The discrete laplace z transform

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DNPb!  c c c  SP DN EN  N Q p p    ccc R ZK H U H W K H SN  N      P D U H NQ R ZQ F R H r F LH Q W V D Q G EN p N    ccc LV D NQ R ZQ V H TX H Q F H  ,Q P R V W F D V H V ZH ZLOO W D NH W K H V H W R E H U H D O E X W W K D W LV Q R W D Q H V V H Q W LD O U H V W U LF W LR Q  -X V W D V W K H / D S OD F H W U D Q V IR U P ID F LOLW D W H V W K H V R OX W LR Q R I OLQ H D U R U G LQ D U \ G LoH U H Q W LD O H TX D W LR Q V ZLW K F R Q V W D Q W F R H r F LH Q W V  D G LV F U H W H YH U V LR Q R I W K H W U D Q V IR U P  IU H TX H Q W O\ F D OOH G W K H ? = W U D Q V IR U P  D V V LV W V LQ V R OYLQ J W K H V H OLQ H D U U H F X U V LR Q H TX D W LR Q V D V ZH OO D V V R P H R W K H U H TX D W LR Q V LQ G LV F U H W H F R Q YR OX W LR Q IR U P  0D Q \ R I W K H IR U P X OD V D Q G U H OD W LR Q V K LS V ZLOO E H V LP LOD U W R W K R V H D S S O\LQ J W R W K H / D S OD F H W U D Q V IR U P E X W W K H U H D U H V LJ Q Lp F D Q W G LoH U H Q F H V  LQ S D U W LF X OD U W K H Q H F H V V LW \ W R G H D O ZLW K G LV F U H W H LQ G LF H V OH D G V W R V R P H P LQ R U F R P S OLF D W LR Q V  7K H U H D U H YD U LR X V ZD \V LQ ZK LF K W K H W U D Q V IR U P F D Q E H G H p Q H G  ZH ZLOO X V H W K H IR OOR ZLQ J  'HpQLWLRQ  / HW Q p p R DN p N       c c c EH D VHTXHQFH RI UHDO RU FRPSOH[ QXPEHUV VDWLVI\LQJ D FRQGLWLRQ MDN M x q N  N       c c c ZKHUH DQG q DUH SRVLWLYH UHDO QXPEHUV 7KH GLVFUHWH / DSODFH WUDQVIRUP DIWHU WKLV ZH ZLOO ERZ WR FRQYHQWLRQ DQG FDOO LW WKH = WUDQVIRUP RI WKH VHTXHQFH IDN J LV WKH IXQFWLRQ DA  ] u  = IDN J  ] RI WKH FRPSOH[ YDULDEOH ] GHpQHG E\ WKH VHULHV  ;  = IDN J  ] DN ] bN  N2 5HPDU NV 7K H J U R ZW K F R Q G LW LR Q R Q W K H V H TX H Q F H J X D U D Q W H H V  D S S O\LQ J W K H UDWLR WHVW IR U V H U LH V F R Q YH U J H Q F H  W K D W W K H V H U LH V F R Q YH U J H V IR U D OO ] ZLW K  M]M ! q X Q LIR U P O\ D S U R S H U W \ LP S OLH V W K D M]M ! q LQ S D U W LF X OD G LoH U H Q W LD W LQ J W K H V H Q G D E V R OX W DA  ] LV U  G H U LYD W U LH V W H U P W H O\ IR U M]M y q LI q ! q 7K LV F R Q YH U J H Q F H D Q DQDO\WLF R U KRORPRUSKLF IX Q F W LR Q R I ] IR U LYH V R I DA  ] W R D Q \ R U G H U F D Q E H F R P S X W H G E \ E \WH UP  7K H LQ YH U V H W U D Q V IR U P   = b  DA  ]  N  LV R E W D LQ H G V LP S O\ E \ H [S D Q G LQ J W K H IX Q F W LR Q DA  ]  D Q D O\W LF LQ V R P H U H J LR Q M]M ! q LQ S R ZH U V R I ] D Q G W K H Q IR U P LQ J W K H V H TX H Q F H R I F R H r F LH Q W V R I S R ZH U R I ] W K X V R E W D LQ H G  / H W X V OLV W V R P H V D P S OH W U D Q V IR U P V  )R U D U H D O R U F R P S OH [ Q X P E H U U W K H V H TX H Q F H IDN J ZLW K DN UN K D V W K H W U D Q V IR U P DA  ]   UN  ; ] N2 ] N UN ; N2 ] N 7K LV IR U P X OD LV YD OLG IR U E R W K U H D O D V K R U W O\ : H V K R X OG Q R W H LQ S D U W LF X / D S OD F H W U D Q V IR U P  W K H =W U D Q V IR U P ZK LOH W K H W U D Q V IR U P R I W K H V H TX ]b W K H W U D Q V IR U P ]    ]  b Q G F OD U W R IW H Q F H   M]M ! U ]bU U ] R P S OH [ U D ID F W ZH ZLOO P K D W  LQ F R Q W U D V W W R W K H  F  K H V H TX H Q F H ZLW K DN K D YLQ J D   DN   D NH X V H R I R Q W LQ X R X V IR U D OO N LV N !  K D V )R U W K H V W X G \ R I R V F LOOD W R U \ V H TX H Q F H V LW LV LP S R U W D Q W W R K D YH W K H W U D Q V  t Q Ru t Q Ru IR U P V = DN F R V EN  ] D Q G = DN V LQ EN  ] IR U U H D O Q X P E H U V D D Q G E : H K D YH  R I F R X U V H D F R V EN N vt   DH LE V LQ J R X U H D U OLH U IR U P X OD IR U W K H F D V H V U t Q = DN F R V EN Ru  ] v   uN  t D Hb uN w D HLE D Q G U    ] b DHLE ] b DHbLE  LE w  D HbLE ZH V H H W K D W ]bD F R V E  ] b D F R V E !  D! V LQ ! E  $ V LP LOD U F R P S X W D W LR Q V K R ZV W K D W t Q = DN V LQ EN Ru D V LQ E  ] b D F R V E !  D! V LQ  ] ! E  : H F R Q F OX G H ZLW K W ZR V LP S OH H [D P S OH V  p U V W ZH Q R W H W K D W LI DN D OO N W K H Q DA  ]  N ;  = INJ  ] N2 ] N   ZLW K M Nb  b H F R Q G O\ V X S S R V H DN  ; DA  ] N2 NÄ    G #; $ G] M2 ] M  N „ G b G] b N2 v G G]  ;  ]b  ]N w    ]b !         7K H Q ZH K D YH  ]bN  ; ] N2 N  N ] N Hb]  M]M !   ] R Z ZH ZLOO OLV W V R P H R I W K H J H Q H U D O S U R S H U W LH V R I W K H W U D Q V IR U P E U LH I YH U Lp F D W LR Q V R I W K R V H S U R S H U W LH V  Q R U RSHU W\ ,  =WU DQVIRU P RI U N DN  U  t Q = U N DN  Ru ; DN v wN N2 U ]   ; DNQ ] bN  ]  ; N2Q DN ]b N  „ ] Q DA  ] b v w ] U DA v w ]  U + H U H ZH K D YH ]Q N2 Q D Q G J LYH + H U H ZH K D YH  = IDN J U RSHU W\ , ,  KH =WU DQVIRU P RI IDNQ J  = IDNQ J  ] N IR U  ; DNQ ] bNQ  N2 b DN ] bN ; Q N2  ] Q DA  ] b b ; Q  N2 DN ] QbN   U RSHU W\ , , ,  =WU DQVIRU P RI IN N b  c c c  N b Q   DN J  Q  LQ W H J H U K D YH  ;  = IN N b  c c c  N b Q   DN J  ] N Nb c c c  NbQ DN ] bN !   N2  ; N N b  c c c  N b Q   DN N2Q  b Q GQ G]Q „  ; „ ]b N  DNQ bN ] Q N2 G b G]  b Q  Q  „ ; N2Q DN bN ] Q b Q GQ  = IDNQ J  ]  G] Q Q 7K H Q X V LQ J U R S H U W \ ,, ZH R E W D LQ  = IN N b  c c c  N b Q   DN J  ]  b Q 5HPDU NV t Q Qb ; GQ Q ] D A  ] b DN ] QbN Q G] N2 „ Q Q GQ Q  ] DA  ]  G] Q Q ,Q S D U W LF X OD U  W K H Q = N N b  c c c  N b Q   UN  b Q  b Q   UN ; GQ Q ] G] Q ] N N2 „  b U Q Q Q GQ G] Q UQ „  N2Q Q Q U NbQ ] N GQ G] Q  b Q  ]  b Q ;  b Q Ru GQ G] Q b „Q ; U ] N Q Q  ]bU bb Q N N2  b U Q b v GQ t Q t Q N Ru u ] = U  ] G] Q Q w Q „ GQ G] Q UQ  ] b U V H TX H Q F H Q N N b  c c c  N b Q   UN R Q   ;  UN ; ] N N2Q b Q UN ] N  Q Q R I b Q LV W K H R N N b  c c c  N b Q   U NbQ  )R U Q  ZH V H H W K D W W K H W U D Q V IR U P R I N U Nb R X U H D U OLH U H [D P S OH IDN J INJ IR U U   U  N2 $ V D Q LP S R U W D Q W F R U R OOD U \ ZH V H H W K D W W K H LQ YH U V H W U D Q V IR U P Q  R LV ] b ] U   7K LV D J U H H V ZLW K U : H 7K LV U H V X OW D OV R D OOR ZV X V W R F R P S X W H  = INQ DN J  ]  b Q GQ  = IDNQ J  Q G] Q 7K H U LJ K W K D Q G V LG H LQ YR LQ G X F W LYH O\ V LQ J W K H IR W X D OO\ LQ YR OYH V R Q O\ N DA  ]  = IDN J  ]  7K X ] b  = I N N b  c c c  N b Q   b NQ DN J  ]  OYH V R Q O\ S R ZH U V N P  P  Q V R ZH F D Q S U R F H H G U P X OD U H S H D W H G O\ ZK H Q W K H U LJ K W K D Q G V LG H H YH Q    H YH U \W K LQ J K D V E H H Q H [S U H V V H G LQ W H U P V R I V  IR U H [D P S OH   = IN DN J  ] : b G  ]D A  ] b D G] b G  ]D A  ]  G] LW K W K LV LQ K D Q G ZH F D Q S U R F H H G W R F R P S X W H t Q = N! DN  Ru  ] u t Qt u Ru G! t ! ! ! ] D A  ] b D b =  N b N b N DN  ]  ] b D  G] !  u G! t ! ] D A  ]   = IN DN J  ]  G]! u G! t ! G  ] DA  ] ] D A  ] b !  G] G]   )R U D S R V LW LYH LQ W H J H U Q ZH G H p Q H W K H  U LJ K W QV K LIW R I W K H V H TX H Q F H IDN J W R E H W K H V H TX H Q F H IDQN J ZK H U H DQN ~    x N  Q DNbQ  N y Q U RSHU W\ , 9 =WU DQVIRU P RI WKH Q6KLIW DA Q  ]  = IDQN J  ]  ; N2Q ]bQ  ; N2Q DNbQ ] bNbQ   ZLW K M NbQ : H F R P S X WH DNbQ ] bN ] bQ  ;  M2  DM ] bM   ] !DA  ]  ] DA  ]  ] bQ DA  ]  Q p p R p p Q R 'HpQLWLRQ  / HW DN p N    ccc  EN p N       c c c EH WZR VH TXHQFHV LQGH[HG DV LQGLFDWHG 7KH F R Q YR OX W LR Q S U R G X F W RI WKHVH VHTXHQFHV LV p Q R p WKH VHTXHQFH FN p N       c c c  ZULWWHQ IFN J IDN J e IEN J ZKHUH N ; FN M2 DM ENbM  U RSHU W\ 9 =WU DQVIRU P RI D &RQYROXWLRQ U RGXFW 2Q W K H E D V LV R I W K H FRU U HVSRQGLQJ /DSODFH WU DQVIRU P U HVXOW ZH V K R X OG H [S H F W V R P H V R U W R I R U G LQ D U \ S U R G X F W R I =W U D Q V IR U P V K H U H  D Q G ZH D U H Q R W G LV D S S R LQ W H G   ;  = IIDN J e IEN JJ  ]  # M2 N2 ]  ; N2  # N ; M2  DM ENbM $ ]bNbM  ]#  ; M2 b  „ DM ]bM  $ N ; ] M   DM ENbM $ ] bN    DM EC ] bC  ] bM ; ;  C2 M2  ; EC ] bC  ] DA  ] A E ]  C2 ZK H U H ZH K D YH W D NH Q C N b M ,W V K R X OG E H Q R W H G W K D W W K H V H TX H Q F H IrN J G H p Q H G E \ r   rN   N !   IR U ZK LF K ZH K D YH V H H Q H D U OLH U W K D W  = IrN J  ]  LV W K H LG H Q W LW \ IR U W K LV F R Q YR OX W LR Q S U R G X F W  F R Q p U P H G E \ ]  = IIrN J e IDN JJ  ]  ]  DA  ] ] D A  ]  ... p p    ccc R ZK H U H W K H SN  N      P D U H NQ R ZQ F R H r F LH Q W V D Q G EN p N    ccc LV D NQ R ZQ V H TX H Q F H  ,Q P R V W F D V H V ZH ZLOO W D NH W K H V H... D O ZLW K G LV F U H W H LQ G LF H V OH D G V W R V R P H P LQ R U F R P S OLF D W LR Q V  7K H U H D U H YD U LR X V ZD V LQ ZK LF K W K H W U D Q V IR U P F D Q E H G H p Q H G  ZH ZLOO... V LQ  ] ! E  : H F R Q F OX G H ZLW K W ZR V LP S OH H [D P S OH V  p U V W ZH Q R W H W K D W LI DN D OO N W K H Q DA  ]  N ;  = INJ  ] N2 ] N   ZLW K M Nb  b H F R Q G O V X S