The ability to manipulate complex numbers is important in the analysis of circuits and systems. In EE 210 you used complex numbers known as phasors to quickly determine the sinusoidal steady-state response of an electric circuit. Please carefully read Background Section B.1 in Lathi before attempting the following review problems. Except for the MATLAB questions, solve the equations by hand, do not use a calculator. To receive credit for the MATLAB questions, include an m-file with your name and section that shows the MATLAB commands used to answers parts 4 and 5. 1. (2 points) Express the complex number 31/3 in rectangular form a + 3b, where a and b are real.
2. (2 points) Without using a calculator, express the following complex number 4/23 (2+1) (1-21) (2+ 2) in polar form re, where r and (in degrees) are real numbers (calculators are not permitted on the exams; you must be able to work with complex numbers by hand).
3. (8 points) Given s, u, v, and w are arbitrary complex numbers with nonzero magnitude, establish the following results:
(a) (2 points) s s* = s12
(b) (2 points) s+s* = 2 Re(s)
(c) (2 points) s-s' = 27 Im(s)
(d) (2 points) su/vw = ($||u/luwe(48+Zu-Zu-Zw)
4. (4 points) You can determine the roots of the equation " + an-13"-1+ ... +213 +0,= 0 using the MATLAB command roots. For example, to find the roots of 3+4x2+2-3 = 0, enter >> roots([1, 4, 1, -3]) and MATLAB will return the roots. First determine the roots of the equation 1+1=0 by hand and sketch their location in the complex plane. Verify your results using the MATLAB command roots.
5. (4 points) The MATLAB command conv finds the product of two polynomials. For example, suppose that p = 14 + 3x3 - 202 + < -5, 9 = x + 4.++10, and we wish to determine the product s = pq. Use the MATLAB commands >>p = >q = >s = (1,3, -2, 1,-5); [1,0,0,4, 1, 10); conv(p, q) In this case s = [1,3, -2,5, 8,5, 32, -39,5, -50] and so s=+ 3.28 - 2x + 5.0 +8.25 + 5x4 + 32c3-392+ 50 - 50. Determine the product (24 + 3c² +2:+1) (4x4 - 3x2 + 2x + 1) (x3 + 2x + 3) using the MATLAB command conv.