Problem Set 11(Practice Problems) Spring 2014
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1 Issued: Wednesday April, ECE Signals and Sysems I Problem Se (Pracice Problems) Spring Due: For pracice only. No due. Reading in Oppenheim/Willsky/Nawab /9/ Secions // Secions See revised schedule on he websie FINAL EXAM: The final exam is comprehensive; i includes maerial from he enire semeser. As discussed in class, he final will consis of wo pars: Par consiss of muliple-choice quesions. This par will be adminisered during he las class period: :-:5am on May 6,. This par will be worh poins (%) of he final exam score. Exra Credi Opporuniy: There are 5 quesions on Par, worh poin each. Thus i is possible o earn up o 5 poins exra credi on he final by doing well on Par. Par consiss of problems o be solved. This par will be adminisered during he final exam period: 9:5-:5am on May,. This par will be worh 8% of he final exam score. You may no use books, noes, calculaors or oher elecronic devices for eiher par of he final exam. Transform ables will be provided for Par. Problem. in he Oppenheim/Willsky/Nawab exbook Problem 6. in he Oppenheim/Willsky/Nawab exbook Do pars a and b only. Problem 6.5 in he Oppenheim/Willsky/Nawab exbook See addiional problems on following pages.
2 Problem - (Old exam problem) Consider he coninuous-ime LTI sysem ha has he impulse response h() shown in Figure. and he frequency response H(j) shown in Figure h()= sin(π( )) π( ) h() Figure.: Impulse response h() of he CT LTI sysem in Problem H(j) / H(j)= +π π +π π +π π Figure.: Frequency response (magniude and phase plos) of he CT LTI sysem in Problem In pars a-d below, you are given signals ha are inpus o he LTI sysem defined above: Deermine and skech he oupu of he sysem corresponding o each inpu. Make sure o label your skeches! (a) Inpu o sysem: x a () = δ(). Deermine he oupu y a (). Provide a skech of y a () and jusificaion of your answer. (b) Inpu o sysem: x b () = cos(π). Deermine he oupu y b (). Provide a skech of y b () and jusificaion of your answer. (c) Inpu o sysem: x c () = p()cos(π), where p() and is Fourier ransform P(j) are shown in Figure.. Deermine he oupu y c (). Provide a skech of y c () and jusificaion of your answer. (d) Inpu o sysem: x d () shown in Figure.. Deermine he oupu y d (). Provide a skech of y d () and jusificaion of your answer.
3 6 p() 6 P(j) π π π.π Figure.: Signal p() and is Fourier ransform P(j). x d () Figure.: Inpu signal x d () for Problem d 8 8
4 Problem - (Old exam problem) Consider a coninuous-ime LTI sysem wih inpu x() and oupu y(), as shown in Figure.: x() Unknown LTI Sysem H(j) y() Figure.: Sysem for Problem. The frequency response H(j) of his sysem is no known. We do know ha when he inpu is he signal x() shown in he plo below, he oupu is y() shown in he same figure. Noe ha x() and y() are periodic (hey repea ouside he inerval shown)..5 x() y() Time (seconds) Figure.: Inpu/oupu pair for he sysem in Problem. Figure. on he nex page shows he frequency responses of six sysems, H (j) hrough H 6 (j). These frequency responses are purely real. (a) Which of he frequency responses shown in Figure. could be he frequency response of he unknown LTI sysem? Hin: There are a leas wo possibiliies. Please provide a brief jusificaion (Answers wihou jusificaion will receive zero credi.) (b) Suppose ha you are old ha he frequency response of he unknown sysem is definiely one of he choices in Figure.. If you could run a second es signal x() hrough he sysem and measure he oupu y(), could you deermine which of your answers from par (a) is frequency response of he unknown sysem? If your answer is yes, explain wha ype of es signal you would use. If your answer is no, explain why you canno deermine which of he choices is he correc one.
5 H (j) H (j) π +π 6π +6π H (j) H (j) +π +5π 8π +8π H 5(j) H 6(j) π +π 6π π +π +6π Figure.: Frequency response choices for Problem.
6 Problem - (Old exam problem) An LTI sysem has he frequency response H(j) shown below: H(j).5π.5π (a) Is he impulse response h() of his sysem real? Why or why no? (Answers wihou jusificaion will receive zero credi.) (b) The inpu o his sysem is he periodic signal x() shown below. x() Deermine he oupu y() of he sysem.
e 2t u(t) e 2t u(t) =?
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