Sample Final Exam (finals03) Covering Chapters 1-9 of Fundamentals of Signals & Systems

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1 Sample Final Exam Covering Chaper 9 (final04) Sample Final Exam (final03) Covering Chaper 9 of Fundamenal of Signal & Syem Problem (0 mar) Conider he caual opamp circui iniially a re depiced below. I LI circui model wih a volageconrolled ource i alo given below. (a) [8 mar] ranform he circui uing he Laplace ranform, and find he ranfer funcion H () V () V (). hen, le he opamp gain A o obain he ideal ranfer funcion A ou in H() lim H (). A A L vx () C R v () in v R ou () L R C vx () v () in Av () x R Anwer: he ranformed circui: L C V Vx () R AVx () () in R

2 Sample Final Exam Covering Chaper 9 (final04) here are wo upernode for which he nodal volage are given by he ource volage. he remaining nodal equaion i where R V () V () AV () V () 0 L in x x x R R C L C C R L RLC L R RL. Simplifying he above equaion, we ge: ( A )( RLC L R) Vin () Vx () 0 R RL R hu, he ranfer funcion beween he inpu volage and he node volage i given by Vx () R Vin () ( A )( RLC L R). RL R RL ( )( ) R A R LC L R R L he ranfer funcion beween he inpu volage and he oupu volage i Vou () AVx () ARL HA () Vin () Vin () R( A )( RLC L R) RL he ideal ranfer funcion i he limi a he opamp gain end o infiniy: RL L R H() lim HA () A RRLC RL RR L ( LC ) R H() (b) [5 mar] Aume ha he ranfer funcion H() ha a DC gain of 50, and ha H() ha one zero a 0 and wo complex conjugae pole wih ω 0 rd/, ζ 0.5. Le L 0H. Find he value of he remaining circui componen R, R, C. Anwer: H() L R DC gain of H() i given by L R 50. L ( LC ) R Componen value are obained by eing H L () L ( LC ) R R n

3 Sample Final Exam Covering Chaper 9 (final04) which yield R 00 Ω, R 0. Ω, C 0.00F (c) [7 mar] Give he frequency repone of H() and ech i Bode plo. Anwer: Frequency repone i H jω jω ( ) ( ) jω 0.( jω ). Bode plo: 60 0log 0 G( jω) 40 (db) ω (log) (deg) 5 H( jω) ω (log)

4 Sample Final Exam Covering Chaper 9 (final04) Problem (0 mar) Conider he caual differenial yem decribed by i direc form realizaion hown below, X () 3 Y() 4 dy(0 ) and wih iniial condiion d uni ep inpu ignal x() u()., y(0 ). Suppoe ha hi yem i ubjeced o he (a) [8 mar] Wrie he differenial equaion of he yem. Find he yem' damping raio ζ and undamped naural frequency ω n. Give he ranfer funcion of he yem and pecify i ROC. Sech i polezero plo. I he yem able? Juify. Anwer: Differenial equaion: d y () dy() 4 y ( ) 3 x ( ) d d Le' ae he unilaeral Laplace ranform on boh ide of he differenial equaion. dy(0 ) Y() y(0) () y(0) 4 () 3 () d Y Y X on he lefhand ide and puing everyhing ele on he righ Collecing he erm conaining Y( ) hand ide, we can olve for Y (). ( ) dy(0 ) 4 Y( ) 3 X( ) y(0 ) y(0 ) d dy(0 ) ( ) y(0 ) 3 X( ) Y() d 4 4 zeroae rep. zeroinpu rep. 4

5 Sample Final Exam Covering Chaper 9 (final04) he ranfer funcion i H 3 (), 4 and ince he yem i caual, he ROC i an open RHP o he righ of he righmo pole. he undamped naural frequency i ω n and he damping raio i ζ p, ζωn ± jωn ζ ± j ± j. herefore he ROC i Re{ } >. he pole are. Syem i able a jwaxi i conained in ROC. Polezero plo: Im{} Re{} (b) [8 mar] Compue he ep repone of he yem (including he effec of iniial condiion), i eadyae repone y () and i ranien repone yr () for 0. Idenify he zeroae repone and he zeroinpu repone in he Laplace domain. Anwer: he unilaeral L of he inpu i given by X (), Re{} > 0, hu, 3 ( ) (4 ) 3 Y () ( 4) 4 ( 4) Re{ } > 0 zeroae rep. Re{ } > zeroinpu rep. Le' compue he overall repone: 5

6 Sample Final Exam Covering Chaper 9 (final04) Le o compue (4 ) 3 Y(), Re{} > 0 ( 4) A B( ) C ( ) Re{ } > Re{ } > 0 A B( ) 0.75 ( ) Re{ } > Re{ } > 0 () (4 )( ) A ( ) 0.75 A, A hen muliply boh ide by and le o ge B 0.75 B.5 : ( ) 0.75 Y () ( ) ( ) Re{ } > Re{ } > Re{ } > 0 Noice ha he econd erm i he eadyae repone, and hu y ( ) 0.75 u( ). aing he invere Laplace ranform uing he able yield y ( ) 0.549e in( u ) ( ).5e co( u ) ( ) 0.75 u ( ). hu, he ranien repone i y e u e u. r ( ) in( ) ( ).5 co( ) ( ) (c) [4 mar] Compue he percenage of he fir overhoo in he ep repone of he yem aumed hi ime o be iniially a re. Anwer: 3 ranfer funcion i H (),Re{} > wih damping raio ζ : 4 6

7 Sample Final Exam Covering Chaper 9 (final04) ς ς OS 00 e % 00 e % 00 e % 4.3% Problem 3 (0 mar) he following nonlinear circui i an ideal fullwave recifier. vin () α he volage are v () e [ u() u( )] [ δ( ) δ( ( ) )] α R, α > 0, and v () v (). in in where (a) [5 mar] Find he fundamenal period of he inpu volage. Sech he inpu and oupu volage vin (), v() for α /. Anwer: We have R v () e vin () e v () e 7

8 Sample Final Exam Covering Chaper 9 (final04) (b) [8 mar] Compue he Fourier erie coefficien a of he inpu volage vin () for any poiive value of α and. Wrie vin () a a Fourier erie. Anwer: DC componen : for 0 : α α ( ) a x() d e d e d 0 0 α ατ e d e d 0 τ 0 0 j a x() e d j 0 j α ( ) α e e d e e d 0 ( ) 0 α j ( α j ) α e d e e d 0 e ( α j ) ( α j ) α α j α j ( e e ) ( e e ) ( α j ) α e (( ) ) ( ) α j (( ) ) α ( ) e α j e ( α j ) α α j α j ( e e ) ( e e ) 0 Fourier erie: (( ) ) α ( e ) vin () e α j j 8

9 Sample Final Exam Covering Chaper 9 (final04) (c) [5 mar] Compue he Fourier erie coefficien b of v () again for any poiive value of α and. Anwer: Here he fundamenal period i. DC componen : for 0 : α 0 () 0 0 α α e e α 0 α b x d e d j ( α j ) b x() e d e d 0 0 α α e ( e ) α j ( α j ) (d) [ mar] Compue he Fourier erie coefficien of he oupu volage ignal v () for he cae α 0 wih held conan. Wha imedomain ignal v () do you obain in hi cae? Anwer: When α 0 we ge a conan ignal for v (). α lim b0 lim e α 0 α 0α α e lim b lim 0 α 0 α 0α j j Problem 4 (5 mar) Syem idenificaion Suppoe we now ha he inpu of a differenial LI yem i and we meaured he oupu o be x () e u (), y () e co in u () e u (). x() H() y () (a) [0 mar] Find he ranfer funcion H() of he yem and i region of convergence. I he yem caual? I i able? Juify your anwer. 9

10 Sample Final Exam Covering Chaper 9 (final04) Anwer: Fir ae he Laplace ranform of he inpu and oupu ignal uing he able: X() ( ), Re{ } > Y () ( ) ( ) ; Re{ } > Re{ } > ( ) ; Re{ } > Re{ } > Re{ } > ( 4 4) ( ), Re{ } > ( )( ) ( ) ( )( ), Re{ } > hen, he ranfer funcion i imply Y () H () X() ( ) ( )( ) ( )( ) ( ) ( ) 3 o deermine he ROC, fir noe ha he ROC of Y() hould conain he inerecion of he ROC' of H() and X (). here are wo poible ROC' for H() : (a) an open lef halfplane o he lef of Re{ }, (b) an open righ halfplane o he righ of Re{ }. Bu ince he ROC of X () i an open righ halfplane o he righ of, he only poible choice i (b). Hence, he ROC of H() i Re{ } >. he yem i caual a he ranfer funcion i raional and he ROC i a righ halfplane. I i alo able a boh complex pole p, ± j are in he open lef halfplane. (b) [ mar] Find an LI differenial equaion repreening he yem. Anwer: I can be derived from he ranfer funcion obained in (a): d y() dy() d x() dx() y () 6 4x () d d d d (c) [3 mar] Find he direc form realizaion of he ranfer funcion H (). Anwer: he ranfer funcion can be pli up ino wo yem a follow: 0

11 Sample Final Exam Covering Chaper 9 (final04) X () W() 6 4 Y() he inpuoupu yem equaion of he fir ubyem i W() W() W() X(), and for he econd ubyem we have Y () W () 6W () 4W (). he direc form realizaion of he yem i given below: W() X () W() W() 6 4 Y() Problem 5 (5 mar) (a) [0 mar] Compue he Fourier ranform X ( jω ) of he following aperiodic ignal x() and give i magniude and phae. x() / Anwer:

12 Sample Final Exam Covering Chaper 9 (final04) Magniude: Phae: jω X( jω ) x( ) e d 0 / jω jω e d e d 0 / jω jω jω e e e jω jω jω e jω jω e jω jω jω jω e e ( e e ) jω jω jω 4 j in ( ω ) e ω 4 4 X jω ω ω 4 ( ) in ( ) ω, ω > 0 X( jω) ω, ω < 0 0, ω 0 (b) [5 mar] Wrie he Fourier erie coefficien erm of ( ) X jω ha you obained in (a) and compue hem. a of he following recangular waveform y () in A y () A Anwer:

13 Sample Final Exam Covering Chaper 9 (final04) We have a A X( j ) j 4 A j in ( ) e 4 A A j e j ( ) j in ( ) (( ) )( ) (( ) ) ja Problem 6 (0 mar) Ju anwer rue or fale. (a) he Fourier ranform Z( jω ) of he produc of a real even ignal x() and a real odd ignal y () i imaginary. Anwer: rue. (b) he yem defined by y () x ( ) i caual. Anwer: Fale. (c) he Fourier erie coefficien a of a purely imaginary even periodic ignal x() have he following propery: a a. Anwer: Fale. (d) he caual linear dicreeime yem defined by yn [ ] 0.4 yn [ ] 0.45 yn [ ] xn [ ] i able. Anwer: Fale. (e) he fundamenal period of he ignal Anwer: rue. 3 x[ n] in( n) i 0. 5 END OF EXAMINAION 3