Partial Fraction Expansion

Transcription

1 Paial Facion Expanion Whn ying o find h inv Laplac anfom o inv z anfom i i hlpfl o b abl o bak a complicad aio of wo polynomial ino fom ha a on h Laplac Tanfom o z anfom abl. W will illa h ing Laplac anfom. Thi can b don ing h mhod of paial facion xpanion PFE, which i h v of finding a common dnominao and combining facion. I i poibl o do PFE by hand o i i poibl o MATLAB o hlp. W will illa hand compaion only fo h impl ca whn h a no pad oo and h od of h nmao polynomial i icly l han h od of h dnominao polynomial. Fi w will how why h by-hand mhod wok and hn w will how how on acally do i. Thn w will illa how o MATLAB o do PFE. Why I Wok Fo ha ca, ppo w hav a LT ha w wih o inv b i i no on o abl. Sppo ha h dnominao of can b facod a D d d d d 0 d p p p

2 Fo h ca w a coniding hi can b xpandd ino h fom p p p wh h i a nmb calld h id of h xpanion. Th goal of doing a PFE i o find h id o yo can fom h igh-hand id of h abov qaion. So w nd a way o olv hi fo ach i. L how o do ha fo h oh i a don h am. W mliply hi qaion on boh id by p and g p p p p p p p p p p p wh in h cond lin w hav cancld h common p in h fi m on h igh hand id. hi i h whol poin of h poc!! W hav iolad!! B h a ill a bnch of m lf in h way b w can now g id of hm a follow..

3 ow no ha if w l p hn all h m conaining h oh id diappa: [ p ] p p p p p p p ow yo migh wond why hi don mak h lf-hand id diappa inc h i now alo a p p m on ha id!!! Rmmb ha h i an p m in h dnominao of ha cancl h mliplid p m!!! So hi giv a nic mahmaical l on how o find h id: i [ p ] i p i Rmmb ha hi qaion only hold fo h ca whn h a no pad oo and h od of h nmao of i icly l han h od of h dnominao of How o Do I Illa wih an xampl. Find h inv LT of Fi w hav o faco h dnominao. Uing h qadaic fomla w g h oo and h wi h facod fom a

4 4 Thn w h qaion w divd abov: [ ] o finally 7. oic ha w MUST cancl h mliplid faco BEFORE w vala a h oo val. Similaly, [ ] o finally -4.

5 5 Paial Facion Expanion via MATLAB Th id fncion of MATLAB can b d o comp h paial facion xpanion PFE of a aio of wo polynomial. Thi can b d fo Laplac anfom o Z anfom, alhogh w will illa i wih Laplac anfom h. Th id command giv h pic of infomaion: h id a givn in op vco, h pol a givn in op vco p, h o-calld dic m a givn in op vco k. Whn h od of h nmao polynomial: i l han h od of h dnominao polynomial h will b no dic m. qal h od of h dnominao h will b on dic m;. i on ga han h od of h dnominao h will b wo dic m. c. Tha i, whn h od of h nmao i p ga han h od of h dnominao wih p 0 h will b p dic m. Th id command qi wo inp vco: on holding h cofficin of h nmao and on holding h cofficin of h nmao.

6 6 Th igh-mo lmn in h vco copond o h 0 cofficin, h nx lmn o h lf i h cofficin, c., nil yo ach h high pow; if a pow i no pn i ha a zo cofficin. I i ai o xplain how o id by giving xampl. Ex. #: o Dic Tm, o Rpad Roo, o Complx Roo In hi ca h will b no dic m bca h nmao od i low han h dnominao od. Th nmao vco i [ ] Th dnominao vco i [ ]. Th command and i l i: o: If h i a pow miing h i how yo handl ha ay h dnominao w hn h dn vco i [ 0 ]» [,p,k]id[ -],[ ] p k []

7 7 o ha k i mpy dnod by [] howing ha h a no dic m. Th nmb in a h id hy a h nmb ha go in h nmao of ach m in h xpanion. Th nmb in p a h pol hy a h nmb bacd fom o fom h dnominao of h m. W hav a pol a - wih a id of 7 and. a pol a - wih a id of 4; h: 7 4 and hn ing h Laplac Tanfom Tabl fo ach of h wo m in h xpanion w g: y 7 4

8 8 Ex. #: On Dic Tm, o Rpad Roo, o Complx Roo» [,p,k]id[ -],[ ] - - p - - k Th, w hav: a pol a: - w/ id - a pol a: - w/ id -;. B in hi ca w alo hav a ingl dic m of, which add a m of o h xpanion: and hn ing h Laplac Tanfom Tabl fo ach of h h m in h xpanion w g: y δ

9 9 Ex. #: Two Dic Tm, o Rpad Roo, o Complx Roo 7 H» [,p,k]id[7 -],[ ] 55-9 p - - k 7-9 Thi giv wo m in h vco k o w know w hav wo dic m: 7 and -9. o: if h w M lmn in vco k h lf mo m wold b h cofficin of a z M od dic m. Th, w hav and ing h Laplac Tanfom Tabl w g: y δ δ G a divaiv of h dla fom h diffniaion popy

10 0 Ex. #4: o Dic Tm, A Dobl Roo, o Complx Roo 5 8 4» [,p,k]id[ -],[ 5 8 4] p k [] o ha p ha wo lmn lid a : ha man ha h i a dobl oo. o alo ha h a wo id fo h pol namly, 4 and -: Fi id i fo h fi-od m fo h dobl oo Scond id i fo h cond-od m fo h dobl oo.

11 Th, h xpanion i and ing h Laplac Tanfom Tabl w g: 4 y o, in gnal: if h w an M h od pol w wold hav a m in h xpanion of fo ach od p o and inclding M fo ha pol.

12 Ex. #5: o Dic Tm, o Rpad Roo, Two Complx Roo Thi how how h givn dnominao polynomial i facod ino on al pol and wo complx pol. o ha h wo complx pol a conga of ach oh hi i a ncay condiion fo h cofficin of h polynomial o b al vald. Th, fo h ca w a ind in, w will alway hav complx pol occ in conga pai. Th of id i no diffn h, b wha w do wih h op i a lil diffn.» [,p,k]id[ -],[ 5 9 5] i i p i i

13 k [] z ow w inv ach of h m ing h am LT pai on h LT abl ing h inv LT fo /b wh b i al fo h al pol and i complx fo h complx pol. So [ ].9 co y Th lin in h abov qaion: com fom ing h abl ny. Th nd lin: conv h ±.5 m ino pola fom Th d lin: pll o h common.69 m and combin h complx xponnial m Th 4 h lin: U El fomla o combin h wo complx inoid ino a al inoid in hi ca wih a fqncy of ad/c and a pha of.9 ad.