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Page 1: (eBook) (Mathematics) the Discrete Laplace Z Transform

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Page 2: (eBook) (Mathematics) the Discrete Laplace Z Transform

7KH 'LVFUHWH /DSODFH �?=�� 7UDQVIRUP

7 U DQVIRU P 'HpQLWLRQ� 3 U RSHU WLHV / LQ H D U UHFXUVLRQ HTXDWLRQV� V R P H �

W LP H V F D OOH G GLoHUHQFH HTXDWLRQV� W D NH W K H IR U P

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ppp N � � � � � � c c cR

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

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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

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DUH SRVLWLYH UHDO QXPEHUV� 7KH GLVFUHWH / DSODFH WUDQVIRUP �DIWHU WKLV ZH ZLOO

ERZ WR FRQYHQWLRQ DQG FDOO LW WKH = WUDQVIRUP� RI WKH VHTXHQFH IDNJ LV WKH

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Page 3: (eBook) (Mathematics) the Discrete Laplace Z Transform

M]M ! q� X Q LIR U P O\ D Q G D E V R OX W H O\ IR U M]M y q� LI q� ! q� 7K LV F R Q YH U J H Q F H

S U R S H U W \ LP S OLH V W K D W AD � ] � LV D Q DQDO\WLF R U KRORPRUSKLF IX Q F W LR Q R I ] IR U

M]M ! q� LQ S D U W LF X OD U � G H U LYD W LYH V R I AD� ] � W R D Q \ R U G H U F D Q E H F R P S X W H G E \

G LoH U H Q W LD W LQ J W K H V H U LH V W H U P E \ W H U P �

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]D Q G W K H Q

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/ 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

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]b�ZK LOH W K H W U D Q V IR U P R I W K H V H TX H Q F H K D YLQ J D� � � DN � � N ! � K D V

W K H W U D Q V IR U P �]�

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IR U P Vt=

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Ru� ] � D Q G

t=

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8 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 D HLE D Q G U D HbLE ZH V H H W K D W

t=

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] b DHbLE

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Page 4: (eBook) (Mathematics) the Discrete Laplace Z Transform

$ V LP LOD U F R P S X W D W LR Q V K R ZV W K D W

t=

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Ru� ] �

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N

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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 �

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Page 5: (eBook) (Mathematics) the Discrete Laplace Z Transform

3 U RSHU W\ , , , � =�WU DQVIRU P RI IN� Nb �� c c c � Nb Q� �� DNJ � Q � LQ W H J H U � ! �� : H

K D YH

� = IN � N b � � c c c � N b Q � � � DNJ� � ] � �;N �

N � Nb� � c c c � NbQ�� � DN ]b�N���

�;N Q

N � N b � � c c c � N b Q � � � DN ]b�N���

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R X U H D U OLH U H [D P S OH IDNJ INJ IR U U � �

Page 6: (eBook) (Mathematics) the Discrete Laplace Z Transform

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 DNJ� � ] � � b� � QQ�

GQ

G]Q� = IDN�QJ� � ] � b � = I� N � N b � � c c c � N b Q� � � b NQ � DNJ� � ] � �

7K H U LJ K W K D Q G V LG H LQ YR OYH V R Q O\ S R ZH U V NP� P � Q V R ZH F D Q S U R F H H G

LQ G X F W LYH O\� 8 V LQ J W K H IR 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 �

W X D OO\ LQ YR OYH V R Q O\ N� � � 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

AD� ] � � = IDNJ� � ] � � 7K X V � IR U H [D P S OH �

� = IN DNJ� � ] � b G

G]� ] AD� ] � b D� � b G

G]� ] AD� ] � � �

: 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=

QN� DN

Ru� ] �

G�

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ub

t=

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uDN

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DQ��N ~� � � x N � Q�DNbQ� N y Q�

3 U RSHU W\ , 9� =�WU DQVIRU P RI WKH Q�6KLIW� : H F R P S X W H

ADQ� � ] � � = IDQ��NJ� � ] � �;N Q

DNbQ ]b�N���

]bQ�;N Q

DNbQ ]b�NbQ��� � ZLW K M NbQ� ]bQ

�;M �

DM ]b�M��� ]bQ AD� ] � �

Page 7: (eBook) (Mathematics) the Discrete Laplace Z Transform

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QEN

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WKH VHTXHQFHQFN

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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

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� = IIDNJ e IENJJ� � ] � �;N �

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� = IrNJ� � ] � �]� 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 \

� = IIrNJ e IDNJJ� � ] � ]�

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