Final Exam Sample Questions Signals and Systems Course Fall 91
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1 1 1. Consider the signal x(t) with the Laplace transform region of convergence a < Real{s} < b where a < 0 < b. Determine the Laplace transform region of convergence for y(t) = x(t)u(t). 2. Given the following fact about a real signal x(t), determine X(s) and specify the region of convergence. a. X(s) has exactly two poles. b. X(s) has no zeros in the finite s-plane. c. X(s) has a pole at s= -1+j. d. e 2t x(t) is not absolutely integrable. e. X(0)=8. 3. We are given the following five facts about a discrete-time signal x[n] with Z-transform X(z): a. x[n] is real and right sided. b. X(z) has exactly two poles. c. X(z) has two zeros at the origin. d. X(z) has poles at z=0.5 e jπ 3 e. X(1) =8/3 Determine X(z) and specify its region of convergence. 4. Consider the following DT pole-zero diagrams, where the circles have unit radius. a. Which if any of the pole-zero plots could represent the z-transform of the following DT signal? b. Which if any of the pole-zero diagrams could represent a system that is stable?
2 2 c. Which if any of the pole-zero diagrams could represent a system that is casual? d. Which if any of the pole-zero diagrams could represent a system that is both stable and casual? 5. Determine which of the following pole-zero diagrams could represent Laplace transforms of even functions of time. Determine expressions for the time functions of those that can represent even functions of time. For those that cannot, explain why they cannot. 6. Consider a discrete-time LTI system with transfer function H(z) = 1 a z, a <1 z a a. Sketch the pole-zero plot of H(z) in the z-plane. b. Is H(z) stable and causal? Why? c. Use the graphic method to show what the magnitude response of the system is. 7. The input x(n) = 2 n [u(n) 3u(n 1)] to an unknown LTI system produces the output y(n) = (3 n 2 n )u(n) a. Determine the impulse response h(n) of the system. b. Is the solution unique? What if the system is known to be unstable? What if it is causal? 8. Consider the digital filter structure shown below. a. Find H(z) for this casual filter. Plot the pole-zero pattern and indicate the region of convergence. b. For what value of k is the system stable. c. Determine y[n] if k = 1 and x[n] = ( 2 3 )n for all n.
3 3 9. Consider the following feedback system in which the box represents a causal LTI CT system that is represented by its system function. a. Determine the range of K for which this feedback system is stable. b. Determine the range of K for which this feedback system has real-valued poles. 10. When the input to an LTI system is The output is x[n] = ( 1 2 ) n u[n] + 2 n u[ n 1] y[n] = 6 ( 1 n 2 ) u[n] 6 ( 3 n 4 ) u[n] Find the system function H(z) of the system. Plot the poles and zeros and indicate the region of convergence. Find the impulse response h[n] of the system. Write the difference equation that characterizes the system. Is the system stable? Is it causal? 11. Questions from Oppenheim Book, 2 nd edition. Chapter 9: 21, 23, 25, 38, 44, 51 Chapter 10: 22, 24, 31, 38
4 4 سیستم LTI علی توصیف شده با معادله تفاضلی زیر را در نظر بگیرید. y[n] + 0.5y[n 1] x[n] = 0 ابتدا پاسخ فرکانسی ) jw H(e را برای این سیستم بدست آورید و سپس پاسخ سیستم را به دو ورودی زیر بدست آورید: الف([ u[n x[n] = (0.5) n ب( X(e jw ) = ejw 0.25 e jw +0.5 )تبدیل فوریه ورودی(.1 با فرض اینکه x(t) سیگنالی با نرخ نایکویست w 0 است نرخ نایکویست را برای سیگنال x(t) cos(w 0 t) + dx(t) dt بدست آورید..2 سیگنال زیر را در نظر بگیرید: x(t) = 2 k 2 cos(40πkt) k=0 الف( می خواهیم یک فیلتر پایین گذر طراحی کنیم که بیشتر از %11 از انرژی سیگنال x(t) را حذف نکند. فرکانس قطع این فیلتر چیست ب( فرض کنید x (t) حاصل خروجی فیلتر قسمت قبل به سیگنال x(t) باشد. پهنای باند و نرخ نایکوییست x (t) را بدست آورید در شکل زیر (t) m 2 و (t) m 1 سیگنال هایی حقیقی با تبدیالت فوریه حقیقی (jw) M 2 و( jw ) M 1 می باشند. اگر داشته باشیم: M i (jw) = 0, for w > 2π 15kHz و اگر فرکانس حامل برابر f c = 100kHz باشد: الف( تبدیل فوریه سیگنال مدوله شده x(t) را بیابید. اندازه تبدیل فوریه و پهنای باند سیگنال x(t) را مشخص کنید. ب( سیستمی طراحی کنید که x را دمدوله و سیگنال های (t) m 2 و (t) m 1 را بازیابی کند. ثابت کنید که سیستم شما کار می کند.
5 5 5. تابع سیستم یک سیستم علی و LTI به شکل زیر است: H(s) = s 4 s 2 + 5s + 4 الف( منطقه همگرایی این تابع سیستم را تعیین کنید. ب( آیا این سیستم پایدار است چرا ج( در حالتی که ورودی سیستم t 2 x(t) = e است خروجی آن را تعیین کنید. 6. درستی و یا نادرستی عبارات زیر را با ذکر دلیل مشخص کنید. الف( اگر X(t) تابعی فرد باشد آنگاه -X(-s).X(s) = e jw 0t ب( تبدیل الپالس در هیچ جای صفحه ی s همگرا نیست. ج( سیگنال مطلقا انتگرال پذیری که در 3=s دارای یک قطب است می تواند راست گرا باشد. H(s) = Y(s) X(s) را در سیستم زیر بدست آورید..7 H 1 (s) = 2 H 2 (s) = 10 s H 3 (s) = 0.1 s + 20 H 4 (s) = 2 s + 4 اگر X(z) تبدیل z مربوط به دنباله x[n] باشد تبدیل z مربوط به دنباله x[k] و ناحیه ی همگرایی آن را بیابید. + k=n+1.8
6 6 نمودار صفر- قطب متناظر با X(z) تبدیل z دنباله علی x[n] در شکل زیر مشخص شده است. نمودار صفر- قطب Y(z) را وقتی [3 + x[ n y[n] = باشد را رسم کنید و ناحیه همگرایی Y(z) را مشخص کنید..9
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