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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 #1 Assigned: Week of 24—May—10
Due Date: 02—Jun—10 (in lecture) The web site for the course uses t—square:
tsguare.gatech.edu => Please check t—square announcements and chat daily. All ofﬁcial course announcements will be
posted there.
ALL of the STARRED problems should be turned in for grading. Some of the problems (unusally, the unstarred ones) may have solutions that can be found on the
CD—ROM. After this assignment is handed in by everyone a solution to all the starred problems
will be posted to the web. Your homework will generally be due in recitation at the beginning of class; however, to avoid the
Memorial Day holiday, this ﬁrst homework (this one) will be due at the start of lecture on June
2. After the beginning of lecture the homework is considered late and will be given a zero. 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 wherejz V71
2 chszermjzf Note that l = V71 in most math courses. The pair (my) can be drawn as a vector, such that m
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: where z = r = V302 + y2 and arg z = 0 = arctan(y/$). In a vector drawing, r is the length and 0
l j. . E ]
Euler’s Formula: ‘
r630 = rcos6+ jrsinH can be used to convert between Cartesian and polar forms.
Before starting this homework, we strongly recommend reading through Appendix A of Signal Processing First. PRQBLEM1.1*: Sinusoidal Signal: X(t) = A 003(0) t + (in) 001 0.005 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.01
t (time in sec) Answer to part (c): y(t) = (2/0) dx(t0.01)/dt
300 200 100 100 200 :300
:0401 —04M15 0 04105 0401 04015 0412 04125 0413 04r35 0414 T UHWBIH SEC (a) The above ﬁgure shows a plot of a sinusoidal wave From the plot7 determine the values
of A 010 and 777 < qb g 71' in the representation miti = Acosiwot + Where appropriate, be sure to indicate the units of the sinusoidal signal parameters. (b) The signal $(t) in part (a) can be written as the real part of a complex exponential. Determine
Z for the complex signal 2(t) = ZBJWOt such that $(t) = %e{z(t)}. (c) Sketch the signal y(t) = %%[$(t i 01)], Where 00(75) is the signal from part (a). Use the axes
provided above or make your own axes covering the same time interval. PRQBLEM 1.2*: Sugpose that MATLAB is used to plot a sinusoidal signal. The following MATLAB code generates
the signal and makes the plot. Draw a sketch of the plot that will be done by MATLAB. Determine
thezunphtude @4),phase($),and.perkxlofthe gnusoklandlabelthe perkxlon.your1ﬂot tt = .10 : dt : .30; z =sqrt(2)*(1+j); XX = rea1( Z*exp( 2j*pi*Fo*tt ) ); % plot( tt, XX ), grid title( ’SECTION of a SINUSOID’ ), Xlabel(’TIME (sec)’) PRQBLEM 1.3*: You can do parts (b) and (c) of this problem without a calculator if you remember your 30—60—90
triangle: (a) (b) V? 300 90° 1
2 60 For V = 1 + j, express the inverse V‘1 in rectangular form. In addition, plot V‘1 as a
vector. (Hint: see Section A—5.5 on p. 437 of the text.) Simplify
ceij2n/3
—1 + j\/§ '
Express your answer in polar form. Assume that C is a positive real number (just leave it
as c; you don’t need to specify a speciﬁc value for c.) In addition, plot the vector U. U: Simplify W = j3(71 + Express your answer in polar form. In addition, plot 3'3 and
M For this one, you’ll deﬁnitely need your calculator. Compute 5e’j0‘657r + 6€j0257r 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 7’ issue discussed
in Section A—2.4 on p. 431 of the text.) Then check your work by typing the commands into MATLAB: a = 5 * exp(j * 0.65 * pi) b = 6 * exp(j * 0.25 * pi)
= + absgc) angleCc) 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 1.4*: 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} + on the right hand side.) (a) %c{z} = —;(z + 2*)
(b) : — z) c 22:22" PRQBLEM 1.5*: Euler’s formula, which says that exp(j9) = c0s(9) + j sin(6), was at the time revolutionary.
From calculus7 you should remember the following Taylor series formulas: 2 4 6
a a a
“05!”! : “Eta—5"“
, a3 a5 a7
sm(a) = (Li—3' +—5' i—7' W"
612 a3 a4 a5
exp(a) : 1+a+—+—+—+— 2! 3! 4! 5! (a) Show that Euler’s formula is correct using these Taylor series formulas. (Hint: plug in jél
for a in the Taylor series formula for exp(a) (b) Using this, show that if z = c0s(9) + j sin(6), then 2:" = cos(9n) + j sin(6n). PRQBLEM 1.6: {a} Suppose A = 7(114 + 45)[21 + 3sin(0.17T)H3 + 2c0s(0.17r)] €Xp(*j7T/3)
B = (133 + 5W9 + 4cos(0.27T)H49 + 4sin(0.27r)](1 + j) Compute the angle of the polar representation of the complex number A X B. (b) Suppose C = 2 exp(j0.37r) exp(j0.1257r) exp(j[0.9 + arctan(0.13)l)
D 3exp(j0.27r)(3 i 43') exp(j0.2737r) exp(j[0.8 + arctan(0.09)l) Compute the magnitude of the polar representation of the complex number C X D. ...
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 Summer '08
 JUANG
 Digital Signal Processing, Signal Processing, The Land, Complex number, rectangular form, polar form, complex number facilities, fancy complex number

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