Instructors Solution for Assignment 3 Chapter 3: Time Domain Analysis of LTIC Systems

Transcription

1 Inrucor Soluion for Aignmn Chapr : Tim Domain Anali of LTIC Sm Problm i a 8 x x wih x u,, an Zro-inpu rpon of h m: Th characriic quaion of h LTIC m i i 8, which ha roo a ± j Th zro-inpu rpon i givn b zi A co B in for, wih A an B bing conan To calcula hir valu, w ubiu h iniial coniion an in h abov quaion Th ruling imulanou quaion ar b A A B ha ha h oluion, A an B Th zro-inpu rpon i hrfor givn b zi Bcau of h zro iniial coniion, h zro-inpu rpon i alo zro Zro-a rpon of h m: To calcula h zro-a rpon of h m, h iniial coniion ar aum o b zro Hnc, h zro a rpon z can b calcula b olving h iffrnial quaion 8 x x wih iniial coniion, an, an inpu x xpu Th homognou oluion of m i ha h am form a h zro-inpu rpon an i givn b h z C co C in for, wih C an C bing conan Th paricular oluion for inpu x xpu i of h form p z K u Subiuing h paricular oluion in h iffrnial quaion for m i an olving h ruling quaion giv K /8 Th zro-a rpon of h m i, hrfor, givn b z C co C in u 8

2 Soluion o Aignmn To compu h valu of conan C an C, w u h iniial coniion, an aum for h zro-a rpon Subiuing h iniial coniion in z la o h following imulanou quaion c C C C 8 wih oluion C /8 an C /8 Th zro-a oluion i givn b z co in u 8 Ovrall rpon of h m: Th ovrall rpon of h m i obain b umming up h zro-inpu an zro-a rpon, an i givn b co in u 8 Sa a rpon of h m: Th a a rpon of h m i obain b appling h limi,, o an i givn b lim co in u 8 iii a x wih x! " co in # $ u,, an Zro-inpu rpon of h m: Th characriic quaion of h LTIC m iii i, which ha roo a, Th zro-inpu rpon i givn b zi A for, wih A an B bing conan To calcula hir valu, w ubiu h iniial coniion an in h abov quaion Th ruling imulanou quaion ar A B A ha ha a oluion, A an B Th zro-inpu rpon i hrfor givn b B b zi u Zro-a rpon of h m: To calcula h zro-a rpon of h m, h iniial coniion ar aum o b zro Hnc, h zro a rpon z can b calcula b olving h iffrnial quaion x wih iniial coniion, an, an inpu x [co in]u Th homognou oluion of m iii ha h am form a h zro-inpu rpon an i givn b h z C C

3 Soluion for, wih C an C bing conan Th paricular oluion for inpu x [co in]u i of h form z p K co K in K co K in Subiuing h paricular oluion in h iffrnial quaion for m iii an olving h ruling quaion giv K co K in K co K in K in K co K in K co K co K in K co K in co in Collcing h cofficin of h coin an in rm, w g K K K co K K K in K K K co K K K in which giv K, K 5, K 6, an K 8 Th zro-a rpon of h m i c C C 5in 6co 8in u z To compu h valu of conan C an C, w u h iniial coniion, an Subiuing h iniial coniion in z la o h following imulanou quaion C C C wih oluion C 6 an C Th zro-a oluion i givn b 6 5in 6co 8in u z Ovrall rpon of h m: Th ovrall rpon of h m i obain b umming up h zro-inpu an zro-a rpon, an i givn b or, u 6 5in 6co 8in u or, 6 9 5in 6co 8in u Sa a rpon of h m: Th a a rpon of h m i obain b appling h limi,, o an i givn b 9 5in 6co 8in u lim 6 or, 5in 6co 8in u v a x wih x u,, an Zro-inpu rpon of h m: Th characriic quaion of h LTIC m v i, which ha roo a ±j, ±j Th zro-inpu rpon i givn b zi j j j j A B C D,

4 Soluion o Aignmn for, wih A an B bing conan To calcula hir valu, w ubiu h iniial coniion in h abov quaion Th ruling imulanou quaion ar A ja A ja B jb B C jc C jc D jd D ha ha a oluion, A j5 Β 5, C j5 an D 5 Th zro-inpu rpon i b which ruc o zi j j j j j5 5 j5 5 u, zi 5in 5 co u Zro-a rpon of h m: To calcula h zro-a rpon of h m, h iniial coniion ar aum o b zro Hnc, h zro a rpon z can b calcula b olving h iffrnial quaion x wih all iniial coniion o an inpu x u Th homognou oluion of m v ha h am form a i zro-inpu rpon an i givn b h z j j j C C C C whr C i ar conan Th paricular oluion for inpu x u i of h form p z Ku Subiuing h paricular oluion in h iffrnial quaion for m v an olving h ruling quaion giv K, or, K Th zro-a rpon of h m i givn b j j j j j z C C C C, for To compu h valu of conan C i, w u zro iniial coniion Subiuing h iniial coniion in z la o h following imulanou quaion A ja A ja B jb B C jc C jc D jd D wih oluion C, C j5, C, an C j5 Th zro-a oluion i givn b c which ruc o z j j j j j5 j5 u, zi co in u Ovrall rpon of h m: Th ovrall rpon of h m i obain b umming up h zro-inpu an zro-a rpon, an i givn b

5 Soluion 5 or, 5in 5 co u co in u or, 5in co in 5 co u Sa a rpon of h m: Th a a rpon of h m i obain b appling h limi,, o an i givn b Problm 5 i co in 5 co u lim 5in Th oupu i givn b Rcall ha Thrfor, h oupu i givn b u u u u u u if if > if u r if < Th aformnion convoluion can alo b compu graphicall iii Th oupu i givn b [ u u u ] [ u u ] Uing h propri of h convoluion ingral, h oupu i xpr a [ u u ] [ u u ] [ u u ] [ u u ] [ u u ] [ u u ] Ba on h rul of par i, i, u * u r, h ovrall oupu i givn b r r r r r r vi Coniring h wo ca < an paral Ca I < : which ruc o or, Ca II : 5 5 5

6 6 Soluion o Aignmn which ruc o or, 5 5 Hnc, h ovrall xprion for i givn b Problm < iii Uing h graphical approach, h convoluion of x wih w i hown in Fig S6, whr w conir ix iffrn ca for iffrn valu of Ca I < : Sinc hr i no ovrlap, Ca II < : Ca III < : w x x a Wavform for z b Wavform for x c Wavform for x w x w x w x Ovrlap bw w an x for < Ovrlap bw w an x for < f Ovrlap bw w an x for < w x w x w x g Ovrlap bw w an x for < h Ovrlap bw w an x for < i Ovrlap bw w an x for >

7 Soluion 6 x*w iii j Convoluion oupu Fig S6: Convoluion of x wih w in Problm 6iii Ca IV < : Ca V < : Ca VI > : Sinc hr i no ovrlap, 9 5 Combining all h ca, h rul of h convoluion x w i givn b Th oupu i plo in Fig S6j 5 9 < < < < lwhr vii Uing h graphical approach, h convoluion of v wih z i hown in Fig 6, whr w conir ix iffrn ca for iffrn valu of v z z a Wavform for v b Wavform for z c Wavform for z

8 8 Soluion o Aignmn v z v z v z Ovrlap bw v an z for < Ovrlap bw v an z for < f Ovrlap bw v an z for < v z v z v z g Ovrlap bw v an z for < h Ovrlap bw v an z for < i Ovrlap bw v an z for > v*z j Convoluion oupu Fig S6: Convoluion of v wih z in Problm 6vii Ca I < : Sinc hr i no ovrlap, Ca II < : [ ] [ ] [ ]

9 Soluion 9 Ca III < < : [ ] [ ] [ ] [ ] Ca IV < < : [ ] [ ] [ ] [ ] Ca V < : [ ] [ ] [ ] Ca VI > : Sinc hr i no ovrlap, Combining all h ca, h rul of h convoluion v z i givn b < < < < lwhr Th oupu i hown in Fig S6j a h n of h oluion of hi problm ix Uing h graphical approach, h convoluion of v wih w i hown in Fig 69, whr w conir ix iffrn ca for iffrn valu of

10 Soluion o Aignmn v w w a Wavform for v b Wavform for w c Wavform for w v w v w v w Ovrlap bw v an w for < Ovrlap bw v an w for < f Ovrlap bw v an w for < v w v w v w g Ovrlap bw v an w for < h Ovrlap bw v an w for < i Ovrlap bw v an w for > 9 w*w j Convoluion oupu 9 Fig S69: Convoluion of v wih w in Problm 6ix Sinc w, hrfor, h xprion for w i w Ca I < : Sinc hr i no ovrlap, 9 if < if >

11 Soluion Ca II < : 5 9 Ca III < < : [ ] [ ] [ ] [ ] [ ] [ ] 9 Ca IV < < : [ ] [ ] [ ] [ ] [ ] [ ] 9 Ca V < : 5 9 Ca VI > : Sinc hr i no ovrlap, 9 Combining all h ca, h rul of h convoluion 9 w v i givn b

12 Soluion o Aignmn 9 Th oupu i 9 hown in Fig S69j 5 5 < < < < lwhr Problm iii Sm h i NOT mmorl inc h for Sm h i caual inc h for < Sm h i BIBO abl inc in in in h π u π π < vii Sm h i NOT mmorl inc h for Sm h i caual inc h for < Sm h i NOT BIBO abl inc h co5 Conir h boun inpu ignal co5 If hi ignal i appli o h m, h oupu can b calcula a: x h co5 5 co5 u co55 co5 Th oupu a i givn b, co { fini valu co 5 co5 co 5 co I i obrv ha h oupu bcom unboun a vn if h inpu i alwa boun Thi prov ha h m i no BIBO abl viii Sm h8 i NOT mmorl inc h8 for Sm h8 i NOT caual inc h8 for < Sm h8 i BIBO abl inc

13 Soluion ln95 ln95 h ln95 [ ] 9 < ln95 ln95

14 Soluion o Aignmn Soluion o Problm of Chapr 6 Laplac Tranform Problm 6 b B finiion X x Ingral I ruc o I I [ ] provi R{ } > ROC R :R{ } < II whil ingral II ruc o II [ ] provi R{ } > ROC R :R{ } > Th Laplac ranform i hrfor givn b 6 X I II wih ROC : R R I R or R : < R{} < 9 B finiion X x co9 u co9 Th abov xprion ruc o [ co9 9 in9 ] X co9 9 or, X 9 9 [ ] provi R{ } > R{ } > f W riv h Laplac ranform for wo ca: an Ca I: B finiion 5 X x

15 Soluion 5 Ca II: B finiion X x Ingral I i givn b I I whil Ingral II i givn b II Th Laplac ranform i hrfor givn b X ROC: Enir -plan II No ha h ca can alo b riv from h ohr rul b appling h limi,, an h L Hopial rul Problm 6 b Uing parial fracion xpanion an aociaing h ROC o iniviual rm, giv X 56 A B ROC:R{}< ROC:R{}< whr conan A, an B wr compu in par a a A, an B Taking h invr ranform of X, giv x u No ha h am raional fracion for X giv iffrn im omain rprnaion if h aocia ROC i chang X 56 A B C ROC:R{}< ROC:R{}< ROC:R{}< f whr conan A, B, an C wr compu in par c a A, B 6, an C Taking h invr ranform of X, giv x 6 u Uing parial fracion xpanion an aociaing h ROC o iniviual rm, giv

16 6 Soluion o Aignmn X whr B C an D A B C R{ } > R{ } > R{ } > [ ] [ ] [ ] [ ] [ ] [ ] To valua A, xpan X a D R{ } > A B C D an compar h cofficin of W g A C D which ha a oluion A 5/ Th Laplac ranform ma b xpr a X 5 R{ } > R{ } > R{ } > R{ } > Taking h invr ranform of X, giv g x 5 u u u u Uing parial fracion xpanion an aociaing h ROC o iniviual rm, giv X whr C 6 D E A B C 6 ROC:R{ } < ROC:R{ } < [ ] [ ] 6 6 To valua A, B, an C xpan X a ROC:R{ } < ROC:R{ } < A 6 B 6 C 6 D E an compar h cofficin of,,, an W g A D A B D E A B C D E A 6B D E cofficin of cofficin of, or, cofficin of cofficin of A A D B D E A B D E A 6B D E which ha a oluion of A 6, B 6, D 6, an E 8 Taking h invr ranform of X, giv

17 Soluion x 6 u 6 u 6 Problm 6 a Calculaing h Laplac ranform of boh i, w g u 6co u 5in u " Y % " % $ ' # $ &' $ Y ' Y # $ &' which ruc o Y or, Y Calculaing h invr Laplac ranform, w g u u Calculaing h Laplac ranform of boh i, w g " % " % $ Y ' # $ &' $ Y ' # $ &' Y, which ruc o Y, or, A B C D E Y, whr A [ ] [ ] Equaing numraor of Y on boh i an ing A, w g B C D E B C B D C E Comparing h cofficin of polnomial of iffrn orr w g Cofficin of : B B Cofficin of : C Cofficin of : B D D D Cofficin of : C E E Th parial fracion xpanion of Y i givn b Y Th invr ranform i hrfor givn b [ ] co 5in u whr w hav u h following ranform pair

18 8 Soluion o Aignmn which i prov in Problm 6b Problm 6 Soluion: a L ω in ω u ω Th Laplac ranform of h inpu an oupu ignal ar givn b an Y X Diviing Y wih X, h ranfr funcion i givn b Y X H Th impul rpon i obain b aking h parial fracion xpanion of H a follow H Taking h invr Laplac ranform, h impul rpon i givn b h δ u u In orr o calcula h inpu-oupu rlaionhip in h form of a iffrnial quaion, w rprn h ranfr funcion a Y X H Cro mulipling, w g Y X which can b rprn a Y 8Y X X X Taking h invr Laplac ranform an auming zro iniial coniion, h iffrnial quaion rprning h m i givn b c x x 8 x Th Laplac ranform of h inpu an oupu ignal ar givn b an Y X Diviing Y wih X, h ranfr funcion i givn b Y X H Th impul rpon i obain b aking h invr Laplac ranform Th impul rpon i givn b h u [ u ] In orr o calcula h inpu-oupu rlaionhip in h form of a iffrnial quaion, w rprn h ranfr funcion a

19 Soluion 9 H Y X Cro mulipling, w g Y Y X X 8X Taking h invr Laplac ranform, h inpu-oupu rlaionhip of h m i givn b x x 8x No ha hr i no ovrlap bwn h ROC of h wo rm xpu an xpu, hrfor, h Laplac ranform for o no xi Problm 65 j j a H c Two zro a j, j Two pol a, Bcau boh pol ar in h lf han i of h -plan, h m i alwa BIBO abl H / On zro a / Two pol a, 6 Bcau boh pol ar in h lf i of h -plan, h m i alwa BIBO abl H Th m o no hav an zro On pol a Thr i onl on pol, which i loca on h imaginar axi Thrfor, h m i a marginall abl m