CALCULATION OF CONVOLUTION PRODUCTS OF PIECEWISE DEFINED FUNCTIONS AND SOME APPLICATIONS

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1 CALCULATION OF CONVOLUTION PRODUCTS OF PIECEWISE DEFINED FUNCTIONS AND SOME APPLICATIONS Abstrct Mirce I Cîru We preset elemetry proofs to some formuls ive without proofs by K A West d J McClell i 99 B L Evs d J H McClell i 99 d J Cvicchi i for the clcultio of the covolutio iterls d sums of piecewise defied fuctios Ulie divide d couer strtey these formuls re of the type couer wht is divided Applictios to differetil eutios probbility theory d lier discrete system theory re ive A emple i coeio with the commuttivity of the covolutio product is lso ive For completeess i Ae we iclude proof of the well-ow elemetry solutio method for solvi o-homoeeous lier differetil eutios with costt coefficiets used i the first pplictio The preset wor is prt of series of the uthor s rticles some of which bei published i this Jourl tht preset the products of covolutio both i discrete d cotiuous cse d some of their pplictios Keywords: covolutio products piecewise defied fuctios o-homoeeous lier differetil eutios elemetry solutio method distributio fuctios of rdom vribles lier discrete system theory Mthemtics Subject Clssifictio: A A 6E 9A Itroductio I mthemtics d its pplictios the fuctios to be used re ot lwys s ood s we would lie So most of them re piecewise defied by differet lyticl formuls Thus c be for emple the coefficiets d the riht side of o-homoeeous lier differetil eutio we wt to solve Of course it c be solved seprtely o ech itervl But this is very slow procedure More useful is method pplicble to the mtter cosidered s whole For the metioed problem such method cosists i computtio by the formuls preseted i this pper of the iterl of covolutio betwee the riht side of the eutio d its elemetry solutio This will be mde i the pplictio ive i Sectio A first method for clculti covolutio iterls d sums of piecewise defied fuctios ws ive i the ppers [] d [] It correspods to below corollries A more compct formul ws ive by T J Cvicchi i [] its result bei preseted i theorem from Sectio i cotiuous vrible cse d i theorem from Sectio i discrete vrible cse The mi purpose of these formuls is to idicte the rel limits of itertio respective summtio Becuse i the cited ppers the proofs re oly setched we will ive i Sectios the elemetry but riorous proofs of these formuls i cotiuous vrible cse I cse of discrete vrible they re similr d will Ph DProfessor t Fculty of Applied Scieces Polytechic Uiversity of Buchrest Romi emil:cirumirce@yhoocom

2 be omitted from Sectio Both specil cse d the cses whe the support itervls of the fctor fuctios hve some ifiite limits re ive i Sectio A rther difficult method for clculti the covolutio iterls bsed o represettios of the fctors by step-fuctios ws ive by I S Goldber M G Bloc d R E Rojs i []As is ow the covolutio product is used i my chpters from mthemtics physics d techoloy Therefore these formuls for fst clculus of the covolutios c be used i these res For emple we ive i Sectio few such pplictios As oted bove i Sectio we determie prticulr solutio of ohomoeeous lier differetil eutio with piecewise costt coefficiets d piecewise cotiuous riht side This is mde by covolutio betwee the riht side of the eutio d its elemetry solutio both piecewise defied fuctios For completeess we ive i Ae the proof of the elemetry solutio method I Sectio the covolutio formul is used to compute the probbility distributio fuctio of the sum of two idepedet rdom vribles s covolutio of their distributio fuctios I Sectio we preset two pplictios to lier discrete system theory I Sectio 6 is ive emple which shows tht lthouh the covolutio is commuttive ( f ) ( f ) clcultio of the two products is differet I ll these pplictios the covolutio iterls d sums betwee piecewise defied fuctios tht rise will be clculted usi the formuls preseted i Sectios d Becuse the formuls preseted i this pper re elemetry mily cosisti i clcultios of iterls or sums they c be used i the techi process s pplictios of the mthemticl lysis i eutios probbility d differet chpters of physics Cotiuous Vrible Cse The covolutio product of two fuctios of rel vrible with comple vlues f d is defied by the well-ow formul ( ) f ( y) ( y) f dy (c) Theorem c (Cotiuous Cse) If The Fuctios f Ad Are Iterble O The Itervls [ l r] d [ ( ) r( ) ] tes the form l d zero otherwise the the covolutio product f mi ( r l( )) ( f ) f ( y) ( y) m ( l r( )) dy [ l( f ) r( f ) ] (c) d zero otherwise where l ( f ) l l( ) d r ( f ) r r( )

3 Proof Whe < l( f ) l l( ) for y < l we hve f ( y) y l we hve y l < l( ) hece ( y) we et ( f ) Sme result is obtied whe > r( f ) Now suppose tht l( f ) r( f ) If y < m ( l r( ) ) followi two cses: d for Therefore i this cse ) If < l r( ) the r( ) < l hece m ( l r( ) ) l y < l hece f ( y) d therefore ( f ) we cosider the It results ) If l r( ) the l r( ) hece m ( l r( ) ) r( ) results y < r( ) hece y > r( ) We obti ( y) ( f ) Sme result is obtied whe y > mi ( r l( ) ) reduced to (c) We deote λ r l d λ ( ) r( ) l( ) itervls of the fctor fuctios f d d It hece Therefore formul (c) is the leths of the two support mi ( l r( ) r l( ) ) M ( l r( ) r l( ) ) m Obviously l( f ) m M r( f ) m Corollry I ssumptios of the theorem C the covolutio product is ive by formuls l ( ) ( f ) f ( y) ( y) dy [ l( f ) m] l ( ) l (c) ( f ) f ( y) ( y) dy [ m M ] if ( ) λ r r( ) λ < (c) ( f ) f ( y) ( y) dy [ m M ] if λ( ) l r ( f ) f ( y) ( y) dy [ M r( f ) ] r ( ) λ < (c) (6c) Proof If < m the < l r( ) d < r l( ) hece r( ) < l d l( ) < r I this cse m ( l r( ) ) l d mi ( r l( ) ) l( ) therefore formul (c) reduces to (c) Aloously if > M formul (c) is reduced to (6c)

4 Now we suppose tht hece m M If ( ) λ r( ) m M r l( ) l λ < the r( ) l( ) < r l I this cse l r( ) d l( ) r m ( l r( ) ) r( ) d mi ( r l( ) ) l( ) (c) is reduced to (c) Aloously if λ( ) hece therefore formul λ < formul (c) is reduced to (c) Remr Throuhout the wor ech fuctio is ive o its support itervl bei zero otherwise Discrete Vrible Cse The covolutio product of two fuctios of iteer vrible with comple vlues f d is defied by formul ( ) f ( ) ( ) f (d) As i Sectio we c show the followi discrete results: Theorem D (Discrete cse) If the fuctios f d [ l r] d [ ( ) r( ) ] tes the form re o the itervls l d zero otherwise the the covolutio product f ( r l( )) mi m ( f ) f ( ) ( ) (d) ( l r( )) Corollry I ssumptios of the theorem D the covolutio product is ive by formuls l l ( ) ( f ) f ( ) ( ) [ l( f ) m] ( ) l (d) ( f ) f ( ) ( ) [ m M ] if ( ) λ r r( ) λ < (d) ( f ) f ( ) ( ) [ m M ] if λ( ) l r ( f ) f ( ) ( ) [ M r( f ) ] r ( ) λ < (d) (6d)

5 Remrs ) If λ( ) λ the m M d the covolutio hs oly cses () d (6) ) Formuls ive i theorem d its corollry both for cotiuous d discrete vrible lso pply if some of the etremities of support itervls [ l r] d [ l ( ) r( ) ] of fuctios f d re ifiite I these cses some of the formuls ()-(6) must to be omitted For emple if r the M r( f ) hece the covolutio hs oly the cses () d () If i dditio r ( ) the m d the covolutio is clculted oly with the formul () I this cse if l l( ) the formul () is reduced to the cusl covolutio f f ( y) ( y) d zero otherwise dy respective f f ( ) ( ) ) A fuctio f is med piecewise defied if f f where f re fuctios m with disjoit support itervls If j j is other such fuctio the m j f f j hece the covolutio product of such fuctios is reduced to the bove-metioed cse Applictios Differetil Eutios A prticulr solutio for lier o-homoeeous differetil eutio with costt coefficiets c be obtied by the covolutio betwee the elemetry solutio of the eutio d its riht side The iterl covolutio formuls preseted i Sectio c be used whe the riht side of the eutio is piecewise cotiuous fuctio d its coefficiets re piecewise costt fuctios Emple Let us determie prticulr solutio of the differetil eutio I u u v [ ) u ( ) u whe stisfyi the iitil coditios

6 π π II [ π ] ( π ) v Solutio We deote v [ π ] d v ( π ) If π the eutio hs the form u u v The elemetry solutio u u is the solutio of the homoeeous eutio coditios u ( ) d u ( ) hece is the fuctio E si [ ) The solutio is v E v E v E u mi π v ( y) E ( y) [ π ] v E m ( ) π v E where dy si( y) dy cos mi ( ) v ( y) E ( y) m ( π ) dy tht stisfies the iitil v E si y dy cos so π π d E v y E y dy π so ( ) v ( y) for [ π ] v E ysi( y) dy π π π If v becuse cos si π the eutio hs the form u u v solutio E si [ ) v E v E where u mi π π with the elemetry The solutio is v E v ( y) E ( y) dy v ( y) E ( y) dy becuse ( y) v E m mi ( ) ( ) v ( y) E ( y) m( π ) Therefore the solutio is dy π ysi ( y) dy π cos v 8 si

7 cos u ( π ) cos si π cos 8 (d zero otherwise) Probbility Theory si [ π ] π π π Becuse the probbility distributio fuctio of the sum of two idepedet rdom vribles is the covolutio of their distributio fuctios it c be clculted by formuls ive i Sectio Emple Let X d Y be idepedet cotiuous rdom vribles hvi the distributios f ( ) [ ] d [ ] Let us determie the distributio of the sum X Y Solutio The reuested distributio is f f ( y) ( y)dy mi m ( ) ( y ) dy [ ] hece ( ) f ( y) dy ( ) [ ] f ( y ) dy [ ] < L( ) f ( y) dy ( ) L fuctio f bei zero otherwise System Theory becuse [ ] the Emple We cosider time-ivrit lier discrete system hvi the output 6 7 ( ) whe the iput is ( ) Determie the output correspodi to iput f [ ] where is comple umber d turl umber Solutio The cosidered system is represeted mthemticlly by covolutio opertor with seuece h med impulse respose or weiht seuece of the system Nmely the reltio betwee rbitrry iput f d its correspodi output f h h c be is ive by the formul The trsfer seuece

8 determied by the iverse opertio of discrete covolutio med decovolutio or lo divisio / / / / / / / / It results h The reuired output is [ ] h f mi m mi m m mi [ ] [ ] m m d zero otherwise For < we et For we et 8 9 For > we et

9 Emple Cosider system s i emple hvi the trsfer seuece β Determie the output correspodi to iput h p f Here p d re comple umbers while d β re iteer umbers Solutio The desired output is ( f h) β I this cse the covolutio product must be clculted oly with formul (d) For p d β usi formul for the sum of eometric proressio we et β For β p p β p p p p d β we et β p p p ( β ) p p p β 6 O The Commuttivity Of The Covolutio Product p β p p As covolutio product is commuttive the products ( f ) d β hve the sme vlue but their clcultio is differet We preset emple tht will show it i differet situtios Emple Clculte the covolutio products ( f ) d ( f ) for fuctios f [ ] d [ ] with > zero otherwise Solutio We hve l r λ l ( ) r ( ) λ ( ) m mi( ) M m( ) l ( f ) l( f ) r ( f ) r( f ) For < m M d λ ( ) < λ hece from (c) (c) d (6c) we hve ( ) f y dy [ ] ( f ) y dy [ ] [ ] d from (c) (c) d (6c) we hve ( ) f ydy [ ] ( f ) ydy [ ] ( f ) ( y) dy [ ] ( ) f ydy

10 For m M hece from (c) d (6c) we hve ( f ) ( y) ( f ) ( y) dy ( ) [ ] dy ( ) ( ) f ydy [ ] ( f ) [ ] d from (c) d (6c) we hve ( ) ydy [ ] For > m M d λ( ) hve ( ) f y dy [ ] ( ) ( y) [ ] λ < hece from (c) (c) d (6c) we f dy [ ] d from (c) (c) d (6c) we hve ( ) f ydy [ ] ( f ) ydy [ ] ( f ) [ ] ydy 7 Ae: Solvi Lier Differetil Eutios By Elemetry Solutios ( ) We preset i the followi theorem the elemetry solutios method of solvi ohomoeeous lier differetil eutios with costt coefficiets Theorem The o-homoeeous lier differetil eutio u ( ) v [ ) ( f ) ( y) (7) with costt coefficiets hs prticulr solutio tht stisfies the iitil coditios u ( ) ( ) ive by the covolutio formul u v E v( y) E( y) where dy dy (8) E is the elemetry solutio of eutio mely the solutio of the homoeeous ssocited eutio

11 ( u ) [ ) (9) tht stisfies the iitil coditios ( ) ( ) E () ( ) E () Proof If u is ive by formul (8) by differetitio of the iterl with respect to its prmeter d ti ito ccout the iitil coditios () d () we obti u ( ) ( v( y) E ) ( y) ( ) ( v( y) E ) ( y) dy v dy () u () Usi reltios (8) () () it results u v y E y dy v ( ) v y ( ) E y dy v v the lst eulity resulti from the reltio (9) pplied for u E( y) fuctio results tht u hs zero iitil coditios Refereces Thus the u ive by formul (8) is solutio of eutio (7) From (8) d () it lso T J Cvicchi Simplified method for lyticl evlutio of covolutio iterls IEEE Trs Educ vol pp - My M I Cîru Solvi differece d differetil eutios by discrete decovolutio UPB Scietific Bulleti A 69 (7) -6 M I Cîru Determitio of prticulr solutios of o-homoeeous lier differetil eutios by discrete decovolutio UPB Scietific Bulleti A 69 (7) -6 M I Cîru Approimte clculus by decovolutio of the polyomil roots UPB Scietific Bulleti A 69 (7) 9- M I Cîru Iitil d boudry vlue problems for differece eutios Jourl of Iformtio Systems d Opertios Memet M I Cîru First order differetil recurrece eutios with discrete uto-covolutio Itertiol Jourl of Mthemtics d Computtio M I Cîru Euleri umbers d eerlized rithmetic-eometric series UPB Scietific Bulleti A 7 (9) - 8 M I Cîru New pplictios of the Abel-Liouville formul Jourl of Iformtio Systems d Opertios Memet 9-9 M I Cîru (with F D Frumosu) Iitil vlue problems for olier differetil eutios solved by differetil trsform method Jourl of Iformtio Systems d Opertios Memet Prt I 9-7 Prt II

12 M I Cîru Lier discrete covolutio d its iverse Jourl of Iformtio Systems d Opertios Memet Prt I Covolutio 9-7 Prt II Decovolutio -7 M I Cîru (with I Bdrlei) New methods for solvi lebric eutios Jourl of Iformtio Systems d Opertios Memet Prt I 8- Prt II M I Cîru Newto s idetities d the Lplce trsform The Americ Mthemticl Mothly 7 () 67-7 M I Cîru (with I Bdrlei) O Newto-Rphso method Jourl of Iformtio Systems d Opertios Memet 9-9 M I Cîru (with G Cîrţlă) Use of spectrl theory of mtrices to study seismic movemets Jourl of Iformtio Systems d Opertios Memet 7-8 M I Cîru Iitil-vlue problems for first-order differetil recurrece eutios with uto-covolutio Electroic Jourl of Differetil Eutios () o - 6 M I Cîru Determitl formuls for sum of eerlized rithmetic-eometric series Boleti de l Asocicio Mtemtic Veezol () o -8 7 M I Cîru A certi iterl-recurrece eutio with discrete-cotiuous utocovolutio Archivum Mthemticum (Bro) 7 () B L Evs J H McClell Alorithms for symbolic lier covolutio Proc Of Asilomr Cof o Sils Systems d Computers 99 9 S Goldber M G Bloc d R E Rojs A systemtic method for the lyticl evlutio of covolutio iterl IEEE Trs Educ vol pp 6-69 Feb K A West J H McClell Symbolic covolutio IEEE Trs Educ vol 6 pp 86-9 Nov 99