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HW1-Ppages - A-8 PROBLEMS 441 elen/7 Figure A—17...

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Unformatted text preview: A-8 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. A-7 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. A-8 Problems P-A.1 Convert the following to polar form: (a) z : 0+ j2 (b) z =(—-1,1) (C) z = —3 — 1'4 (d) z = (0, A1,) P-A.2 Convert the following to rectangular form: (a) z = Vigil-3””?! (b) z = 1.64 (71/6) (c) z : 3€~j<n/2) (d) z = 7.: (7n) P-A.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 P-A.4 Simplify the following complex-valued 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. P-A.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 P-A.6 Simplify the following complex-valued 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). P-A.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”} P-A.8 Solve the following equation for z: z = j Be sure to find all possible answers, and express your answerlfs) in polar form. P-A.9 Let m = ej2”/N. Prove that 23"] = l/zo. P-A.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. P-2.4 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 —- +2Xd-ETd—ET‘f'” 92 04 cos(6)=l—E+E+~- . 93 ()5 P-2-5 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. [3-2.6 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 P-2.7 Simplify the following expressions: (3) gejfl/3 + 4e-jn/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). P-2.8 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) ’) P-2-9 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 complex-valued signal z(t) such that x(t) : me{z(t)}. ...
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