Inverse Systems

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INVERSE SYSTEMSDSP

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GROUP MEMBERS  Muhammad Raza (12063122-043)

  Salah Ud Din (12063122-029)

  Mubashar Naeem (12063122-03)

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INVERSE SYSTEM  !he s"s#em $i(z) is #he in%erse s"s#em #& $(z)

i' 

    (z) $(z)*$i(z) 1

  +hi,h imlies #ha#

  $(z) 1*$i(z)  !he #ime-d&main e.ui%alen# is

  !he .ues#i&n &' +hi,h R/ #& ass&,ia#e +i#h

$i(z) is ans+ered b" #he ,&n%&lu#i&n #he&rem

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IN TIME DOMAIN

  &r #he #ime d&main e.ua#i&n #& h&ld #he

rei&ns &' ,&n%eren,e &' $(z) and $i(z)

mus# &%erla

  For example:

  Inverse sysem !or !"rs#or$er sysem

  5e# $(z) be

  +i#h R/ z 7 09

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E%AMP&E!hen $i(z) +ill be

  Sin,e #here is &nl" &ne &le #here are &nl"

#+& &ssible R/s  Imp'lse response o! "nverse sysem

  !he ,h&i,e &' R/ '&r $i(z) #ha# &%erlas

+i#h z 7 09 is z 7 08

  !here'&re #he imulse res&nse &' #hein%erse s"s#em is

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MINIMUM P(ASE SYSTEM

  n #his ,ase #he in%erse is b&#h ,ausal and

s#able

  : 5! s"s#em is s#able and ,ausal +i#h as#able and ,ausal in%erse i' and &nl" i' b&#h

#he &les and zer&s &' $(z) are inside #he

uni# ,ir,le

  Su,h s"s#ems are als& ,alled m"n"m'm p)ases"s#ems

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FRE*UEN+Y RESPONSE

!he 're.uen," res&nse &' #he in%erses"s#em i' i# e;is#s is

  N&# all s"s#ems ha%e an in%erse  &r e;amle #here is n& +a" #& re,&%er #he

're.uen," ,&m&nen#s ab&%e #he ,u#&''

're.uen," #ha# +ere se# #& zer& b" #he a,#i&n

&' #he l&+-ass 'il#er

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IMPU&SE RESPONSE FOR RATIONA&SYSTEM FUN+TIONS

  ' a s"s#em has a ra#i&nal #rans'er 'un,#i&n+i#h a# leas# &ne &le #ha# is n&# ,an,elled

b" a zer& #hen #here +ill al+a"s be a #erm

,&rres&ndin #& an in'ini#e len#h se.uen,e

in #he imulse res&nse  Su,h s"s#ems are ,alled "n!"n"e "mp'lse

response ,IIR- s"s#ems

  /n #he &#her hand i' a s"s#em has n& &les

e;,e# a# z0 #hen

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IMPU&SE RESPONSE  n #his ,ase #he s"s#em is de#ermined #&

+i#hin a ,&ns#an# mul#ilier b" i#s zer&s S&

#he imulse res&nse has a 'ini#e len#h

  <hen #he imulse res&nse is 'ini#e in

len#h !hen #he s"s#em is ,alled a !"n"e"mp'lse response ,FIR- s"s#em

  Example: A !"rs#or$er ,IIR- sysem i%en a,ausal s"s#em sa#is'"in #he di''eren,e

e.ua#i&n

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E%AMP&E

  #he s"s#em 'un,#i&n is

  <here #he ,&ndi#i&n '&r s#abili#" is

.a. / 0

  !he in%erse z-#rans'&rm is

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ANOT(ER E%AMP&EA s"mple FIR sysem1 ,&nsider #he #run,a#ed

imulse res&nse

  !he s"s#em 'un,#i&n is

  !he zer&s &' #he numera#&r are a#

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RO+

  <i#h a assumed real and &si#i%e he &le a#

za is ,an,elled b" a zer& !he &le-zer& l&#

+ill be d&ne '&r #he ,ase &' M

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DIFFEREN+E E*UATION  !he di''eren,e e.ua#i&n sa#is'ied b" #he

inu# and &u#u# &' #he 5! s"s#em is #he,&n%&lu#i&n

  !he inu# and &u#u# als& sa#is'" #he

di''eren,e e.ua#i&n