MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer scic 6.003: Signals and Systems--Fall 2003 PROBLEM SET 1 Issued: September 4. 2003 Due: September 12, 2003 Reading Assignments: Lectures #1-2& PS#1: Chapter 1 of Oppenheim and Willsky(O& W), including (see p. 71) Lectures #3-4 PS#2: Chapter 2 of Oppenheim and Willsky(O&w) Exercise for home study to be turned in, although we will provide solutions) (El) Problem 1.54 Problems to be turned in: Problem 1 Answer the questions asked in Problem 1.49 of O& W for each of the following complex numbers a)(√3+j)5c-i3 Problem 2 Consider the signal a(t)depicted in Figure P1.21 on p 60 of O&W. Sketch and carefully label each of the following two signals (a)m(1 (b)x(t-2)[6(t-2)+(3-切] Problem 3 Consider the signal rn depicted in Figure P1.22 on p 60 of O&W. Sketch and carefully label each of the following two signals )x[2-] (b)x[2n+1]
✂✁☎✄✆✄✝✁✟✞✡✠☞☛✌✄✎✍✑✏✒✏✟✄✔✓✖✕☞✄✎✏✒✓✗✏✘☛✙✏✘✍✛✚☎✜✢✏✘✍✣✞✡✠✙✕✌✚☎✤✥✚✧✦✌★ ✩✌✪✬✫✮✭✰✯✲✱✴✳✵✪✬✶✷✱✘✸✰✹✺✍✼✻✽✪✬✾✿✱✴✯❁❀✽✾❂✭❃✻❄✍✼✶✆❅❆❀✽✶❇✪✬✪✬✯❁❀✽✶✆❅✵✭✰✶❇❈✔✞❉✸❆✳✵✫✆❊❇✱❋✪✬✯✟✄✎✾●❀✽✪●✶✆✾✬✪ ❍❏■▲❑▼❑▼◆❏❖◗P❄❘❚❙▼❯✥❱✆❲❨❳❩❱✮❯✥❬❭P❫❪❄❳●❴❛❵✝❜❝❳❡❞❣❢❄❱✮❲❨❲❉❤✮❑▼❑▼◆ ✐✌❥❧❦✺♠❫♥♣♦rqts❏♦✮✉✛✈ ✓✗✇❋✇❁❊✆✪✬❈❫①✡✄✎✪●✫❇✱✴✪●✳◗②▼✪✬✯✟③✆④r⑤✰⑥❆⑥✰⑦ ✩✌❊❇✪✰①⑧✄✎✪●✫❇✱✴✪●✳◗②❧✪●✯⑩⑨❛⑤✎④r⑤✰⑥❆⑥✰⑦ ❶❵♣❱✮❬✥❘❨❯❷❙❝❸❹❳✬❳❂❘❺❙▼❯✥❜❻❵✝❯❧❴❼❳✰❖ ❽❵✝❾✰❴❼❿❷➀➁❵✝❳➃➂➅➄✰➆➇❤❝➈➊➉❩P➋➂✢➄r❖ ✞❉➌✆✭✰✫❇✱✴✪●✯✌⑨✌✸✰✹❷✚☎✫❇✫❧✪●✶✆➌✆✪✬❀➍✳➎✭❃✶✆❈➃➏➐❀✽✻➍✻✽✇❁➑♣➒✂➓❨✚☎➔☎➏❭→❡④✎❀➍✶✆✾✬✻➍❊✆❈✆❀➍✶✆❅ ✯❁✪●➣✝❀✽✪✿↔↕✸✰✹➋✾✬✸✰✳➙✫✆✻➍✪✿➛➜✶♣❊✆✳➝②❧✪●✯❋✇⑩➓❺✇❁✪✬✪➞✫❏➟⑧➠✎⑨❼→ ❽❵✝❾✰❴❼❿❷➀➁❵✝❳➃➂➡◆✎➆❋➢➅➈➊➉❩P➋➂➤❤❄❖ ✞❉➌✆✭✰✫❇✱✴✪●✯✌⑤◗✸✰✹➥✚✌✫✆✫▼✪✬✶✆➌✆✪●❀✽✳❣✭❃✶✆❈➦➏➐❀➍✻✽✻➍✇❋➑✷➒➧➓❨✚☎➔☎➏❭→ ➨☞➩❵♣➀➁❾✷❘❺❳✬❵⑩➫➯➭▼➀✙➲❷➭❧❜➳❵◗❳●❴❼❿✥❬❄❪❝➵❁❯✥➭✮❴✒❴❛➭✵➸✼❵⑩❴❼❿❷➀➁❯✥❵✝❬✔❘❺❯✑➺✮❱✮❲❚❴❛➲✥➭▼❿❷❙❧➲➜➻◗❵➞➻➃❘❺❲❺❲r➼✥➀❼➭✎➽❷❘❺❬❷❵➝❳✬➭▼❲❺❿❷❴❛❘❺➭▼❯✥❳➁➾❼❖ ➓❺✍✘⑨❛→➦➚✼✯❋✸✰②✆✻✽✪●✳➪⑨❆➟➹➶❃③ ➉➞➀➁➭r➸➋❲❺❵✝❜❝❳➝❴❂➭❻➸✼❵➘❴❼❿❷➀➁❯✥❵✝❬➴❘❺❯✑❖ ➉➞➀➁➭r➸➋❲❺❵✝❜ ➄ ✁✙✶✆✇❁↔✣✪✬✯✣✱✴➌✆✪✙➷✝❊❇✪✬✇❁✱❋❀✽✸❆✶❇✇✡✭✰✇❁➑✰✪✬❈➃❀✽✶➬➚✼✯❁✸❆②✆✻✽✪●✳❣⑨❆➟➮③✷➱➞✸❃✹✥✚☎➔☎➏ ✹❚✸❆✯✑✪❂✭✰✾➇➌➬✸❃✹❏✱✴➌✆✪✟✹❚✸❆✻✽✻➍✸❛↔✘❀✽✶✆❅ ✾✬✸✰✳➙✫✆✻➍✪✿➛➬✶✝❊❇✳◗②❧✪●✯❋✇●① ➓❨✭❆→➳➓❐✃⑦✟❒➳❮✝→✲❰✴Ï❃Ð❃Ñ✲Ò❛Ó❐Ô ➉➞➀➁➭r➸➋❲❺❵✝❜ ❤ ✞❉✸❆✶✆✇❋❀➍❈✆✪✬✯⑩✱✴➌✆✪➬✇❁❀✽❅❆✶✆✭✰✻✣Õ➋➓×Ö❁→⑩❈✆✪✬✫❇❀✽✾●✱❋✪✬❈➅❀✽✶❝✜✺❀✽❅❆❊❇✯❋✪➬➚✒⑨❆➟➹⑤✎⑨➬✸✰✶✢✫❏➟➹Ø✰⑥✔✸✰✹✙✚☎➔☎➏Ù➟➥✄✎➑✰✪✿✱✴✾➇➌ ✭✰✶✆❈➦✾❂✭❃✯❋✪●✹❚❊❇✻✽✻➍➒➙✻Ú✭❃②❧✪●✻❏✪❂✭✰✾➇➌✂✸✰✹✥✱✴➌✆✪☎✹❚✸❆✻➍✻✽✸❛↔✘❀✽✶✆❅⑩✱✗↔❉✸✵✇❋❀➍❅❆✶✮✭✰✻➍✇✬① ➓❨✭❆→❹Õ➬Û✴⑨✒ÜÞÝÔ❼ß ➓❺②✮→❹Õ➋➓×Ö➥Ü➅⑤❆→✒àâá✎➓❚Ö➋Üäãå →❄❒çæ➋➓❨⑦☎Ü❻Ö❁→✗è ➉➞➀➁➭r➸➋❲❺❵✝❜ ◆ ✞❉✸❆✶✆✇❋❀➍❈✆✪✬✯⑩✱✴➌✆✪➬✇❁❀✽❅❆✶✆✭✰✻⑧Õ➋éëê❧ì✡❈✆✪✬✫✆❀➍✾●✱❋✪✬❈✢❀✽✶❝✜✺❀✽❅❆❊❇✯❋✪➬➚✒⑨❆➟➹⑤❆⑤➘✸❆✶❻✫❏➟➹Ø✰⑥✔✸✰✹✙✚☎➔☎➏Ù➟➥✄✎➑✰✪✿✱✴✾➇➌ ✭✰✶✆❈➦✾❂✭❃✯❋✪●✹❚❊❇✻✽✻➍➒➙✻Ú✭❃②❧✪●✻❏✪❂✭✰✾➇➌✂✸✰✹✥✱✴➌✆✪☎✹❚✸❆✻➍✻✽✸❛↔✘❀✽✶✆❅⑩✱✗↔❉✸✵✇❋❀➍❅❆✶✮✭✰✻➍✇✬① ➓❨✭❆→❹Õ➋é➮⑤✌Ü✢ê❧ì ➓❺②✮→❹Õ➋é➮⑤❃ê➙❒✛⑨✬ì ⑨
Problem 4 Determine and sketch the even and odd part of the following signal. Label ur sketch carefully .(t) Problem 5 Determine whether or not each of the following signal is periodic. If the signal periodic, determine its fundamental period (a)x(t)=[sin(4t-1)2 (b)xln]=cos(4n+丌/4 (c)an=(1)"cos(2Tn/7) Problem 6 Answer the same set of system property questions as described at the bottom of page 61 of O&W(i.e, determine which of the listed properties hold) for each of the following systems (a)y(t)=x(t+3)-x(1-t) (b)yn 1)"r{n],xm]≥ 2c{n] cm]<0 (c)yIn c[k
í➞î➁ïrð➋ñ❺ò✝ó ô õ✌ö●÷❋ö✬ø❁ù➙ú➍û✆ö➃ü❃û✆ý❻þ❁ÿ✰ö●÷✁✄✂✢÷☎✂✆ö➙ö✝✆❆ö●û➅ü✰û✆ý✟✞✝ý✆ý✡✠✮ü✰ø❁÷☛✞✌☞✡÷✁✂✆ö✍☞✎✞✌✏✑✏✑✞✓✒✘ú➍û✕✔➜þ❋ú✑✔✰û✮ü✌✏✗✖✙✘❷ü✌✚▼ö✛✏ ✜✞✌✢✆ø✘þ❋ÿ✰ö✿÷☎✄✂✣✬ü✰ø❋ö✤☞✎✢✕✏✑✏✜✖ ✥ ✦✪✫ ✦✪✩ ✦★✧ ✧ ✩ ✫ ✧ ✩ ✬✮✭✎✯✱✰ ✯ í➞î➁ïrð➋ñ❺ò✝ó ✲ õ✌ö●÷✴ö●ø❋ù✵ú✽û✆ö✳✒✴✂✆ö✿÷☎✂✆ö●ø✵✞❆ø✑û✕✞❃÷✣ö✬ü✌✄✂✶✞✷☞❫÷☎✂❇ö✸☞✎✞✹✏✺✏✑✞✓✒✘ú✽û✻✔✌þ❋ú✺✔❆û✮ü✌✏✮ú➍þ✼✠▼ö✬ø❁ú✑✞✝ý✆ú✑✷✖✮✽✾☞❫÷☎✂✆ö✟þ❋ú✺✔❆û✮ü✌✏ ú✽þ✳✠▼ö✬ø❁ú✑✞✝ý✆ú✑✷✿❇ý✆ö✿÷✴ö●ø❋ù✵ú✽û✆ö☎ú▲÷✴þ❀☞✎✢✆û✆ý✆ü✰ù➙ö●û✷÷➇ü✌✏❁✠▼ö✬ø❁ú✑✞✝ý❂✖ ✭ ü ✰❃✬✮✭❄✯✱✰✼❅❇❆ þ❁ú✽û ✭✎❈✌✯ ✦❉✧ ✰❋❊✺● ✭ ✚ ✰❃✬✮❆■❍❏❊❏❅ ✝✞❆þ ✭✎❈✹❍▲❑◆▼P❖◗❈❘✰ ✭ ✰❃✬✮❆■❍❏❊❏❅❙✭ ✦☛✧ ✰❋❚ ✝✞❆þ ✭ ✩▼❁❍❁❖✌❯✹✰ í➞î➁ïrð➋ñ❺ò✝ó ❱ ❲☞û❇þ✱✒✣ö✬ø✡÷☎✂❇ö✌þ✴ü✰ù✵ö☞þ❁ö●÷✳✞✷☞❷þ✜þ❁÷❋ö✬ù❳✠❇ø✁✞✹✠▼ö✬ø✲÷✜✶❨✢✆ö●þ❁÷✴ú✺✞❆û✆þ✘ü✰þ✣ý✆ö✬þ✱✬ø❁ú✑✚▼ö✬ý➦ü❃÷❉÷✁✂✆ö✪✚❩✞✰÷❁÷☎✞❆ù ✞✌☞❬✠✮ü✌✔❆ö❪❭ ✧ ✞✌☞❴❫❛❵❝❜ ✭ ú✗✖➮ö✌✖❞✿✣ý✆ö✿÷✴ö●ø❋ù✵ú✽û✆ö❃✒✴✂✆ú✺✄✂❡✞✌☞✙÷☎✂✆ö✙✏➍ú✽þ❁÷❋ö✬ý◆✠✆ø✱✞✹✠❧ö●ø❁÷❋ú✽ö✬þ❢✂✕✞✌✏✽ý ✰ ☞✎✞❆ø❩ö✬ü✌✄✂❉✞✷☞☞÷☎✂❇ö ☞✎✞✹✏✺✏✑✞✓✒✘ú✽û✕✔➝þ✜þ❁÷❋ö✬ù✵þ✛❣ ✭ ü ✰✙❤✐✭✎✯✱✰❥❅❦✬✮✭✎✯❧❑ ✫ ✰ ✦ ✬✮✭ ✧✴✦ ✯✱✰ ✭ ✚ ✰✙❤✐❆♠❍❩❊❩❅ ♥ ✭ ✦☛✧ ✰ ❚ ✬✮❆■❍❏❊♣♦ ✬✮❆■❍❏❊Pq ✥ ✩✬✮❆■❍❏❊♣♦ ✬✮❆■❍❏❊Pr ✥ ✭ ✰✙❤✐❆♠❍❩❊❩❅tst✉✈✄✇❚ ✬✮❆②①❘❊ ✩
Problem 7 Consider an LTI system whose response the to signal a(t) is the signal g(t) where these signals are depicted below (t) yi(t) Determine and provide a labeled sketch of the response to the input 2(t) depicted below The following problem is from Computer E.xplorations in Signals and Systems Using MAT LAB Second edition by Buck, Daniel, and Singer(BDS). You may want to look at BDS Section 1.I to be familiar with Matlab. Please submit your code as a part of your solution Problem 8 BDS Section 1 Reminder: The first 20 problems in each chapter of o& w have answers included at the end of the text. Consider using these for additional practice, either now or as you study f tests
③★④◗⑤⑦⑥✮⑧⑩⑨❷❶ ❸ ❹✵❺✹❻✕❼✱❽✑❾✕❿✝➀✴➁✌❻✙➂➄➃✳➅✼❼✱➆❷❼✱➇✁❿✛➈❳➉✴➊✻❺✹❼✁❿➋➀✁❿✝❼✁➌❩❺✹❻✕❼✱❿❴➇☎➊✕❿❴➇✁❺➍❼✁❽✑➎✌❻➏➁✌➐❏➑❂➒✛➓❄➔✱→➣❽✑❼➣➇☎➊✕❿❛❼✱❽✑➎✹❻➏➁✷➐✐↔❷➒✤➓✎➔✱→✤↕ ➉✴➊✕❿✝➀✁❿➋➇☎➊✕❿✝❼✁❿★❼✁❽✑➎✌❻➏➁✌➐✑❼❀➁✌➀✁❿➋❾✕❿✝➌✕❽✑➙✤➇☎❿✛❾❪➛❩❿✛➐✑❺✓➉➋➜ ➝ ➞ ➑❂➒✛➓❄➔✱→ ➟ ➝ ➝ ➝ ➔ ➝ ➞ ➠ ➝ ➟ ➝ ↔❘➒✝➓✎➔✱→ ➔ ➡❴❿✝➇✁❿✛➀✁➈▲❽✑❻✻❿✳➁✌❻✕❾➍➌✕➀✱❺✓➢➄❽✺❾✕❿✳➁❴➐✑➁✌➛❏❿✝➐✑❿✝❾➤❼✁➥✷❿✝➇☎➙✄➊✍❺✌➦➏➇☎➊✻❿✳➀✁❿✛❼✱➌❏❺✌❻✕❼✁❿✳➇☎❺❴➇✁➊✕❿❀❽✺❻✕➌✕➧✻➇➨➑⑦➩◗➓❄➔✱→✮❾✕❿✝➌✕❽✑➙✤➇☎❿✛❾❢➛❩❿✛➐✺❺➫➉➋➭ ➝ ➟ ➝ ➝ ➞ ➟ ➝ ➟ ➞ ➟ ➠ ➟✸➯ ➞ ➠ ➑⑦➩◗➓❄➔✱→ ➔ ➃✳➊✕❿★➦✎❺✹➐✑➐✺❺➫➉✴❽✺❻✕➎➍➌✕➀✱❺✹➛✕➐✑❿✝➈➲❽✑❼✳➦✎➀✱❺✹➈ ➳➣➵◗➸✪➺❩➻➄➼➾➽✤➚❴➪➹➶✝➺❩➘✑➵◗➚✁➴✌➼⑩➷✗➵◗➬✻➮★➷❄➬✃➱⑦➷❞❐✌➬❩➴✌➘❒➮❮➴✌➬❩❰✍➱⑦Ï➫➮✤➼➾➽✤➸❮➮ÑÐ❂➮✤➷❄➬❷❐✶Ò❃Ó☛Ô❧Õ Ö❂Ó❴×❳➱❂➽✄Ø✄➵◗➬❩❰Ù➽✁❰✌➷❄➼⑩➷✗➵◗➬Ú➛Û➆◆Ü✵➧✻➙✄➥❩↕❀➡❛➁✌❻✕❽✑❿✝➐✗↕Ý➁✌❻✻❾ßÞ➄❽✺❻✕➎✹❿✝➀à➓⑩Ü❀➡➋Þ✕→á➭✮â✼❺✹➧ã➈✶➁ä➆Ú➉✵➁✌❻Û➇▲➇☎❺å➐✑❺❷❺✹➥✟➁◗➇▲Ü❀➡➋Þ Þ➄❿✝➙✝➇☎❽✺❺✹❻ ➝ ➭ ➝ ➇✁❺▲➛❩❿❛➦⑩➁✌➈▲❽✑➐✑❽✑➁✌➀❥➉✴❽✺➇✁➊àæÑ➁✷➇☎➐✑➁✌➛❂➭Ýç❥➐✑❿✛➁✌❼✁❿➋❼✁➧✻➛✕➈✍❽❞➇❀➆✹❺✌➧✕➀❬➙✝❺➄❾✻❿☛➁✷❼✴➁▲➌➏➁✷➀✱➇✴❺✌➦P➆✹❺✹➧✻➀✸❼✱❺✹➐✑➧➄➇☎❽✑❺✌❻❂➭ ③★④◗⑤⑦⑥✮⑧⑩⑨❷❶ è Ü❀➡➋ÞÑÞ❷❿✛➙✝➇✁❽✑❺✹❻ ➝ ➭ ➠ éê⑨❷❶✟ë⑩ì✮í❧⑨❘④❘î➍➃✳➊✕❿▲ï✕➀✁❼✱➇ ➞✌ð ➌✕➀✱❺✹➛✕➐✑❿✝➈✍❼➋❽✺❻ñ❿✛➁✌➙✄➊✟➙✄➊➏➁✷➌✻➇☎❿✝➀☛❺✌➦✳ò❛ó❛ô ➊➏➁ä➢✹❿✍➁✌❻✕❼õ➉✵❿✝➀✁❼➤❽✺❻✕➙✛➐✺➧✕❾✕❿✛❾✡➁✷➇➋➇☎➊✻❿ ❿✛❻✻❾Ñ❺✌➦❧➇✁➊✕❿❴➇☎❿áö➄➇✛➭➣❹✵❺✹❻✕❼✁❽✺❾✕❿✛➀✴➧✕❼✱❽✑❻✕➎➤➇✁➊✕❿✛❼✱❿❛➦✎❺✹➀✴➁✌❾✕❾✕❽❞➇☎❽✺❺✹❻➏➁✌➐❩➌✕➀☎➁✷➙✝➇☎❽✺➙✛❿✷↕✕❿✛❽❞➇☎➊✕❿✝➀✳❻✕❺✓➉÷❺✹➀✴➁✷❼❀➆✹❺✹➧✙❼✱➇✁➧✕❾✻➆ê➦✎❺✌➀ ➇☎❿✝❼✱➇✁❼✛➭ ➠
