DIFFERECE EQUATIOS Lier Cos-Coeffiie Differee Eqios Differee Eqios I disree-ime ssems, esseil feres of ip d op sigls pper ol speifi iss of ime, d he m o e defied ewee disree ime seps or he m e os. These ssems re lso lled he seqeil ssems.,, or ) re sed o represe he ip. The re desried differee eqios. A geerl h-order lier os-oeffiie differeil eqio e wrie s h e wrie i omp form Solios of differee eqios re ver similr o hose of he differeil eqios oerprs. If we ssie p h, h ieldig p The omplee solio reqires ddig he prilr solio o he homogeeos oe d evlig he oeffiies of he homogeeos pr sig iiil ilir) odiios. We will e mil fosig o ssems h re iiill res ieldig d for <. These ssems re lso lled s sl ssems. The eqio e rerrged o epress he rre op s he sm of rre d previos ips d ps vles of he op. The eqio lerl idies he eessi for iiil odiios -, -,, - o evle. The "" Operor We epress he i del shif sig he operor s d - -. The, he differee eqio e epressed s
) ) Cllig ) ) D d ) ), he op e epressed s ) ) D D) is he hrerisi polomil of he ssem. Is roos re sed i oiig he homogeeos solio hee he implse respose) of he ssem. The roos e rel or omple. Roos migh e disi or repeed roos. A o-repeig roo ) will orie o he solio ) fio. Repeed roos lso orie similr fios, hvig s mliplier for repeiios. For ssem h ws iiill res, H) )/D) is defied s he rsfer fio of he ssem rsform of he implse respose h) Emple Fid he omplee solio of he ssem represeed 4 4- - - for ssmig h he ssem is iiil res i.e. - - ) D) 4 4 - -, mliplig oh sides : 4 4 ields, /. Hee he homogeeos solio is h The ip is ). The prilr solio is 4 4 p Complee solio Coeffiies d ms e evled sig iiil odiios - - The eqio e rewrie s 4-4- - - 4-4- - - / 4 4 - - 3 3/4 / -/ /) /) -/4 /) 3/4
3 Therefore {-/)/) } Comple roos re epressed i polr form s α±β re ± d he orrespodig solio is defied s r osω). Repeed oge roos will prode sme fios wih mliplier dded for ever repeiio. The Implse Respose The implse respose of he ssem esil fod sig he homogeos solio d ig he ip s δ Emple Fid he implse respose of he ssem i emple - h {/) /)} 4h δ δ- 4h- - h- The firs implse is pplied. Therefore ll vles of d h prior o re osidered s ero. 4h h / 4h - h /4 /)/) /) Therefore, h /)/). This is ifiie implse respose IIR) ssem. Implse d Sep Resposes Implse d sep resposes for lier ssems e drive esil form eh oher. For oios-ime ssem, he i implse e oied from i sep d he i sep from i implse differeiio d iegrio respeivel. For disree-ime ssem, he i implse smple plse) e oied from i sep d he i sep from i implse differeig d smmio respeivel. Tle- smmries he operios. Ssem pe Coiosime Tle Derivio of implse d sep resposes from eh oher Ip Op Sep o implse Implse o Sep o implse Implse o sep sep d δ ) δ τ ) dτ d h dg g ) hτ ) dτ d Disreeime δ δ h g g g h
4 Emple 3 Deermie he sep respose of he ssem i emple-. g h The resl is he sme s wh we oied i emple- from dire solio o Sili The oded-ip oded-op BIBO) sili of he ssem reqires h < This odiio is sisfied ol of he roos of he hrerisi polomil remis wihi i irle s illsred i Fig.. Roos o he irle iself re sle provided h he ssem is o disred oe of is rl modes i.e. he ip does' hve oe of he fios formig he implse respose). rgil sili - Sle Regio Imgir is - Fig. Illsrio of sili for disree-ime ssems -ple Rel is The Freqe Respose Fio If he ip is defied s X m e, he e fod s H ) θ θ θ e ) θ X e me H e ) X me H e ) X me Yme e ) Ω H e ) H e ) e is lled he freqe respose fio of he disreeime ssem. e ) Ω H e ) H e ) e is lled he freqe respose fio of he disreeime ssem. Qesios o Disree-Time Ssem Desripios. For disree-ime ssem represeed he differee eqio: 6. Drw he represeive lo digrm sig os mliplier, i del d smmig io elemes. Fid he implse respose seqee of he ssem d epress i i implii form.
5. Is he ssem sle? Does i hve fiie implse respose FIR)? d. Fid he op seqee sig ovolio operio for -4 e. Drw refll he seqees for d.. Fid d seh he op for he followig ssems sig ovolio:. --4) ; he -. ) - ) ; h) 3* - *3 - ) 3. A disree-ime is represeed he differee eqio.-.35-.4-. Drw represeive lo digrm of he ssem sig del elemes, os mlipliers d smmig io. Fid o d drw he implse respose of he ssem firs 5 oero elemes ol). Is he ssem sle? Wh? d. Fid he op seqee for sig ovolio Hi: 4. A disree-ime filer is desiged so h he op is oied ddig he ip d 3% of he op wo seps efore d srig % of he op sep efore.. Wrie dow he differee eqio represeig he ssem d drw represeive lo digrm of he ssem sig del elemes, os mlipliers d smmig io. Fid o d drw he implse respose of he ssem firs 5 oero elemes ol). Is he ssem sle? Wh?