ECE 314 Summer 2008 Homework 5: Signals and Systems, Assignments of Signals and Systems

Download ECE 314 Summer 2008 Homework 5: Signals and Systems and more Assignments Signals and Systems in PDF only on Docsity! ECE 314 Summer 2008 HW 5 Due July, 15, 2008 (Tuesday) Textbook problems: Problems 3.48(b, d); 3.49(d); 3.50 (a, d); 3.51 (d); 3.52 (a,c), 3.53 (c), 3.54 (a,b,f), 3.55(c,d) Special Problem 1: Let x(n) = 2(u(n)−u(n−11)) and y(n) = (n+2)[u(n+2)−u(n−17)]. (a) Determine z(n) = x(n) ∗ y(n) analytically. (b) What are the value of n for which z(n) ≠ 0? (c) Compute z(n) using the conv command in MATLAB. Plot z and compare it to your results from part (a). Does you answer agree with part (a)? Suggestion: Define X(1) = X(2) = 0 and X(3) = X(4) = … = X(13) = 2, and Y (k) = (k − 1); for 1 ≤ k ≤ 19. Then plot Z = conv(X, Y ) as a function of the actual time (and not the array index) K = [−2,…, 28]. Special Problem 2: Consider the difference equation 4 1 1( ) ( 1) ( 2) ( ) ( ) 3 3 3 n y n y n y n u n u⎛ ⎞= − − − + + ⎜ ⎟ ⎝ ⎠ n with initial conditions y(−2) = 1 and y(−1) = 3. (a) Find the zero-input response y0x(n). (b) Find the impulse response h(n). (c) Find the zero-initial-condition response y0ic(n). (d) Find and plot the total response y(n) using the results obtained in (a) - (c). Evaluate y(n) at n = 0; 1; 2; 3; 4; 5. Compute y(n) iteratively (directly from the difference equation) for n = 0; 1; 2; 3; 4; 5, and compare to the values obtained from the total solution. (e) Is the LTI system described by the above difference equation stable? Justify your answer.