1.5*: Prove the following three convenient relations involving complex conjugates (See p. 433 of the
text for a deﬁnition of complex conjugation). They will come in handy at various times during the
course. (Hint: Substitute z : ﬂak} + j$m{z} on the right hand side.) (a) %e{z} = —%(Z + 2*)
(b) smc} : 24142 — 2*) c 22:22" PROBLEM 1.6*: Use Euler’s formula to prove DeMoivre’s formula:
[cos(6) + j sin(6)]" : cos(n6) + j sin(n6) Then use it DeMoivre’s formula to evaluate (4/5 + j3/5)200. (Hint: You’ll need to use your
calculator or MATLAB to ﬁnd the arctangent of 3/4.) ...
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 Spring '08
 JUANG

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