Sample Exam 1: Chapters 1, 2, and 3

L L 1 ' ] ^, % ' ) 3 Sample Exam 1: Chapters 1, 2, and 3 #1) Consider the lineartime invariant system represented by Find the system response and its zerostate and zeroinput components What are the response steady state and transient components #1b) Using the linearity and time invariance principle (note that you have to assume zero initial conditions) find the response of the system defined in #1a) due to the input signal equal to #2a) Using the properties of the impulse delta function simplify the following expressions!!! 21 %,!"$#&%(')* #%43+) 5 687 :( * = 1+<!!+, % > +!? %' +* 0/ #&%@'+)BA>C5( /* =D/+ #2b) Plot the graph of the signal represented in terms of unit step and unit ramp signals as E F and find its generalized derivative G H JI KG =HL #3a) Find the Fourier series of a periodic signal represented by E F E J =NM LOPLQH RQS 2TQH UQVL #3b) Find the system response due to the periodic input signal given in #3a) of a linear system YX&ZU[ whose transfer function is W #3c) Using the tables of common pairs 1\ ]K^ and properties of the Fourier transform find Fourier transforms of the following signals:!+_ +# %@3a`)` +!!b# %@3+) KBA>C5( @c!!! #3d) Find the inverse Fourier transform of the signals!" XZ A>C5( ZU# %!!", d dh efe g!? Z @c 1

n n Ž n ~ Sample Exam 2: Chapters 1, 2, and 3 #1) Consider the lineartime invariant system represented by ibjklkm i l jklmfoqp ryskltumvgjkxwymfoqw Find the system response and its zerostate and zeroinput components What are the response steady state and transient components #2a) Using the properties of the impulse delta signal simplify the following expressions kz+mk l t{m }k l=th{mvhkz z+m~ kz m ~ Ž ~ƒ 4 p r sku&l>m}kl=tdum+ilv pfk lkm+} ˆŠ " "kl ts{&m+i l>vhkzz z+m k mq klm}ˆ K "kl p rysk+ (lm}ku&l=tm+i l #2b) Plot the graph of the signal represented in terms of unit step, rectangular, and unit ramp signals as and find its generalized derivative klmfoqškltdum klœt #3a) Find the Fourier series for the sawtooth signal with Ÿ Hint: lyprysk œl>milo { prysk œlmœt m td ktužl l ym nh o {vk on{ mil Note it is an odd signal #3b) Find the liner system response due to the periodic input > signal defined in 3a) The system k Um o transfer function is K #3c) Using the tables of common pairs and properties of the Fourier transform find Fourier transforms of the following signal: kz"mp rysk+ lkm"ª«f kl=t m t f kl=ts{ m vgk+zz m_l @ k lkm>v kz zz m$p r s ku&l=tdžm p(k œl>m #3d) Find the inverse Fourier transform of the signal ± ky² ³mFoq k Um tµ k t{m #3e) Find the response of the system defined in 3b) due to the input signal > pf "{awl ņ @¹ 2

Ê Ð Ð Ê Sample Exam 3: Chapters 4 and 5 1) Find the Laplace transform of the following signals º¼» ½ ¾K FV½¾ ÁHÂ Ã Ä ÅÆ ½¾ ÇS& º&Ê ½¾ FqË>ÌBÍ ½Î(¾> Æ ½¾=Ç >2º Ï&½¾K FqоÃÄ ÅyÍ Ñ8Ò½Ó¾K +Æ(½ ¾K 2) Find the inverse Laplace transform of the following function Ô ½Õ F ÂÕUÁà Ä4 Õ Ê ½ÕGÁq 3) Find the transform of the following signals º»&Ø«ÙÚ qóûü ½ Ù ÁHÂ Æ Ø«Ù ÇDÂ Ú º Ê&Ø0ÙÚ Ù Ë>ÌBÍ Ù Î Â Æ Ø«ÙÚ _º Ï ØàÙÚ âá ã ä  ٠qå æ Ù qç è Ì&éK@ ìí³ñîí" 4) Find the inverse transform of the following function Ô ½Kï æ ½ïÁ ½ïÁ K½ïÁJÓy +½ïÁåy Using the initial and final values theorems verify the values obtained for º Ø è Ú and º Ø ðú 5) Consider a continuoustime system ñ Ê>ò ½¾ ñ ¾ Ê ÁÐ ñ ò ½ ¾K ñ ¾ ÁJç ò ½¾ FqÓ ñ ºœ½ ¾K ñ ¾ ÇDºœ½+¾ òôó è Ä õ ö ò»kø ó è Äùõ è ºù½ ¾K FÃ Ä Å Æ ½¾ Find its transfer function (1pt), impulse response (1pt), step response (1pt), zerostate (1pt), and zeroinput (1pt) responses 6) Consider a discretetime system ò Ø0ÙÚ Ç ò Ø0Ù ÇS Ú Ç ò Ø0Ù Ç Â Ú qº Ø«Ù ÇH Ú Áº Ø«Ù ÇDÂ Ú ò Ø ÇÂ Ú Ç ò Ø Çú Ú è º Ø0ÙÚ û½kçp& Û Æ Ø«ÙÚ Find the system transfer function (1pt), impulse response (1pt), zerostate (1pt), and zeroinput (1pt) responses What is the system steady state response due to º Ø«ÙÚ öå Æ Ø«ÙyÚ (1pt)? Hint: Ë>ÌBÍ(½üþýÿ ú Ë>ÌÍ(½ü yë>ìí@½ ÿ Í Ñ Ò ½+ü yí ÑYÒ½ÿ 3

! 4 B <! 4 < H B Sample Exam 4: Chapters 4 and 5 #1a) Find the Laplace transform of the following signals! "#%$&(') *",+ /0) 1234 5 "#768!:% < ) 1 #1b1) Find the time domain component of the inverse Laplace transform that corresponds to the pole located at the origin = ># > @> A " >!/? 23@ > 3/ @> @ @> 0B #1b2) Find the inverse Laplace transform of the function =! ># 5? > >! A #2a) Find the C transform of the following signals +ED34FGIHKJ D 2L@M+DL4N!+DL4F$&('POQD * 2R M+DS 4T 5I+UDL4MWV X Y DZ [B \]NDZ%2I^ _ &a`/bckd/egf'@c #2b) Find the inverse C transform of the following function = /hl h h A 2L hi Can you apply the initial and final values theorems to the function = @hl If yes, find j+ _ 4 and j+lkm4 #3a) Consider a continuoustime system n!ko " n! pb n o / n o /#% q " o 6 _ G2 or @s 6 q " " Find its transfer function, impulse response, step response, zerostate, zeroinput, and steady state responses #3b) Consider a discretetime system o +EDL4 o +EDt o +wx2 4)G2 o +l o +UDu24MG +Dt 4F j+edl4mtyj Mz j+dvm24 M+DL4 Find the system transfer function, impulse response, zeroinput, and zerostate responses Hint: $&(' {} 0~i7$&L' {3$&L' ~# A'fQ ƒ @{3'fQ x ~ 4

¼» ˆ ¼ à ˆ Sample Exam 5: Chapters 6, 8, and 12 1) Convolve graphically continuoustime signals /ˆ and Š K 1 pˆ 2) Using the sliding tape method find the discretetime convolution IŽ E L g Ž E L, where Ž U 3 ) š œ x Z% Z% Z ZA Z%ž Ÿa / [ @ g, 3) Consider the continuoustime linear system represented in the state space form by L ª Ž "ˆ ª "Ϋª Ž "ˆ P «ª /ˆ«u )«"ˆ E ª Ž L ˆ ª @ Ϋœ]«a) Find the system transfer function b) Find the system impulse response c) Find the system transition matrix using the Laplace transform d) Find the system output response ( "ˆ ) due to the system input q "ˆ±³² /µ /ˆ 4) Consider the discretetime linear system represented in the state space form by ª Ž U ª U «ª Ž E L œ x «ª E 3 «L F w j «ª Ž U L ª U L K«q "ˆ ª Ž "ˆ ª /ΫL ª Ž E ª ««a) Find the system transfer function b) Find the system transition matrix c) Find the system output response ( U L ) due to the system input L @¹Ežaˆº E 3 5) The openloop system transfer function is given by 8¼ˆ# Assuming the unity feedback system, find the sensitivity function of the closedloop system with respect to parameter ½ p½ Hints: ² F¾/µ "ˆ: p Á ² ¾/µ "ˆ: ¼  º E L à à º U L à 5

ì ÛÙÙÙ ø î Î Ô Î ÛÙÙÙ î Sample Exam 6: Chapters 6, 8, and 12 #1a) Convolve graphically continuoustime signals ÄÅ[ÆÇpIÊËLÌ and ÄÍ ÆÇ/ÌjÎÂÏŠÍ ÆÇ"Ì #1b) Using the sliding tape method find the discretetime convolution Ð ÑÒUÓÔÕ±Ð Í ÒEÓ3Ô, where ÓÜ% ÓÜ Ú ÓÜ Ú ÓÜ Ð Ñ ÒEÓLÔØ ÙÙÙ ÓÜ ÐÍÒEÓLÔØ ÙÙÙ ÓÜAà ÓÜAà œ ÓÜ%Ë áiâ/ãä åæèçé@äm áiâ"ãäkå"ægçé@ä #2A) Consider the continuoustime linear system represented in the state space form by Lì í Ñ ÆÇ/Ì í Í ÆÇ/Ìî í Ñ ÆÇ"Ì í Í ÆÇ"ÌIî ð ÆÇ"Ì Ò jô )î ÐMÆÇ/Ì í Ñ ÆÇ"Ì í Í ÆÇ/Ì î í Ñ ÆLïÌ í Í Æ3ïÌ î a) Find the system transition matrix ñ\æòì, and obtain ñpæç"ì using the Laplace inverse transform b) Find the system transfer function c) Find the system state response for ÐqÆÇ"Ì\³óaÆÇ"Ì and the given initial conditions d) Find the system output response ( ð ÆÇ/Ì ) due to ÐqÆÇ"Ì#õô ï Å/ö ÆÇ/Ì and the given initial conditions #2B) Consider the discretetime linear system represented in the state space form by í Ñ ÒUÓvÎA Ô í ÍÒUÓtÎA Ô î Ñ í Ñ ÒEÓLÔ úù ø î í ÍÒEÓ3Ô î ð ÒÓLÔF ÒE î ÐÒÓLÔ í Ñ ÒUÓLÔ í Í ÒUÓLÔ î í Ñ ÒUÔ í ÍÒUÔ î jî a) Find the system transition matrix ñ\æû3ì Obtain ñxòeó3ô via the ü transform b) Find the system transfer function c) Find the system output response ( ð ÒEÓ3Ô ) due to ÐjÒEÓLÔM7ý" Ñ þ(ÿ ÒÓLÔ and the given initial conditions d) Given the system represented by ð ÒEÓœÎÂËÔÎ ð ÒÓiÎmàIÔ ð ÒEӜΠFind the system state space form Ô ð ÒEÓtÎ ÔÎ ð ÒUÓiÎGKÔ #3) The system openloop system transfer function is given by Æ@òÌ# òæògîgì"æò\î Ì"Æò\Î0Ë3Ì ð ÒÓLÔM%ÐjÒEӜΠÔ(Îmà(ÐjÒEÓLÔ Assuming the unity feedback system and the system asymptotic stability for some values of the static gain, find the steady state step, ramp, and parabolic errors in terms of Hint: Some common pairs: ÆÇ/Ì ò ô ï ö ÆÇ"Ì ògî Ç@ô ï ö ÆÇ"Ì ÆògÎÌ Í ÒÓLÔ û ûœ Ó ÒÓLÔ Lû Æ@ûƒÌ Í 6