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ECE111_2005SPRING_EXAM - v The Cooper lJnion Department of...

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v .\, The Cooper lJnion Department of Electrical & Computer Engineering ECE111 Signal Processing & Systems Analysis Exam I February 24, 2005 Time: 90 min. Closed book, closed notes. No calculators, Note: cos Acos B : ,l cos (,4 + A) +|cos(A-A). 1. f10 pts.] The impedance of a circuit at 1Mffz is Z = 4- j3O. In this problem, given the appropdate units with all answers. (a) Determino th" r"sistance, reactance, admittance, conductance, susceptance. Also provide the standard letter (e.g., Z for irnpedance) used for all of these quantities. (b) Is the impedance equivalent to a resistor in seies with an inductor or capacitor? (c) Find the value of the inductance or capacitance- leave the answer as an explicit numerical expression (i.e., ready to be plugged into a calculator, no symbols). 2. 16 pts.] Express the following in the form Acos (1011 |d) bywingphasorc, specillcally complex 0, thmetic- not trigonomet c identities, and not phasor diagrams: 3 cos (10?rl + ?r/4) + \cos(t,rt-r12) 'JZ 3. f9 pts.l A chi.rp signal is given by: t(t): Acos (2r J.t + dt+ Bt') (a) Find the instantaneou,s frequency j"", (l). (b) The input voltage to a VCO rl,hich generates this chirp signal must be: 1. a step function; 2. a ramp function (grows linearly with time); 3. a quadratic function (grows - i2). (c) Assume o, B and the tine span we axe examining are small enough that the banci- widthof r(t) is srnall compared with l" (o<< l",P<<f.,and j""1(t) -l"l << /. for the time-span of interest). Suppose e (l) is mixed (multiplied) with another chirp given by: b (t) : B cos (2r f.t - ci - A*) and the result is iowpass fllteredto reject terms near /" and above. That is: y (t) : [z (t) .u (t)]*:"". ."". ., - ,_*, i. Show that y(l) has the lorllr 1lcos(P(l)), i.e., its em,elope is constant. Find K and the instantaneous frequency of g (t). s
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I 4. [10 pts.] A signal c (t) is transmitted tbtou'gh a multipath Jad'r'ng channel, testittng in a received signal g (t) given byr Y (t) = aP lt - rr) + a2r lt -'t2) where or, (1,2 axe constant attenuatiorN (positive real values) and 7r, r, axe f.xeddelays' For example, the first term may be a line-of-sight (LOS) path' which is a direct path from transmitter to receivet, and the second term may repreient a reflected path lf the input signal has a baseband equivalent rse (l) and ca.rrier frequency f., find the baseba.nd representation of g (i), yes (t). Specifically, show that: ltee ft) : AxsB (t - 11) + A2t BB (t - 12) where.4t are complex valued but do not depend on t (they may depend on 7i, oi, f", but not i). Thus, the baseband model is similar to the bandpass model,except that the attenuation factors become complex valued. 5. l8 pts.] A real periodic signal with period 5sec is expanded in a complex exponential Fourierseries, resulting in coefficients {c.}. It is known that .{ : 2, cr:5+ j and c2-4 2j. (a) (b) (") Find the ftindamental and second harmonic frequencies, in ,42- Find the DC power. You have enough information to determine certain other c," coeficients, but not all of them. Write the values for a.llother coeficients which are uniquely detexmined fi-onl the above information.
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