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ECE 351 1/8 Homework Set 1 ECE 350 Review, Mostly (No lecture material required for problems 1 – 9, 12 and 13.) 1. The Fourier Transform of a continuous, periodic signal is always a. continuous b. periodic c. discrete d. analog e. none of the above 2. Briefly explain what we mean by the system response, H(j ) or H( ), of a linear, time-invariant system. 3. Consider the signal : x(t) = 3 sin(8 t) + 3, for t (0, 2). a. What is the frequency in rad/sec? b. What is the frequency of x(t), in Hz? c. What is the period, in seconds? d. Graph the signal by hand. e. Use MATLAB to graph the signal on the interval (0, 2). f. Use MATLAB to graph the discrete time signal, x[n], that results if the time between samples is T s = .05. (Remember to plot x on an n-axis, at integer values of n ; use stem , not plot .) g. Use MATLAB’s subplot command to plot the answer for (f) beneath the answer for (e), on a single page. 4. Consider the ideal, low-pass filter with transfer function as shown below: H( ) 5. Write the formula for the sum of an infinite geometric series whose first term is a and whose ratio is r , where | r | < 1: a + ar + ar 2 + … = __________ 6. By definition, impulse response h(t) is the output of a linear, time-invariant system when the input is: __________ 7. Use partial fraction expansion to find the coefficients A, B, and C in the equation given below: , rad/sec -10 10 5 Find the output y(t) if the input to the filter is: x(t) = 2 sin(30 t) + 4 cos(5 t) + 5
ECE 351 2/8 3 2 1 ) 3 )( 2 )( 1 ( 11 9 2 2 x C x B x A x x x x x (Hint: See Lathi, p. 29) 8. Consider a linear time-invariant (LTI) system with input x(t) and output y(t). Find the output (in terms of y(t)) if the input is: a. 4 x(t) b. 3 x(t – 2) c. 4 x(t) + 3 x(t – 2) 9. Consider the signal x(t) shown below: a. the period is T 0 = _______ sec. b. the fundamental frequency is 0 = ___________ rad/sec., or f 0 = ________ Hz. c. If the Fourier Series is given by x(t) = a 0 + 1 n {a n cos(n 0 t) + b n sin(n 0 t) }, calculate a 0 , the dc term. 10. (Lecture 1) True or false – give a counterexample or an explanation for statements that are false: a. If a signal is continuous in time, it is analog. b. If a signal is discrete in time, it is digital. c. The equation y[n] = 3 x[n] + 2 x[n – 1] represents a time-invariant system. d. Taking time samples from a continuous-time signal results in a digital signal. e. The signal e -.2n is periodic. f. Analog signals have a finite number of amplitude levels. g. Causal signals are negative for negative values of time. 11. (Lecture 1) Consider the system with input x[n] and output y[n] = 3 x[n]. Is this system linear? (Show why or why not.) 12. Find and sketch the Fourier Transform, X( ), of the signal: x(t) = sinc(2t), where sinc(t) (sin(t))/t. Hint: See Lathi, FT Table, p. 702. Ans: ( /2) rect( /4) x(t) -5/2 -3/2 -1/2 1/2 3/2 5/2 10 t … , sec. … 4
ECE 351 3/8 13. Find the Inverse Fourier Transform of: X( ) = rect( T/(2 )) where rect is as defined on Lathi, p. 687. 1 Ans: x(t) = (1/T) sinc( t/T) 14. First use the internet to read something about the specification for USB 2.0. Then answer the following (with 1 or 2 sentences per question). a. What is the functional difference between the Standard A and Standard B connections on a USB connector – i.e., which plug goes where? b. What is the maximum amount of power (based on the standard voltage and the maximum current ) that can be drawn from the computer (to “power up” a device) using a USB connection? c. Explain the terms upstream and downstream as used in the USB specs. Mostly Following Lecture 2 15. Use MATLAB to graph the following; use MATLAB’s subplot command to put parts a, b, and c on a single page, and d and e on another single page: a. x[n] = (.9) n , n = - 4, …, 4 b. y[n] = x[-n], where x[n] is given in part a, n = - 4, …, 4 c. y[n] = x[n – 2], where x[n] is given in part a, n = - 4, …, 4 d. x[n] = 5 { u[n] – u[n – 3] }, n = - 5, …, 10 e. y[n] = (2n) { u[n -1] – u[n – 4] } n = - 5, …, 1 0 16. True or false – give a counterexample or an explanation for statements that are false: a. Every energy signal is of finite duration. b. (Lectures 2 and 3) [n] = u[n] – u[n – 1] c. The signal u[n] is causal. d. The signal u[n] is even. e. (Lecture 1) The equation y[n] = 3 x[n] + 2 x[n – 1] represents a time-invariant system. 17. (Lectures 2 and 3) Simplify, writing your answer as a sum of (possibly shifted, possibly weighted) delta functions. a. u[n] – u[n – 3] b. 3u[n + 2] – 3u[n – 1] c. u[n – 2] – u[n] 18. (Lectures 2 and 3) Rewrite the following expressions as a sum of (possibly shifted, possibly weighted) unit-step functions. 1 Note that this X( ) can also be written in the f domain: X(f) = rect( T f ). Also see hint for #12.
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ECE 351 4/8 a. [n - 1] + [n – 2] b. 4 { [n] + [n - 1] + [n - 2] } c. 2 [n] + 3 [n – 1] 19a. Find the energy in x[n] shown below: 19b. Find the energy in y[n] = - x[n - 2] where x[n] is given in part a. 19c. Find the energy in y[n] = 3 x[n] where x[n] is given in part a. 20. Lathi: Problem 8.1-1 21. Lathi: Problem 8.1-2abc 22. Lathi: Problem 8.1-7a 23. (Lecture 2) Suppose that a signal x(t) is sampled at rate f s = 5 (sample/sec.), and the sampled signal is: x s (t) = 1 (t) + 3 (t – .2) + 5 (t – .4) + 2 (t – .6) + 3 d(t - .8) a. The time between samples is T s = _______ sec. b. Use the interpolation equation to find x r (t), the signal that would be recovered by an ideal LPF with input x s (t) and cut-off frequency f s /2. (Hint: See pp. 14 – 16 of Lecture 2). c. Use MATLAB to plot both x s (t) (as circles) and the signal recovered by the LPF, x r (t) (as a line), on a single graph, for t values from -.2 to 1. 24. Problem 8.3-1 25. Problem 8.3-2 26. Problem 3.1-1 27. Problem 3.1-2 (Remember to use the fact that the signals are periodic.) 28. Problem 3.2-1 29. Problem 3.2-2, power only 30. Problem 3.2-3abc 31. Problem 3.3-1 (Hint for e: use the shortcut for periodic signals if the signal is periodic) 32. Problem 3.3-3abc (for n values from 0 to 10) Following Lecture 3 33. Problem 3.3-5ac 34. Problem 3.3-6ac x[n] … … n -3 -2 -1 0 1 2 3 4 5 2 2 2 2
ECE 351 5/8 35. For each of the signals below, determine whether or not they are periodic. For each of the periodic signals, find the period, N (in # of samples). a. 4 cos[2.4 n] b. 3 cos[2n] c. 5 sin[n/2] d. 5 cos[3 n/4] e. exp(j 2 n) 36. (Lecture 2) Use MATLAB to generate the following plots. In each case, plot the function for n values from -8 to 8. Use the subplot command to plot parts a, b, and c on one page, and parts d and e on another. a. x[n] = 2 [n] - [n + 2] b. y[n] = x[n - 3], where x[n] is defined in part a c. y[n] = x[-n], where x[n] is defined in part a d. x[n] = 4{ u[n] – u[n – 4] } e. x[n] = e -.3n { u[n] – u[n – 5] } Verify that your MATLAB plots are predictable according to the graph theory covered in class. 37. True or False – give an explanation or counterexample for the statements that are false: a. (Lecture 1) The system with difference equation: y[n] = (n + 3) x[n] is causal. b. (Lecture 1) The system with difference equation: y[n – 2] = x[n] is causal. c. (Lecture 2) A discrete-time signal with finite power is always a power signal and not an energy signal. d. (Lecture 2) If a discrete-time signal has infinite energy, then it is a power signal. e. (Lectures 2 and 3) u[n] – u[n – 1] = [n - 1] From Lathi: 38. Problem 3.4-9 c, e 39. Problem 3.5-1 40. Problem 3.5-2 Following Lecture 4 41. Problem 3.6-1 42. Problem 3.6-2 43. Problem 3.6-3 44. Problem 3.6-5 Following Lecture 5 45. Problem 3.7-1
ECE 351 6/8 Following Lecture 6 46. Find x[n] * y[n] if x[n] = { 1, 1, 1} and y[n] = { 1, 1, 1, 1}. 47. Evaluate: a. { [n] + 3 [n – 2] } * sin(3 n/10) b. { [n] + 3 [n – 3] } * [n + 1] c. 2 [n] * { cos(3 n/10) + sin(3 n/10) } 48. Let x[n] = {2, 2, 2} and let h[n] = {1, 3}. Find y[n] = x[n] * h[n]. 49. Let x[n] = u[n] – u[n – 2]. Let h[n] = n {u[n – 2] – u[n – 4]} Find y[n] = x[n] * h[n]. 50. Let x[n] = (.3) n u[n] and let h[n] = (.5) n u[n]. Find y[n] = x[n] * h[n]. Following Lecture 7 51. Consider the system with input/output relationship as shown: a. Find the impulse response, h[n]. Ans: h[n] = 3 [n] + 2 [n – 3] b. Find the output if the input is x[n] = .2 n Ans: 253 (.2) n c. Is the system causal? d. … linear? e. … time-invariant? 52. Lathi 3.8-1 53. Lathi 3.8-3 54. Lathi 3.8-10 55. Lathi 3.8-11 56. Find the over-all impulse response, say h[n], for the system modeled below if h 1 [n] = {1, 2, 1 }, h 2 [n] = 4[u[n] – u[n-2]], and h 3 [n] = 2 [n] – 3 [n-3] : 57. A certain mobile communications channel is modeled as having impulse response: h[n] x[n] y[n] = 3 x[n] + 2 x[n – 3] Justify your answers. h 1 [n] x[n] h 2 [n] h 3 [n] y[n]
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ECE 351 7/8 h[n] = [n] + .5 [n – 1] + .2 [n – 3] Find the output, y[n], if the input is x[n] = u[n] – u[n – 4]. 58. Find the characteristic modes of the system with difference equations: a. y[n] – (5/6) y[n – 1] + (1/6) y[n – 2] = x[n] b. y[n + 2] + (2/3) y[n + 1] + (1/9) y[n] = x[n + 1] 59. Convolve x[n] with y[n]. (Call the output z[n]) a. x[n] = {1, 2, 3}, y[n] = [n] + 2 [n – 3] Ans: {1, 2, 3, 2, 4, 6} b. x[n] = {1, 2, 3}, y[n] = u[n] – u[n – 4] c. x[n] = {1, 2, 3}, y[n] = {1, 2, 3} d. x[n] = {1, 1, 2, 1}, y[n] = {2, 1, 2} Ans: {2, 3, 7, 6, 5, 2 } e. x[n] = 2n u[n], y[n] = 3n u[n] f. x[n] = y[n] = u[n] - u[n – 4] Use the MATLAB function conv to check your answers. Syntax: conv(a, b) convolves the causal sequences a and b when a and b are given as vectors. 60. Consider the systems a. … with impulse response: h[n] = (1/4) n u[n]. Is this system BIBO stable? b. … with difference equation: y[n + 2] – 5.5 y[n + 1] – 3 y[n] = 2 x[n] Is this system BIBO stable? Following Lecture 8 61. A particular “running average” system outputs the average of the most recent 4 terms of the input sequence, and has impulse response h[n] = { [n] + [n – 1] + [n – 2] + [n – 3] } / 4 a. Find a closed-form expression for output y[n] for an arbitrary input x[n]. y[n] = ___________________ b. Find the output if the input is x[n] = (-.9) n u[n]. c. Find the output if the input is: {5, 5, 5, 5, 5, 5, 5}. d. Execute the following MATLAB code to generate two signals (sinusoid x[n] and the noisy version of the sinusoid, x_noisy[n]), the latter of which will be input to the filter described above: >> n = 0: .2 : 20; >> x = 4 * cos(2*pi*n/5); % a clean sinusoid >> x_noisy = x + normrnd(0, .5, 1, length(n)); % sinusoid with noise
ECE 351 8/8 Find the output y[n] from the running average filter given above in part (a), using x_noisy[n] as input. Use subplot to graph x[n], x_noisy[n], and your answer y[n] on a single page. 62. Consider the difference equation y[n + 2] – (1/4) y[n + 1] + (1/8) y[n] = 2 x[n + 1] – 3 x[n] a. Find the dc gain of the system. b. Find the steady-state output if the input is x[n] = 2. c. Find the steady-state output if the input is x[n] = .7 n . d. Find the steady-state output if the input is x[n] = cos( n/8). e. Use the MATLAB “filter” function to find the output for parts b, c, and d, for n = 0 : 30. Use the subplot command to plot all 3 graphs on the same page. Note: these answers will not necessarily agree with your answers obtained analytically - with MATLAB, you are only inputting a “chunk” of 31 values rather than the entire set of x[n] values, starting at n = - . (You are inputting x[n] values seen through the window u[n] – u[n - 31].) However, the steady-state answers should agree with your analytical results. 63 . Both the “filter” function and the “conv” function can be used to find the zero - state output of a discrete-time LTI system in MATLAB. In what sense are the two functions different? (Focus on the form in which the problem is given. What should be the givens to use the filter function? What should be the givens to use the convolve function?) 64. (Resonance) Consider the difference equation: y[n] - 1.5 y[n-1] + .5 y[n-2] = x[n] Without actually finding the outputs for the inputs given below, predict which of the inputs would yield the largest (magnitude) output: a. x[n] = (.25) n u[n] b. x[n] = (4) n u[n] c. x[n] = cos( n/4) u[n] d. x[n] = (.6) n u[n] 65. (Lecture 2) Consider the signal x(t) with spectrum X(f) as given below. Find the spectrum of the sampled signal if the sampling rate is 200 samples/sec; i.e., find the spectrum of: x(t)   (t - .005 n) . (Hint: See Lecture 2, pp. 7 - 9; sketch your answer for frequency values from -500 to 500 Hz.) -100 100 X(f) f

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