EEL3135: Homework #4

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Mervyn McGee
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1 EEL335: Homework #4 Problem : For each of the systems below, determine whether or not the system is () linear, () time-invariant, and (3) causal: (a) (b) (c) xn [ ] cos( 04πn) (d) xn [ ] xn [ ] xn [ 5] (e) x[ n ] + xn [ + ] x[ n ] (f) 3x[ n ] + n Problem : Assume the impulse response of an LTI system is hn [ ] given below: hn [ ] δ[ n + ] + δ[ n ] 3δ[ n 4] (-) (a) Give the difference equation for this LTI system (b) Compute the output for an input of: xn [ ] 4δ[ n] + δ[ n ] + 4δ[ n 4] δ[ n 5] + 6δ[ n 6] (-) using the convolution operator for the range of n where 0 (set up a table similar to one of the two tables in the notes) (c) Verify your answer in part (b) by direct substitution into the difference equation for part (a) Problem 3: Assume that the response of an LTI system to the input xn [ ], xn [ ] δ[ n] δ[ n ] (3-) is given by, δ[ n] δ[ n ] + δ[ n 3] (3-) For this system, compute the output y [ n] to the following input: x [ n] 5δ[ n] 5δ[ n 3] (3-3) Problem 4: Assume that the response of an LTI system to the input xn [ ] un [ ] (discrete-time unit step) is given by, δ[ n] + δ[ n ] δ[ n ] (4-) (a) For this system, compute the output y [ n] to the following input: x [ n] 3u[ n] u[ n 4] (4-) (b) Derive the impulse response hn [ ] for this system (c) Give the difference equation for this system - -

2 Problem 5: Assume an LTI system of the following form: M b k xn [ k] k 0 with impulse response, hn [ ] 3δ[ n] + 7δ[ n ] + 3δ[ n ] + 9δ[ n 3] + 5δ[ n 4] (a) Give numeric values for M and b k, k { 0,, M} (b) Compute, n, for input xn [ ] given by, (5-) (5-) xn [ ] 0 n n even odd (5-3) Problem 6: (a) (b) (c) For each part below, give an expression for When applicable, assume xn [ ] 0, n < 0 ; also, un [ ] denotes the discrete-time unit step function, and * denotes the convolution operator (d) a n un [ ] * u[ n], a < Problem 7: un [ ] * u[ n] xn [ ] * u[ n] un [ ] * x[ n] * δ[ n] (a) Compute the impulse response hn [ ] for a filter with frequency response, as illustrated in Figure below, using the inverse DTFT integral Simplify your answer as much as possible 3 Figure π π θ (b) Using L Hopital s rule, simplify the formula for a general filter h g [ n], derived in the notes, for 0, where, h g [ n] ( a) cos( n( θ c ) ) + cos( n( θ c + ) ) cos( nθ c ) cos( nθ c ) (7-) n + δ[ n] π (c) Explain how you can use your result in part (b) to derive the impulse response hn [ ] for part (a), without explicit computation of the inverse DTFT integral - -

3 Problem 8: Given an IIR filter defined by the difference equation: -- yn [ ] + xn [ ] (8-) (a) Determine the transfer function Hz ( ) for this system (b) Compute the system poles (c) Compute hn [ ] for this system Is this system BIBO-stable? (d) Determine for xn [ ] δ[ n] δ[ n ] + 4δ[ n 4] Problem 9: Given an IIR filter defined by the difference equation: y[ n ] yn [ ] + xn [ ] (9-) (a) Determine the transfer function Hz ( ) for this system (b) Compute the system poles (c) Compute hn [ ] for this system Is this system BIBO-stable? (d) Determine for xn [ ] δ[ n] 3δ[ n ] + δ[ n 4] Problem 0: Determine the discrete-time signals x a [ n] and x b [ n], respectively, corresponding to the following z - transforms: X a ( z) z z 6 --z 6 [part (a)] (0-) X b ( z) + z z 07z [part (b)] (0-) Problem : An LTI system has the transfer function Hz ( ), Hz ( ) 3z 4z 4 (-) The input to this system is given by, xn [ ] 0 0δ[ n] + 0cos( 05πn + π 4), < n < (-) Determine the output of the system for all n - 3 -

4 Problem : (a) True or False: For the discrete-time system a n xn [ ] the output for an arbitrary input xn [ ] is given by xn [ ] * h[ n], where hn [ ] is the impulse response of the system (b) True or False: For the discrete-time system yn [ ] + xn [ + ] the output for an arbitrary input xn [ ] is given by xn [ ] * h[ n], where hn [ ] is the impulse response of the system (c) True or False: For the discrete-time system y[ n ] + xn [ ], the frequency response of the system is given by, e jθ (d) True or False: For the discrete-time system ( )yn [ ] + xn [ ], the frequency response of the system is given by, ( )e jθ (e) True or False: An LTI system with frequency response ( ) u[ θ + ] u[ θ ], θ [ π, π], is noncausal (f) Give the difference equation for an LTI system with transfer function Hz ( ): (-) (-) Hz ( ) + z + 3z z 5 (g) Give the difference equation for an LTI system with transfer function Hz ( ): (-3) Hz ( ) z z z + (-4) Problem 3: (a) Give the difference equation of an LTI system with frequency response: Hint: This problem, when approached correctly, does not involve a lot of computation (3-) (b) Is your answer in part (a) causal? If not, specify a difference equation for a causal LTI system with the same magnitude frequency response as in equation (3-) Problem 4: ( ) + cos( θ) + cos( 5θ) Consider the following discrete-time system: (a) Compute the transfer function Hz ( ) of this system (b) Is this system stable? --y[ n ] + x[ n] (c) Give an expression for the output for the input xn [ ] un [ ], assuming that 0, n < 0 (d) Compute the impulse response hn [ ] of this system (e) Compute the frequency response ( ) of this system (f) Give an expression for the output for the following input: (4-) xn [ ] 0 + 5δ[ n 3] + sin( 05πn + ), < n < (4-) - 4 -

5 Problem 5: Consider the LTI system below: --x[ n] --x[ n ] (5-) (a) Compute the frequency response ( ) for this system (b) Give an analytic expression for (c) Give an analytic expression for (d) Compute the output of this system for the following two inputs: Problem 6: Be sure to simplify your answer as much as possible ( ) Be sure to simplify your answer as much as possible x [ n] cos( πn), < n < (5-) x [ n] cos( πn ), < n < (5-3) Assume that you apply the following input xn [ ] to an LTI system: xn [ ] cos( πn ) un [ ] and observe the following output : (6-) 3 ( ) n un [ ] + cos( πn )u[ n] (6-) (a) Using the z -transform pair, a n un [ ] show that (6-3) az Xz ( ) z (b) Compute the transfer function Hz ( ) for this LTI system (c) Compute the impulse response hn [ ] for this LTI system (d) Specify the difference equation that describes this LTI system Is this an FIR or an IIR system? (e) For the input in equation (6-), explain why, θ π cos( πn + He ( jθ) θ π ) does not give the output for this LTI system (f) Specify ( ) θ π and ( ) Hint: This part requires no computation θ π (6-4) - 5 -

ECE-314 Fall 2012 Review Questions for Midterm Examination II

ECE-314 Fall 2012 Review Questions for Midterm Examination II First, make sure you study all the problems and their solutions from homework sets 4-7. Then work on the following additional problems. Problem