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### exam3-solutions-s12

Course: ECE 350, Spring 2012
School: Boise State
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Word Count: 829

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Spring ECE350 2012 - Exam 3 25 April 2012 Name: ___Solutions ____ ECE 350 Signals and Systems Spring 2012 Exam #3 One 8 x 11 sheet of notes, and a calculator are allowed during the exam. Write all answers neatly and show your work to get full credit. Rationalize all complex fractions. A calculator is allowed during the exam, but should not be needed. Write neatly and WATCH YOUR ALGEBRA! Problem 1 Problem 2...

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Spring ECE350 2012 - Exam 3 25 April 2012 Name: ___Solutions ____ ECE 350 Signals and Systems Spring 2012 Exam #3 One 8 x 11 sheet of notes, and a calculator are allowed during the exam. Write all answers neatly and show your work to get full credit. Rationalize all complex fractions. A calculator is allowed during the exam, but should not be needed. Write neatly and WATCH YOUR ALGEBRA! Problem 1 Problem 2 Problem 3 Problem 4 Total /32 /15 /28 /25 /100 ECE350 Spring 2012 - Exam 3 Name: 25 April 2012 1. (35 points) Given the following system diagram ___Solutions ____ c1(t)=cos(240t) x1(t) + a1(t) X y1(t) 2 x1(t) + b(t) d(t) H2(j) H1(j) a2(t) X y2(t) y3(t) X c3(t)=cos(200) c2(t)=cos(120t) The spectrum for the real signal x1(t) is shown below: 2 X1(j) -40 40 and the filters H1(j) H2(j) 1 1 -220 30 -30 220 (a) (4 points)Draw and fully label A1(j). 4 X1(j) 2 -40 40 (b) (4 points) Draw and fully label A2(j). 2 X1(j) -30 30 (c) (5 points) Draw and fully label Y1(j). 2 1 Y1(j) 2 1 -280 -240 -200 200 240 280 (d) (5 points) Draw and fully label Y2(j). Y2(j) 1 1 -150 -120 -90 90 120 150 ECE350 Spring 2012 - Exam 3 25 April 2012 (e) (4 points) Draw and fully label B(j). B(j) 2 Name: 1 1 1 ___Solutions ____ 2 1 -280 -240 -200 -150 -120 -90 90 120 150 200 240 280 (f) (5 points) Draw and fully label D(j). D(j) 1 1/2 -220 -200 1 -150 -120 -90 1/2 90 120 150 200 220 (g) (5 points) Draw and fully label Y3 (j). Y3(j) 1/4 -420 -400 1/2 -350 -320 -290 1/2 1/4 1/4 1/2 -110 -80 -20 0 20 50 80 -50 1/2 1/4 110 290 320 350 400 420 ECE350 Spring 2012 - Exam 3 25 April 2012 2. (15 points) A signal is Name: ___Solutions ____ x(t ) cos2 (30t ) (a) (9 points) What is the equation describing X(j)? Which properties did you use to decide this? Draw X(j). 2 methods: 1) trig identities: cos(2) = 2(cos2()-1) cos2() = (1+cos(2))/2 X(j) = () + /2 (+60) + /2 (-60) 2) modulation * -30 30 -30 30 = /2 /2 -60 0 60 (b) (6 points) According to Nyquist, what is the smallest sampling frequency, s, that may be used to sample x(t) so that full reconstruction is possible? Nyquist states that s > 2m s > 2*60 = 120, so 120.00001 Here because the signal has impulses at the larges frequencies, the strictly greater matters. ECE350 Spring 2012 - Exam 3 25 April 2012 3. (28 points) For the transfer function Name: ___Solutions ____ 1 s s s 20 a) (5 points) Sketch the pole-zero plot for this transfer function H ( s) 2 Complex Plane Im 6 4 2 0 Re -2 -4 -6 -6 -4 -2 0 2 4 6 b) (4 points) How many and what are the possible ROCs for this transfer function? Sketch the possible ROCs. Three possible ROCs: Re{s}<-4, < -4 Re{s} < 5, Re{s}>5 Complex Plane Im Complex Plane Im Complex Plane Im 5 5 5 0 Re 0 Re 0 -5 -5 ROC -5 0 5 -5 ROC -5 0 Re 5 ROC -5 0 5 c) (6 points) For each ROC in (b) above, determine stability and causality of the impulse response. ROC Re{s}<-4 -4 < Re{s} < 5 Re{s}>5 Stability not stable stable not stable Causality not causal not causal causal ECE350 Spring 2012 - Exam 3 25 April 2012 d) (8 points) If h(t) is stable, find h(t). H (s) Name: ___Solutions ____ 4/9 5/9 1 s s s 20 s 5 s4 2 h(t) = -5/9e-4tu(t) +4/9e5tu(-t) e) (5 points) Sketch each term of h(t) from (d). -5/9*exp(-4*t)u(t) 4/9*exp(5*t)u(-t) 0 0.35 -0.05 0.3 -0.1 0.25 -0.15 0.2 -0.2 0.15 -0.25 0.1 -0.3 0.05 -0.35 -0.4 -5 0 t 0 -5 5 0 t 5 -5/9*exp(-4*t)u(t) + 4/9*exp(5*t)u(-t) 0.3 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 -5 -4 -3 -2 -1 0 t 1 2 3 4 5 ECE350 Spring 2012 - Exam 3 25 April 2012 4. (20 points) For the transfer function Name: ___Solutions ____ s 2 25 s 2 s 1.25 a) (5 points) Sketch the pole-zero plot for this transfer function. H ( s) Complex Plane Im 6 4 2 0 Re -2 -4 -6 -6 -4 -2 0 2 4 6 b) (8 points) Sketch the magnitude of the Fourier Transform associated with H(s). limit approaches a constant maxima near =1 -5 -1 1 5 touches zero ECE350 Spring 2012 - Exam 3 Name: ___Solutions ____ 25 April 2012 c) (7 points) Draw the magnitude of the straight line Bode plot for this transfer function. Does it agree with your answer in (b)? Why/why not? s 2 25 1 2 5 s 25 H (s) 2 s s 1.25 s 2 s 1.25 1 1.25 1.25 Both numerator and denominator are second order with complex (or imaginary) roots. Therefore keep in second order form.\ The numerator bends upward 40dB/decade at =5. The denominator bends downward at -40dB/decade at =sqrt(1.25)~1.1 20log10(25/1.25)=26 20 0 -20 -26 -40 1 5 10 ECE350 Spring 2012 - Exam 3 25 April 2012 Name: ___Solutions ____ Bode Diagram Magnitude (dB) 50 0 -50 -100 - 150 45 Phase (deg) 0 -45 -90 -135 -180 0 1 10 10 Frequency (rad/sec) It is the same but in part (b) we didnt know what the gain of the constant attenuation would be for frequencies greater than =5. The bode plot tells us that it is a lower gain than the attenuation at low frequencies. d) (5 points) What type of filter is this? This is a low pass filter. It passes low frequencies. The high frequencies are passed with a constant gain, but it is much lower than the gain of the low frequencies. In part (b) we didnt know what the gain of the constant attenuation would be for frequencies greater than =5. The bode plot tells us that it is a lower gain than the attenuation at low frequencies.
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