ECE 301 Fall 2015 Midterm I 10/5/2015 Professor: Aly El Gamal Name: PUID: TA: Xianglun Mao This exam contains 13 pages (including this cover page) and 6 questions. Total of points is 110, (100 basic points + 10 bonus points). Coverage: Chapters 1-3 in the textbook. Closed Book and Closed Notes. Calculators NOT allowed. This test contains Six problems. Show your work in the space provided for each problem. You must show all work for each problem to receive full credit. Always simplify your answers as much as possible. Grade Table Question Points Score 1 15 2 15 3 20 4 25 5 20 6 15 Total: 110 ECE 301 Midterm I - Page 2 of 13 10/5/2015 1. (15 points) Determine the values of P and E for each of the following signals: (a) (5 points) x1 (t) = et (b) (5 points) \u001a x2 (t) = (c) (5 points) x3 (t) = e0 et , 0 t 5000 0, otherwise ECE 301 1.(cont.): Midterm I - Page 3 of 13 10/5/2015 ECE 301 Midterm I - Page 4 of 13 10/5/2015 2. (15 points) A continuous-time signal x(t) is shown in Figure 1, where A = 1, T = 2. Please answer the following questions. Note that you shall only give the resulting figures as answers. Please mark the corresponding values in your answers. Figure 1: The continuous-time signal x(t). (a) (5 points) Calculate x(t 1) + x(t + 1). (b) (5 points) Calculate x(2t) x(t/2). (c) (5 points) Calculate x(t + 1) x(t 1). ECE 301 2.(cont.): Midterm I - Page 5 of 13 10/5/2015 ECE 301 Midterm I - Page 6 of 13 10/5/2015 3. (20 points) Determine the fundamental period of the following continuous/discrete time signals. (a) (5 points) x1 (t) Figure 2: The continuous-time signal x(t). t) + sin( 5 t) (b) (5 points) x2 (t) = cos( 10 3 4 2 3 (c) (5 points) x3 [n] = ej( 3 )n + ej( 4 )n (d) (5 points) x4 (t) = 1 for t R. ECE 301 3.(cont.): Midterm I - Page 7 of 13 10/5/2015 ECE 301 Midterm I - Page 8 of 13 10/5/2015 4. (25 points) Let two continuous-time signal be 4 (t) and r4 (t), which are shown in Figure 3. Figure 3: The continuous-time signal 4 (t) and r4 (t). (a) (15 points) Calculate lim S1,4 (t) 40 , where S1,4 (t) = 4 (t) r4 (t). (b) (10 points) Calculate lim S2,4 (t) 40 , where S2,4 (t) = S1,4 (t) S1,4 (t). ECE 301 4.(cont.): Midterm I - Page 9 of 13 10/5/2015 ECE 301 Midterm I - Page 10 of 13 10/5/2015 5. (20 points) Consider three continuous-time periodic signals whose Fourier series representations are as follows: 100 X 2 1 x1 (t) = ( )k ejk 50 t 2 k=0 x2 (t) = 100 X 2 cos(k)ejk 50 t k=100 x3 (t) = 100 X k=100 jsin( k jk 2 t )e 50 2 Use Fourier series properties to help answer the following questions: (a) (10 points) Which of the three signals is/are real valued? Explain the reasons. (b) (10 points) Which of the three signals is/are even? Explain the reasons. ECE 301 5.(cont.): Midterm I - Page 11 of 13 10/5/2015 ECE 301 Midterm I - Page 12 of 13 10/5/2015 6. (15 points) Given a discrete-time LTI system h[n], please answer the following questions. (a) (5 points) If this system is causal, then what is the condition that h[n] should satisfy? (b) (10 points) (Extra bonus question) Please show that this condition is a necessary and sufficient condition of a causal LTI system. In other words, show that if this given discrete-time LTI system h[n] is causal, then this condition should be satisfied. Meanwhile, show that if this condition is satisfied on this given LTI system h[n], then this system is causal. ECE 301 6.(cont.): Midterm I - Page 13 of 13 10/5/2015