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Unformatted text preview: A8 PROBLEMS 441 elen/7 Figure A—17: Graphical display of the 7th roots of unity.
Notice that the sequence aﬂﬂﬂ repeats with a period
equal to 7. A7 Summary and Links This appendix has presented a brief review of complex
numbers and their visualization as vectors in the two—
dimensional complex plane. Although this material
should have been seen before by most students during
hi gh—school algebra, our intense use of complex notation
demands much greater familiarity. The labs in Chapter 2
deal with various aspects of complex numbers, and also
introduce MATLAB. In Lab #2, we also have included a
number of MATLAB functions for plotting vectors from
complex numbers (zvect, zcat) and for changing
between Cartesian and polar forms (zprint). The complex numbers via MATLAB demo is a quick
reference to these routines. ‘ In addition to the labs, we have written a MATLAB
GUI (graphical user interface) that will generate drill
problems for each of the complex operations studied here:
addition, subtraction, multiplication, division, inverse, and conjugate. A screen shot of the GUI is shown in
Fig. A—6. A8 Problems PA.1 Convert the following to polar form: (a) z : 0+ j2 (b) z =(—1,1)
(C) z = —3 — 1'4
(d) z = (0, A1,) PA.2 Convert the following to rectangular form:
(a) z = Vigil3””?! (b) z = 1.64 (71/6) (c) z : 3€~j<n/2) (d) z = 7.: (7n) PA.3 Evaluate the following by reducing the answer to
rectangular form: (a) 1'3
(b) ej(”+2”ml (m an integer)
(c) jg” ()1 an integer) (d) 1'”2 (find two answers) 442 APPENDIX A COMPLEX NUMBERS PA.4 Simplify the following complexvalued expres
sions: (a) 3ei27r/3 — 4e—W6
(b) (ﬂ — 12)8
(c) M2" — 1'2)—l
(d) N5 — 12W
(6) %m{je”j”/3}
Give the answers in both Cartesian and polar form. PA.5 Evaluate each expression and give the answer in
both rectangular and polar form. In all cases, assume that
21 = —4+j3 andz2 =1—j. (a) Z1“ (0 Zl/Zz
(b) z§ (g) 6“ (C) Z1+z§ (h) ZIZT
(d) jzz (i) an PA.6 Simplify the following complexvalued sum: Z = ej97r/3 +e—j57r/8 +ejl37'r/8 Give the numerical answer for z in polar form. Draw a
vector diagram for the three vectors and their sum (z). PA.7 Simplify the following complex—valued expres
sions. Give your answers in polar form. Reduce the
answers to a simple numerical form. (a) For 2: = —3 + j4, evaluate l/z.
(b) For z = —2 + j2, evaluate z5.
(c) Forz = ——5 + j13, evaluate M? (d) For z = —2 + j5, evaluate maze—1'77”} PA.8 Solve the following equation for z:
z = j Be sure to ﬁnd all possible answers, and express your
answerlfs) in polar form. PA.9 Let m = ej2”/N. Prove that 23"] = l/zo. PA.10 Evaluate (— j )1/ 2 and plot the resu1t(s). 32 (c) Sketch cos (2m / To) for values of t such that three
periods of the function are shown. Label the
horizontal axis in terms of the parameter To. (d) Sketch cos (2m / To + 7r/2) for values oft such that
three periods of the function are shown. P2.4 Use the series expansions for ex, cos(6), and
sin(6) given here to verify Euler’s formula. 2 3 x_1 x x
6 — +2XdETd—ET‘f'”
92 04
cos(6)=l—E+E+~
. 93 ()5 P25 Use complex exponentials (i.e., phasors) to show
the following trigonometric identities: (a) cos(61 + 62) = cos(61) cos(62) — sin(61) sin(62) (b) cos(61 — 92) = cos(91) cos(92) + sin(61) sin(62) Hint: Write the left—hand side of each equation as the real
part of a complex exponential. [32.6 Use Euler’s formula for the complex exponential
to prove DeMoivre’s formula (cos6 + j sin 6)" = cos 119 + j sin n6 Use it to evaluate (% + j 100
‘3‘) . CHAPTER 2 SINUSOIDE P2.7 Simplify the following expressions:
(3) gejﬂ/3 + 4ejn/6 (b) (ﬂ — 1'3)10 (c) (J3 — 13)—1 (d) (ﬂ — 1'3)“3 (6) Sieve—1W3} Give the answers in both Cartesian form (x + j y) ant polar form (reje). P2.8 Suppose that MATLAB is used to plot a sinusoida
signal. The following MATLAB code generates the signa
and makes the plot. Derive a formula for the signal; the]
draw a sketch of the plot that will be done by MATLAB. dt = 1/100;
tt = —l : dt r 1;
F0 = 2; 22 = 300*exp(j*(2*pi*Fo*(tt — O.75)));
XX = real( 22 ); 9 plot( tt, xx ), grid on
title( ’SECTION of a SINUSOID’ )
Xlabel ( ’TIME (sec) ’) P29 Deﬁne x(t) as
x(t) = 2sin(w0t + 45°) + C0s(w0t)
(a) Express x(t) in the form x(t) = A cos(a)0t + 4’). (b) Assume that coo = 511'. Make a plot of x(t) over th
range ~l 5 t 5 2. How many periods are include
in the plot? (c) Find a complexvalued signal z(t) such that x(t) :
me{z(t)}. ...
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 Fall '08
 JUANG
 Signal Processing

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