MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science 6.003: Signals and Systems--Fall 2003 COMPUTER LAB 1 Issued: September 4, 2003 Due: October 8. 2003 Problems to be handed in: In this lab, you will complete the Basic, Intermediate, and Advanced Exercises for the Echo Cancellation problem considered in Section 2.10 on pages 44-46 of Buck, Daniel, and Singer(BDS). For all of the exercises, please include your Mat LAB code. As stated in the General Information Sheet, we do not expect a formal lab report but it is still important that you present your results clearly and in an organized manner Specific items which should be turned in are listed below Note: Signals y, y2 and y3 are contained in column vectors. Thus, using the second to last line of code given on page 46 of BDS will give an INCORRECT answer, since the fiplro command operates on rows. Thus, substitute"Ryy=conv(y, fliud(y)); " into the second to last line of code on page To load the appropriate data file into Matlab, type >>cd/mit/6.003/ data > load lineup Alterna tively you can obtain the data file from the course web page Basic Problems (a) turn in a plot of the impulse response of the echo system, contained in the vector he (b)In this part, you are asked to verify that the echo removal system is indeed the inverse of the echo system. For the derivation of this, do not deri ve the overall difference equation Instead, analytically compute the impulse response of the echo system and the impulse response of the echo removal system, and convolve the two impulse responses to get se of the cascaded system The result of this convolution should be the unit sample 8[n] Intermediate Problems (c) Turn in a plot of the approximated impulse response, contained in the vector her (d) Turn in a plot of the filter output, contained in the vector z￾✂✁☎✄✆✄✝✁✟✞✡✠☞☛✌✄✎✍✑✏✒✏✟✄✔✓✖✕☞✄✎✏✒✓✗✏✘☛✙✏✘✍✛✚☎✜✢✏✘✍✣✞✡✠✙✕✌✚☎✤✥✚✧✦✌★ ✩✌✪✬✫✮✭✰✯✲✱✴✳✵✪✬✶✷✱✘✸✰✹✺✍✼✻✽✪✬✾✿✱✴✯❁❀✽✾❂✭❃✻❄✍✼✶✆❅❆❀✽✶❇✪✬✪✬✯❁❀✽✶✆❅✵✭✰✶❇❈✔✞❉✸❆✳✵✫✆❊❇✱❋✪✬✯✟✄✎✾●❀✽✪●✶✆✾✬✪ ❍❏■▲❑▼❑▼◆❏❖◗P❄❘❚❙▼❯✥❱✆❲❨❳❩❱✮❯✥❬❭P❫❪❄❳●❴❛❵✝❜❝❳❡❞❣❢❄❱✮❲❨❲❉❤✮❑▼❑▼◆ ✐☎❥✺❦✌❧♥♠❄♦❄♣♥qsr✉t✥✈①✇ ✓✗②❋②❁❊✆✪✬❈❫③✡✄✎✪●✫❇✱✴✪●✳◗④▼✪✬✯✟⑤✆⑥♥⑦✰⑧❆⑧✰⑨ ✩✌❊✆✪❃③✉✚☎✾✿✱✴✸❆④▼✪✬✯✟⑩❶⑥▼⑦✰⑧❆⑧✰⑨ ❷❹❸❻❺♥❼❲❽❵✝❜❝❳◗❴ ❺❾❼ ❵➀❿✥❱✮❯✥❬➁❵✎❬✛❘❽❯✉❖ ✓✗✶✂✱❋➂✆❀✽②✘✻✽✭✰④❏⑥✆➃❆✸✰❊✂➄✘❀✽✻➅✻❏✾✬✸✰✳➆✫✆✻➅✪●✱❋✪✧✱❋➂✆✪➈➇✡✭✰②❋❀➅✾✰⑥♥✓✗✶✷✱✴✪●✯❋✳✵✪✬❈✆❀✽✭❃✱✴✪❃⑥➉✭❃✶✆❈ ✁☞❈✎➊✰✭❃✶✆✾✬✪●❈➋✍✺➌❇✪●✯❋✾●❀✽②❋✪●②☞✹❚✸❆✯✘✱✴➂✆✪➈✍✼✾➍➂✆✸➎✞✡✭✰✶❇✾✬✪✬✻➅✻➏✭❃✱❋❀✽✸❆✶➎✫✆✯❁✸❆④✆✻➅✪✬✳❣✾✬✸✰✶✆②❋❀➅❈✆✪✬✯❁✪✬❈✔❀➅✶✔✄✎✪●✾●✱❋❀✽✸❆✶✔⑦✎➐➅➑❂⑧✵✸❆✶✂✫✮✭✰❅❆✪●② ⑤❆⑤❻➒➓⑤✷➔➈✸✰✹✥➇✣❊✆✾➍→♥⑥♥✩☎✭✰✶✆❀✽✪●✻➣⑥✝✭✰✶✆❈❶✄✎❀✽✶✆❅✰✪✬✯☎↔❽➇✡✩✧✄✆↕❡➐✆✜✮✸❆✯✣✭✰✻✽✻▼✸✰✹❄✱✴➂❇✪✌✪✿➌✎✪✬✯❁✾✬❀➅②❋✪✬②●⑥✮✫✆✻✽✪✬✭✰②❋✪✙❀✽✶✆✾●✻✽❊✆❈❇✪✙➃❆✸✰❊✆✯✘➙t✆♦▼➛ ➜t✥✈ ✾✬✸✝❈✆✪✰➐➝✁✙②✑②❁✱➍✭❻✱✴✪✬❈✵❀✽✶❩✱❋➂✆✪✙✦✧✪●✶✆✪✬✯❋✭✰✻✆✓✗✶❇✹❚✸❆✯❁✳➞✭❻✱✴❀✽✸✰✶✵✄✎➂✆✪✬✪✿✱❂⑥✷➄✣✪✟❈❇✸➈✶✆✸❃✱✑✪✿➌✎✫➉✪●✾●✱❉✭✌✹❚✸❆✯❋✳➆✭✰✻❇✻✽✭✰④❩✯❋✪●✫➉✸✰✯❁✱❂⑥ ④✆❊❇✱❹❀▲✱☎❀✽②☎②✲✱✴❀✽✻➅✻✺❀➅✳➆✫▼✸❆✯❁✱✴✭✰✶✷✱☞✱❋➂✮✭❃✱☎➃❆✸✰❊➟✫✆✯❁✪✬②❁✪✬✶✷✱➈➃✰✸❆❊✆✯✧✯❋✪●②❋❊✆✻▲✱✴②✧✾✬✻➅✪❂✭✰✯❁✻➅➃➠✭✰✶✆❈➡❀✽✶➀✭✰✶➀✸❆✯❁❅✷✭✰✶✆❀➅➢✬✪●❈➡✳➆✭✰✶✆✶❇✪✬✯✬➐ ✄✎✫▼✪✬✾●❀➅➤✮✾✧❀➅✱❋✪✬✳✵②✡➄✘➂✆❀✽✾➍➂➥②❋➂❇✸❆❊✆✻✽❈➎④▼✪☎✱✴❊✆✯❁✶✆✪✬❈➥❀✽✶➥✭✰✯❁✪❹✻➅❀✽②✲✱✴✪✬❈❶④▼✪✬✻➅✸❛➄✧➐ ➦✂❺❴❛❵♥❖ ✄✎❀✽❅❆✶✆✭✰✻✽②✉➃❆⑥✆➃❇⑦◗✭✰✶✆❈❶➃✎⑨➧✭✰✯❋✪✌✾●✸❆✶✷✱➍✭✰❀➅✶✆✪✬❈➎❀➅✶❶✾✬✸❆✻➅❊✆✳➆✶➆➊❆✪✬✾✿✱✴✸❆✯❁②✬➐✉✏✒➂➓❊✆②✬⑥✮❊✆②❁❀✽✶✆❅❩✱❋➂✆✪☎②❋✪●✾✬✸❆✶✆❈➎✱✴✸❩✻✽✭✰②❁✱ ✻✽❀➅✶✆✪❩✸✰✹✒✾●✸✎❈❇✪✵❅✰❀➅➊❆✪●✶❾✸❆✶➟✫✮✭✰❅❆✪✵⑤✷➔➎✸✰✹✒➇❉✩✧✄➋➄✘❀➅✻✽✻✼❅❆❀▲➊❆✪✵✭✰✶❾✓✖✕☞✞✒✚☎➨✙➨✟✍✉✞❉✏➩✭✰✶✆②❁➄✣✪✬✯●⑥✺②❁❀✽✶✆✾●✪➆✱✴➂❇✪✵➫✮❀➅✫✆✻✽✯❂↔➣↕ ✾✬✸✰✳➆✳➆✭✰✶✆❈✔✸✰✫➉✪●✯✴✭❃✱❋✪✬②☎✸❆✶➀✯❋✸❛➄✘②●➐➭✏✒➂➓❊✆②✬⑥➁②❁❊✆④✆②❁✱❋❀➅✱❋❊❇✱✴✪➡➯❋➨✒➃➓➃❇➲☎✾●✸❆✶✷➊❫↔❚➃❆⑥➳➫✮❀➅✫✆❊✆❈➁↔➵➃✆↕❁↕✿➸➻➺➋❀✽✶✷✱✴✸➼✱❋➂✆✪◗②❁✪✬✾✬✸✰✶✆❈➡✱❋✸ ✻➏✭❃②❁✱✘✻✽❀➅✶✆✪✌✸✰✹➽✾✬✸✝❈✆✪✧✸❆✶➥✫✮✭✰❅❆✪☎⑤❆➔❇➐ ✏✥✸➆✻➅✸✷✭✰❈➼✱✴➂❇✪➈✭❃✫✆✫✆✯❋✸✰✫✆✯❋❀✽✭❃✱✴✪☎❈✮✭❻✱➍✭◗➤✮✻✽✪☎❀➅✶✷✱✴✸◗➾✆➚❆➪✮➶❆➚❃➹➉⑥♥✱✗➃✝✫▼✪✰③ ➘➓➘➷➴❻➬➱➮❐✃❫❒ ➪ ➮❆❮✉❰✖Ï➓Ï✷Ð❇➮✰➬➚✰➪❇➚ ➘➓➘ ➶➓Ñ❆➚➬ ➶ ❒❂Ò✮Ó❃Ô➓Õ ✁☞✻▲✱✴✪●✯❋✶✮✭❃✱❋❀➅➊✰✪✬✻➅➃➆➃❆✸❆❊➥✾❂✭❃✶✂✸❆④✎✱➍✭✰❀➅✶➼✱✴➂✆✪✧❈✮✭❃✱✴✭◗➤✮✻✽✪☎✹❚✯❋✸✰✳Ö✱✴➂❇✪❹✾●✸❆❊✆✯❁②❋✪✧➄❉✪●④✂✫✮✭❃❅❆✪✰➐ ×❱✮❳❂❘❽Ø ❷➈❸❐❺♥❼❲❚❵✝❜❝❳ ↔❨✭❆↕➎✏✥❊✆✯❋✶➥❀✽✶➎✭❩✫✆✻➅✸✰✱✒✸✰✹✥✱✴➂❇✪☎❀✽✳✵✫✆❊✆✻✽②❁✪✌✯❋✪●②❋✫▼✸❆✶✆②❁✪❹✸❃✹➽✱❋➂✆✪✧✪✬✾➍➂✆✸✵②❁➃✝②❁✱❋✪✬✳➥⑥✮✾✬✸❆✶✷✱✴✭✰❀✽✶✆✪●❈➎❀✽✶❶✱✴➂✆✪✌➊✰✪✬✾✿✱✴✸❆✯✡ÙÓ ➐ ↔❽④✮↕➎✓✗✶✧✱✴➂✆❀➅②➁✫✮✭✰✯❁✱✬⑥❻➃✰✸❆❊◗✭❃✯❋✪✉✭✰②❋→❃✪✬❈❹✱✴✸✘➊❆✪✬✯❁❀➅✹➵➃✌✱❋➂✮✭❃✱➁✱✴➂❇✪✣✪●✾➍➂✆✸☞✯❁✪✬✳✵✸❐➊❃✭✰✻✰②❁➃✝②❁✱❋✪✬✳➷❀✽②➁❀✽✶❇❈✆✪✬✪●❈❹✱❋➂✆✪✣❀➅✶✷➊❆✪✬✯❁②❋✪✉✸✰✹ ✱❋➂✆✪✑✪✬✾➍➂✆✸✙②❁➃✝②❁✱❋✪✬✳➥➐➝✜✮✸❆✯❏✱✴➂❇✪✣❈✆✪●✯❋❀▲➊✰✭❻✱✴❀✽✸✰✶☎✸✰✹✎✱❋➂✆❀✽②●⑥❃❈❇✸✙✶✆✸❃✱➽❈❇✪✬✯❋❀▲➊❆✪✼✱✴➂❇✪✣✸❛➊❆✪●✯✴✭✰✻➅✻❆❈✆❀▲Ú❫✪●✯❋✪✬✶❇✾✬✪✉✪✬Û➓❊✮✭❃✱❋❀✽✸❆✶❏➐ ✓✗✶✆②✲✱✴✪✬✭✰❈❏⑥✝✭✰✶✮✭✰✻▲➃➓✱✴❀✽✾✬✭✰✻✽✻▲➃❹✾✬✸❆✳✵✫✆❊❇✱❋✪✡✱✴➂✆✪✘❀✽✳✵✫✆❊✆✻➅②❋✪✡✯❁✪✬②❁✫➉✸❆✶❇②❋✪✘✸✰✹▼✱✴➂✆✪✘✪✬✾➍➂❇✸➈②✲➃✎②✲✱✴✪●✳➩✭✰✶✆❈➧✱✴➂✆✪✒❀➅✳➆✫✆❊❇✻✽②❋✪ ✯❁✪✬②❋✫▼✸❆✶✆②❁✪➞✸✰✹✉✱✴➂❇✪➆✪●✾➍➂✆✸➠✯❁✪✬✳✵✸❐➊❃✭✰✻✼②✲➃✝②❁✱✴✪●✳✂⑥✺✭✰✶✆❈➡✾✬✸❆✶✷➊✰✸❆✻➅➊✰✪✵✱❋➂✆✪✵✱✗➄✣✸➠❀➅✳➆✫❇❊✆✻✽②❁✪❩✯❋✪✬②❁✫➉✸✰✶✆②❋✪●②➈✱✴✸➥❅❆✪✿✱ ✱❋➂✆✪✌❀✽✳✵✫✆❊✆✻➅②❋✪✙✯❋✪✬②❁✫➉✸✰✶✆②❋✪✧✸✰✹❄✱❋➂✆✪☎✾❂✭✰②❁✾❂✭✰❈❇✪✬❈➥②❁➃✝②❁✱❋✪✬✳➥➐✉✏✒➂❇✪☎✯❋✪✬②❁❊✆✻➅✱✡✸✰✹❄✱✴➂✆❀➅②✡✾✬✸✰✶➓➊✰✸❆✻✽❊✎✱✴❀✽✸✰✶❶②❋➂✆✸❆❊❇✻✽❈❶④▼✪ ✱❋➂✆✪✧❊✆✶✆❀➅✱✘②❋✭✰✳✵✫✆✻✽✪✧Ü✎Ý➳Þ▼ß➣➐ à❯➉❴❛❵❸❜❾❵✝❬➽❘❚❱❇❴❛❵ ❷➈❸❐❺♥❼❲❽❵✝❜❝❳ ↔❽✾❂↕➎✏✥❊✆✯❋✶➥❀✽✶➥✭❩✫✆✻➅✸✰✱✘✸✰✹✥✱✴➂✆✪❹✭✰✫✆✫❇✯❋✸❂➌✎❀✽✳➆✭❃✱✴✪●❈➥❀➅✳➆✫❇❊✆✻✽②❁✪✌✯❋✪●②❋✫▼✸❆✶✆②❋✪❃⑥♥✾✬✸❆✶✷✱➍✭❃❀✽✶✆✪●❈✂❀➅✶➼✱✴➂✆✪☎➊❆✪●✾●✱❋✸❆✯✘ÙÓ❆á ➐ ↔❽❈✮↕➎✏✥❊✆✯❋✶➥❀✽✶➥✭❩✫✆✻➅✸✰✱✘✸✰✹✥✱✴➂✆✪☎➤✮✻▲✱✴✪✬✯✒✸✰❊❇✱✴✫✆❊✎✱❂⑥✮✾✬✸✰✶➓✱✴✭✰❀✽✶❇✪✬❈➎❀✽✶❶✱✴➂❇✪✧➊✰✪✬✾✿✱✴✸❆✯✟â♥➐ ➑