Q1) [20 points] answer for the following questions (ON THIS SHEET):

Transcription

1 Dr. Anas Al Tarabsheh The Hashemie Universiy Elecrical and Compuer Engineering Deparmen (Makeup Exam) Signals and Sysems Firs Semeser 011/01 Final Exam Dae: 1/06/01 Exam Duraion: hours Noe: means convoluion means muliplicaion Q1) [0 poins] answer for he following quesions (ON THIS SHEET): ( )d 1.1) Calculae he funcion sinc() 1.) Plo Y ( ω) (i.e. he frequency-domain of y ( ) ) given ha y() = ( sinc() ) j 1.) Deermine he FT of sgn( ) e : j 1.4) Deermine he FT of sgn( ) e : jπ d d 1

2 1.5) For x( ) = cos ( ) cos( 4 ), hen C ox 1.6) For x( ) = cos ( ) cos( 4 ) and y ( ) = x( ) 4, hen C oy 1.7) Calculae he inegraion δ ( π ) δ ( π ) sin( π )d : 1.8) Find he period of he signal x( ) = cos( ) + cos(π ) (in seconds): 1.9) The Laplace ransform of he funcion f ( ) = sin( 5 ) is : 1.10) The Laplace ransform of he funcion f () = u( τ). dτ is: 0

3 Q) [0 poins] A coninuous-ime LTI sysem has he inpu ( ) response h ( ) as shown in Fig.1..1 Wrie he funcion of ( ) x. h.. Wrie he funcion of ( ). Find and plo he sysem oupu ( ).4 Find and plo X ( ω).5 Find and plo H ( ω) y. x and he impulse

4 The Hashemie Universiy Final Exam Elecrical Engineering Deparmen Signals and Sysems Dae: 1/06/01 Exam Duraion: hours All work should be done on he shees provided. You mus show all work for each problem o ge full credi. Schedule your ime carefully. Noe: means convoluion and means muliplicaion. Q1) [4 poins] Answer he following quesions wih shor answers: 1. The FT of δ +1 is: sinc. The ()d. If he FT of ( ) rec( a ) h = equals H ( ω), hen he FT of ( h ( ) ) j 4. The FT of sgn ( ) e is: 5 of 1Page

5 j 5. The FT of sgn ( ) e is: 6. The FT of cos( ) cos( ) 7. For x( ) = cos ( ) cos( ) and y ( ) = x( ) 4, hen C oy 8. Consider a sysem wih he relaionship of is inpu and oupu given by y() = x( a) Wha are he values of a such ha he sysem is causal? + 5 τ dτ. 5 of Page

6 Q) [4 poins] Consider he LTI sysem shown in Figure 1. If he inpu x( ) = e u( ) produces he oupu y( ) = e u( ) e u( ). Deermine h( ) of he sysem.... Q) [ 8 poins] for he sysem shown in Figure, Given ha he inpu x () = sin(8 ) + cos(4 ), and he impulse response h () = sin c(5 ) answer he following : Figure a) Check if x() is periodic or no.. b) Skech X ( W ), H ( W ), and Y ( W ).. 5 of Page

7 c) Find y().. Q) [8 poins] for he following inpu signal x ( ) x () + 1 = 1 0,,, elsewhere (a) Find energy of he signal. (b) Find and plo ( ) = x( ) + x( 1) y of 4Page

8 Q4) [6poins] Find rigonomeric Fourier series coefficiens (A 0, A K, B K ) of x( ) 5cos( π) = of 5Page

9 The Hashemie Universiy Final Exam Elecrical Engineering Deparmen Signals and Sysems Dae: 4/05/01 Exam Duraion: hours All work should be done on he shees provided. You mus show all work for each problem o ge full credi. Schedule your ime carefully. Noe: means convoluion and means muliplicaion. Q1) [0 poins] Fill in he following able he mos appropriae answer for he following quesions (IN CAPITAL LETTER): ω The FT of δ +1 is: (a) δ (b) sinc. The ()d jω e (c) j e ω (d) jω e jω (e) e (a) 0 (b) 1 (c) π (d) π (e) h = equals H ( ω), hen he FT of ( h ( ) ) H (b) ( H ( ω) ) d H (c) H ( ω) ( ω) (d). If he FT of ( ) rec( a ) (a) ( ω) dω (e) ( ω) dh dω j 4. The FT of sgn ( ) e is: (a) j (b) j π (c) jπ (d) jπ (e) jπ j 5. The FT of sgn ( ) e is: j jπ (a) (b) ω ( ω 1) 4 jπ (c) ( ω 1) 4 jπ (d) ( ω + 1) j π (e) ( ω + 1) 6. The FT of cos( ) cos( ) (a) 0 (b) 1 (c) j (d) j (e) 1 5 of 1Page

10 7. For x( ) = cos ( ) cos( ) and y ( ) = x( ) 4, hen C oy (a) 4 (b) (c) (d) 0 (e) 4 8. If a sysem is characerized by he inpu-oupu relaion y ( ) = x( a + b) where a > 0 and > 0 hen he sysem is: (a) Linear (b) Time-invarian (c) Causal (d)memoryless (e) Asable 9. If ( ) π π + x is a coninuous signal hen he inegraion δ δ x()d (a) (b) 1 (c) 0 (d) 1 (e) b, 10. The odd par of x ( ) = u( ) is: (a) 1 x( ) (b) x( ) (c) x ( ) 0. 5 (d) 1 + x( ) (e) x( ) 11. The period of he signal x( ) = cos( ) + cos(π ) is (in seconds): (a) 0 (b) π (c) 1 (d) 1. The Laplace Transform of he funcion f () = sin is : a) s + 9 b) ( ) s + c) s + 9 d) s + 9 π (e) Aperiodic e) ( s + 9) 1. The Laplace ransform of he funcion a) s s 4s + b) s s 4s + 5 c) f () e cos s = is: s 4s + 5 d) s s 4s + e) ss ( ) s 4s Consider a sysem wih he relaionship of is inpu and oupu given by y + () = x( τ a) dτ where a is a consan. The sysem is 5 (a) inverible and sable (c) sable bu no inverible (b) inverible bu no sable (d) neiher inverible nor sable 15. Consider again he sysem in (14), he sysem is causal if: (a) a 15 (b) a (c) a 6 (d) a 1. 5 (e) a of Page

11 Q) [ poins] Consider he LTI sysem shown in Figure 1. If he inpu x( ) = e u( ) produces he oupu y( ) = e u( ) e u( ). Deermine h( ) of he sysem.... Q) [ 8 poins] for he sysem shown in Figure, Given ha he inpu x () = sin(8 ) + cos(4 ), and he impulse response h () = sin c(5 ) answer he following : Figure a) Check if x() is periodic or no.. b) Skech X ( W ), H ( W ), and Y ( W ).. 5 of Page

12 c) Find y().. Q) [5 poins] for he following inpu signal x ( ) x () + 1 = 1 0,,, elsewhere (a) Find energy of he signal. (b) Find and plo ( ) = x( ) + x( 1) y of 4Page

13 Q4) [6poins] Find rigonomeric Fourier series coefficiens (A 0, A K, B K ) of x( ) cos( π) = of 5Page

14 The Hashemie Universiy Elecrical Engineering Deparmen Signals and Sysems Final Exam 10/1/01 Exam Duraion: Hour Insrucors: Dr. Anas Tarabsheh, Ashraf A. Ali Name :.. Number:... Secion. Noe: means convoluion means convoluion Q1) [0 poins] Choose he mos appropriae answer for he following quesions: 1) The ordinary differenial equaion (ODE) of he following simulaion diagram is: (a) 0.5y y + 4x = 0 (b) 0.5y + y 4x = 0 (c) 4 y + x = 0 (d) 8 y + y + 4x = 0 ) The FT of δ +1 is: ω (a) δ +1 (b) e jω ω (c) e j (d) e jω jω (e) e ) The sinc ()d (a) 0 (b) 1 (c) π (d) π (e) 1 jπ d d 4) The signal y() = ( sinc() ) can be represened in frequency-domain as: h = equals H ( ω), hen he FT of ( h ( ) ) H (b) ( H ( ω) ) d H (c) H ( ω) ( ω) (d) (e) 5) If he FT of ( ) rec( a ) (a) ( ω) dω ( ω) dh dω 6) The FT of j ( ) e (a) j (b) j π (c) jπ (d) jπ (e) jπ sgn is: 1

15 j 7) The FT of sgn ( ) e is: j jπ (a) (b) ω ( ω 1) 4 jπ (c) ( ω 1) 4 jπ (d) ( ω + 1) j π (e) ( ω + 1) 8) The FT of cos( ) cos( ) (a) 0 (b) 1 (c) j (d) j (e) 1 9) For x( ) = cos ( ) cos( ) and y ( ) = x( ) 4, hen C oy (a) 4 (b) (c) (d) 0 (e) 4 10) If a sysem is characerized by he inpu-oupu relaion y ( ) = x( a + b) a > 0 and b > 0, hen he sysem is: (a) Linear (b) Time-invarian (c) Causal (d)memoryless (e) Asable where 11) If x( ) = [ u( ) u( π )] sin( ) hen x( ) is: π π (a) x() = rec sin() π + π (d) x() = rec sin() π + π π = rec sin π (b) x() = rec sin() (e) x() () π π (c) x() = rec sin() x is a coninuous signal hen he inegraion π π δ δ + sin()d (a) (b) 1 (c) 0 (d) 1 (e) 1) If ( ) 1) The odd par of x ( ) = u( ) is: (a) 1 x( ) (b) x( ) (c) x ( ) 0. 5 (d) 1 + x( ) (e) x( ) 14) The period of he signal x( ) = cos( ) + cos(π ) is (in seconds): (a) 0 (b) π (c) 1 (d) π (e) Aperiodic

16 =. If ( ) 15) The oupu signal in Fig.1 is expressed as y ( ) x( ) * h( ) h 1 is causal and asable while h ( ) is non-causal and sable, hen h ( ) is always: (a) causal and asable (b) causal and sable (c) non-causal and asable (d) non-causal and sable (e) answers (a) or (c) (f) answers (b) or (d) 16) The ramp funcion shown in Fig. can expressed as: (a) ( )[ u( ) + u( + ) ] (b) ( ) [ u( + ) + u( ) ] (c) u( ) u( ) (d) ( ) u( ) + u( + ) (e) ( ) u( + ) (f) u( ) [ ] [ ] [ ] [ ] 17) The Laplace ransform of he funcion f () = sin is : 9 9 a) b) c) d) s + ( s + ) s + 9 s ) The Laplace ransform of he funcion f () = e cos is: s s s s a) b) c) d) s 4s + s 4s + 5 s 4s + 5 s 4s + e) ( s + 9) ss ( ) e) s 4s + 19) The Laplace Transform of he funcion 1 a) s b) ( s ) f () c) ( s 1) = e is: s d) ( s ) e) s ( s ) 0) The Laplace ransform of he funcion a) 1 s 1 b) s f () = u( τ). dτ is: 0 c) s + δ ( s) d) δ ( s ) e) s δ ( s)

17 Q) [8 poins] A coninuous-ime LTI sysem has he inpu x ( ) and he impulse response h ( ) as shown in Fig.. Find and plo he sysem oupu y ( ). Q) [8 poins] For he periodic signal x ( ) shown below; 1) Deermine he Fourier coefficiens of y( ) = x( 4 ) ) Skech he specrum of ( ) x.. Fig. 4 Q4) [8 poins] For he following ime funcion f ( ) : f ( ) u( ) 4 ( ) = e sin( ) + e sin( 5) 4.1 Find he Laplace ransform F ( s) 4. Plo he region of convergence (ROC) for he signal f ( ) Wishing you a good luck 4