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simata kai systimata - users.auth.grusers.auth.gr/~hadjidem/simata kai systimata.pdf · @)A B C D E F G H I H A (TIME DOMAIN ANALYSIS) J LKNMO P Q R4SUTWVYXZV\[^]`_badcfe S ThgjikS

Sep 26, 2020

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Page 1: simata kai systimata - users.auth.grusers.auth.gr/~hadjidem/simata kai systimata.pdf · @)A B C D E F G H I H A (TIME DOMAIN ANALYSIS) J LKNMO P Q R4SUTWVYXZV\[^]`_badcfe S ThgjikS

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http://users.auth.gr/hadjidem/simata kai systimata.pdf

Page 2: simata kai systimata - users.auth.grusers.auth.gr/~hadjidem/simata kai systimata.pdf · @)A B C D E F G H I H A (TIME DOMAIN ANALYSIS) J LKNMO P Q R4SUTWVYXZV\[^]`_badcfe S ThgjikS

=

Page 3: simata kai systimata - users.auth.grusers.auth.gr/~hadjidem/simata kai systimata.pdf · @)A B C D E F G H I H A (TIME DOMAIN ANALYSIS) J LKNMO P Q R4SUTWVYXZV\[^]`_badcfe S ThgjikS

> ? > @)A B C D E F G�H I C B I F H ? H A

(TIME DOMAIN ANALYSIS)

J �LKNMO�P�������

Q R4SUTWVYXZV\[^]`_badcfe S ThgjikS T`lnm cpoqcsrbm tZlvu�uqw`uhexu`lycswhg{z | }�rYcvikS T~r�t��qTx(t) � mfu~u`w`uhe�u�u�S uh�q}`� u`�Z�qc

�s� V�� � �P�hThgYmfu��qcpmfT`rY���Z�qTqmUe{� ckgYr�c�ikS T�}qoZoqu�r�t��qTy(t) � mfu�uqw`u�e�u�u�S uh�q}`� u`�Z�qcd�s�p� �U� =

�Pw`}h| ��uh��S�z �Zu�zUg{T`�\u`| cpmUg��hu�e�m | �qw`u�g��Yg{T�m ���qcpoqism ��csS �`ldr���rYm t��qT`mku`l � ��aq�^���� �Vj]¢¡ �¤£j¥ �§¦�¨�©¦ £Yª �f�f ¬«®­

time domain analysis)�hThg2�"aq� �f�b �V\]�¡ �¤£�¥ �§¦P¨h©�V� ¯¦®�f��[^¨�[±°4��«

(frequecy domainanalysis). ² T`SU}qoq��r��Pw`cs|¬g�u���tZl<��|U��S u`�T`SUT`��ik| cpmfThghrbm �PrY�Z�Zw`ck|Ug{��u`| }�csSU�`l6rYtZ�qTqmfu`l0³4l0w`| u`lm ����| u�S¬g´��t~mkuh��csµ iso`g�µU� � ��Thg¶��T`S }`oZ�ZrY��w`ck|Ugxu���tZldrY�Z��S u¬m t�mk³4S·T`S Th��is| csmfThg¶rbm ��r����Zw`ck|Ug{��u`|U}mfu`�N��}`r��qTqmkuhlvckS �`lrYtZ�qTqmfu`l =¸ mfu�w`T`| �ZS~�hcs�\}qoZT�g�uW¹\T��qcsoZcsm tZrYu`�Z�qc�mUg�lN�qcp¹\�`z uh��lNw`u`����| �Zr�g{�quqw`uhgxu`�ZS^mfThg%rYm ��S�T`S }qoq�ZrY�

w`cs|¬g�u���tZlv��|U��S u`� =»º ud¼hT`r\g´���½w`| ��¼`oq�Z�qT½cke SUThgY�d�qcpoqism ��m ��l®csµ �`z uh�y(t)

csSU�`l®rY�Zrbm tZ�qTqmfu`lH ���mkThS¾��ckexrYu`z uhl�cke S Thg2mku"rYtZ�qT

x(t) =�¿ lH

w`T`|UT`rbm tZrYT`�qcNmkuZS�m cpoqcsrYm t¾w`u`�WT`S^mUg�w`| u`rb³'w`ck�Zckgmfu~r���rYm ���qT � �hT�gjr����¬¼hu¬o`g��h}~m �½rY��ikrY�½ckgxrY�`z uh�½��Thgjckµ �`z uh��m ��Syw`Th|Ug�rYm }`S u`�Z�qcnrbmfu�w`T`| Tq�h}`mk³rY��tZ�qT

x(t) → ² → y(t),

t·³4ly(t) = H [x(t)] .

¸ m �6�bcsSUg��ht<w`cs|¬e´wqmk³4rY� � isS T�r���rYm ���qT$is��ckg^w`u¬oZoZikl'ckgxrY�`z uh��l'�hT�g^w`u¬oZoqisl'ckµ �`z u`�Zl � �hT�g rY�ZS cswqÀ4l �rbmUgxlPw`Th| Tqw`}`S^³Ár���ikrYcfg�l®mkT�rYtZ�qTqmfTx(t) � y(t) T`S^mUg�w`| uhrb³'w`cs�Zu`�ZSyz¬g�ThS �ZrY�qTqmfTbÂ

x(t) =

x1(t)x2(t)...

xn(t)

,

�hThg

y(t) =

y1(t)y2(t)...

ym(t)

,

Ã

Page 4: simata kai systimata - users.auth.grusers.auth.gr/~hadjidem/simata kai systimata.pdf · @)A B C D E F G H I H A (TIME DOMAIN ANALYSIS) J LKNMO P Q R4SUTWVYXZV\[^]`_badcfe S ThgjikS

�qw`u`�n � m Tq�hik| ThgxuhgjT`|Ug�¹\�qu�e � �hT�g�csS��bisS ckg

n 6= m =Ä�ÅjÆ�Å�Ç�ÈZÉ´Ê%ËÌÅ�ÍqÅÏÎ�Ð�ÑÒÍqÓ�Ë�Ô\ÍZÕ®Ö× T`| }`z cfg �b�qTWظ mfu�rY�Zrbm �Z�qT®mfu`� ¸ ��t��`Tqmkuhl6À4l6ckexrYu`z uyu`|Ue{� u`�Z�qc0m �ZS<m }`r��x(t)

��Thg<³4lOikµ u`z uÙm ��S¢m }`rY�y(t) = ² rY��isrY��ckgxrY�`z uh�ÛÚ

ckµ �`z u`�½zUe{z csmkT�g�Tqw`��m �·z¬g�Th��u`|Ug���t�csµUexrb³4rY�

RCdy

dt+ y = x(t).

Ü S^mUe%�Yg�T"m �ZS�m }`rY�y(t)

¹\T"�Zw`u`|Uu`�ZrYT`�qc�S T"uh|Ue�r�u`�Z�qcN³4likµ u`z u�m ��S�isS mkT`r��

i(t) =

++

_ _

x(t) y(t)i(t)

R

C

× T`| }`z cfg �b�qT¾Ý¸ mfu�rY�Zrbm �Z�qT®mfu`� ¸ ��t��`Tqmkuhl6³4l6ckexrYu`z uyu`|Ue{� u`�Z�qc0m �ZS<m }`r��vs(t)

�hT�gÒ³4l�isµ u`zUuWm �ZS~ikS^mkThrY�i(t) = ² rY��isrY��cfg�r��`z u`�¾Ú

ckµ �`z u`�½zUe{z csmkT�g�Tqw`��m �·z¬g�Th��u`|Ug���t�csµUexrb³4rY�

Ldi

dt+Ri = vs(t).

++

_ _

i(t)

R

v (t)s L

¸ mfTyz �Zunw`Th| Tqw`}`S^³Þr���rYm t��qT`mkTn�T`S^mUexrbmfT`rY�R

�Zw`uh| ckehS Tycke S ThghrYmkTq¹\cs| t � tvS Tyckµ T`|^m }qmfThg�Tqw`�m �ZSdisS mkT`r��

i(t) � R = R0 + α i(t) = ¸ m �ZSyw`|^À'm �½w`ck|Ue�wqms³»rY�½mku�r���rYm ���qT�cfe S Thgj�b| T`�q�`g��h� � ckS^Àrbm �·z cs��m ck| �·cke S Thg��q�·��|UT`�q�`g��h� � oq�¬��³Ïmku`�N�`|Uu`�α i(t) =

× T`| }`z cfg �b�qT ø mfu�rY�Zrbm �Z�qT�mkuh� ¸ ��tZ�qTqmfu`l·��ckexrYu`z uhl·cke S ThgÌ��m }`r��

x(t)�hT�gh�ikµ u`z u`lßcke S Thg��vm }`rY�y(t) = ² rY��isrY�vcfg�r��`z u`�®ÚÒckµ �`z uh�

zUe{z csmkT�g�Tqw`��m �NzUg�Th��u`|Ug���t�csµUexrb³4rY�

LCd2y

dt2+ y = x(t),

�qw`uh�Pw`}`oqgquhgZw`T`|U}`�qcpm | u�gR, L, C

�Zw`u`| ckeqS Tncfe S Thg`rbmfTZ¹\ck| isltN�qcpmfT�¼`oq�¬m ikl =

++

_ _

x(t) y(t)C

L

à á �MÛ�â��ã6ä M¢KN�n�åP�P��Mçæ$�¯ä

¸ c¾w`u¬o�oqikl¾whcs|Ug�wqm^À4rYcfg�l�isSUT�T`SUTqoquU�Yg����ÏrYtZ�qTÞT`S T`oZ�ZcpmfThg0r�c�rbmfuhgx��ckg�À4z ckgxl�rY�ZS T`|^m tZrYcfg�l =éè g�h�Z|Ug{��m cs| cklnT`w`��T`��m islncke S T�g\mku�_j� �pa`� ¥ a � �$ê�¨`_ba¾ë

unit step),��V� ��^� £ [^]�V\]��U�ì�Y[±a¾ë

delta func-tion)

�hThg\�unit ramp =4º g�lr���S Th|^m tZrYckgxl®T`��m islP¹jT��qcpoqcpm tZrYuh���qc�rbmfT�cpw`�h�qcsS T =

í

Page 5: simata kai systimata - users.auth.grusers.auth.gr/~hadjidem/simata kai systimata.pdf · @)A B C D E F G H I H A (TIME DOMAIN ANALYSIS) J LKNMO P Q R4SUTWVYXZV\[^]`_badcfe S ThgjikS

0

t

1

u(t)

¸ ��tZ�qT�ØZ ² rY�ZS }`| m ��r��u(t)

î u�SUT`zUg{Thexu�¼`tZ�qTº u��quZS T`z¬g�T�e�u�¼htZ�qT

u(t) � ë unit step) uh|Ue{� cpmfThg�³4l

u(t) =

0�Yg{T

t < 0w`ck| Tqmk³4�qisSUu �Yg{Tt = 0

1�Yg�T

t > 0.

ë�Ø^ï

² �b| T`�\g��ht½w`T`| }`rbmfT`r��·m ��lvrY�ZS }`|^m �ZrY�Zl®T`��m t�lvzUe{z csmkT�g�rbmfu ¸ ��tZ�qT�Ø =ð �Z�`gxu`�Z|^�Ye�T½¼hT`r\g´�hÀ4SyrY�ZS T`|^m tZrYcs³4Sn�qcPT`S T`zU| u`�`g��hiklnrY��ikrYckgxlî cñm �$¼hu`t¬¹\cfg�Tm �Zl0rY�ZS }`|^m �ZrY�Zl

u(t)�Zw`uh| u`�Z�qcòS Tnu`|UexrYu`�Z�qcò�`g{TnT`�hu¬oqu`�U¹�e{Tv¼hT`r�g��`À4S�rY�ZS T`|^m tUÚ

rYcs³4S � ³4lvTq�hu¬oqu`�¬¹�³4l¬Â è |Ue{� u`�Z�qcPm �·r���SU}`|^m �ZrY�u−1(t)

³4lu−1(t) ≡ u(t),

�hThg\rbm �·rY�ZS is��ckg{T�u`|Ue{� u`�Z�qc®mUg�lrbmfuhg���ckg�À4z cfg�lnrY�ZS T`|^m tZrYcfg�lui(t)

³4l

ui−1 =∫ t

−∞ui(λ)dλ, i = . . .− 2,−1, 0, 1, 2 . . . ëóÝ�ï

tui(t) =

dui−1

dt.

ë Ã ï

² rY�ZS }`|^m �ZrY�unit rampR4}`Sdrbm �ZSnR4µUexrb³4rY�½Ý�¹\ikrYu`�Z�qc

i = −1 � w`T�e�|US u`�Z�qcPm �NrY�ZS }`|^m �ZrY�u−2(t) ≡ r(t)

Â

r(t) =∫ t

−∞u(λ)dλ,

ë í ï�Nuqw`uhe{T~cfe S Thg\�

u(t) =

{

0�bg{T

t < 0t

�bg{Tt ≥ 0

ëóô�ï

² �b| T`�\g��ht½w`T`| }`rbmfT`r��·m ��lr(t)

zUe{z cpmfThg�rYmku ¸ ��tZ�qT~Ý =ô

Page 6: simata kai systimata - users.auth.grusers.auth.gr/~hadjidem/simata kai systimata.pdf · @)A B C D E F G H I H A (TIME DOMAIN ANALYSIS) J LKNMO P Q R4SUTWVYXZV\[^]`_badcfe S ThgjikS

0

t

r(t)

¸ ��tZ�qT~Ýh ² r���SU}`|^m �ZrY�r(t)

0

t

ä(t)

¸ ��t��qT à  ² rY�ZS }h|^m �ZrY�δ(t)

² rY�ZS }`|^m �ZrY�~� �ì�Y[±aõ®ismkuh���qcPrYm ��SnR»µUe�rY³4rY� Ã

i = 0��Thg�ik��uh���qc

u0 =du−1dt�hThg�w`The{| S u`�Z�qc®m �·rY�ZS }h|^m ��r��dz ipoZmfT�ëöcke S Thg

u−1(t) ≡ u(t)�hThg\u`|Ue{� u`�Z�qc®³4l

u0(t) ≡ δ(t)ïsÂ

δ(t) =du(t)

dt,

ë¤÷qï�Nuqw`uhe{T~cfe S Thg\�

δ(t) =

Tqw`| u`rYz¬g��h|Ug�rYm � �Yg{Tt = 0

0�Yg{T

t 6= 0.ëóø�ï

² �b| T`�jg��ht¾w`T`| }`rbmfT`r��~m �ZldrY�ZS }`|^m �ZrY�ZlyzUe{z cpmfThgÌrY�Z�¬¼hu¬oqg��h}�rbmfu ¸ ��t��qT à = õ®T�z cke{µ u`�Z�qcdrbmfTcpw`�h�qcsS T¢�¬mUg2��rY�ZS }h|^m �ZrY�

δ(t)z ckS�cfe S Thg'rY�ZS }h|^m �ZrY� � �qcNm ��r���S t¬¹\��isS S u�g�T¢mfu`���`| uh� � T`o�oq}m cpoqckrbm tZl =

ù ú:û ��üÞån�MWý MWý�������þÿ� ��æ �Pý � ½���Ïæ$� �Pþ � MO�ß ����"æ$�±þ��� �M¢KN�n�åP�P��Mçæ<þÿ�

è gbrbmfuhg���cfg´À»z ckgxl®r���SUT`|^m tZrYckgxlu(t) � r(t) �hThg δ(t) �Zw`u`| u`�ZS�S T½��| �Zr�g{�quqw`u�gx�¬¹\uh��Sn�Yg{T½SUT½cp����| }�Ú

rYu`�Z�qc4isS T®m �����ZS6rY�Z�qTf(t)

rY�ZS Th|^m t�r�ckgUT`��m^À4S =%¿ lñisSUTPw`T`| }hz ckg����qTnT`S T`�\is| u`�Z�qc4mfuZSß_\� �sa`� ¥ a � �÷

Page 7: simata kai systimata - users.auth.grusers.auth.gr/~hadjidem/simata kai systimata.pdf · @)A B C D E F G H I H A (TIME DOMAIN ANALYSIS) J LKNMO P Q R4SUTWVYXZV\[^]`_badcfe S ThgjikS

0

t

1

-1/2

Ð(t)

1/2

¸ ��tZ�qT í  ² r���SU}`|^m �ZrY�Π(t)

0

t

1

u(t- )á

á

¸ ��tZ�qT~ôh ² r���SU}`|^m �ZrY�u(t− α) = Q R4��u`�Z�qcPw`T`|U}qoZoZ��oq�N�qTqmfTqm �qwhgxrY�·m �Zl

u(t)�hTqm }

α =¡ af��_ ª ë

unit pulseïΠ(t) � u�uqw`uhexu`lnu`|¬e��UcpmfThg�³4l

Π(t) =

0�Yg{T

t < −12

1�Yg{T −1

2≤ t ≤ 1

2

0�Yg{T

t > 12

ë�Zï

² �b| T`�jg��htWw`Th| }`rbmfT`rY�Wmkuh�W�qu�S ThzUg{Thexu`�"w`Tqoq�qu`�Wz¬e�zUcpmfThg'rYmku ¸ ��tZ�qT í = R»e S T�g4cs���hu¬oquçS T¢zUgxÚTqwhgxrbmk³2¹\cfe��¬mUg\�½rY�ZS }`|^m �ZrY�

Π(t)T`S Tqoq�ZcpmfThg\³4lv�b| T`�q�`g��h�`l�r���SUz �ZT`rY�q�`l�qu�SUT`zUg{The�³4Sy¼h���q}`ms³4S

³4lΠ(t) = u(t+

1

2)− u(t− 1

2).

× |Ug SOw`| u���³4| tZrYu`�Z�qc¾rYc¾}qoZoqTÞw`T`|UT`z cke��b�qTqmfT � ¹jT�z �`rYu`�Z�qc¾�qck|Ug��hisl¾¼hT`r�g��hislWg{zUg{�¬m ��m csl�rbm �rY�Z�Zw`ck|Ug��\u`| }çmk³4S�rY�ZS Th|^m tZrYcp³4S��¬mfT`S��OThS csµ }h|^m �¬m �"�qcsmkT�¼`oq��m t

t�qcpmfT`rY���Z�qTqmUgxr�¹\cfe2rYc

t →(t− α)

tNrYct → kt � �qw`u`� α � k rbmfTZ¹\cs|Uisl =

è �qcpmfT`r����Z�qTqmUgxrY�q�`lt → (t− α)Q R4rYms³ÏrY�ZS }`|^m �ZrY�

f(t)��ThgY�·�qcsmkThrY�����qT`mUg�rY�`isS �

f(t− α) = ² rY�ZS }`|^m �ZrY�f(t− α)

w`| u`�h�¬w`m ckgTqw`�Om �ZS

f(t)�qcdw`T`|U}qoZoZ��oq���qcsmkT`m �qwhg�r���rbmfu�S~}hµ u�SUT

t�hTqm }OmfuOzUg{}`rbm �Z�qT

α � �qwq³4l���T�e S cpmfThgrbmfu ¸ ��t��qT~ôd�Yg{T�m �·rY�ZS }h|^m �ZrY�

u(t) =è �qcpmfT`r����Z�qTqmUgxrY�q�`l

t → ktQ R4rYms³ rY�ZS }`|^m �ZrY�f(t)

�hThg�qcpmfT`rY���Z�qTqmUgxrY�qisSU�f(kt)

�hThgòT`l���w`uZ¹jisrYu`�Z�qc~��mUgk > 1 = ²

ø

Page 8: simata kai systimata - users.auth.grusers.auth.gr/~hadjidem/simata kai systimata.pdf · @)A B C D E F G H I H A (TIME DOMAIN ANALYSIS) J LKNMO P Q R4SUTWVYXZV\[^]`_badcfe S ThgjikS

- 1 , 0 - 0 , 5 0 , 0 0 , 5 1 , 00 , 0

0 , 2

0 , 4

0 , 6

0 , 8

1 , 0

x

y

f ( 2 x )

f ( x )

(a)

- 1 , 0 0 , 0 1 , 0 2 , 0 3 , 0 4 , 00 , 0

0 , 2

0 , 4

0 , 6

0 , 8

1 , 0

5 / 2

f ( 2 x - 5 )

x

yf ( x - 5 / 2 )

f ( x )

(b)

¸ ��tZ�qT¢÷�ÂWëöThï ² r���S }h|^m �ZrY�f(x) = 1 − x2

�hT�g4�W�qcsmkThrY�����qT`mUg�rY�qikS �f(2x) = 1 − (2x)2 =Q R4��u`�Z�qcPrY�Z�Zwhe{csr��n��Tqm }½mfu�Sw`T`|U}q�bu�S^mfT~Ý = ë ¼�ï ² rY�ZS }`|^m �ZrY�

f(x) � �drY�ZS }`| m ��r��f(x− 5/2) ���uqw`uhe{T½cke S Thg��

f(x) � �qcpmfTqmfuqwhgxrY�qisS �n�hT`m }5/2 � ��Thg���r���S }h|^m �ZrY�

f(2x− 5) = f([2(x− 5/2)] ��Nuqw`uhe{T�w`| u`�h��wqm ckg�Tqw`��m �ZSf(x− 5/2) �`cPrY�Z�Zwhe{csrY�d�hTqm }�mfuZSnw`Th| }q�bu�S mkT¾Ý =

rY�ZS }`| m ��r��f(kt)

w`| uq���¬wqm cfg�Tqw`�ym �ZSf(t)

�qc6r����Zwhe{ckrY�®�hT`m }ymfu�SPw`T`| }q�buZS^mfTk � �qwq³4lß��The SUcpmfThg

rbmfu ¸ ��t��qT�÷ZT��Yg�T�m �·rY�ZS }`|^m �ZrY�f(x) = 1− x2 =

è �qcpmfT`r����Z�qTqmUgxrY�q�`lt → (kt− α)Ü w`u¬m csoZcfe�r���SUz ��ThrY�q�yms³4S®zU��u·T`S^³'m ik|^³Þw`cs|Ug�wqm^À4rYcs³4S =4è �qcsmkThrY�����qT`mUg�rY�q�hl<�b| }`�\cpmfThg�³4l

t→k(t− α

k)�hThg���r���SU}`|^m �ZrY�

f(kt−α)w`|Uuq�h��wqm ckg�Tqw`�·m �ZS

f(t)�qc6w`Th| }qoZoq�¬oq�y�qcpmfTqm �qwhgxrY�n��Tqm }

αk

�hThgÒrY�Z�Zwhe{csrY�~�hTqm }k = Q R»S T�w`T`| }`z cfg �b�qTWw`T`| uh��r\g�}h� cpmfThg%rbmfu ¸ ��tZ�qTW÷ ¼��Yg�T�m �¾rY�ZS }`|^m �ZrY�

f(x)�hThg�m �·�qcpmfT`rY���Z�qTqmUgxrY�qisSU�

f(2x− 5) = f [2(x− 5/2)] =è �qcpmfT`r����Z�qTqmUgxrY�q�`l

t → −tQ R4rYms³ rY�ZS }`|^m �ZrY�f(t) = ² rY�ZS }`|^m �ZrY�

f(−t) ik��cfg���|UT`�\g��htdw`T`| }`rYmkThrY�~ëörbmfu·cpwhe�w`csz ut f(t)

ïñ�uqw`uhe{T¾cke S Thgj�½rY�Z�q�qcpm |Ug��htNm ��lrY�ZS }`| m ��r���l

f(t)³4lvw`|Uu`lmfu�Sd}`µ uZS T

f(t) = Ü ��m �~��The S cpmfThgjrbmfu¸ ��tZ�qT~ø =× T`| T`z cfe �b�qTqmfT¸ mfT ¸ ��tZ�qTqmfT�UÚ±Ø� �w`T`| u`�Zr�g{}`� uh�qcdzUg{}`��uh| T�w`T`| Thz cke����qT`mfT�rY�Z�q}qmk³4S·mfT�u`w`uhe{T�T`S Tqoq�ZuZS^mkT�g¶³4l�b| T`�q�`g��h�`lnr���SUz �ZT`rY�q�`l®ms³4Sv¼hT`r�g��`À4SdrY�ZS T`|^m tZrYcp³4S

u(tïò�hT�g

r(t) =

Page 9: simata kai systimata - users.auth.grusers.auth.gr/~hadjidem/simata kai systimata.pdf · @)A B C D E F G H I H A (TIME DOMAIN ANALYSIS) J LKNMO P Q R4SUTWVYXZV\[^]`_badcfe S ThgjikS

0

t

f(t)

á-á

f(-t)

¸ ��tZ�qT~ø� ² rY�ZS }`| m ��r��f(t)

�hT�g��N�qcpmfT`rY���Z�qTqmUg�r��qisS �f(−t)

0

t

1

1

¸ ��tZ�qT�� ² rY�ZS }`| m ��r��dcp�h�\| }`� csmkT�gY³4lr(t)− r(t− 1)

0

t

1

1

2

¸ ��tZ�qT��� è m |Ug���³4SUg��h�`l~w`Tqoq�q�`l~cp�h�\| }`� csmkThg'³4lr(t) − r(t − 1) − [r(t − 1) − r(t − 2)] =

r(t)− 2r(t− 1) + r(t− 2)

0

t

1

1

¸ ��t��qT�Ø� � ² rY�ZS }`|^m �ZrY�·cp�h�\| }`� cpmfThg�³4lu(t)− u(t− 1)

Page 10: simata kai systimata - users.auth.grusers.auth.gr/~hadjidem/simata kai systimata.pdf · @)A B C D E F G H I H A (TIME DOMAIN ANALYSIS) J LKNMO P Q R4SUTWVYXZV\[^]`_badcfe S ThgjikS

0

t

1

1 2 3 4 5

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¸ ��tZ�qT�ØZØZÂ

0

t

1

1 2 3

2

-1

¸ ��tZ�qT�Ø Ý�Â����� ��������� �"!$#R'�h�\| }`rbm c$mkT·rYtZ�qTqmfTymk³4S ¸ �����q}`ms³4SdØ� � Ø ÝrY�ZS Th|^m t�r�ckg`mk³4S®rbmfuhgx��ckg�³4z^À4Svr���SUT`|^m tZrYcp³4S

u(t)�hThgr(t) =

���&% ��������� �"!$#¸ ��cszUg{}`rYm cPmfT�rYtZ�qTqmfTbÂΠ(t + 5) � Π(6t) � Π(4t − 3) � Π(4t + 3) � −Π(2t) � r(−0.5t + 3) � u(t − 1) � u(t) × u(t − 1) �sin(2t)× u(t) =

' ( M¢KN�n��åP�Pý~M�ý �nãñ�·�ß

² V� �� � £ [^]�V\] �U�ì�b[pa � δ(t) � ��| ��r\g��`uqw`uhg{cke�mkT�gß�Yg{TÙm � �qcpoqipm � �\Thg S u`�`isS^³4SÞmfTÙuqw`u�e�T �Ye S uZS^mfThgrYcWw`u¬oq�Ï�hg´��| � zUg{}`rbm �Z�qT ��|U��S uh�Ï�hThgPik��uh��SÞ³4lçTqw`u�m ipoqcsrY�qTÁw`csw`cs| T`r��qisS �Ï�qcsmfT�¼`u¬oqtÏcsSU�`l�qcp�bip¹\u`�Zl =*) g�T mfu�oq�¬��u T`��m � uZS u`�q}`�UcpmfThg6�hThg�+ £ �� �Vj[ ¥ +�¨ÏV¶ ¬� � £ [^]hVj]Ùë

impulse function).× T`| }�mfu���mUgÌuZS u`�q}`� csmfThg-, rY�ZS }`| m ��r��., � zUcsS�cke S ThgÒr���S }h|^m �ZrY��TqoZoq}�m csoZckrbm tZl � �qwq³4l·¹\T�z u`�Z�qcw`T`| T`�h}qmk³ = R»e S T�gÌ��| tZr�g{�qu��`�Z³4l � rYmUg�l·cs�\T`| �qu¬�bisl � S T�m �ZS½w`T`| Thrbm tZrYu`�Z�qc��bcp³4�qcsm |Ug´��}�³4ldmfu�`|UgxuÞ�hg�Thl¾rY�ZS t¬¹\u`�Zl~r���SU}`|^m �ZrY�Zl � �hTZ¹�À4l¾�`g{TÛw`T`| }`�qcsm | u`l¾m cfe S cfg»rbmfu

0 = �Pw`}`| ��u`�ZS�w`u¬oZoqislm ipmfuhg{ckl��bcp³4�qcsm |Ug��hisl�w`| u`r�cp���Ye�rYcfg�l � �hThg<rbmfu ¸ ��tZ�qT Ø Ã ¹jT z �`rYuh���qc�m �ZSOwhg��ÏrY�ZS �¬¹jg�r��qisS ��Â

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¸ ��t��`T�Ø Ã Â ² rY�ZS }`|^m �ZrY�δε(t) � ³4l®�bcp³»�qcpm |Ug��ht·w`|Uu`rYip���Yg�r��·m �Zl

δ(t) � �¬mfT`S�mfuεm cke SUckgYrYmku

0

è |Ue{� uh���qcPm �·r���SU}`|^m �ZrY�δε(t)

³4l

δε(t) =1

2εΠ(

t

)

=

12ε

�Yg{T |t| < ε

0�Yg{T |t| > ε

ë��Zï

² �bcs³4�qcpm |Ug��htw`T`| }hrbmfT`rY�m ��lδε(t)

zUe{z csmfThg�rbmfu ¸ ��tZ�qT�Ø Ã � �hThg�cke S Thg�isSUT`l$u`|^¹\u¬��À4SUg�uhl$w`Tqoq�q�`l�qc$¼h}`rY�

2�Thg��0/�uhl

1/2ε = R»e SUThg\��T`S ck| ����mUg�gxrY���ZckgY�NrY��isr��∫ +∞

−∞δε(t)dt = 1,

ë Ø� qï�Yg{T¾m ����u`�ZrYT�mUg{�qt�mfu`�

ε = õ®cp³4| uh���qcm^À4|UT¾�¬mUgε → 0 � uqwh�¬m cnmfu¾�1/\u`l�mfu`��w`Tqoq�qu`��m cfe S cfg�rbmku

}qw`cfg�|Uu � csS^À��v¼h}`rY��mku`��m cke S ckg�rYmku0 � ism r�ghÀ4rbm c<mkuNcs�¬¼hT`z �ZS®mfu`�yS TNcke SUThgbe�r�udw`| u`l�m ���quZS }`z T =

) g{T�w`u¬oZ�½�`g´��| tNmUg��qt·mfu`�ε � �NrY�ZS }h|^m �ZrY�

δε(t)¹\cp³4|Ucke�mkT�gb³4l�`e{T��bcp³4�qcpm |¬g´��t·w`| uhrYip���Yg�r��·m �Zl

δ(t)ëö���bg��`�Z³4l®³4lvu`|¬g�rY�q�hlfï =

2 ��� 3546!7� 8�9�#;:=<>#��@?�A6B>4=:C<��D<�#δ(t)

² δ(t)u`|Ue{� csmfThgY³4lvu�m csoZckrbm tZlP�qc�m �ZSyg�z¬g���m ��mfT

∫ +∞

−∞x(t)δ(t)dt = x(0),

ë�Ø�Ø ï�Yg{T��h}Z¹\cvrYtZ�qT

x(t) � r���SUcs��islvrbmfut = 0 = R4}hSyrYm ��S�csµ¬e�rb³4r��~ØZØvoq}�¼hu`�Z�qcP³4lrY�ZS }`|^m �ZrY�

x(t)m �ZSx(t) = 1 � ik��uh���qc

∫ +∞

−∞δ(t)dt = 1.

ë Ø ÝZï² rY��isrY�~ØZØ�Zw`u`|Ucke�S T�w`}`| ckg��`g{T��bcsS¬g´���¬m ck| �½�qu`| �\t

∫ +∞

−∞x(t)δ(t− t0)dt = x(t0).

ë�Ø Ã ï× T`| Tqm �Z| u`�Z�qc6�¬mUg`unm csoqcsrbm tZl

δ(t− t0)cswhg{z | }yrbm �®rY�ZS }h|^m ��r��

x(t)�hT�g`z¬e S cfgqm �ZS$mUg��qt®m �Zl

x(t0)rbmfut = t0 =

ØZØ

Page 12: simata kai systimata - users.auth.grusers.auth.gr/~hadjidem/simata kai systimata.pdf · @)A B C D E F G H I H A (TIME DOMAIN ANALYSIS) J LKNMO P Q R4SUTWVYXZV\[^]`_badcfe S ThgjikS

º uNu¬oquq�hoZtZ|^³4�qTNmfu`�yT`|Ugxrbm cs| uh�y�qipoqu`�Zlßm �ZlØ Ã uZS u`�q}h� cpmfThg�� ���E+f�j¨ £ °Ì_baN� ¥ ¡ �-F4V � °4©®ëconvo-

lution integral). ² g{zUg{��m ��mkT¾Ø à w`|Uuq�h��wqm ckg�Tqw`�Nm ��rY��isrY�½ØZØPcs}`Sv³4l�rY�ZS }`|^m �ZrY�x(t)

oZ}Z¼`uh���qcm �ZS

x(t+ t0) �∫ +∞

−∞x(t+ t0)δ(t)dt = x(t0),

�hThg\rbm �·rY�ZS is��ckg{T��h}`SUu`���`cPm �ZS�TqoZoZT`��t½�qcpmfT�¼`oq�¬m tZlt → t′ − t0

Â∫ +∞

−∞x(t′)δ(t′ − t0)dt

′ = x(t0).

2 �&% GIHJ!K9L:C<M:6�N#5:=<>#��@?�A6B>4=:C<��D<�#δ(t)

•δ(αt) =

1

|α|δ(t).ë�Ø í ï

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α > 0�hThg��bg{T

α < 0ï =

•δ(−t) = δ(t).

ë�Ø ôZï² g{zUg{��m ��mkT~T`��m tdw`| u`�h��wqm ckg�Tqw`��m �·rY��isr���Ø í ck}`Sn¹\isrYuh���qc

α = −1 =•

∫ +∞

−∞x(λ)δ(t− λ)dλ = x(t).

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t → λ�hThg

t0 → t ��h}`S uZS^mfT`lncpw�ebwqoqisuZS���| tZrY�d��ThgYm �Zlvg{zUg{��m �¬mfT`lNØ^ô =•

∫ t2

t1x(t)δ(t− t0)dt =

x(t0)�bg{T

t1 < t0 < t2

0�Yg�T

t0 < t1tt0 > t2.

ë�Ø øZï

² g{zUg{��m �¬mfTçTh�¬m tWw`| uq�h��wqm cfg'T`w`�¢m �ZS�g{zUg{��m �¬mfT Ø Ã � cs}`S�oq}�¼hu`�qc���w`�O/\�WmkuZS�u`|¬g�rY�q�¢m �ZlrY�ZS }`|^m �ZrY�Zl®z ipoZmfT =

•x(t)δ(t− t0) = x(t0)δ(t− t0).

ë�Ø�Zï² g{zUg{��m ��mkT~T`��m tdw`| u`�h��wqm ckg�T`�qikrb³4lvTqw`��mfuZS�u`|¬g�rY�q��m �ZlvrY�ZS }`|^m �ZrY�ZlPzUipoZmkT =

• è m ��w`u`lnØ Ã cke SUThg�w`u¬oq�y��| tZr�g{�qu`l��Yg�TNmfu�S��w`u¬oqu¬�bgxrY�q�½csS �hlPu¬oZu`�`oq��|^À»�qTqmkuhl�m �Zl��qu`| �\t�l∫ +∞

−∞F (t)δ(t− t0)dt.

Ø Ý

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× | u`�h��wqm ckg�T`�qikrb³4lv�¬mUg\g�r����ZcfgY�·rY��ikrY�∫ +∞

−∞F (t)δ(t− t0)dt = F (t0).

ë�Ø��Zï

² g{zUg{��m ��mkT�Ø��N�Zw`u`| cfeYSUT½w`}h| ckgY�hT�gYm �·�qu`|U��t∫ t2

t1F (t)δ(t− t0)dt = F (t0), t1 < t0 < t2.

ëóÝ1 qï

P ÑRQNSjÑ2È0TVU�Pw`u¬oqu¬�Ye�rYm cPmkT�u¬oZu`�`oq��| À4�qTqmfT∫ +∞

−∞e−αtδ(t− 10)dt,

∫ +∞

−∞e−t2δ(t)dt,

∫ 10

5cos(2πt)δ(t− 2)dt,

∫ 5

2(x2 − 1)δ(t)dt.

W ( M¢KN�n��åP�Pý~M�ýδ(t)

² r���S }h|^m �ZrY�δ(t)

cfe S T�g�cswhe�r���lisS T`lm cpoqcsrYm t�l � u�uqw`uhexu`l�u`|Ue{� csmfThg\T`�Zrbm �Z| }�³4lvu�m cpoqcsrbm tZl®�qcg{zUg{�¬m ��mfTbÂ

∫ +∞

−∞x(t)δ(t− t0)dt = −x(t0).

ëóÝ�Ø^ï× T`| Tqm �Z| u`�Z�qcy��mUg¶u¾m csoqcsrbm tZl

δ(t − t0)cswhg�zU| }�rbm ��r���SU}`|^m �ZrY�

x(t)�hThg�z¬e S cfg�m �ZS·w`T`|U}q��³'���

m �Zl®rbm �y¹\ikrY�t = t0 � x(t0) � �qcPT`| S ��mUg��h��w`| �`r����qu =

î w`u`| uh���qc®S T�is��u`�Z�qc��Yg�T�mfuZSm csoqcsrbm tδ(t)

�`g{T½�bcp³4�qcsm |Ug��htNckg��h��SUT � T`S }`oZu¬�b�½�qc�cs�hcke S �·w`u`�z cfe�µUT`�qc��Yg{T½mfu�Snm cpoqcsrYm t

δ(t)rbmfu·mfu ¸ ��tZ�qT�Ø Ã = ² �bcp³4�qcpm |¬g´��tdw`T`| }`rYmkThrY�dmkuh�drY��tZ�qTqmfu`lyØ Ã

z ckS��Zw`u`| cfeÒS TW��| �Zr�g{�quqw`uhg´�¬¹\ckeÌ�Yg{T�m �ZS�w`T`| }q��³'�bu � oq�¬��³.mk³4S�T`rY�ZS ck��cfg�À4S�w`uh�¾w`T`| u`�Zr�g{}`�Uckg =R4z^À � �bg{T�m ��Sw`T`| }`��³'�bu � ¹jT½��| ��r\g��`uqw`uhg{tZrYu`�Z�qcP�`g{T½}`o�oq�d�bcp³4�qcpm |¬g´��tdw`T`| }`rbmfT`r��dm ��lδ(t) � �uqw`uhe{T¾is��ckgYw`}`oqg�m �y¼hT`r�g��ht�g{zUg{�¬m ��mfT�w`u`�½zUe{z csmfThg\Tqw`��m � ¸ ��isr���Ø Ý = õ®cp³»| u`�Z�qcPmfu�S�m |¬g ��³4SUg��h�

w`Tqoq�q�fε(t)

mfu`�NrY��t��qT`mkuhlyØ í � �Yg{T�mfu�Syuqw`uhexu�gxrY���ZckgY�NrY��isrY�∫ +∞

−∞fε(t)dt = 1,

�Yg{T��h}Z¹jcPmUg��qt·mfu`�ε = ² w`T`| }`��³'�bu`lnm �Zl®rY�ZS }`| m ��r���l®T`��m tZl®cfe S Thg\�

fε(t) =

0�Yg{T |t| > ε

1/ε2�Yg{T −ε < t < 0

−1/ε2 �Yg{T0 < t < ε

ëóÝZÝZï

�hThg\�¬mfT`Sε → 0 � ik��uh���qc®�`g{T�w`| u`rYis���bgxrY�·m �Zl

δ(t) =Ø Ã

Page 14: simata kai systimata - users.auth.grusers.auth.gr/~hadjidem/simata kai systimata.pdf · @)A B C D E F G H I H A (TIME DOMAIN ANALYSIS) J LKNMO P Q R4SUTWVYXZV\[^]`_badcfe S ThgjikS

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t

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δ(t) � �¬mfT`S�mfuεm cke S cfgYrYmku

0

² rY��isrY�·�qcpmfT`µU�dmk³4S�u`|UgxrY�ZÀ4S�Ø Ã �hThg\Ý�Ø�\The SUcpmfThg�T`w`��mfT½w`T`|UTq�h}qmk³6Â∫ t2

t1

d

dt[x(t)δ(t− t0)] dt =

∫ t2

t1x(t)δ(t− t0)dt+

∫ t2

t1x(t)δ(t− t0)dt,

t[x(t)δ(t− t0)]

t2t1= x(t0) +

∫ t2

t1x(t)δ(t− t0)dt,

�hThg\cpw`cfg�zUtNcke S Thg[x(t)δ(t− t0)]

t2t1= 0,is��u`�Z�qcPm cpo`g��h}�m �·r���ikrY�NÝhØ�ë:¹\cs³4|^À4S^mfT`lvcpwheYwqoqisuZS���mUg

t1 → −∞ �hT�gt2 → +∞ ï =

X>��� Y[Z>46Z\:C��4C<>��<

è g'rY�ZS Th|^m t�r�ckgxlδ(t)

�hThgδ(t)

z ckS�cke S Thg4rY�ZS T`| m t�r�ckgxl��qc½m �WrY�ZS t¬¹\��isS S u�g�Tçmfu`�O�`| u`� � �qwq³4lT`S Th��is|UT`�qc��hThg4w`| u`���bu`�Z�qisS^³4l � TqoZoZ}Ûcke S T�g'm csoZckrbm isl = î g�TÛrY�ZS t¬¹\�Zl�rY�ZS }`|^m �ZrY�WTqw`ckg��huZSUe{� cfgisSUTÞrY�ZS u¬oquÛ�`g{T`l��qcsmfT�¼`oZ��m tZl"ë w = � = mfuÞ��| ��S u

tïdr�c~isS T�}qo�oquÞr���SUu¬oZu

f(t) = Ü S^mUe�¹\csmfT � uhgrY�ZS T`| m t�r�ckgxl

δ(t)��Thg

δ(t)T`w`ckg��huZSUe{� uh�ZS~ikS T�rY�ZS u¬oqu�rY�ZS T`|^m tZrYcp³4S�ë w = � = x(t) ïPrYcdisS T�}qoZoqu

rY�ZS u¬oqu � �qwq³4l·�\The SUcpmfThgÌTqw`��mfu`�Zl·u`|UgxrY�qu`�Zl~Ø Ã �hThgÒÝ�Ø = ¸ m ��rY��isr���Ø Ã � u�m cpoqckrbm tZlδcswhg{z | }

rbm �·rY�ZS }`|^m �ZrY�x(t)

m �·rYmUg �b�qtt0

�hThg\zUe S cfgYm ��SnmUg{�qtdm �Zl � x(t0) � m �NrbmUg �b�qtt0 =

² rb³4rbm t���| tZrY��ms³»Sδ(t)

�hT�gδ(t)

cfe S Thg'�qisrYTOrbmfTOu¬oZuq�hoZ�Z|^À4�qT`mkT Ø Ã t�Ý�Ø = Q è mfT`S���| �Zr�gxÚ�quqw`u�g�u`�ZS^mfThg\cp�`m �hlvu¬oZu`�`oq��|^³»�q}qms³»S � ³4l®rY�ZS t¬¹\cfg�lPr���S Th|^m tZrYckgxl � ��| �Z}`� cpmfThg\g{zUg{The m ck| �Nw`| u`rYu���t× = � = � r�c��`g{T�is�h��| ThrY�·m �Zl®�qu`| �\tZl

F (t) = G(t)δ(t),ëóÝ Ã ï

¹\T½w`|Uipw`ckg�S T�ik��uh���qc®��w`�O/��N��mUgYrbm �y¹\ikrY�ym �Zlδ(t)

is��u`�Z�qcPm �d�bcp³4�qcpm |¬g´��t·m ��l®w`| u`r�ip���Yg�rY�

δε(t) =1

2εΠ(

1

)

,

�Yg{Tεw`u¬oq�N�`g��h| � =

Ø í

Page 15: simata kai systimata - users.auth.grusers.auth.gr/~hadjidem/simata kai systimata.pdf · @)A B C D E F G H I H A (TIME DOMAIN ANALYSIS) J LKNMO P Q R4SUTWVYXZV\[^]`_badcfe S ThgjikS

Ä�ÅjÆ�Å\Ê^]�ÊLT{Ñ�Ó) g{T�S TÞw`T`| Tq��³'�Ye�r�u`�Z�qc�m �¢rY�ZS }`| m ��r��

F (t)w`u`�çzUe{z csmkT�gñrYm ��SWÝ Ã � w`|^À'mfTÛm �"��|U}`��u`�Z�qc�³4l

F (t)δ(t) = F (0)δ(t)�hT�g��qcpm }½w`Th| Tq��³'�Ye��Uu`�Z�qc �

d

dt[G(t)δ(t)] = G(0)δ(t).

ëóÝ í ï

_ P ÑRQ\Ó�Ñ�Óõ®cs³4| tZrYTqm cP�¬mUg��·r���S }h|^m �ZrY�

F (t)rbm t ¸ ��isrY�½Ý à zUe{z csmkThg�w`| u`rYcs���YgxrbmUg��h}�³4lF (t) = G(t)fε(t),

�qw`u`�fε(t)

cfe S T�gUuPm |Ug���³4S¬g´���`l»whTqoZ�`�`l4mfu`� ¸ ��tZ�qTqmfu`l$Ø í � �$u`w`uhe{Tvcke SUThg¬�`g{TPrY�ZS t¬¹\�Zl'rY�ZS }`|^m �ZrY� =× T`| Tq��³'�Ye�rYm c�m �ÛrY�ZS }h|^m �ZrY�çTh�¬m tÛ�hT�g6rYm �ÛrY�ZS is��ckg{T�¹\cs³4| tZrbm c¾��mUgε → 0 = î c�¼h}`r��çmfu

Tqw`u�m ipoqcsr��qT�T`��m � � zUg��hThg�oqu¬�bt�rYm c¾mkuZSWm �¬w`u Ý í �Yg{T�m �ZS�w`T`| Tq��À'�Yg�r��çm �Zl�rY�ZS }`|^m �ZrY�ZlF (t) ��qwq³4lzUe{z cpmfThg\Tqw`��m �·r���ikrY�NÝ Ã =

` �a�Oã6MWý ��æñ�ßcbed �Pþ �u(t)

����δ(t)

Ü w`��mkuZSnm ��w`u�u`|Ug�r��qu`��ë¤m ��w`u`lÝZï mk³4Sv¼`Thr�g��`À4SyrY�ZS T`|^m tZrYcs³4Sui(t) �

ui−1(t) =∫ t

−∞ui(λ)dλ,

is��u`�Z�qc � �Yg{T i = 0 � �hThgY¹\ipmfuZS^mkThlu−1(t) ≡ u(t)

��Thgu0(t) ≡ δ(t),

u(t) =∫ t

−∞δ(λ)dλ,

ëöÝZôZït

δ(t) =du

dt.

ëóÝ�÷qï) g{T½S T½z u`�Z�qc$wqÀ4lPrY�ZS z iscsmfThg�uNm cpoqcsrYm t�l

δ(t) � �qwq³4l®zUe{z cpmfThgbrbmfu�Svm ��w`u½Ý�÷ � �qc$mfuZSu`|UgxrY�q�Nw`u`�zUe{z csmkT�g�Tqw`��m �·r���ikrY�~Ø�Ø � ck|^�bT`� �`�qT`rYm cP³4lvcsµU��l¬Âñõ®cp³4| uh���qcPmfu�u¬oquq�`oqtZ|^³4�qT

∫ +T

−Tx(t)

du

dtdt

�hThg\u¬oquq�`oq�Z|^À4S u`�Z�qc Â∫ +T

−Tx(t)

du

dtdt = [x(t)u(t)]T−T −

∫ +T

−Tu(t)x(t)dt =

Ø ô

Page 16: simata kai systimata - users.auth.grusers.auth.gr/~hadjidem/simata kai systimata.pdf · @)A B C D E F G H I H A (TIME DOMAIN ANALYSIS) J LKNMO P Q R4SUTWVYXZV\[^]`_badcfe S ThgjikS

x(T )−∫ +T

0x(t)dt = x(T )− [x(t)]T0 = x(0),

�hThg�m cpo`g��h} � �bg{T T → ∞ �∫ +∞

−∞x(t)

[

du

dt

]

dt = x(0),

�Yg{T·m �Z����SrYtZ�qTx(t) = ¸ �����h|Ue SUu�S^mfThlPm �ZSPm csoZck��mkT�e�TNT`��m t�r���ikrY�n�qc$mkuZS®m ��w`u~ØZØ�w`Th| Tqm �Z| u`�Z�qc

��mUg�uvm csoqcsrbm tZldu/dt

rY�Z�Zwhe�wqm ckg��qc4mfuZS6m csoqcsrbm tδ(t) � z ��oqT`z tPu�g�m ��w`uhgZÝZôß�hThg`Ý�÷�cfe S ThgqrY�Z�¬¼hTqmfuhe

�qcP�¬oqT��`rYT�cs�q¹\isr�T`�qc�w`| u`���bu`�Z�qisS^³4l =

f ��������ß æ6�ã6å��Þæ$� �ä ����çMW������� �ìMg�hdN ¾ä

õ®cs³4| u`�Z�qc<mfu·rYtZ�qTx(t) � mkuNuqw`uhe�u��Zw`u`| cfe�S TNcke S Thg��hThgb�`g �bT`zUg��h� =òè |Ue{� u`�Z�qc$³4l� � ¥ +�¨ � � � £ji��s¥ a

Emkuh�·r�t��qT`mkuhl

x(t)m ��Snw`u`r��¬m ��mfT

E = limT→∞

∫ +T

−T

∣x(t)2∣

∣ dt,ëóÝZø�ï

�hThg�³4l ¥ V ¦$Xvmfu`�NrYtZ�qTqmfu`lx(t)

m �ZSnw`u`rY��m �¬mfT

p = limT→∞

1

2T

∫ +T

−T

∣x(t)2∣

∣ dt.ëóÝ1Zï

Q R4SUT"r�t��qTx(t)

u`|Ue{� cpmfThg2³4l�rYtZ�qT�csS ik|^�bckg{T`l T`S��hThg'�q��SUu"T`S¾��ckS is| ��cfg{}Omfu`�E

cke SUThg%w`ck|fÚTqmk³4�qisSU�d�hT�g

p = 0 =Q R4SUT�rYtZ�qT

x(t)uh|Ue{� cpmfThg̳4ldrYtZ�qT�g�r����Zu`l T`S½��ThgÌ�q��SUu�ThS��¾gxrY���Zlymfu`�

pcke S Thg�w`ck| Tqmk³4�qisS �

�hThgE =∞ =

Ä�ÅjÆ�Å�Ç�ÈZÉ´Ê%ËÌÅ�ÍqÅQ R4rYms³ mfu�r�t��qT

x1(t) = Ae−αtu(t), α > 0.Q R4��u`�Z�qcE =

a2

2α, p = 0.

Ü lv��w`uZ¹\isr�u`�Z�qcßm^À4|UT���mUgα→ 0 � uqw`��m c®ik��uh���qcPmfu�r�t��qT

x2(t) = Au(t).

¸ m �ZSnw`cs|¬e´wqmk³4rY�NT`��m t·cfe S Thg

E =�q�dw`cs|UTqmk³4�qisS u � �hThg p =

A2

2.

Ø^÷

Page 17: simata kai systimata - users.auth.grusers.auth.gr/~hadjidem/simata kai systimata.pdf · @)A B C D E F G H I H A (TIME DOMAIN ANALYSIS) J LKNMO P Q R4SUTWVYXZV\[^]`_badcfe S ThgjikS

õ®cs³4| u`�Z�qc�m^À4| T�mfu½w`ck|Ug�u`z¬g´����rYtZ�qT

x(t) = A cos(ω0t+ θ),

w`cs|¬g��hz u`�T = 2π/ω0 = ² �qisr��·gxrY���Zl®mkuh�·cfe S Thg\�

p = limT→∞1

2T

∫ +T

−TA2 cos2(ω0t+ θ)dt =

A2

2.

kml�ÑÌÓnT{Ñ6oCpqUsr¶È¬ÆtTVu�ÇvTwQCuxpÞÑCS�Ë�Å\Í0u"Uõ®cs³4| u`�Z�qcNw`cs|Ugxu`zUg��h�ÛrYtZ�qT

x(t)w`cs|Ug{�`zUu`�

Tëx(t + T ) = x(t)

�Yg�T"m �Z���ZStï =Oè |Ue{� uh���qc½³4l

_�� V\] ¥ V^¦$Xvmfu`�·w`ck|Ug�u`z¬g´��u`��r�t��qT`mku`lx(t)

m ��Snw`u`r��¬m ��mfT

p =1

T

∫ t0+T

t0|x(t)|2 dt. ëóÝ1�Zï

² mUg{�qtdmfu`�pcfe S Thg\T`S ckµ }`|^m ��m �·m �ZlPmUg{�qtZlPmfu`�

t0 =Ä�ÅjÆ�Å�Ç�ÈZÉ´Ê%ËÌÅ�ÍqÅõ®cs³4| u`�Z�qc�mfu��`g��bT`zUg��h�~w`cs|¬g�u`zUg��h�¾rYtZ�qT

x(t) = Aej(ω0t+θ)

= A [cos(ω0t+ θ) + j sin(ω0t+ θ)] ,

w`cs|¬g��hz u`�T = 2π/ω0 � �`w`u`� j = √−1 ����ThS^mkThrbmUg��ht~�quZS }`z T = ² �qikrY��gxrY���Zl�mku`��w`ck|Ug�uhzUg´��u`�

rYtZ�qTqmfu`lvcke S Thg��p =

ω02π

∫ t0+2π/ω0

t0

∣ej(ω0t+θ)∣

2dt = A2.

y�Å�ÍqÅ�ÖIu�ËNSÛÍ`ÓNU�ȬÖjl�Æ`Ê2È0T Å"U�ÑÒÍzu|{'ÔjÑ2Ë�ÅÜ lv��w`uZ¹\isr�u`�Z�qc���mUgY�NcsSUis|^�bckg{T~csS �`lrYtZ�qTqmfu`lvcp����| }`�UcpmfThg�³4l

E =∫ +∞

−∞G(f)df,

ë à qï�qw`u`���Û�qcpmfT�¼`oq��m t

fcfe S T�g0�ÞrY�Z��S ��m ��mfT = ² rY�ZS }`|^m �ZrY�

G(f)uZS u`�q}`�UcpmfThg�¡� 0+�� ª [^]�[paÏ[^]�©

� � � £ji��s¥ a`©nVj[±�~}��`V�_Ya~ëenergy spectral density

ï =Ü S��½g�rY���ZlvcsS �hlvr�t��qT`mku`lvcs�h��| }`�UcpmfThg�³4l

p =∫ +∞

−∞S(f)df,

ë Ã Ø ï�·r���SU}`|^m �ZrY�

S(f)u�SUu`�q}`� csmkT�g2¡Z 0+�� ª [^]�[±a�[^]�© ¥ V ¦$XZ� ©®V\[p�}\�`V�_ba~ë

power spectral densityï =

Ø ø

Page 18: simata kai systimata - users.auth.grusers.auth.gr/~hadjidem/simata kai systimata.pdf · @)A B C D E F G H I H A (TIME DOMAIN ANALYSIS) J LKNMO P Q R4SUTWVYXZV\[^]`_badcfe S ThgjikS

� � �y���d�PýN�næ6ä �LKNMO�Pý�������þÿ�

º T®rY�Zrbm tZ�qTqmfT��Zw`u`| u`�Z�qc4S TPmfT��hTqmfTqm }hµ u`�Z�qc4�hTqm }®w`u¬o�oqu`�Zl»m | �qw`u`�Zl � T`SU}qoqu¬��T�qc2mUg�l»z¬g�}h��u`| cklg{zUg{�¬m ��m ikl®mfu`�Zl =»è g�¼`Thr�g��h�¬m ck| cslnTqw`��mUgxl®�hTqm ���bu`|Ue{cslnTh�¬m iklvzUe{z u�S^mfThg�w`Th| Tq�h}qmk³6Â

• ¸ ��rYm t��qT`mkT�rY�ZS ck��u`�Zl®��| �ZS u`�N��Thg�rY�Zrbm tZ�qTqmfT½T`rY�ZS ck��uh��l��| ��SUu`� =¸ mkTyr���rYm t��qT`mkTnrY�ZS ck��uh��l<��| �ZS u`�n�ckexrYu`z u`l<��Thgh�isµ uhz u`l$cfe S Thg�T`S Tqoqu¬�Yg��h}·rYtZ�qTqmfT � csS^ÀrbmfT½r���rYm t��qT`mkT½T`r���SUcs��u`�Zlv��| �ZS u`�½�NckexrYu`z u`lv��Thg��Nisµ uhz u`lcfe S Thgx/��Z�\g{Tq�h}~r�t��qT`mkT =• ¸ ��rYm t��qT`mkT�rbmfTZ¹jcs| }�tN�qcpmfT�¼hTqoZoq�`�qcsS T =Q R4S T~rY�Zrbm �Z�qT½u`|¬e��UcpmfThg�³4ldVj[±a�� �ö£Yª T`Sd�·ikµ u`z uhlrYm ��S�ckexrYu`z u

x(t)ckµ T`|^m }qmfThg\�q�ZS u�T`w`�

mkuçrYtZ�qT"m �Zl~ckgxrY�`z u`�O�hT�g4���bg4T`w`�¢m �OrbmUg �b�qtWw`u`��¹jT¢T`| �bexrYckg»S Tçcs�\T`| �q�`� csmfThg = ¸ m �ZSw`cs|Ue�wqmk³4rY�·Th�¬m t½g�r�����cfgY�Ng{zUg{��m �¬mfT

H x(t− τ) = y(t− τ),�bg{T��h}Z¹\cPmUg{�qtdmfu`�

t.

ð ��oZThz t � rbm �ZS�w`cs|Ue�wqmk³4rY��Th�¬m t���isµ uhz u`l�rbm �ZS�ckexrYu`z ux(t − τ)

cfe S T�g%rY�ZS }`|^m �ZrY�¾�`g{T`l�q��S u~�qcpmfT�¼`oq�¬m tZlP�hThg\�·rY��isr��

y(t) = Hx(t)w`cs|Ug��b| }`��cfgjwqoZtZ|^³4lm �½rY�Z�Zw`ck|Ug{��u`| }~mfu`��rY�Zrbm tZ�qTqmfu`l = R»}`S·��ckexrYu`z u`lx(t)

cs�\T`| �qu`rb¹\cke�qcpmfTqmfuqwhgxrY�qisS �$rbmfuß��| �ZS uP�hTqm }�mfu�zUg{}`rYm ���`T

τ � x(t−τ) � �<ikµ u`z uhl»cke SUThgU�$e{zUg{T®rY�ZS }`|^m �ZrY� ��qcpmfTqmfuqwhgxrY�qisS �·�hTqm }�mfu�e{zUgxu���| uZSUg��h��zUg{}`rbm �Z�qTτ =Ü S½isS T�rY�Zrbm �Z�qT~cfe S Thg��qcpmfT�¼hTqoZoq�`�qcsSUu � �~isµUu`z u`l·rbm �ZS·ckexrYu`z u

x(t − τ)csµ T`|^m }`mkT�g¶T`w`�

z �Zu·�qcsmfT�¼qoq��m isl � mfuN��| ��S ut�hThg�m �yrbmUg �b�qt�ck��T`|U�qu¬��tZl�m �Zlßcfg�rY�hz u`�

τ =6� r�����cfg�cpw`u`�qikS^³4lrbm �ZSw`cs|Ue�wqmk³4rY�NT`��m t·�NrY��isrY�H x(t− τ) = y(t, τ),

�bg{T��h}Z¹\cPmUg{�qtdmfu`�t.

• Ü g�mUg�u`�h| TqmUg���}�rY�Zrbm tZ�qTqmfT =Q R4S T~rY�Zrbm �Z�qTNuh|Ue{� cpmfThg�³4lda ¥ [ ¥ �E+ £ a�[ ¥ + ª � ThSy�·rY��ikrY�x1(t) = x2(t),

�Yg{T���}Z¹\ct ≤ t1,

rY�ZS cpw`}q�bcsmkT�gb�hT�gYm �·rY��ikrY�Hx1(t) = Hx2(t),

�bg{T��h}Z¹\ct ≤ t1.

• ¸ ��rYm t��qT`mkT�z �ZS Th�`g´��}��hThg�r���rYm t��qT`mkT½rbmUg��b�`g�T�e�T =Q R4S T¾rY�Zrbm �Z�qT�u`|Ue{� cpmfThgj³4lNVj[ ¥ i _ ¥ a � �T`S·��ikµ u`z u`l�m �½rbmUg��b�qttckµ T`|^m }qmfThg��q�ZS u�Tqw`��m �ZS

ckexrYu`z uOm ��rbmUg��b�qtt � �hThg2���bg%Tqw`�Ow`| u`���bu`�Z�qcsS ckl½mUg{�qislNm �ZlNcfg�r��`z u`� = Ü �¬m �Ocp�h�\| }`� csmfThg

�qTZ¹\�Z�qTqmUg��h}�³4ly(t) = f(x(t), t).

�

Page 19: simata kai systimata - users.auth.grusers.auth.gr/~hadjidem/simata kai systimata.pdf · @)A B C D E F G H I H A (TIME DOMAIN ANALYSIS) J LKNMO P Q R4SUTWVYXZV\[^]`_badcfe S ThgjikS

Q R4S T¾rY�Zrbm �Z�qT�u`|Ue{� csmkThgj³4l½�U ¬�sa�_ ¥ + ª � T`S·��isµ uhz u`l�m �½rbmUg��b�qttckµ T`|^m }qmfThgj�hThg�Tqw`�¾�¬oqcsl

mUg�lNw`|Uu`����uh���qckS csl½mUg{�qislNm �Zl½ckg�r��`z u`� = Q R4S TOm ipmfuhgxuWw`T`| }`z cfg �b�qT¢cke S ThgÒmfuOrY�Zrbm �Z�qT�csSU�`lw`��S¬e�u`� � �`w`u`�½�·cfe�rYuhz u`lcke S Thg��·m }`rY�

v(t)��Thg��Nisµ u`zUu`lcfe S Thg�mku�| cs�Z�qT

i(t) �i(t) =

1

L

∫ t

−∞v(λ)dλ,

�qw`u`�½�½isµ u`z uhli(t)

csµUT`|^m }qmfThg\Tqw`�~�¬oqckl®mUgxlvmUg��`isl®m �Zlvcfg�rY�`zUu`�v(t)

T`w`��mku −∞ is³4l®mfut = ð �¬oqT`z t � rbm �ZSw`ck|Ue�wqms³4r��·T`��m tdmfu�rY�Zrbm �Z�qTNis��cfgY�qS tZ�q� =

• ) | T`�q�`g��h}��hT�g î � ) | T`�q�`g��h} ¸ �Zrbm tZ�qTqmfT =Q R4S T~rY�Zrbm �Z�qTNuh|Ue{� cpmfThg�³4l if£ a�_�_ ¥ + ª ThSygxrY���Zu`�ZS�uhg�z �Zu~g{zUg{�¬m ��m cslH [αx1] = αH [x1] ,

ë à Ý�ï�hThg

H [x1 + x2] = H [x1] +H [x2] .ë ÃZÃ ï

Ü ��m ��r����qT�e S ckg\�¬mUg\�¬mUg\T`SdrYc®isS T��b| T`�q�`g��h�¾rY�Zrbm �Z�qT½�½ckexrYu`z u`lncfe S T�g��b| T`�q�`g��h�`l�rY�ZS zU� ÚT`rY�q�`lz �Zu�rY�ZS Th|^m tZrYcp³4S

x1�hThg

x2 �x = λ x1 + µ x2,

�·ikµ u`z u`lncke SUThg�u~e{zUg�uhlv�b| T`�q�`g��h�hlnrY�ZS z �ZT`rY�q�`l®mk³4S�csµ �`z^³4S`Ây = H [λx1 + µx2] = λ H [x1] + µ H [x2] .

Q R»g SUThg$��T`S ck| � � �¬mUg<�qc�cpw`Tq��³'�bt��Zw`u`|Uu`�Z�qc�S TÏTqw`uhz cke{µ u`�Z�qcW��mUg6�Ûw`Th| Tqw`}`S^³ g{zUg{��m �¬mfTg�rY���Zckg��Yg{T��b| Th�q�`g´���OrY�ZS z �ZT`rY�q��w`u¬oZo�À4S~�qcpmfT�¼`oq�¬m^À»S = Ü S~�¾cfe�r�u`z u`l

x(t)cp�h�\| }`� csmkThg

³4l®�b| T`�q�`g��h�`lnrY�ZS z �ZT`r��q�`l®r���S Th|^m tZrYcp³4Sxi(t) �

x(t) =n=N∑

n=1

λi xi(t),ë Ã í ï

�·ikµ u`z u`lncp�h�\| }`� csmkT�gY³4l

H

[

n=N∑

n=1

λi xi

]

=n=N∑

n=1

λi H [xi] .ë à ô�ï

º u�wqoqt¬¹\u`l®ms³4S��`| ³4Syrbmfu½ThS }qwqm ���b�qT à í �Zw`u`|Ucke�S T�cke SUT�}qw`cfg�|Uu � N =∞ =² g{zUg{��m ��mkT�m ��l®�b| T`�q�`g��h��m ��mkT`lnT`S Th��is|UcpmfThg\rbmUg�lvckgxrY�`z uh��l =è g�rY�ZS T`|^m tZrYcfg�l

xi(t)rbmUgxlßuqw`u�e�cklPT`SUTqoq��csmkT�g�mkuNrYtZ�qT

x(t)cfe S Thg�uhgh¼hT`r�g��hikl�rY�ZS Th|^m t�r�ckgxl

u(t) � δ(t) rbm �ZSThS }qoq��r��dw`cs|Ugxu�����l��| ��SUu`�·�hT�gYu�gcos(t) � sin(t) rbm �ZSnT`S }qoq�ZrY�dw`cs|¬g�u���tZl

rY�Z��SUu¬m t�mk³4S =

��

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× T`|UT`z cke��b�qTqmfT

º u�rY�Zrbm �Z�qT·w`uh�·w`cs|Ug��b| }`�\cpmfThg\Tqw`��m �·zUg{T`��u`|¬g´��t�csµUexrb³4rY�dy

dt+ ty = x(t)

cke SUThgq�b| T`�q�`g��h�~ëöcpw`T`oZ�¬¹\ck��rYm c4mfut�§ï � ckS^À"mfunrY�Zrbm �Z�qT®w`uh�®w`cs|Ug��b| }`�\cpmfThghTqw`�nm �®zUg{T`�\u`|Ug���tcsµUexrb³4rY�

dy

dt+ αy + β = x(t)

z csSycke SUThgY�b| T`�q�`g��h�Oë¤�Yg{TqmUe;ï =

Ä�ÅjÆ�Å\ÍzSjÆ`ÓjÑÌÓ¸ mfu�Sßu`|UgxrY�q�m �Zlñ��|UT`�q�`g��h�¬m ��mfT`l � �qwq³4l0z �Z¹j�¬��c»w`T`| T`w`}`S^³ � ¹\cs³4| u`�Z�qc»��mUgZ�Pcfe�rYuhz u`l6cs��Th| �q�`� c±ÚmfThgÌrYcd��| �ZS u

t = −∞ �hThgÒ��mUg�mfu�rY�Zrbm �Z�qT��Z| ck�qu`�ZrYcym �¬m c � z �¬oqT`zUt¾z ckS�ckex��c·Tqw`uZ¹\���hck���qikS �csSUis|^�bckg{TOë �Yg{TqmUe�cfe S Thg\Tqw`T`| The�m ��m �½T`��m t·�N�¬wh�Z¹\csr��

;ï =

¸ m ��rY��ikrY� à í mfu~r�t��qTx(t)

T`SUTqoZ�¬¹\���hcnrYcvT`|Ug�¹\�qtZr�g{�qu~wqoqt¬¹\u`ln�`|^³4S � �hThg���rY��isrY�½�qcpmfT`µU�ckgxrY�`zUu`���hT�g�csµ �hz u`��z¬e�zUcpmfThg�Tqw`��mfuZSdm ��w`u à ô = Ü ��m ���Zw`u`| cfejSUT~�bcsS¬g´��cs�¬¹\ckej�hThg���mfT`S·mfu�rYtZ�qTT`S T`oZ�ZcsmkT�gY�hTqm }~rY�ZS cs��tdm | �qw`u~rYcPrbmfuhg���cfg´À4zUckg�lnrY�ZS T`|^m tZrYcfg�l � ��w`�½m �N�qu`| ��t

x(t) =∫ +∞

−∞α(λ)w(t;λ)dλ.

ë à ÷qï¸ m �ZS<w`Th| Tqw`}`S^³Ûip����| T`r��Pmfu`�®rYtZ�qTqmfu`l

x(t) � u�gqrbmfuhg���cfg´À4zUckgxl6r���S Th|^m tZrYckgxlòrYmUg�l0uqw`u�e�ckl<T`S Tqoq�Zc ÚmfThg¶mfu�rYtZ�qT

x(t)cke S T�g�u�g

w(t;λ) � �qw`u`� λ�`g{T�r���S ck��t�lyw`T`|U}`�qcpm | uhl � ��uqw`u�e�T�w`T�e��Uckg�mfu�| �¬oZu

mfu`�¾z cfe´�hm �λi

rYmkuZS½m ��w`u à í = ¸ m �ZS�w`cs|Ue�wqmk³4rY�¾T`��m t��¾rY��ikrY��ckgxrY�`z u`�~Úßckµ �`z uh��cke SUThg � oq�¬��³m �ZlP�b| T`�q�`g��h��m ��mfT`lvmfu`�NrY�Zrbm tZ�qTqmfu`l � �

y(t) = H∫ +∞

−∞α(λ)w(t;λ)dλ =

∫ +∞

−∞α(λ)H [w(t;λ)] dλ.

ë à ø�ï

è m ��w`u`lvT`��m �`lcke S Thg�ThS^mUe�rbmfuhgx��u`lmkuh�dm ��w`u`� à ô =è gÌm �¬whuhg à ô��hThg à ø�cfe S T�g¶¼hT`r\g´��uhe%rbm �ZS�T`S }`oZ�ZrY��w`cs|Ugxu���tZl���| ��SUu`��mk³4S��b| T`�q�`g��`À4S¾rY�Zrbm �UÚ

�q}qmk³4S � zUg{�¬mUg���isµ uhz u`l�rYmkuym �Z����SvrYtZ�qTx(t)

cs�h��| }h� cpmfThg�rY�ZS T`|^m tZrYcfg`m �Zl$csµU�`z u`��mk³4Sß¼hT`r�g��`À4SrY�ZS T`| m t�r�cp³4S

xi(t)tw(t;λ) � ³4l H [xi] � H [w(t;λ)] � T`S^mUe�rYmku�g���T � �qwq³4lß¹\TNz uh���qc�rbmfT·csw`�`�qc Ú

S T =ñè gYrbmfuhg���cfg´À4zUckgxlvrY�ZS T`| m t�r�ckg�lw(t;λ)

mUgxl®uqw`uhe{cklP¹jT½��| �Zr�g{�quqw`uhg{tZrYu`�Z�qcPrbmfT½cpw`�h�qcsS T�cke S T�guhgu(t− λ)

�hThgδ(t− λ) =

Ý1

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� �a�Oã6MWý æ$�ìMg�s�n ~K ��æ�b �c� ~K M¢æ �Ûån����������MO�ß�� æ6ån� M¢KNMO�P�������

¸ mfu��hcs�\}qoqThg�u"T`��m ��¹\TWT`rY��u¬oq�U¹ju`���`cy�q�ZS uW�qcdr���rYm t��qT`mkT�rbmfT�u`w`uhe{TO�¾ckS is| ��cfg{TWcke S ThgÒr��¬��Ú�hckS^m |^³4�qisSU��rYc·u`|UgxrY�qikS TOr����qcfe{T��hThg2z ckS~cfe S T�gÒ�hT`mkThS cs�q�Z�qisSU��rY�ZS ck��À»lNr�c·�¬oquWmfuWr���rYm ���qT =¸ m �ZSNw`cs|Ue�wqmk³4rY�~T`��m t � �~rY��ikrY��cfg�rY�hz u`�UÚ¯ckµ �`z u`�¾zUe{z csmkT�g¶T`w`��zUg{T`��uh|Ug��hiklNcsµUgxrbÀ4rYcfg�ld�qc�rY�ZS t Ú¹\cfg�l®w`Th| Tq��À'�bu`�Zl = Ü S^mUe�¹\cpmfT � rbmfT�rY�Zrbm tZ�qTqmfT½�qw`uh�½�·ckS is| ��cfg{T~cke S T�g�r���S ck��À4lP��TqmfT`S cs�q�Z�qikS � ��¾r���ikrY��cfg�r��`z u`�UÚ csµ �`zUu`��zUe{z cpmfThgÒTqw`�OzUg{T`��u`|¬g´��isl�ckµUgxrbÀ4rYckgxlN�qcd�qcs|Ug��hikl·w`Th| Tq��À'�bu`�Zl =¾è gÌzUgxÚT`�\u`|Ug��hisl�T`��m islvcsµUgxrbÀ4rYcfg�lvcfe S Thg��b| T`�q�`g��hisl � ck��Q`�hrYu�S�mfu�r���rYm ���qT½cke S ThgY�b| Th�q�`g´��� =õ®T�z �`r�u`���`c�m^À4| T�m �d�bcsSUg��htNoq�ZrY�dzU��uN¼hT`r�g��`À4S��b| T`�q�`g��`À4SdzUg{T`�\u`|Ug��`À4S·csµUgxrbÀ4r�cp³4SqÂ) | Th�q�`g´��tNzUg{T`�\u`|Ug���t½ckµUexrb³4rY�dw`|^À'm �Zl®m }`µ cp³4lõ®cs³4| u`�Z�qc�m �d�b| T`�`�`g´��tNz¬g�Th��u`|Ug���t�csµUexrb³4rY�dw`|^À'm �Zl®m }`µ cs³4lv�qcP�qcpmfT�¼`oq��mkuh��lvrY�ZS^m cpoqcsrYm isl

a(t)dy

dt+ b(t)y = x(t),

ë à qï�qw`u`�

a(t)��Thg

b(t)�qcpmfT�¼`oq�¬m ikl¾w`T`| }h�qcpm | uhg = ² �bcsSUg��htÛoq�ZrY�¢m �Zl�csµUexrb³4rY�Zl�Th�¬m tZl � �bg{T mUgxl

T`| �bg��hiklnrY�ZS�¹\t��hcslt = t0, y = y(t0),cke S Thg��

y(t) = e−∫ t

t0

b(λ)a(λ)

{

y(t0) +∫ t

t0

x(ξ)

a(ξ)e∫ ξ

t0

b(λ)a(λ)

dλdξ

}

.ë Ã �qï

) | Th�q�`g´��tNzUg{T`�\u`|Ug���t½ckµUexrb³4rY�·z cs��m ik| T`l®m }`µUcp³4lõ®cs³4| u`�Z�qc�m �d�b| T`�`�`g´��tNz¬g�Th��u`|Ug���t�csµUexrb³4rY�·z ck�¬m ik| T`l®m }hµ cp³4lv�qc®rbmfTZ¹\cs|Uu`�Zl®rY�ZS^m csoZckrbm isl

d2y

dt2+ Ay = x(t),

ë í qï�qw`u`�½u�rY�ZS^m csoZckrbm tZl

Acke S Thg�rbmfTZ¹\ck| �`l = ² ��ckSUg��ht·oq�ZrY�ym �Zl®zUg{T`��uh|Ug��htZlncsµUexrb³4rY�Zl í ·cfe S ThgYmfu

}Z¹\|Uuhg�r��qT�m ��l®�bcsSUg��htZlvoq�ZrY�Zl�m �Zl®uh�qu¬��ckS u`�Zl �d2y

dt2+ Ay = 0,

�hThg��`g{T`lß�qck|Ug��htZl$oq��r���l6m �Zl<wqoqtZ| u`�Zl$csµ¬e�rb³4r���l = ² ��ckSUg��htnoq�ZrY�vm �Zl<uh�qu¬��ckS u`�Zl�csµ Th|^m }qmfThg�Tqw`�mfu½w`| �hrY�Z�qu½mfu`�NrY�ZS^m cpoqcsrYm t

A � �hT�g�is��u`�Z�qcPmUg�lvckµ tZlvzUg{T`��u`|UcpmUg��hislnw`cs|Ug�wqm^À4rYcfg�l¬Â

• A > 0² oq�ZrY�dm ��lvu`�qu¬�bcsS uh��lncke S Thg��

y = C1 cos(√A t) + C2 sin(

√A t)

ë í Ø ïÝ�Ø

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�·uqw`u�e�T��Zw`u`| cfe�S T�w`}`| ckgY��ThgYm �ZSyg�rYuhz �ZS T`�q�N�qu`|U��t= D cos

(√A t+ θ0

)

,ë í ÝZï

�qw`u`�C1 � C2 � t D � θ0 cfe S ThgZw`T`| }h�qcpm | uhg � u�g�u`w`uhe{csl6csµ Th|^m^À4S^mfThgZTqw`�mUg�l0T`| �bg��hisl6rY�ZS�¹jt¬��csl =R»e S T�gò��ThS cs| ����mUgñrbm �ZS�w`cs|Ue�wqmk³4rY�¢T`��m t¢�"oq�ZrY�"cke S Thgñw`cs| T`ms³4�qikS � � �hT�gò�q}qo`g�rbmfTÞcke S Thg

T`| �quZSUg��htPmfTqoq}`S^mk³4rY���qc4wqoq}qmkuhlD

�hThgZr�����S ��m ��mfT √A =^� rY����cfg¬��r���isrY�

D =√

C21 + C22ë¤Tqwhu`z cke{µ^m c®mfut�{ï =• A < 0

² oq�ZrY�dm ��lvu`�qu¬�bcsS uh��lncke S Thg��y = C1e

√−A t + C2e

−√−A t.

ë í Ã ï¸ m ��S�w`cs|Ue�wqmk³4rY�·Th�¬m tN�doq�ZrY�·z csSycke S T�gYw`cs|UTqmk³4�qisS � � ��Thg��q}qo`gxrbmfT�T`�Zµ }`SUckg�cs�q¹\csmUg´��} =

• A = 0² �bckSUg��htNoZ�ZrY�dm �Zl®u`�qu¬�bcsSUu`�Zlcke S Thg��

y = C1t+ C2.ë íZí ï

¸ c®Th�¬m tNm ��Syu`|Ug{Tq��t½w`ck|Ue�wqms³4r�� � �·oq��r��·cke S Thg\cswhe�rY�Zl�q�·w`cs| T`ms³4�qikS � � T`o�oq}~�N�qcsmkTZ¼`u¬oqtcke SUThgY�b| T`�q�`g��htN�hT�g����bg�cp�q¹jcpmUg��ht =

Q è wq³4l0��The S cpmfThgqTqw`�mfTvw`T`| Tqw`}hS^³ � ��w`uhg�u�mUg��htvrY�Z�Zw`ck|Ug{��u`| }m ��lñoq�ZrY�Zl»m �ZlòzUg{T`��u`|¬g´��tZl6ckµUe�rY³%ÚrY�Zl í nckµ T`|^m }`mkThg�T`w`�·mkudw`|U�`rY�Z�qudmfu`��rY�ZS^m csoZckrbm t

A � �hT�g���oZ�ZrY�ncfe S Thg�w`ck| Tqmk³4�qisS �y�q��SUud�Yg{TA < 0 =² �qcs|Ug��htNoq��r��ym �Zl®zUg{T`�\u`|Ug��htZlnckµUe�rY³4rY�Zl í ·csµ Th|^m }qmfThg�T`w`��m �ZS�ckexrYu`z u

x(t) =

J�� � �����KNM�ý MW������� �ä M¢æ MO�ß ��&�Oæ$�����æ$�¯ä M"KN�n�å®�P��M¢æ$�¯ä

Q è wq³4l¢T`S T`�\is| T`�qc"w`| u`���bu`�Z�qisS^³4l � rbm �ZSçThS }qoq��r�� w`cs|Ugxu���t�lç��| �ZS u`� � isS TÁm �����ZSÞrYtZ�qTx(t)T`S T`oZ�ZcsmkT�g�rYcPrbmfuhg���cfg´À4zUckgxlrY�ZS T`|^m tZrYcfg�l =»è gb�h�Z|Ug{��m cs| cklTqw`�~T`��m islvcke S Thg���V� �� � £ [^]�V\]~� �ì�Y[pa �

δ(t) � �hThg�mku¾_\� �pah� ¥ a � �<ê�¨`_ba � u(t) =�=����� ��A6B\�\?��D<��D��8�Z\:C�>#*� ���@?�A6Z>4=:C��� �"!$#�H6�-��:CZ¸ m �ZS�w`Th| }q�b| T`�\u à = Ø�z �`r�T`�qc6mfuZSPuh|Ug�r��q�ymfu`�m csoZckrbm t

δ(t) � �qc0mfuZSPm �¬whu�Ø Ã � �hT�g�rbm �rY�ZS is��cfg�T�qcpmfT`r����Z�qTqmUexrYT`�`cßmfuZSnm ��w`u�T`��m ��rbmfu�Snm ��w`u�Ø^÷ �

x(t) =∫ +∞

−∞x(λ)δ(t− λ)dλ.

ë í ôZïè m ��w`u`l í ô"cfe S Thgòm ��l�e{zUg{T`l��quh| ��tZl��qc~mfuZS�m ��w`u à ÷ � �qw`uh�çThS Tqoq��csmfThgñmfuÞrYtZ�qT

x(t)r�c~r�Ú

mfuhgx��cfg�À4z ckgxldrY�ZS T`|^m tZrYcfg�lw(t;λ) = ¸ �ZS cswqÀ4l � ��[fXh¡ � ©�z� �s� �pa ¥ �<ê�a`V ¥ + ª ©·[fX�¡ � ©��·�q¡^� � � ©y� � � �k¥[^]���aq�^���� �Vj]¾[p�� ·V\¨`_ba�[p� ©x(t)

V � V� ��sa £ [^¨�V �s¥ © δ =ÝZÝ

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-Äë 0 Äë 2Äë nÄë ( +1)Äën

t

x(t)x(nÄë)

1t-nÄë

Äë( )ÐÄë

Äë

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trYcPzUg{T`rbm tZ�qTqmfT∆λ � �hThg\rbmfu�rY�Z�qckexu

t = n∆λ¹\cp³4|Uu`�Z�qc�mkuZS�uh|�¹\u¬��À4SUgxu�w`Tqoq�q�

Π

(

t− n∆λ

∆λ

)

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1�Yg{T

n∆λ− ∆λ2< t < n∆λ+ ∆λ

2

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.

² w`T`| }`rYmkThrY�x(n∆λ) Π

(

t− n∆λ

∆λ

)

w`| u`r�cp���Ye��UckgYm �NrY�ZS }`|^m �ZrY�x(t)

rbmfu�zUe{T`rbm �Z�qT

n∆λ− ∆λ2

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�qc4m �ZS0mUg{�qt$m �Zlñrbmfu®�hisS^m | uvmfu`��z¬g�Thrbm tZ�qTqmfu`lòT`��mkuh� � x(n∆λ) � ckS^ÀW�bg{TvmUg{�qikl»mfu`�tcp�`m �hlòT`��mfu`�

mfu`�NzUg{T`rbm tZ�qTqmfu`lv�·w`T`| T`w`}`S^³Ùw`T`|U}`rbmfT`rY�NzUe{z cfgYmUg��qt½e�r��dw`| u`l�q�Zz isS = ¸ �ZS cpwqÀ4l � �NrY�ZS }`|^m �ZrY�x(t)

cp�h�\| }`� csmkThg � �hTqm }�w`| u`rYis���YgxrY� � ��w`��m �·�qu`| �\t

x(t) =+∞∑

n=−∞x(n∆λ) Π

(

t− n∆λ

∆λ

)

.

² w`| uhrYip���Yg�r��ßcfe S Thg¬m �`r�u���Tqoq�¬m ck| � � �`rYuv�`g��h| �¬m ck| uvcfe S T�gUmfu®zUg{}`rYm ���qT∆λ

rYmku®uqw`uhexu��³»|Ue��UcpmfThgu�}`µ uZS T`l

t = ² m cpoqcs��mfThe{T�T`��m tNrY��isrY�d�b| }`��csmfThgY�hThg�³4l

x(t) =+∞∑

n=−∞x(n∆λ)

[

1

∆λΠ

(

t− n∆λ

∆λ

)]

∆λ.

Ü S∆λ→ 0 � u��h| u`l

1

∆λΠ

(

t− n∆λ

∆λ

)

Ý Ã

Page 24: simata kai systimata - users.auth.grusers.auth.gr/~hadjidem/simata kai systimata.pdf · @)A B C D E F G H I H A (TIME DOMAIN ANALYSIS) J LKNMO P Q R4SUTWVYXZV\[^]`_badcfe S ThgjikS

w`| u`r�cp���Ye��Uckg�m �yrY�ZS }`|^m �ZrY�δ(t− n∆λ) � rY�Z�q�Y³4SUT·�qcß�`rYTNThS T`��ik| T`�qcPrbm �ZSPw`T`|U}q�b| T`��u í � �hThgm cpo`g��h}�u�w`Th| Tqw`}`S^³ m ��w`u`lw`The{| S cfg\m �½�qu`| ��t½mkuh�Nm ��w`u`� í ô�ë¤ck}`Sym �½zUg{}q�h|¬g m ���qcsmkT�¼`oq��m t

n∆λm �ZSnThS^mUg��hTqmfT`rbm tZrYu`�Z�qc®�qc�m �·rY�ZS ck��t·�qcpmfT�¼`oq�¬m tλï =

�=���&� �[4^��?M�@:J!���� ZM�>9M�D4J!$�D<î e{TW¼`Thr�g��htWrY�ZS }`|^m �ZrY��w`u`���hTZ¹\u`|¬e��Uckg2m ��rY�Z�Zw`cs|¬g��\u`| }"ckS �`l��b| T`�q�`g��huh�OrY�Zrbm tZ�qTqmkuhl � rYm ��ST`S }`oZ�ZrY�dw`cs|Ugxu���tZlv��|U��S uh� � cfe S Thg���+ £ �f ZV\[ ¥ +�¨~aq¡ ª + £j¥ V\]¾ë

impulse responseï � h(t, t0) � �·uqw`uhe{T

cke S Thg¬�$ikµ u`z u`lñrYm ��Sòcfe�rYuhz uδ(t−t0) � z ��oqT`z tßrbm �<rY�ZS }`| m ��r��

δck��T`|U�qu`rY�qisSU�<m �$��| u�SUg��htPrbmUg �b�qt

t0 � �hT�gYm �·rbmUg��b�qt·cp�hcfe S �Nmfu�rY�Zrbm �Z�qTN�Z| ck�qu`�ZrYcdë¤�Nisµ uhz u`lt�mfT`SyexrY�dw`|Uu`l�q��zUisS � y(t0) = 0 ïsÂ�qcpmfT�¼hTqoZoZ�h�qcsS u�rY�Zrbm �Z�qT Â

δ(t− t0) → ² → h(t, t0).

² w`T`| Tqw`}hS^³�rY��isr��0T`S Th��is|UTqmfThgUrbm �0�bcsSUg��ht<w`cs|¬e´wqmk³4rY�<�qcpmfT�¼hTqoZoq�`�qcsSUu`�<rY�Zrbm tZ�qTqmfu`l = × Th| Tqm �Z| tUÚrYTqm c���mUgb�y�h| uh��rYmUg´��t·Tqw`�q�h|¬g�r�� � �du`w`uhe{T�cke S ThgY�disµUu`z u`lvrYm �yrY�ZS }`|^m �ZrY�yz iso�mfT½�du`w`uhe{T�cs�\T`|fÚ�q�`� csmfThgYm �·rbmUg����qt

t = t0 � ckµ T`|^m }qmfThg\Tqw`���UX��®�qcpmfT�¼`oZ��m ikl � t �hT�g t0 =¸ m �ZS�ckg{zUg��ht¾w`ck|Ue�wqmk³4rY�~w`u`�¾mku���|UT`�q�`g��h�WrY�Zrbm �Z�qT�cke S T�gÌrbmfTZ¹\ck| � � ��ikµ u`z u`l � �qwq³4lNis��u`�Z�qctZz �¾T`S T`�\is| cfg � ckµ T`|^m }qmfThgÒ�q�ZS u�Tqwh��mfu�rYtZ�qT�m ��l·ckgxrY�`zUu`�¾�hThgÒ����gÒTqw`��m ��rbmUg��b�qt�cs�\T`| �qu¬�btZlmfu`� = ¸ m �·w`cs|¬e´wqmk³4rY�NT`��m t·ik��uh���qcPm �·rY��ikrY�

rYmkTq¹\cs| ��r���rYm ���qT Âδ(t− t0) → ² → h(t− t0).

× T`| Tqm �Z| u`�Z�qc��¬mUg»rYc�rbmfTZ¹jcs| �çrY�Zrbm �Z�qT¢�W��| u`�ZrbmUg��ht¢Tqw`�q�h|¬g�rY�çcke S ThgñrY�ZS }`|^m �ZrY�W�`��S uÞ�`g�Thl�qcpmfT�¼`oq��m tZl � �hThg�rY�ZS cpw`À4lP�d�h| u`�Zrbm^��t·Tqw`�q��|Ug�r��NzUe{z csmkT�gYTqw`��rY�ZS }`|^m �ZrY�ym ��l®�qu`| �\tZl

h(t) � rYcT`S^mUe�¹\ckrY�6�qc%m �0�bcsSUg��ht<w`ck|Ue�wqms³4r��vëö�qcpmfT�¼hTqoZoq�`�qcsSUu`�<rY�Zrbm tZ�qTqmfu`l�ï � �`w`u`�$�0�h|Uu`�ZrbmUg��ht$Tqw`�q�h|UgxrY�cke S Thg�rY�ZS }`|^m �ZrY��z �Zu��qcpmfT�¼`oZ��m^À4S � h(t, t0) � csµ T`| m }qmfThg¶zU�¬oqT`z t~�hThg�Tqw`��m ��rbmUg��b�qt�cs�\T`| �qu¬�btZlm �Zl®cfg�rY�hz u`�

t0 =¸ �Z�qckg�À4S u`�Z�qc½�¬mUg%u"u`|UgxrY�q�`l�m ��lN��| u`�ZrbmUg��htZl�Tqw`�`�h|UgxrY�Zl�w`| uO¡�w`uZ¹\ism ckg%��mUgÒm ��rbmUg �b�qt�cs�\T`|fÚ�qu¬�btZlym �ZldckgxrY�`z uh�

δ(t − t0) � mku�rY�Zrbm �Z�qT¾�Z| cs�qcfe � y(t0) = 0ë �Yg{TqmUeÌcke S T�g�T`w`T`| The�m ��m �¾T`��m t��

��w`�Z¹\ckrY�;ï =

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x(t)�Zw`u`| cke�SUT�T`S Tqoq�¬¹\cke�rYc�rbmfuhgx��ckg�À4z ckgxl·r���SUT`|^m tZrYckgxlyz isoZmkT ��qc�mfuZSnm ��w`u í ô = ² ikµ u`z uhl

y(t)cfe S Thg\�

y(t) = H [x(t)] = H∫ +∞

−∞x(λ)δ(t− λ)dλ,

Ý í

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�hThg\cs�\T`| �q�`� uZS^mfT`lm �·rY�ZS�¹jt¬���y�b| T`�q�`g��h��m ��mkThl � w`The{| S u`�Z�qc®m �·rY��isr��

y(t) =∫ +∞

−∞x(λ)H [δ(t− λ)] dλ.

© T`�¬¼h}`S u`�Z�qcPm^À4| T��¬w`��/��N�¬mUgH [δ(t− λ)] = h(t, λ),

ë í ÷qï�hThg\is��u`�Z�qcPm �·rY��isr��

y(t) =∫ +∞

−∞x(λ)h(t, λ)dλ,

ë í øZï�qw`u`�

h(t, λ)cfe S Thg\�d�h| uh��rYmUg´��tNTqwh�q�h|UgxrY� =

² �h| uh��rYmUg´��tÏT`w`�q�h|UgxrY�h(t, λ)

cke S ThgP� ikµ u`z u`lçmfu`�ÏrY�Zrbm tZ�qTqmfu`l"��mkT`S �ÏckexrYu`z u`lÛcke SUThg��rY�ZS }`| m ��r��

δ(t− λ) � �Nuqw`uhe{T~cs�\T`| �q�`�UcpmfThgYm �·rYmUg �b�qtt = λ

��ThgYmfu�rY�Zrbm �Z�qTN�Z| ck�qu`�ZrYcPcp�hcfe S �m �·rbmUg��b�qt = Ü ��m ��rY�Z�qThe S ckgY��mUg

h(t, λ) = 0�bg{T

t ≤ λ.ë í qï

¸ �ZS cpwqÀ4lu�m �¬w`uhl í øyw`The{| S cfgYm csoqg��h}�m �N�qu`| ��t

y(t) =∫ t

−∞x(λ)h(t, λ)dλ.

ë í �qïÜ �¬m �hl<cfe S Thg�u¼hT`r\g´���`l$m �¬w`uhl<rbm �ZS�T`S }qoq�ZrY�vw`ck|Ug�u���tZlß��| ��S uh� =»è m ��w`u`l$T`��m �`lßg�r����Zcfg`��w`��m �ZSw`| uO¡�w`�Z¹jcsrY����mUgÌ��cfe�rYuhz u`l

x(t)cs�\T`| �q�`�UcpmfThgÒr�cd��| ��SUu

t = −∞ �hT�gÌmfuWrY�Zrbm �Z�qT��Z| ck�qu`�ZrYcm ��m cdë¤z¬g���mUg�cke S Thg � rY�Z�q�Y³4SUT½�qc�m �ZSyg{zUg{�¬m ��mfT í � h(−∞, λ) = 0

�bg{Tλ > −∞ ï =

î cWm �Û¼hu`t¬¹\ckg{TÁT`��mku`� mfu`� m ��w`u`� �Zw`u`| u`�Z�qcOS TÁcp����| }`r�u`���`c�m �ZSÛisµ uhz uÁrYc�m �Z����SÞrYtZ�qTckgxrY�`zUu`�

x(t)r���SUT`|^m tZrYckgUm �Zl»�h| u`�ZrbmUg��htZl0Tqw`�q�h|¬g�rY�Zlòmfu`�PrY�Zrbm tZ�qTqmfu`l = R'w`u`�qikS^³4l0�ß�h| u`�ZrbmUg��ht

Tqw`�q��|Ug�r��¾cke S Thg��`g{T�¼hT`r�g��ht¾rY�ZS }`|^m �ZrY��w`u`�~w`cs|¬g �b| }`�\ckg�m �~rY�Z�Zw`cs|¬g��\u`| }�mfu`��rY�Zrbm tZ�qTqmfu`ln�Yg{Tm �Z��u`�ZrYT�ckexrYu`z u =è m �¬w`uhl í �dg�r�����cfg��Yg{TNm ����ckSUg��htyw`ck|Ue�wqmk³4rY�y�qcpmfT�¼hTqoZoq�`�qcsS uh�yr���rYm t��qT`mku`l = Ü SvmfuNrY�Zrbm �Z�qT

cke S ThgZrYmkTZ¹jcs| � � m ��m c»�ß��| u`�ZrbmUg��htPTqwh�q�h|UgxrY�h(t, λ)

cke SUThgZt�rY�ZS }h|^m �ZrY�ß�`g{T`lò�`cpmfT�¼`oZ��m tZlh(t−λ)�hThg\u�m �¬w`uhl í �dw`The{| S ckg�m �ZS�Tqwqoqu`��rYm cs| �N�qu`| �\t

y(t) =∫ t

−∞x(λ)h(t− λ)dλ.

ëóô1 qï

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t = t0 � �hT�g�mku�r���rYm ���qT½�Z| ck�qu`�ZrYcdëy(t0) = 0

ï = ²isµUu`z u`l»rY�Z�¬¼hu¬o`e��UcpmfThg^³4la(t−t0)

�hThgUuZS u`�q}`� csmfThghaq¡ ª + £j¥ V\]P_\� �pa`� ¥ a � �� Òê�¨`_ba�[p� ©ñë step response)

rYmkTq¹\cs| ��rY�Zrbm �Z�qT Âu(t− t0) → ² → a(t− t0).

ÝZô

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× T`| Tqm �Z| tZrYTqm c��mUg\��isµ uhz u`l�cfe S T�gjrY�ZS }`| m ��r��N�`g{T`l��q��SUu��qcpmfT�¼`oq��m t�l � a(t) � zUg{�¬mUg\mfu�rY�Zrbm �Z�qTcke S Thg�rYmkTq¹\cs| � =² rY�ZS }`|^m �ZrY�

a(t)cfe S Thg�cpwhexrY�Zl4�`g{Tß¼hT`r�g��htßrY�ZS }`| m ��r�� � �ßuqw`uhe{Tv��T`| Tq�hm ��|¬e��Uckg¬m �$rY�Z�Zw`ck|Ug��\u`| }

csSU�`lvrYmkTq¹\cs| uh�d�b| T`�q�`g��hu`��rY�Zrbm tZ�qTqmfu`l =

�=���&­ ®n¯@�N�D<°8��-:CZ>§R�¦:±�hA5�e?MA6Z>4=:C��� �N�hAh(t)

�DZ�!a(t)

�NA69>#n��:CZLª²�x46�M�³ 46Z>8�8�!��D�M���@?���:=�>8�Z\:C�>#

è g�r���SUT`|^m tZrYckgxlδ(t)

�hThgu(t)

rY�ZS zUisuZS^mkT�g��qcsmkThµ ��mfu`�Zly�qcmfu`�Zl�m ��w`u`�ZldÝZô�t�Ý�÷ = õ®T~¼`| uh���qcm^À4| T�wqÀ4lvrY�ZS z iku�S^mfThg��qcpmfT`µ �·mfu`�Zlvuhg�T`S^mUexrbmfuhg���cklTqw`uq�h|UexrYcfg�l

h(t)�hThg

a(t) =´ | �Zr�g{�quqw`uhgxu`�Z�qcPmfuZSnm ��w`u�ô1 d�Yg{T�m �ZS�ckexrYu`z u

x(t) = u(t)�hT�g�is��u`�Z�qc

a(t) =∫ t

−∞u(λ)h(t− λ)dλ =

∫ t

0h(t− λ)dλ

�Yg{Tt > 0.

µ }`SUu`�Z�qcPm^À4| T½m �ZS�TqoZoZT`��t½�qcpmfT�¼`oq�¬m tZlλ→ t− λ

��ThgYw`The{| SUu`�Z�qcPm csoqg��h}�m �·rY��isrY�

a(t) =∫ t

0h(λ)dλ

�Yg�Tt > 0.

ëöô�Ø ï

è T`S^mUexrbm | uh��u`lT`��mfu`�·mfu`�·m ��w`u`�·cfe S Thg\u�m �¬w`uhl

h(t) =da(t)

dt

�Yg{Tt > 0.

ëóôZÝ�ï¸ �Z�q�Y³4S TW�qcymkuZS�u`|UgxrY�q��mk³4S�r���SUT`|^m tZrYcp³4S

h(t)�hThg

a(t)g�rY���Zckg

h(t) = 0�hThg

a(t) = 0�bg{T

t < 0 =è gÌm ��w`uhg%ô�ØN�hT�g2ô�ݾcs�h��| }h� u`�ZS~m ��rY��isrY�¾�qcsmfT`µ �¾m �Zl·�h| u`�ZrbmUg��htZl½T`w`�q�h|UgxrY�Zl

h(t)�hThgÒm �Zl

Tqw`�q��|Ug�r���l��quZS T`zUg{Thexuh�O¼`tZ�qTqmfu`la(t) � �Yg{TÛrYmkTq¹\cs| �¢�b| T`�q�`g��h��rY�Zrbm �Z�qT = × T`|UTqm �Z| tZrYTqm c�m ��S

u`�quhg{��m ��mkT�mk³4S�rY��isrYcs³4SnT`��m^À4S��qcPmUg�lvrY��ikrYckgxlvÝZôy��Thg\Ý�÷·�qcsmkThµ �dmk³4Sδ(t)

�hT�gu(t) =

�=����¶ ¢¤£ �D¥�46Z>�D<�:C<�#¦�N§^9�H^��?y(t)

�e?MA6Z>4=:=�>��N!�:=<�#a(t)

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∫ +∞

−∞x(λ)h(t− λ)dλ.

º uvz ckµUg{�v�qipoqu`lñcke SUThg�isS Tvu¬oquq�`oqtZ|^³4�qTvzUg�wqoZÀ4rYcs³4lßëconvolution integral)

�hThg��Zw`uh| cke�S TP�b| T`�\cke³4l

y(t) =∫ +∞

−∞x(t− λ)h(λ)dλ.

Ý�÷

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ë Ü w`u`zUckg��h�ZcpmfThgÌThS½�h}`S uh���qcym �ZS½TqoZoqTq�bt��qcsmfT�¼`oZ��m tZlλ → t − λ

ï =�º u�m cpoqcs��mfThexu�T`��m ��uUoquq�qÚoqtZ|^³4�qT½mfu�u¬oquq�`oq�Z|^À4S u`�qc®�hT`m }�w`T`|U}q�bu�S^m ckl = õ®ismkuh���qc

dv = h(λ)dλ

�hThg\is��u`�Z�qc

y(t) =∫ +∞

−∞x(t− λ)dv = [x(t− λ)v(λ)]λ=+∞λ=−∞ +

∫ +∞

−∞v(λ)x(t− λ)dλ.

© T`�¬¼h}`S u`�Z�qcPm^À4| T��¬w`��/��·m �·rY��isr��·ô�Ø �v(λ) = a(λ) =

∫ λ

−∞h(ξ)dξ,

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h(t) = 0�Yg{T

t < 0ïN�hThg

��w`uZ¹\ismkuh���qcN��mUg%��ckexrYu`z uhlx(t)

ck��T`|U�q�`� cpmfThg2r�c·��| �ZS ut = −∞ �hT�g%mkuOr���rYm ���qT��Z| ck�qu`�ZrYc

m ��m c = R4w`ckg{z t½cke SUThga(−∞) = 0, x(t− λ)|λ=+∞ = 0,w`The{| S uh���qc®m �·r���ikrY�

y(t) =∫ +∞

−∞x(t− λ)a(λ)dλ.

R'�`m csoqu`�Z�qc»rYm �ßrY�ZS ik��ckg{T®m �ZS<TqoZoqTq�btP�qcpmfT�¼`oq��m t�lλ→ t−λ �hThg�oqT`�¬¼h}`SUu�S^mfT`l6cpwhg�wqoqisuZS$��w`�O/��

��mUgYcfe S Thga(t− λ) = 0

�Yg{Tt < λ,w`The{| S uh���qc®m �ZSnm cpo`g��htNrY��isrY�

y(t) =∫ t

−∞x(λ)a(t− λ)dλ.

ëóô Ã ï

è m �¬w`uhl�T`��m �`l~cfe S Thg4T`S^mUexrbmfuhgx��uhl~mkuh��m ��w`u`�Oô1 � ��Thg4cp�h�\| }`� cfg2m �ZS¾ikµ u`z uçckS �`l~rbmfTZ¹jcs| u`��b| T`�q�`g��hu`��r���rYm t��qT`mkuhlyrYcnm �Z����S�rYtZ�qT�cfg�r��`z u`�

x(t)rY�ZS T`| m t�r�ckg�m ��ldTqw`�`�h|UgxrY�ZlN�qu�S ThzUg{The�uh�

¼htZ�qTqmfu`l =2è m ��w`u`l4ô à cfe S ThgUTqwqoqu`�Zrbm cs|Uu`l'rYm ��S»ck��T`|U�qu¬��t<mfu`�<Tqw`�ßmfuZS»m ��w`u�ô1 � rbm �ZS4w`cs|Ue�wqmk³4rY�w`u`�N�NrY�ZS }`|^m �ZrY�x(t)

cfe S Thg\Tqwqoqu`�Zrbm cs| �·m �Zl®rY�ZS }`| m ��r���lx(t) =

× T`| Tqm �Z| tZrYTqm c�¬mUgj�¬mfT`S·cfe S Thg\�bS^³4rYm tN�½�h| u`�ZrbmUg��ht�Tqw`�q�h|¬g�rY�h(t) � m ��m cv�Zw`u`|UckejT`�qisrb³4lnS T

¼h| c±¹jcke\�hThg¶�~Tqw`�q�h|UgxrY���quZS T`zUg{Thexuh��¼htZ�qTqmfu`la(t) � �hThg¶T`S mUe�rbm |Uu`��T � �qcm �N¼hu`t¬¹\ckg{T�mk³4S·m �¬wq³4S

ô�Øt½ô�Ý =

ÝZø

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J·J ·¹¸ y�� ����¯Mç�s��ä �Pý~ä ��ån ~KNMO�y���~�~äÿ ¸ ����åy�¯M�ý~ä.�Þå��������º�ÿ�M¢KNMO�Pý������Pþÿ�

�\�L��� ®¨?���:C��8�ZM:CZ»��:CZ¹�\�����KZ¼<½�¯@�x��<½�"!$� 9�H^��?6¾z�N§69>H6�M?¿�-�D¥�4^B�À^�-:CZ�!8���HJ!�Z�¥~��46!������N§J�$��h�D<��>46ÁÂ:=<>#;:6B�§^�N�h#

Ü l<��w`uZ¹\ikrYu`�Z�qc0��mUgh�rY��isrY�vcfg�rY�`zUu`�UÚ¯ckµ �`z u`��csS �hl<��|UT`�q�`g��hu`��rY�Zrbm tZ�qTqmfu`lvëöcsS���ikS ckgh�qcsmfT�¼`T`oUÚoq�`�qcsSUu`�qï6cp�h�\| }`� csmkThg\Tqw`��m �·zUg{T`��uh|Ug��ht�csµ¬e�rb³4r��dw`|^À'm �ZlPm }`µUcp³4l

a(t)dy

dt+ b(t)y = x(t),

ëóô í ï

² �bcsSUg��htNoq�ZrY�ym �Zl®ckµUe�rY³4rY�Zl®T`��m tZl®z¬e�zUcpmfThg�Tqw`��m �·rY��isrY� à � �y(t) = e

−∫ t

t0

b(λ)a(λ)

{

y(t0) +∫ t

t0

x(ξ)

a(ξ)e∫ ξ

t0

b(λ)a(λ)

dλdξ

}

.ëóôZôZï

õ®Tv��| �Zr�g{�quqwhuhg{t�r�u`���`c»T`��m tßm �ßoq�ZrY�$�Yg{TS TP¼`| uh���qc4m �ZS6�h| u`�ZrbmUg��htPT`w`�q�h|UgxrY�Pmfu`��rY�Zrbm tZ�qTqmfu`l =õ®cs³4| u`�Z�qc�³4l"ckexrYu`z uÙm ��rY�ZS }h|^m ��r��Þz isoZmkT � cs�\T`| �qu`rY�qikS ��m ��rbmUg��b�qt

t = t0 � δ(t − t0) =�Pw`uZ¹jipmfu`�Z�qc0Tq�h�h�qTd��mUgqmfu�r���rYm ���qTn�Z| cs�qcfeqm �vrbmUg��b�qtt = t0 � y(t0) = 0 ë:¹j���q�¬¹\t�m c»mfu�SPu`|UgxrY�q�

m �Zl��h| u`�ZrbmUg��htZl�Tqw`�`�h|Ug�r���lÃ�{ï = Ü S�rbmfu�S�m ��w`uçôZô�¹\isrYuh���qcy(t0) = 0

�hThgx(t) = δ(t − t0) �is��u`�Z�qcPm �ZS�h|Uu`�ZrbmUg��htNT`w`�q�h|UgxrY�

h(t, t0) = e−∫ t

t0

b(λ)a(λ)

dλ ×∫ t

t0

δ(ξ − t0)

a(ξ)e∫ ξ

t0

b(λ)a(λ)

dλdξ,

�Yg{Tt ≥ t0.

ëóô�÷qï× T`| Tqm �Z| u`�Z�qc�m^À4| T½��mUg�mfu½u¬oquq�`oqtZ|^³4�qT�rbmfuNz ckµUg{���qipoqu`ldëö���bgbrbmfuZScs�q¹\ipm �qïñcfe S Thgbm �ZlP�qu`| �\t�l

G(t) =∫ t

t0F (ξ)δ(ξ − t0)dξ,

ëóô�øZï�qw`u`�

F (ξ) =1

a(ξ)e∫ ξ

t0

b(λ)a(λ)

dλ,

�hThg\rY�ZS cpwqÀ4l � r����q��³4S T��qc�m ��Syg{zUg{��m �¬mfT�Ø�� � G(t) = F (t0) �

G(t) =1

a(t0)× u(t− t0).

Q Ü | T � m cpo`g��h} � u½m ��w`u`lnô¬÷·w`The{| S ckg�m �·�qu`|U��t

h(t, t0) =1

a(t0)e−∫ t

t0

b(λ)a(λ)

dλ × u(t− t0).ëóô1Zï

Ý1

Page 29: simata kai systimata - users.auth.grusers.auth.gr/~hadjidem/simata kai systimata.pdf · @)A B C D E F G H I H A (TIME DOMAIN ANALYSIS) J LKNMO P Q R4SUTWVYXZV\[^]`_badcfe S ThgjikS

Ü S~mku"rY�Zrbm �Z�qT�cke S Thg2rbmfTZ¹\ck| � � uhg2rY�ZS^m cpoqckrbm isl a(t) b(t) rbm ��zUg{T`��u`|¬g´��tWckµUexrb³4rY��ô í cfe S ThgrbmfTZ¹\ck| uheb�hT�g��d��| u`�ZrbmUg��ht½Tqw`�q�h|UgxrY�½w`| uq�h��wqm ckgYm^À4|UT���mUgYcfe S Thg\�

h(t, t0) =1

a(t0)e−

ba(t−t0) × u(t− t0).

ëóô1�Zï× T`| Tqm �Z| tZrYTqm c���mUg�rYm ��Svw`cs|Ue�wqmk³4rY�drbmfTZ¹\cs|Uu`�yrY�Zrbm tZ�qTqmfu`l � �y�h| u`�ZrbmUg��ht·Tqw`�q�h|¬g�r��·csµ Th|^m }qmfThg�q�ZS uOT`w`�Om ��zUg{T`�\u`| }

(t − t0)�hThg2����g2��³4|Ugxrbm }OTqw`�Omfu

t�hThg%mfu

t0 = ¸ �ZS cpw`À4l½���h|Uu`�ZrbmUg��htTqw`�q��|Ug�r��½cke S Thg�r���SU}`|^m �ZrY�d�`g{T`l�q�ZS u�T`S ckµ }`|^m ��m �Zlv�qcpmfT�¼`oq��m t�l � mkuh� t �

h(t) =1

ae−

bat × u(t).

ëö÷1 qï

Ä�ÅjÆ�Å\ÍzSjÆ`ÓjÑÌÓÅÄÜ w`��mkuZS�m ��w`u~ôZô·cfe S Thg\�\T`S ck| �~��mUg�uhg\rY�ZS�¹\t��hckl®��| Th�q�`g��h�¬m ��mfT`l à Ýd��Thg ÃZà z ckSdg�r����Zu`�ZSyThS�m �rbmUg��b�qt

t0mfu�r���rYm ���qT½z csSy�Z| cs�qu`�ZrYc � y(t0) 6= 0 =Ä�ÅjÆ�Å\ÍzSjÆ`ÓjÑÌÓ¨Æ

è m ��w`u`l�ô1¢zUe S cfg»m �¢r����Zw`ck|Ug{��u`|U}Ûmfu`�çrY�Zrbm tZ�qTqmfu`l���mkThSWm �"rYmUg �b�qtt = t0

mfu`�çz uZ¹\cfeñ�`g{TÀ2¹\�ZrY� � �qisrb³Ám �Zl�rY�ZS }h|^m �ZrY�Zl

δ(t − t0)�hThg¶mfu¾rY�Zrbm �Z�qT~�Z| cs�quh��r�cvw`| u`���bu`�Z�qisS ³4l~ëöz ckSNcke���c

Tqw`uZ¹j�¬��cs�Z�qisS ��csS ik|^�bckg{Thï = Ü lN�¬w`uq¹\isrYuh���qc�m^À4| T���mUg¶mfu�r���rYm ���qT�z ckS�z ik��csmfThg��hT`�`e{TWcfe�rYu`zUu �x(t) = 0

�bg{T���}Z¹\ct � TqoZoq}�m �NrYmUg �b�qt

t = t0ckex��c®T`w`uZ¹\���hcs�Z�qikS �NckS is| ��cfg{T � �qc®Tqw`u¬m isoqcsrY�qT�SUT

��w`}`| ��ckgbcp�hcfe S �ym �yrbmUg����qty�`g{TNikµ u`z u`ly(t0) 6= 0 =2º ��m c���ikµ u`z uhl � �Yg{TN�h}Z¹\c

t ≥ t0 � zUe{z cpmfThgbTqw`�m �d�bcsS¬g´��t·oq�ZrY�Nô�ô � �hT�g�is��u`�Z�qc

h(t, t0) = y(t0) e−∫ t

t0

b(λ)a(λ)

dλ × u(t− t0).

Ü Snm ipoqu`lP¹jisrYu`�Z�qcy(t) =

1

a(t0),

is��u`�Z�qc$m �ZSP�h|Uu`�ZrbmUg��htyT`w`�q�h|UgxrY�·ô1 = µ TqmfTqoqt���uh���qcßcpw`uh�qisS^³4lPrbmfu·rY�Z�Zw`is| T`r��qTd��mUg�mfu·T`w`u¬m i±ÚoqcsrY�qT m �Zl�À4rYcs³4l � �Þu`w`uhe{TÙzUe{z csmkT�g<rYc�ikS TÏrY�Zrbm �Z�qTÞmfu u`w`uhe�uÁ�Z| cs�qcfe � Tqw`�Ïm �ÞrY�ZS }`|^m �ZrY�δ(t− t0) � cke SUThghmfude�z¬g�ud³4l<T`SPmfuyrY�Zrbm �Z�qTnz ckSPckex��c<z ck�h¹jcke`�hT`�`e{T·ckexrYu`z u � TqoZoZ}ymfuyrbmUg����qt

t = t0ckex��c®isµ uhz u~exrY�dw`|Uu`ly(t0) = 1/a(t0) =

�\�L��% ®¨?���:C��8�ZM:CZ»��:CZ¹�\�����KZ¼<½�¯@�x��<½�"!$� 9�H^��?6¾z�N§69>H6�M?¿�-�D¥�4^B�À^�-:CZ�!8���HJ!�Z�¥~��46!������N§J�$��h�D<�H^�q?�:6�N4^Z�#Ç:JB>§6�x�h#

�Pw`uZ¹jipmfu`�Z�qc'��mUg¬�ßrY��ikrY�$ckgxrY�`zUu`�UÚ¯ckµ �`z u`�PrYc4isS T®��|UT`�q�`g��h�vr���rYm ���`T�zUe�zUcpmfThg�Tqw`�v�`g{T®zUg{T`�\u`|Ug���tcsµ¬e�rb³4r��·z cs��m is| Thl®m }`µ cs³4lv�qcPrbmfTZ¹jcs| u`�Zl®r���S m cpoqcsrbm ikl �

1

ω2d2y

dt2+ y = x(t).

ë¤÷�Ø ïÝ1�

Page 30: simata kai systimata - users.auth.grusers.auth.gr/~hadjidem/simata kai systimata.pdf · @)A B C D E F G H I H A (TIME DOMAIN ANALYSIS) J LKNMO P Q R4SUTWVYXZV\[^]`_badcfe S ThgjikS

² �bcsSUg��htNoq�ZrY�dcfe S Thg�mku�}Z¹\| u�g�rY�qT�m �Zl®��ckSUg��htZlvoq��r���lPm �Zl®u`�qu¬�bckS u`�Zl �1

ω2d2y

dt2+ y = 0,

ëö÷ZÝZï�Nuqw`uhe{T~cfe S Thg\�

y(t) = A cos(ωt) + B sin(ωt),ë¤÷ à ï

�hThg¬�`g{T`l»�qck|Ug��htZl4oq��r�cp³4l2m �Zl'wqoqt�|Uu`�Zl4csµ¬e�rb³4r���l4÷�Ø = ² oq��r��6Th�¬m t$ckµ T`|^m }qmfThg¬Tqw`�Pm �$rY�ZS }`|^m �ZrY�x(t) =õ®Tß¼h| u`�Z�qc'm �ZSò�h| u`�ZrbmUg��ht�T`w`�q�h|UgxrY�PT`��mfu`�$mfu`�ßr���rYm t��qT`mkuhl � z ��oqT`z t$m �$oq�ZrY�<m ��l»zUg{T`�\u`|Ug���t�l

csµ¬e�rb³4r���l½÷�Ø � �¬mfT`S¾��ckexrYu`z u`lx(t)

cfe S T�g2��rY�ZS }`| m ��r���z ipoZmfT � x(t) = δ(t)�hT�g

y(t) = 0�Yg�T

t ≤ 0 = ë × T`| Tqm �Z| tZrYTqm cN��mUgÌw`tZ| T`�qct0 = 0 � �Yg{TqmUe ;) ¸ �ZS cpwqÀ4l � ��zUg{T`��uh|Ug´��tWcsµUexrb³4rY�¾w`uh�¾¹\T

z �`r�ckgbm �ZSn�h| u`�ZrbmUg��ht½Tqw`�q��|Ug�r��½cke S Thg��1

ω2d2y

dt2+ y = δ(t).

ë¤÷ í ïõ®isoZuh���qc$S T�¼h| u`�Z�qc6m �noZ�ZrY�nT`��m tZl<m �ZlßzUg{T`��uh|Ug´��tZlPcsµ¬e�rb³4r���l$�Yg�T

t > 0 = R»e S Thg���ThS cs| �N��mUgh�Yg{Tt < 0

�½oq�ZrY��cke SUThg¶exrY�½wh| u`ly�q�Zz isS =�º u¾z ckµUg{�¾�qisoZuhlnm ��lnwqoqtZ| u`�Zl�csµUexrb³4rY�Zln÷�Ø·g�rYuh�¬mfThgjw`| u`l�q�Zz isSN�Yg{T

t > 0 � �hT�g�rY�ZS cswqÀ4ly�~oZ�ZrY��m ��ly�Yg�Tt > 0

cke S ThgÌ��oq�ZrY��m �Zlyuh�qu¬��ckS u`�Zl·ckµUexrb³4rY�Zl÷qÝ � �·uqw`u�e�T�cke S Thg��

h(t) = (A cos(ωt) + B sin(ωt))× u(t),�qw`u`�A � B rbmfTZ¹\ck| isl0w`u`�®¹jTnw`| u`rYzUgxu`|Ugxr�¹\uh��SPT`w`��mUg�l<Th| �bg´��islßrY�ZS�¹\t��hcsl = ² oq�ZrY�®Th�¬m tvw`| ipw`cfg

S TWcpw`Tqoq�¬¹\cs�Zcfg�cp��mfT`��m ��m �¬mfu`l·m ��zUg{T`�\u`|Ug���t�csµUexrb³4rY�¾÷ í = Ü S mUg´��TZ¹jgxrbmfu`�Z�qcy�hThgÌoqT`�¬¼h}`S uZS^mfT`l��w`�O/��ç�¬mUg

du/dt = δ(t) � w`The{| S u`�Z�qc � oq�¬��³ÿ��Thgñm ��l�g{zUg{�¬m ��mfT`l�Ý í �Yg{TÛm �ZS�w`T`| T`��À4�bgxrY� � m �rY��isr��

A

ω2δ(t) +

B

ωδ(t) ≡ δ(t).

Ü w`�m ��S<m cpoqcs��mfThe{TTh�¬m t®rY��isz ��w`|Uuq�h��wqm ckg � csw`ckg{z tPmfTδ(t)

�hT�gδ(t)

cke S T�gqT`S ckµ }`|^m ��mfuhgZm cpoqckrbm isl ���mUgA = 0 � B = ω � �hThg�m cpo`g��h}��·�h| u`�ZrbmUg��ht½Tqw`�q�h|¬g�rY�½mkuh�·r���rYm t��qT`mkuhlP÷�Øcke S Thg��

h(t) = ω sin(ωt)× u(t).ë¤÷qôZï

�\�L�&� �;��9\�D4J!$�D<a8���A6Z>HJ!KZ��$��?Ȭ\��8�Z\:C�>#��@:CZLª��N46�M� ³ 46Z>8�8�!��D�M��e?M�@:C��8�ZM:±��#

�Pw`uZ¹jipmfu`�Z�qc��¬mUg6�Ûckµ ipo`g{µ �ÞcsSU�`l�r���rYm t��qT`mkuhl¾�hTZ¹ju`|Ue{� cpmfThg$Tqw`� zUg{T`�\u`|Ug���t�ckµUexrb³4rY�çw`|^À'm �Zlm }`µ cs³4l � m �Zlß�qu`| �\t�lPô í � �qw`u`�duhg�w`Th| }`�qcpm |Uuhg

a�hThg

bcke S Thg�rbmfTZ¹\ck| isl = õ®cp³»| u`�Z�qc$�¬mUg��ycke�r�u`z u`l

cke S Thg��x(t) = u(t)

��Thg���mUgy(0) = 0

�hThg\�d�bcsS¬g´��t·oq�ZrY�Nô�ô·zUe S cfg

a(t) = e−bat ×

∫ t

0

u(ξ)

aebaξdξ,

�Yg{Tt > 0.

Ã

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© T`�¬¼h}`S uZS^mfT`l��w`�O/��N��mUgYcfe S Thgu(ξ) = 0

�Yg�Tξ < 0

�hThgu(ξ) = 1

�bg{Tξ > 0 � is��u`�Z�qc

a(t) =1

ae−

bat ×

∫ t

0ebaξdξ,

�Yg{Tt > 0,

�hThg�m cpo`g��h}��NTqw`�q�h|¬g�rY���qu�S ThzUg{Thexu`�·¼htZ�qTqmfu`lvcke S Thg��

a(t) =1

b

[

1− e−bat]

× u(t).ëö÷�÷qï

² ��| u`�ZrbmUg��ht¢T`w`�q�h|UgxrY�h(t)

�hThgñ�"T`w`�q�h|UgxrY�ç�quZS T`zUg{Thexuh�O¼htZ�qTqmfu`la(t)

��T`| Tq�`m �Z|Ue{� uh�ZSWm �rY�Z�Zw`ck|Ug��\u`| }�mfu`�·�b| T`�q�`g��hu`��rbmfTZ¹\cs|Uu`�·rY�Zrbm tZ�qTqmfu`l � �qw`uh�Nik��u`�Z�qcP�`e{T�ckexrYu`zUu��hThg\�`e{T�isµ u`zUu =è gßrY�ZS T`| m t�r�ckgxl

h(t)�hT�g

a(t) � �qwq³»l¢zUe SUu�S^mfThgPTqw`�Ámfu`�ZlOm ��w`u`�Zl¢÷1 ��Thg�÷Z÷ � T`S^mUexrbmfuhgx��T �rY�ZS z iku�S mkT�gh�qcpmfT`µU�vmfu`�Zl$�qc0mfu`�Zl<�bcsSUg��hu`�Zlßm ��w`u`�ZlßôhØß�hThg�ô�Ý = Ü l$��w`uZ¹\ikrYu`�Z�qc6�¬mUgh�bS^³4|Ue{� u`�Z�qcm �ZSPTqw`�q�h|UgxrY�y�qu�SUT`zUg{Thexu`�¼htZ�qTqmfu`l

a(t)�hThgh¹\isoZuh���qc6S Tn¼h| uh���qc6m �ZSß��| u`�ZrbmUg��ht�Tqw`�q�h|¬g�rY�

h(t) =R»e S Thgh(t) = da/dt

�hThg\rY�ZS cpwqÀ4l

h(t) =1

ae−

batu(t) +

1

b

[

1− e−bat]

u(t).

R»e S Thg\�`�Z³4lu(t) = δ(t),

�hThgG(t)δ(t) = G(0)δ(t).Q Ü | T�m cso`g´��} � �·�h| u`�ZrbmUg��ht½Tqw`�q�h|¬g�rY��cke SUThg��

h(t) =1

ae−

bat × u(t),

z ��oqT`z ty¼h|Ue�rY�hu`�Z�qcPm �·rY�ZS }`|^m �ZrY�d÷0 =_ P ÑRQ\Ó�Ñ�ÓÉ | cke�m cPm ��S

a(t)T`w`��m �ZS

h(t) =

Jvà Ê å ~K½MW�y����� ¸ ����å��¯MWý ����ç ¸ ����å��¯MWý �¾ ½�s�y� Â˯ ~K� �~����� �ä ýN��æñ���nåy���º�ÿ� ��Ky���~þ �¾����þÿ�

�6%���� �°���>�>�h8�ZRC

õ®cs³4| u`�Z�qcnmfu¾�h���`oZ³4�qTRC

mfu`��rY��tZ�qTqmfu`l�Ø^÷ � �qw`u`��³4lycfe�r�u`z u�u`|Ue{� u`�Z�qcym ��SNm }`rY�x(t)

�hT�g³4lvisµUu`z u�m ��Snm }`rY�

y(t) = ² rY��ikrY�dcfg�r��`z u`�dÚñcsµ �hz u`�½zUe{z cpmfThg�T`w`��m �·zUg{T`�\u`|Ug��ht�ckµUe�rY³4rY�

RCdy

dt+ y = x(t).

à Ø

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++

_ _

x(t) y(t)i(t)

R

C

¸ ��tZ�qT�Ø^÷� º u��h���`oZ³4�qTRC

µ | uh��rYmUg´��t·T`w`�q�h|UgxrY�² ��| u`�ZrbmUg��ht�Tqw`�`�h|UgxrY��cs�Z|Uexrb�hcpmfThgÒTqw`��mfu�S�m ��w`u�÷0 ¾T`S�oq}�¼hu`�Z�qcd��w`�O/\�¾��mUg

a(t) = RC�hT�g

b(t) = 1 = Q R»��uh���qch(t) =

1

RCe−t/RC × u(t).

ëö÷ZøZï

) ckSUg��ht·oq�ZrY�d�bg{T�m ����u`�ZrYT�cfe�r�u`z ux(t)

Ì ��mfu`�Z�qc6m �n�bcsSUg��htyoq��r��n��w`uZ¹\ismkuZS^mfT`l���mUg��yckexrYu`z u`lx(t)

cs��Th| �q�`� csmkT�g�rYc$��| ��SUut = −∞ �hThg

mfu�rY�Zrbm �Z�qT½��|Ucs�qu`�ZrYc�m ��m c � y(−∞) = 0 = R4�\T`| �q�`�Uu�S^mfT`lnmkuZSnm ��w`u�ô1 Nis��u`�Z�qc

y(t) =1

RC

∫ t

−∞x(λ)e−(t−λ)/RCdλ.

ëö÷0Zï

Ü w`�q�h|UgxrY�½�qu�SUT`zUg{Thexuh�d¼htZ�qTqmfu`l² Tqw`�q��|Ug�r����quZS T`zUg{Thexuh�½¼ht��qT`mku`l�cs�Z|Uexrb�hcpmfThgjTqw`��mfu�Sdm ��w`u~÷0�T`Sy¹\isr�u`�Z�qc

x(t) = u(t)ë¤oqT`�¬Ú

¼h}`S uZS^mfT`lcswhebwqoqisuZS���w`�O/\�·��mUgYmku��h}qmk³Ù�`|Ug�u�mfu`�Nu¬oquq�`oq�Z|^À4�qTqmfu`lcfe S T�g\e�r�u½w`| uhl0ï = Q R4��u`�Z�qc

a(t) =1

RC

∫ t

0u(λ)e−(t−λ)/RCdλ,

�bg{Tt > 0

�hThg�m cpo`g��h} � cpw`cfg�z t½cke SUThgu(λ) = 1

�bg{Tλ > 0 �

a(t) =(

1− e−t/RC)

× u(t).ëö÷1�qï

× T`| Tqm tZ| �ZrY�è gbm ��w`uhg�÷Zød�hThg\÷1�NrY�ZS z isuZS^mfThgY�qcsmfT`µ �·mkuh��lv�qc�mUgxlvrY��ikrYckgxlvô�Øv�hT�g\ôZÝ =�6%���% �°���>�>�h8�Z

LC

õ®cs³4| u`�Z�qc$mfu·�h���`oZ³4�qTLC

mfu`� ¸ ��t��qT`mkuhl�Ø ø = ² zUg{T`��u`|¬g´��tNcsµUexrb³4rY��mfu`�����¬�ho�À4�qT`mkuhlPTh�¬mfu`�cke S Thg��

LCd2y

dt2+ y = x(t),

à Ý

Page 33: simata kai systimata - users.auth.grusers.auth.gr/~hadjidem/simata kai systimata.pdf · @)A B C D E F G H I H A (TIME DOMAIN ANALYSIS) J LKNMO P Q R4SUTWVYXZV\[^]`_badcfe S ThgjikS

++

_ _

x(t) y(t)C

L

¸ ��tZ�qT�Ø ø� º u��h���`oZ³4�qTLC

�Nuqw`uhe{T~cfe S Thg�m �Zl®�qu`| �\t�l÷�Ø � �Yg{T ω2 = 1/LC = R4w`u`�qisS^³4l � �d��| u`�ZrbmUg��htNT`w`�q�h|UgxrY�½cfe S T�g��

h(t) =1√LC

sint√LC

× u(t).ëóøÃ qï

J®ù ·¹¸ y�� ����¯Mç�s��ä �Pý�ä æ�b �s�n ~K �Ûå¾�������h� � M¢KNMO�Pý������Pþÿ�

õ®T"�¬w`uq¹\isr�u`�Z�qc·csz À �¬mUgÒ�bS^³4|¬e��Uu`�Z�qcNm �ZS��h|Uu`�ZrbmUg��ht�Tqw`�`�h|UgxrY� � t�m �ZS~Tqw`�`�h|UgxrY�W�quZS T`zUg{Thexuh�¼htZ�qTqmfu`l � ckS �`lñ��|UT`�q�`g��hu`�®rY�Zrbm tZ�qTqmfu`l'�hT�g�� ��mfu`�Z�qc'm ��S6isµUu`z uvr�c2m �Z��u`�ZrYT®r���SU}`|^m �ZrY�$ckgxrY�`zUu`�x(t) =õ®T"��| �Zr�g{�quqw`uhg{tZrYu`�Z�qc½mkuO�h���`oZ³4�qT

RCmfu`� ¸ ��tZ�qTqmfu`l�Ø^÷�³4l½w`Th| }`z ckg��b�qT = ² �h|Uu`��rYmUg��ht

Tqw`�q��|Ug�r��Nmfu`�NrY�Zrbm tZ�qTqmfu`lPT`��mfu`�Ncfe S Thg\�

h(t) =1

RCe−t/RC × u(t),

�hThg\�·T`w`�q�h|UgxrY�½�quZS T`zUg{Thexuh�·¼ht��qT`mkuhlv�

a(t) =(

1− e−t/RC)

× u(t).

õ®TN¼h| u`�Z�qc�m �ZS�isµ u`z u~rYcPT`��m ��mku��h���`oZ³4�qT��Yg{T�zUg{}`�\u`| cslw`ck|Ug´w`m^À4rYckgxlvckgxrY�`zUu`� =Ä�ÅjÆ�Ô�Ç�È0T´Ê%ËÌÅÅIJ ckexrYu`z uhl$cke S Thg��

r(t)ëunit ramp),

�uqw`u�e�T·zUe{z csmkThg�rbmfu ¸ ��tZ�qTdÝ = ² cfe�r�u`z u`lr(t)

cp�h�\| }`� csmkThgrY�ZS T`| m t�r�ckg�m �

u(t)³4l

r(t) = tu(t),�hThg�cpw`ckg{z t~��w`T`| }q��³'�b�`ldm �Zl�cfe S Thg¶T`wqoZuh��rYm cs| ��m �Zlr(t) � ¹\T¾��| �Zr�g{�quqw`uhg{tZrYu`�Z�qcnmkuZS·m ��w`u�ô Ã

�Yg{T�S TN¼h| u`�Z�qc�m �ZS�isµ uhz u � T`S^mUeYmkuh�dm ��w`u`�½ô1 = Q R4��u`�Z�qcr(t) = u(t) + tu(t)

= u(t) + tδ(t)= u(t) + 0δ(t)= u(t),

ÃZÃ

Page 34: simata kai systimata - users.auth.grusers.auth.gr/~hadjidem/simata kai systimata.pdf · @)A B C D E F G H I H A (TIME DOMAIN ANALYSIS) J LKNMO P Q R4SUTWVYXZV\[^]`_badcfe S ThgjikS

�hThg\rY�ZS cpwqÀ4ly(t) =

∫ t

−∞u(λ)a(t− λ)dλ =

∫ t

0a(λ)dλ.

´ | �Zr�g{�quqw`uhgxu`�ZS^m cslmkuZSnm ��w`u�÷0�·�bg{T�m ��S�Tqw`�q��|Ug�r��½�qu�SUT`zUg{Thexuh�d¼htZ�qTqmfu`la(t) � ik��uh���qc

y(t) =∫ t

0

(

1− e−λ/RC)

dλ× u(t) = r(t)−RC(

1− e−t/RC)

× u(t).

Ä�ÅjÆ�Ô�Ç�È0T´Ê%ËÌŨƲ ckexrYu`z uhlcke SUThg�u�m |Ug���³4S¬g´���`lw`Tqoq�q�`l

x∆(t) � u�uqw`u�e�u`l�zUe{z cpmfThg�rYmku ¸ ��tZ�qT�� =J� rY���ZckgY�·r���ikrY�

x∆(t) = r(t)− 2r(t− 1) + r(t− 2),

�hT�g�rY�ZS cpw`À4l®�·ikµ u`z uhl � oq�¬��³ m �Zl®��|UT`�q�`g��h�¬m ��mfT`lmfu`�d���¬�ho�À4�qT`mkuhl � zUe{z csmfThg�Tqw`��m �·rY��isrY�

H[x∆(t)] = H[r(t)]− 2H[r(t− 1)] +H[r(t− 2)].

î c0m �P¼`uhtU¹jckg{TyT`��m tZl6m �Zl<rY��isrY�Zl0�hT�ghm ��l<ckµ �`z u`�y(t) = H[r(t)]

w`u`�®¼`|Ut¬��T`�qc0w`| u`���bu`�Z�qisS�Ú³4l � �Zw`u`| u`�Z�qc®cs���hu¬oqT�SUTN¼`| uh���qcPm ��S�ikµ u`z u~rbm �ZSnckexrYu`z u

x∆(t) =_ P ÑRQ\Ó�Ñ�Ó�Pw`u¬oqu¬�Ye�rYm c<m �ZS®isµUu`z u·mfu`�n���¬�ho�À4�qT`mkuhlnØ^÷��qc$ckexrYu`z u·mfuZS®m |Ug ��³4SUg��h�Nw`Tqoq�q�·mfu`� ¸ ��tZ�qTqmfu`lD� ��Yg{T½m �ZSvmUg{�qt

RC = 1�hThg

RC = 2 = R'whexrY�Zl��Yg{T½m �ZSmUg��qtRC = 0.01 ='º gbw`T`| Tqm �Z| cfe m c��Yg{T½m �ZS

m cpoqck�¬mfThe{T�w`cs|Ue�wqmk³4rY� � �qw`u`�NmfuRC

cke SUThg��`g��h| � � r�c�r��¬���h|¬g�rY�N�qc�mUgxlvz �Zu�}qoZoqcsl®w`ck|Ug�wqm^À4rYcfg�l;

Ä�ÅjÆ�Ô�Ç�È0T´Ê%ËÌŨͲ ckexrYu`z uhlcke SUThg�u�u`|�¹\u¬��À4SUgxu`lw`Tqoq�q�`l

Π(

t

)

=

1�Yg{T − ε < t < ε

0�Yg{T |t| > ε.

× T`| Tqm �Z| u`�Z�qc®��mUgYu�u`|^¹\u¬��À4SUg�uhlw`Tqoq�q�`lΠ(

t2ε

) T`S Tqoq�ZcpmfThg�r�c�r���SUT`|^m tZrYckgxlu(t)

³4l

Π(

t

)

= u(t+ ε)− u(t− ε)

�hThg\rY�ZS cpwqÀ4lv�·ikµ u`z uhlcke S Thg��

H[

Π(

t2ε

)]

= H[u(t+ ε)]−H[u(t− ε)]

= a(t+ ε)]− a(t− ε).

à í

Page 35: simata kai systimata - users.auth.grusers.auth.gr/~hadjidem/simata kai systimata.pdf · @)A B C D E F G H I H A (TIME DOMAIN ANALYSIS) J LKNMO P Q R4SUTWVYXZV\[^]`_badcfe S ThgjikS

² Tqw`�q�h|¬g�rY���qu�S ThzUg{Thexu`�·¼htZ�qTqmfu`lvzUe{z cpmfThg\Tqw`��m �·r���ikrY�·÷1�·�hThg\is��u`�Z�qc

y(t) =(

1− e−t+εRC

)

u(t+ ε)−(

1− e−t−εRC

)

u(t− ε).

× T`| Tqm �Z| u`�Z�qc®��mUgY�NisµUu`z u`lncke S T�g\e�r��dw`| u`l�q�Zz ikSn�bg{Tt < −ε �hT�g\exrY�dw`| uhl

y(t) = e−t/RC[

eε/RC − e−ε/RC]

, t > ε.

) g{T��`g��h| �εik��uh���qcPm �ZSnw`| u`rYcs���YgxrbmUg��ht½rY��isrY�

y(t) = e−t/RC × 2ε

RC+

�`| uhg�m }`µ cs³4lε2, t > ε.

Ä�ÅjÆ�Å\ÍzSjÆ`ÓjÑÌÓÜ w`u`z cke{µ^m c��¬mUgj��mfT`S

ε → 0�hThg���cfe�r�u`z u`lycke SUThg�u�u`|^¹\u¬��À4SUg�uhl�w`Tqoq�q�`l 1

2εΠ t2ε� ��isµ u`zUu`l�m cke S ckg

w`| u`lvm �ZSn�h| u`�ZrbmUg��ht½Tqw`�q�h|¬g�rY��÷Zø =P ÑRQNSjÑ2È0TVU_ P ÑRQxS�Ñ�ÓÎĸ mfu�rY�Zrbm �Z�qT�mku`� ¸ ��tZ�qTqmfu`l·��ckexrYu`z uhl·cke S Thg��~m }`r��

x(t)�hThg��isµUu`z u`lßcke S Thg��vm }`rY�y(t) = ² rY��isrY�vcfg�rY�hz u`�®ÚÒckµ �`z uh�

zUe{z csmkT�g�Tqw`��m �·z¬g�Th��u`|Ug���t�csµUexrb³4rY�dy

dt+R

Ly = x(t),

�qw`u`��u�gÌwhT`| }`�qcsm | uhgR, L, C

cke S T�g%rbmfTZ¹jcs| ikl = Ü w`u`z cfe{µ^m c��mUgY�·�h| u`�ZrbmUg��ht½Tqw`�q�h|¬g�rY��cke SUThg��h(t) = δ(t)− R

Le−(R/L)t × u(t) =

++

_ _

i(t)

R

x(t) L y(t)

_ P ÑRQxS�Ñ�ÓÏƸ mfu�rY�Zrbm �Z�qT�mku`� ¸ ��tZ�qTqmfu`l·��ckexrYu`z uhl·cke S Thg��~m }`r��

x(t)�hThg$�Þisµ uhz u`lOcke S Thg<�Ûm }hrY�y(t) = É |Ucke�m c�m �ZSO�h|Uu`�ZrbmUg��ht

Tqw`�q��|Ug�r��Omkuh�WrY�Zrbm tZ�qTqmfu`l � �hTZ¹�À»l~cpwhexrY�Zl���Thg'm �ZS¾ikµ u`z urbm �ZSncfe�rYuhz u

r(t)�hThg\rbm �ZSnckexrYu`z u

Π( t−12) =

++

_ _

x(t) y(t)

R

CR

1

2

J ' û KNMW�n��� æ$� �Ûån��������º�ÿ� M"KNMO�Pý�������þÿ�

�Pw`}`|U��uh��S�w`u¬oZoZu�eZm |U�qw`uhgq�Yg{T�S T�uh|Ue�r�u`���`cñm �ZSßcs�Zrbm }Z¹jckg{TncsS �hl6rY�Zrbm tZ�qTqmfu`l = R4z^À"¹\TnzU�`rYu`�Z�qcmfu�SyTq�h�¬oqu`�¬¹\u~u`|UgxrY�q�YÂ

à ô

Page 36: simata kai systimata - users.auth.grusers.auth.gr/~hadjidem/simata kai systimata.pdf · @)A B C D E F G H I H A (TIME DOMAIN ANALYSIS) J LKNMO P Q R4SUTWVYXZV\[^]`_badcfe S ThgjikS

Ð ÆvT{Ñ2ËCÑ"U ÂeÒ Ó��pa¾VYXZV\[^]`_Ya �s� �sa ¥¶�  �V\[paO�Z�s©naq�Ô+`a ¥ _ ª �f�daq�·V � +`��� � ¡ �¤£ a�[ °Ì_��^�¬] �s� V�� � �daq�¬[ ¥ VIÕ[p� ¥ ¦ �s� ¡ �¤£ a�[±°Ì_��^�¬]½�s�p� �U� © = Q R4S TNm ismku�g�uNr���rYm ���qTduZS u`�q}`�UcpmfThgBIBO,

Tqw`�NmfT·Th| �bg´��}Bounded

Input Bounded Output.õ®Tn�qcsoZcsm tZrYu`�Z�qc4mUgxl0rY�ZS�¹jt¬��cslñÀ4rYm cñisS Tn��|UT`�q�`g��h��rYmkTq¹\cs| �nrY�Zrbm �Z�qTvS Tncfe S Thg`ck��rYmkTq¹\isl � ��Tqm }mfu�SyT`S^³'m ik|^³Ïu`|¬g�rY�q� =² rY��isrY�·ckgxrY�`zUu`�UÚ¯ckµ �`z u`�½z¬e�zUcpmfThg�Tqw`��m �d�bcsSUg��ht½rY��ikrY�·ô1 �

y(t) =∫ +∞

−∞x(λ)h(t− λ)dλ,

�qw`u`�h(t−λ) cke S Thg¬�6��| u`�ZrbmUg��htßTqw`�q�h|¬g�rY�ßmfu`�$rbmfTZ¹\ck| u`�<r���rYm t��qT`mkuhl = × The{| SUu`���`c'mUg�l4T`w`�¬oZ��m ckl

mUg{�qisl®mk³4S�z �Zu��qcsoZÀ4S �|y(t)| =

∫ +∞

−∞x(λ)h(t− λ)dλ

,

t|y(t)| ≤

∫ +∞

−∞|x(λ)| |h(t− λ)| dλ.

² ckexrYu`z uhlx(t)

cfe S Thg�w`cs| Tqmk³4�qikS � �|x(λ)| ≤M ≤ +∞,

�hThg\�dw`Th| Tqw`}`S^³Ár���isrY�·zUe S cfg

|y(t)| ≤M∫ +∞

−∞|h(t− λ)| dλ,

t|y(t)| ≤M

∫ +∞

−∞|h(λ)| dλ.

Ü w`��m ��Snm csoZck�¬mfThe{T�T`��m t·rY��ikrY�y��TqmfTqoqt¬�bu`�Z�qcvrbmfu½r����Zwhis| T`r��qT½��mUg ÂÖ +`aq�¬¨"V¶ ¬�E�`¨I+�] i�¥ aç�sa �s� �pa ¥ �^�sa i�£ a�_j_ ¥ + ª Vj[±a�� �ö£Yª VYXZV\[^]`_Ya �  �V\[paO�Z�k©p«\+`a�[f�"[p�^��aq�s°ñ[f� £ °� £�¥ V�_ ª « �s� �pa ¥ [p��� ���E+f�j¨ £ °Ì_ba

∫ +∞

−∞|h(λ)| dλ

�sa �s� �sa ¥ ¡ �ö£ a�[±°�_�� �k�Ø×× T`| Tqm �Z| u`�Z�qc���mUg�����| u`�ZrbmUg��htdT`w`�q�h|UgxrY�dw`cs|¬g �b| }`�\ckg�wqoqtZ|^³4l�m �yrY�Z�Zw`cs|Ug{��uh| }·mfu`�drY�Zrbm tZ�qT�Ú

mfu`l®³4l®w`|Uu`l®m �ZS�cs�Zrbm }Z¹jckg{}½mfu`� =Ä�ÅjÆ�Ô�Ç�È0T´Ê%ËÌÅ) g{T�mfu�r���rYm ���qTNmfu`� ¸ ��tZ�qTqmfu`l·Ø ø � �qc��h| uh��rYmUg´��tNTqw`�`�h|UgxrY�

h(t) =1

RCe−t/RC × u(t),

à ÷

Page 37: simata kai systimata - users.auth.grusers.auth.gr/~hadjidem/simata kai systimata.pdf · @)A B C D E F G H I H A (TIME DOMAIN ANALYSIS) J LKNMO P Q R4SUTWVYXZV\[^]`_badcfe S ThgjikS

gxrY����cfg∫ +∞

−∞|h(λ)| dλ = 1

RC

∫ ∞

0e−λ/RCdλ× u(t) = 1,

�hThg\rY�ZS cpwqÀ4l®mfu�rY�Zrbm �Z�qT½cke S T�g�cs�ZrbmfTZ¹jisl =

à ø

Page 38: simata kai systimata - users.auth.grusers.auth.gr/~hadjidem/simata kai systimata.pdf · @)A B C D E F G H I H A (TIME DOMAIN ANALYSIS) J LKNMO P Q R4SUTWVYXZV\[^]`_badcfe S ThgjikS

> ? > @)A B C D E F GPH I C B B A I ? H Ù C Ù Ú ?

(FREQUENCY DOMAIN ANALYSIS)

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Laplace .

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J@` � �ß ����"æ>˱ ¸ � �Pý �Ïæ6þÿå�Ë �Pþÿ� Mçæ$� å~� �Fourier

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rYcfg�|U}Fourier,

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x(t) = a0 +∞∑

n=1

an cos(nω0t) +∞∑

n=1

bn sin(nω0t),ëóø�Ø^ï

�qw`u`�ω0 =

T0,

cke S Thg��y¼hT`r�g��htNr�����S ��m ��mfTNmfu`�NrYtZ�qTqmfu`l �a0 =

1

T0

∫ T0

0x(t)dt

ëóø�ÝZïÃ

Page 39: simata kai systimata - users.auth.grusers.auth.gr/~hadjidem/simata kai systimata.pdf · @)A B C D E F G H I H A (TIME DOMAIN ANALYSIS) J LKNMO P Q R4SUTWVYXZV\[^]`_badcfe S ThgjikS

cke S Thg��N�qisr��ymUg{�qt·mkuh�·rYtZ�qTqmfu`l®�Yg{T��`g{T�w`cs|¬e�u`z u � �hThg

am =2

T0

∫ T0

0x(t) cos(mω0t)dt, bm =

2

T0

∫ T0

0x(t) sin(mω0t)dt,

ëóø à ïuhgjrY�ZS^m csoZckrbm isl®mfu`��T`S Tqwqm ���b�qTqmfu`l

Fourier. ² r���S }h|^m �ZrY�x(t)

¹jcp³4| cfe mfThg\��mUg\cfe S Thg�rY�ZS cs��tZl ��qcPrY�ZS ck��tdw`|^À'm �d�hT�g�z ck�¬m ck| �dw`T`| }`��³'�bu � tNis��ckgYw`cpw`ck| T`rY�qikS u�wqoqtU¹ju`l®ThrY�ZS cs��ckg�À4S =Ü S·��rY�ZS }`|^m �ZrY�

x(t)cke S T�g�rY�Z�q�qcsm |Ug��ht � uqw`��m cyg�rY���Zckg���r���ikrY�

x(t) = x(−t) � �¬oquhg¶uhg�rY�ZSfÚm cpoqckrbm islPms³4S��Z�`g�m ��S ³4S�rbmfu�T`S }qwqm ���b�qT~ø�Øncke S T�g\e�r�uhgYw`| u`l�q�Zz isS � bm = 0 =Ü S¾�WrY�ZS }`| m ��r��x(t)

cke S Thg4T`S^mUg�r����q�qcsm |Ug��ht � uqw`��m c�gxrY���Zckg2�Wr���ikrY�x(t) = −x(−t) � �¬oquhguhg2rY�ZS^m csoZckrbm isl·mk³4S�rY�ZS ���`g�m �ZS^³4S~rYmkuOT`SU}qwqm �����qTçøhØ~cke SUThg'exrYuhg%w`| u`l��q��zUisS � am = 0 � m =

0, 1, 2, . . . =î g{T~gxrYu`zU��S Th�q�Nip�h�\| T`rY�·m �ZlvrYckg{| }hl

Fouriercfe S T�g��

x(t) = A0 +∞∑

n=1

An cos(nω0t+ θn),ëóø í ï

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an � bn mfu`�~T`S Tqwqm ���b�qTqmfu`l·ø�Ø�qc�mUgxlvrY��isrYcfg�l

An =√

a2n + b2n, θn = − tan−1(bn/an).î g{TN}`o�oq�dg�r�u`z �ZS T`�q�dip�h�\| T`rY���Yg{T·mfuNT`S }qwqm ���b�qT½ø�ØPm ��l�rYckg{| }hl

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�Yg{T��`g���ThzUg��h�~w`cs|Ugxu`zUg��h��r�t��qTx(t) � cke S Thg\�·cp�`¹\cpmUg��htN�quh| ��t

x(t) =+∞∑

n=−∞Xne

jnω0t,ëóøZôZï

�qw`u`�j =

√−1cke S Thg��N��ThS^mfT`rbmUg��htN�quZS }`z T � �hThg\uhg�rY�ZS^m cpoqcsrYm isl Xn

zUe{z uZS^mfThg\Tqw`��mUg�lvrY��ikrYckgxl

Xn =1

T0

∫ T0

0x(t)e−jnω0tdt.

ëóø�÷Zï

Ü Snmfu�rYtZ�qTx(t)

cfe S T�gYw`| Tq�b�qTqmUg��h� � g�rY���ZckgY�NrY��isr��

Xn = X∗−n,

ëóøZøZï�qw`u`�½u�T`rbm ck|Uexrb�hu`lrY�Z�¬¼hu¬oqe{� cfgbmku�rY�Z� ����ikl®�`g��bT`zUg��h� =

à �

Page 40: simata kai systimata - users.auth.grusers.auth.gr/~hadjidem/simata kai systimata.pdf · @)A B C D E F G H I H A (TIME DOMAIN ANALYSIS) J LKNMO P Q R4SUTWVYXZV\[^]`_badcfe S ThgjikS

�^­>��% Ý �N�D<a!7� ¯D��#Ç���x4J!$��H6!��D��� �D��8�Z\:C�>#ßÞ.ª²�xÁh4C<�8�ZParceval)

² �qisrY�NgxrY����l � P � csS �hlvr�t��qT`mkuhlx(t)

rbmfu���| uZSUg��h��zUg{}`rbm �Z�qT[0− T0]

u`|Ue{� cpmfThg�³4l

P =1

T0

∫ T0

0|x(t)|2 dt ëóø1Zï

tP =

1

T0

∫ T0

0x(t)x∗(t)dt,

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x(t) =Ü w`��mku�T`S }qwqm ���b�qT�ø�ôdw`The{| S u`�Z�qc

x∗(t) =+∞∑

n=−∞X∗

ne−jnω0t,

�hThg\T`S^mUg��hTZ¹�g�rbm^À4S mkThlrbm �dr���isrY�NøÃ�Nis��u`�Z�qc

P = 1T0

∫ T00 x(t)

∑+∞−∞X∗

ne−jnω0tdt,

=∑+∞

n=−∞X∗n1T0

∫ T00 x(t)e−jnω0tdt,

=∑+∞

n=−∞XnX∗n.

) g{T�w`| T`���qT`mUg´���~r�t��qT½w`| uq���¬wqm cfg � oq�¬��³Ïm ��lvrY��isrYcs³4lvøZø �

P = X20 + 2

+∞∑

n=1

|Xn|2 .ë�0 qï

è m ��w`u`lNT`��m �`lNzUe S ckgÌm �¾�qisr��¾gxrY���¾ckS �`l·w`ck|Ugxu`zUg��huh��r�t��qT`mkuhlx(t)

�Yg{T�mfu���| uZSUg��h�Oz¬g�}hrbm �Z�qT�`g{T`lvw`cs|Ug{�`zUu`�

T0 � rY�ZS T`|^m tZrYcfg�ms³4S�rY�ZS^m csoqcsrbm^À4SFourier Xn =Ä�ÅjÆ�Ô�Ç�È0T´Ê%ËÌÅ

õ®cs³4| u`�Z�qc�mfu½w`ck|Ug�uhzUg��h��rYtZ�qTx(t) = 4 sin(50πt),�qc�w`ck|Ue�uhz uT0 =

50π= 0.04.

² �qisrY�NgxrY����l®mfu`�Ncke S Thg��

P =1

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í

Page 41: simata kai systimata - users.auth.grusers.auth.gr/~hadjidem/simata kai systimata.pdf · @)A B C D E F G H I H A (TIME DOMAIN ANALYSIS) J LKNMO P Q R4SUTWVYXZV\[^]`_badcfe S ThgjikS

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x(t)T`S Tqoq�ZcpmfThgÒrYcyrYckg{| }

Fourier,��w`�½m �·�`u`| ��t

x(t) =+∞∑

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jnω0t,

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−n=<è g�r���S m cpoqcsrbm ikl

Xn

cfe S Thgj�`g��bT`zUg��hu�ej�hThgj�Zw`u`|Uu`�Z�qcvS T~mfu`�Zlcp����| }`r�u`�Z�qc®�¬w`��m �·�qu`|U��t

Xn = Anejθn ,�qw`u`�

An

w`| T`���qT`mUg´��uhebrY�ZS^m cpoqcsrbm ikl = × T`|UTqm �Z| u`�Z�qc$��mUghmfuyw`ck|Ug�u`z¬g´���½rYtZ�qTx(t)

cp�h�\| }`� csmkT�g�³4lmfu½}q¹\| uhgxrY�qT½mfTqoqT`S^m^À4r�cp³4S

ejnω0t � �qcPrY�Z��SU�¬m ��m cslPw`u¬o�oqTqwqoq}`r\g�T�m �Zlß¼hT`r�g��htZlvrY�Z��SU�¬m ��mfT`lω0 =

º u�r�t��qTx(t)

cke S T�gYw`| Tq�b�qTqmUg��h�~�hT�gY�h}Z¹jc®� cs���bu`lvmkuh�·T`SUTqwqm �����qT`mkuhlvm �Zl®�qu`|U��tZl

X−ne−jnω0t +Xne

jnω0t, n = 1, 2, 3, ...ëá�Ø ï

cke S ThgYw`|UTq�b�qTqmUg��h�"ë oq�¬��³ m �Zl®r���ikrY�ZlXn = X∗

−n

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An

[

e−j(nω0t+θn) + ejnω0t+θn]

= 2An cos(nω0t+ θn).ë�qÝ�ï

× T`| Tqm �Z| u`�Z�qc0��mUg`u�rY�ZS^m cpoqcsrbm tZlAn = |Xn|

cfe S Thg`mfuwqoq}qmfu`l6ms³4S$mfTqoqT`S m^À4rYcp³4Sß�qcòr�����S ��m ��mkTωn = nω0 =Ä�ÅjÆ�Å\ÍzSjÆ`ÓjÑÌÓÅÄ× T`| Tqm �Z| u`�Z�qcç�¬mUg��h}q¹\c¢� ck�¬�bu`lçm �Zlç�qu`| �\t�lâ�ØÛrbmfu T`SU}qwqm �����qT

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x(t)mku·cp�h�\| }`� u`�Z�qcß�¬w`�·�`g �bT`z¬g´��t

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nω0T`S mUg�rbmfuhgx��ckej�N�\}`rY�

+θn � �hThgjrbm �ZS�Th| S ��mUg��htrY�Z��S ��m �¬mfT −nω0 � T`S^mUg�rYmku�g���cke\�·�\}`rY� −θn =Ä�ÅjÆ�Å\ÍzSjÆ`ÓjÑÌӨƸ mfTÞw`| u`���bu`�Z�qcsS T uZS u`�q}`r�T`�qc¾m �ZS�w`uhrY�¬m ��mfT

ω0 = 2π/T0V� ¯¦®� ª [^]�[pa = ² w`u`rY��m ��mkT�T`��m t

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fn =1

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t

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n = ...−3,−2,−1, 0, 1, 2, 3, ... = Q R'm r�g � mkuzUg{}q�b| T`�q�qT�rY�Z��S ��m �¬mfTfÚ�wqoq}qmfu`l

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x(t)cke S T�g4w`cs|¬g�u`zUg��h�Þ��Thg»rY�ZS cswqÀ4l~uhg»rY�Z��S �¬m ��m ckl~rbmUg�l�uqw`uhe{csl

T`S T`oZ�ZcsmkT�g�cke S Thg�zUg{}q�h|Ug�m csl = Ü SPmkudrYtZ�qTx(t)

cfe S Thg�w`| Tq�b�qTqmUg��h� � mfTyr����`cke{Tyrbmfuyz¬g�}`��|UT`�q�qT·Th�¬m �cke S Thg�rY�Z�q�qcpm |¬g´��}�³4lyw`| u`lymfuZS½}`µ uZS T |Xn| =º u�zUg{}q�b| T`�q�qT�T`��m ��uZS u`�q}`� csmfThgJ}\�`V�_baO¡ ����[±�� Z©­amplitude spectrum) =äyÔ�Ñ2Ë�Ũ{'Ô�Ñ2ȬÕ@Uâæ

phase spectrum)

º u¢T`S }qw`m �¬�b�qTçøZô�cke S Thg'}Z¹\| u�g�rY�`T"�`| ³4S�m �Zl��qu`| �\tZlçqÝ��hThg'rY�ZS cswqÀ4l�rbm ��rY�Z��S ��m ��mkTfn =

nω0/2πT`S mUg�rbmfuhgx��cke\�N��}`r��

θn�hThg\rYm �·rY�Z��SU�¬m ��mfT

f−n = −nω0/2πT`S¬g�rbmfuhgx��ckg\�N�\}`rY� −θn =Q R'm r\g � mku�zUg{}q�b| T`�q�qT���}`r��

θnÚñwqoZ}`mkuhl |Xn| � zUe S ckgYmfu�wqoZ}`mkuhlvw`u`�NT`S mUg�rbmfuhgx��cke\rYc��h}Z¹jc®��}`r�� =

) g{T�w`| T`���qT`mUg´���~r�t��qT½mfu�zUg{}q�b| Th�q�qT~T`��m ��cke S T�g�T`S^mUgxrY�Z�q�qcpm |¬g´��� =Ä�ÅjÆ�Ô�Ç�È0T´Ê%ËÌÅõ®cs³4| u`�Z�qc�mfu�r���| �`�½uh|�¹\u¬��³4SUe�³4S�w`T`oZ�ZÀ4S

x(t) =n=+∞∑

n=−∞A× Π

(

t− (t0 +mT0)

τ

)

, τ < T0ë� í ï

�qw`u`�T0

cfe S Thg\�dw`cs|Uexu`z u`l�hThgτ�`g{T½whT`| }`�qcsm | u`l � �Nuqw`uhe{T~zUe{z ckgYmfu�cs�Z| u`l®mfu`�·�h}Z¹\cPw`Tqoq�qu`� = ²�b| T`�jg´��tNw`T`| }hrbmkThrY�·mfu`�·r�t��qT`mkuhl®Th�¬mfu`�NzUe{z csmfThg�rbmfu ¸ ��tZ�qT�Ø� =

º uNrYtZ�qTx(t)

T`S T`wqm ��r�cpmfThg�r�c<rYcfg{| }Fourier,

rY�Z�q�Y³4S T·�qc<mkuZS®m ��w`uNøZô � �hThgbuhg�rY�ZS^m cpoqcsrbm iklmfu`�y¼`|¬e�rb��u�S^mfThg\�qc�m ��¼hu`t¬¹\cfg�T�mk³4Snm �¬wq³»S�ø�÷�Â

Xn =2A

nω0T0e−jnω0t0 sin

nω0τ

2, n = 1, 2, 3, ...

ëáqôZï

X0 =Aτ

T0,

ë�Z÷qïí Ý

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-5 -4 -3 -2 -1 0 1 2 3 4 5

1

0

sinc(ö)

ö

¸ ��tZ�qT�Ø��� ² rY�ZS }h|^m �ZrY�sinc(φ)

�qw`u`�ω0 = 2π/T0 = ¸ m �ZS�cfg�zUg��ht½w`cs|Ue�wqmk³4rY�

t0 = τ/2ik��u`�Z�qc

Xn =Aτ

T0sinc(nf0τ)e

−jπnf0τ ,ë�qøZï

�qw`u`�f0 =

1

T0cke S Thg��y¼hT`r�g��htNr�����S ��m ��mfT � �hThg\�·rY�ZS }h|^m �ZrY�sinc

u`|Ue{� cpmfThg�³4l

sinc (φ) ≡ sinπφ

πφ.

ë�0qï

² �b| T`�\g��htçw`T`| }hrbmkThrY�¢T`��m tZl�m ��l¾rY�ZS }`|^m �ZrY�Zl�zUe{z cpmfThgòrbmfu ¸ ��tZ�qTÙØ�� = × T`| Tqm �Z| tZrYT`m c~��mUg|sinc (φ)| → 0

��mfT`Sφ→ ±∞ =

º u��\}`rY�qT�mkuh�dw`ck|Ug�uhzUg��huh�½r�t��qT`mkuhl@ í � �Yg�T t0 = τ/2 � ¼`|Uexrb�hcsmfThg�Tqw`��m �·r���ikrY�èZô = R»e S Thg

|Xn| =Aτ

T0|sinc(fnτ)| , fn = nf0 =

n

T0

ë�0�qï�hThg�uhgbmUg{�qislPmk³4SwqoqTqm^À4S |Xn| � �bg{T n = ...− 3,−2,−1, 0, 1, 2, 3, ... ¼h|Uexrb�huZS^mfThgYrYm ��S�hT`�Zw`��oq�

y =Aτ

T0|sinc(τf)| ,

�ßuqw`uhe{T�q��zUcsSUe{� cpmfThg¬rYmkTPrY�Z�qcfe�Tτf = ...−3,−2,−1, 0, 1, 2, 3, ... � z �¬oqT`zUtßrbmfTPrY�Z�qcke{T

f = ν/τ �ν = ...− 3,−2,−1, 0, 1, 2, 3, ... =î c¼h}`rY�~mkT�w`T`|UTqw`}`S^³ � mfu���}`r��qT¾wqoq}qmfu`�Zldmkuh��rY�Z| �qu`��mk³4S½u`|^¹\u¬��³4SUe�³4S�w`Tqoq�ZÀ4S² í � �Yg{T

t0 = τ/2 � zUe{z cpmfThg�rbmfu ¸ ��tZ�qT~Ýà =í Ã

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|Xn|

0

0 1/ 2/-1/-2/

1/ T0

f

¸ ��tZ�qT~Ý1 � º u��\}`rY�qT½wqoq}qmfu`�Zl®mfu`�NrY�Z| �quh�dms³4S�u`|^¹\u¬��³4SUe�³4S�w`T`oZ�ZÀ4S� í× T`| Tqm �Z| tZrYTqm c��mUg\ik��u`�Z�qcvz¬g�}`�h|Ug�m csly��ThrY�qTqmUg��hisln�b| Th�q�qisl � uhgjuqw`uhe{ckl�T`w`is��u`�ZS·�`cpmfT`µ �½mfu`��l

Tqw`�`rYmkThrY�1/T0

rbmfu�SW}`µ uZS T�mk³4SWrY�Z��S u�m t¬mk³4S = R'whexrY�Zl¾mkuÞwqoq}qmfu`l¾m �Zl¾mkT`oZ}hS^mk³4rY�Zl¾�b¹je S ckg�hTZ¹\À4l"T`�Zµ }`SUckg$mfu |f | =�¿ l"��ThrY�qTqmUg��h� cs�Z| u`l"�Zw`uh| u`�Z�qcOS TÙuh|Ue�r�u`���`c�m �ZSçw`u`rY��m �¬mfT

1/τ �z ��oqT`z t·mfu�rY�Z�qcfe�u½mfu`�·w`|^À'mfu`�N�q�Zz csS¬g�rY�quh� =× T`| Tqm �Z| tZrbm c'�¬mUgUrbmfu ¸ ��tZ�qT®Ý1 $cs�q�\T`SUe{� u�S mkT�gU�hThg¬T`|US ��mUg´��isl»r�����S ��m ��m csl � zUg���mUg¬��|U��r\g��qu`w`uhg{t ÚrYT`�`cvm ���`g��bT`zUg��ht��qu`| �\t�m �Zlnr�ckg{| }`l

Fourier�Yg�T�S T¾cs�h��|U}`rYu`�Z�qcmfu¾rY�Z| �q�~mk³4S·u`|�¹ju¬��³4S¬e´³4S

w`Tqoq�ZÀ4Sé í = R'w`cfg�zUt��`�Z³4lNmkuWrYtZ�qT�Th�¬m �Wcfe S T�gÌw`|UTq�b�qTqmUg��h� � mfuW��}hrY�qT�mkuh��cke S Thg%rY�Z�q�qcpm |Ug��h�ëtwo sided).

º u��\}`rY�qT���}`rYcs³4l®mkuh�dw`Th| Tqw`}`S^³ rYtZ�qTqmfu`l®cfe S Thg

θn(f) = −πfτ,�Yg{T

f = nf0, n = ..− 2,−1, 0, 1, 2, ..�hThg4zUe{z cpmfThg'rbmfu ¸ ��tZ�qTçÝ�Ø = × T`| Tqm �Z| u`�Z�qc���mUg%mfu"zUg{}q�b| T`�q�qTçcfe S T�g'T`S^mUgxrY�Z�q�qcpm |Ug��h� � zUg{�¬mUg2mfurYtZ�qTç í cke S ThgYw`| T`���qT`mUg´��� =Ä�ÅjÆ�Å\ÍzSjÆ`ÓjÑÌÓÜ Sß��}qw`uhgxu·rYtZ�qT�is��cfg`�quZS �qwqoqck��|Uu~ë

one sided)��}hrY�qTyrbm �ZSßip����| T`r��nøZô � z ��oqT`z tck�q��T`SUe{� uZS^mfThg

�q�ZS u�uhgb¹\csmUg´��islvrY�Z��S ��m ��m csl � mfu�rYtZ�qT½Th�¬m ��z ckS��Zw`u`| cfe�S T�cke S T�gYw`| Tq�b�qT`mUg´��� =

J f �LKN��¾åP�Pý�MWý ��æñ�ß¾üÞ �å�¾ä MW��è�Ïæ6å éd �Þå��������� édM¢KNMO�P������� �ä

õ®cs³4| u`�Z�qcdrbmfTZ¹\ck| ���b| T`�q�`g��h�OrY�Zrbm �Z�qT¾�hThgÒckexrYu`zUu�m �ZS½mfTqoq}`S^mk³4rY�ejωt = �Pw`uZ¹jipmfu`�Z�qcy��mUgÌ�

ckexrYu`zUu`lNck��T`| �q�h� cpmfThgÒrYcy��| ��SUut = −∞ �hThgÌmku�rY�Zrbm �Z�qT��Z| cs�qcfe¶m �¬m c = õ®T�Tqw`u`z cfe�µ uh�Z�qc·��mUg

íZí

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f

èn

-2ð

1/ô 2/ô-1/ô-2/ô

¸ ��tZ�qT~Ý�ØZ º u���}hrY�qT��\}`rYcs³4lPmfu`�NrY�Z| �qu`�·ms³4S�u`|^¹\u¬��³4SUe�³4S�w`T`oZ�ZÀ4S� í�dikµ u`z uhlvcke S T�gbmfTqoq}`S^mk³4rY�d�qcßm �ZSne{zUg{T�r�����S ��m ��mkT � TqoZoq}½zUg{T`�\u`| csmUg´����wqoq}qmfu`l®�hThgYz¬g�Th��u`| csmUg´��t��}hrY� � m �Zl®�qu`| �\t�l

ejωt → ² → H(ω)ejωt =² rY�ZS }`|^m �ZrY�

H(ω)uZS u`�q}`� csmfThgòV� �� � £ [^]�V\]W_ � [paO}\� £ �`© (transfer functionê = ² r���SU}`|^m �ZrY�

T`��m tNcke S T�g��`g �bT`z¬g´��t =P r±ÑjÇ�ÈzTìë`Ó´ | �Zr�g{�quqw`uhgxu`�Z�qcPmfu���ckSUg��h��m ��w`u½w`u`�Ncs�h��|U}`� ckg�m �·rY��ikrY�dcfg�rY�hz u`�UÚ¯ckµ �`z u`� �

y(t) =∫ +∞

−∞x(t− λ)h(λ)dλ,

�Yg{T�m �ZS�ckexrYu`z ux(t) = ej ωt = Q R»��uh���qc

y(t) =∫ +∞

−∞ej ω(t−λ)h(λ)dλ = ejωt

∫ +∞

−∞e−j ωλh(λ)dλ.

è |Ue{� uh���qcPmfu�u¬oquq�hoZtZ|^³4�qT�mfu`�Nz csµUgxu`�½�qipoqu`�Zl®³4l

H(ω) =∫ +∞

−∞e−j ωλh(λ)dλ,

ëá�0 qïuqw`��m c®�Nisµ u`zUu`lrbm �ZSncfe�rYuhz u

ejωtcfe S Thg\�

y(t) = H(ω)ej ωt.ë���Ø ï

² rY��isrY���0 6zUe{z cfg�m �<rY�ZS }`|^m �ZrY�0�qcsmfT`��u`|U}`lH(ω) = ² rY�ZS }`|^m �ZrY�6T`��m t6cke S Thg �`g��bT`zUg��htßrY�ZS }`|^m �ZrY� �m �Zl®�qu`|U��tZl

H(ω) = A(ω) ej θ(ω).ë��qÝZï

í ô

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Ü w`�ymfT�w`T`|UTqw`}`S^³ cke S T�g��\T`S cs| �·��mUgh�nrY�ZS }`| m ��r��v�qcpmfT`�\u`| }`lßw`cs|Ug���|U}`��cfghm �nrY�Z�Zw`cs|Ug{�\u`| }dcsSU�`l�b| T`�q�`g��hu`�~rYmkTZ¹jcs| u`��rY�Zrbm tZ�qTqmfu`lv³4lw`|Uu`lnm ��rY��isrY�½�qcsmkT`µU�½mkuh����}`r��qTqmfu`lm �ZlnckgxrY�`zUu`���hThgmfu`�<��}hrY�qTqmfu`l'm ��l'ckµ �`z u`� = × T`| T`m ��| uh���qc'��mUg^�<r�����S ��m ��mkTßm ��l'ckgxrY�`zUu`�<w`T`| T`�qikS ckgUT`S T`o�oquhe�³'m � �TqoZoq}Û�qcpmfT�¼h}qoZoqcpmfThg'mfu¢wqoq}qmfu`l�m �Zl�mfTqoq}`S ms³4r���l � ��Tqm }çmfuZS�w`Th| }q�bu�S mkT |H(ω)| = A(ω)

�hThg���\}`rY�¾m �Zl·mfTqoq}`S ms³4r���l � �hTqm }Om �¾��³4SUe{T

θ(ω) =~º �`rYuO���qcpmfT�¼hu¬oqt¾mfu`��wqoZ}`mku`�Zl � �`rYuW�hThg2��qcpmfT�¼hu¬oqtPm �Zlò�\}`rY�Zl0csµUT`|^m^À4S^mfThgqT`w`�nm �PrY�Z��S ��m �¬mfT

ω � �hThg`rY�ZS cswqÀ4l0rbm �ZS<ikµ u`z u��qcpmfT�¼h}qoZoqcpmfThg�·�hTqmfT`S u`�qt½m ��lvcsS ik|^�bckg{T`lw`u`�Nckex��T`�qc®rYmku��\}`rY�qT½mfu`�NrYtZ�qTqmfu`l®m �Zl®ckgxrY�`zUu`� =õ®T¢z cfe�µUu`�Z�qc�rbmfT"cpw`�`�`csS T¢��mUg'��r���SU}`|^m �ZrY���`cpmfT`��u`|U}`l~cke S Thg4u¢�qcpmfT`rY���Z�qTqmUgxrY�q�`l

Fourierm �ZlP�h|Uu`�ZrbmUg��htZlvT`w`�q�h|UgxrY�Zl =Ä�ÅjÆ�Å\ÍzSjÆ`ÓjÑÌÓ² ckexrYu`z uhl

ejωt�b| }`��csmfThgY³4l

ejωt = cos(ωt) + j sin(ωt),�hThg\�·ThS^mUe�rbmfuhgx���Nikµ u`z u`lncke S Thg��H(ω) ejωt = A(ω) ej(ωt+θ(ω) = A(ω) [cos(ωt+ θ(ω)) + j sin(ωt+ θ(ω))] .

× T`| Tqm �Z| u`�Z�qcN�¬mUgÒrbm �¾�`g��bT`zUg��ht�ckexrYu`z ucos(ωt) + j sin(ωt)

T`S^mUgxrbmfuhgx��cfeÒ���`g��bT`zUg��ht�isµ uhz u`lA(ω) [cos(ωt+ θ(ω)) + j sin(ωt+ θ(ω))] = © �¬��³ m �Zl��b| T`�q�`g��h��m �¬mfT`l~mfu`�OrY�Zrbm tZ�qTqmfu`l � rbmfuw`| Tq�b�qTqmUg��h���qis| u`l�m �Zl�ckgxrY�`z uh�dT`S^mUgxrbmfuhg���cfe�mfu·w`| Tq�b�qTqmUg��h���qis| u`l�m �Zl�csµ �hz u`��ë¤mfu·Th�¬m ��e�rY���Zckg�hThg��Yg�T�mfu���ThS^mfT`rbmUg��h�~�qis| u`l�ï = ¸ �ZS cpwqÀ4l � T`Sy�NckexrYu`z uhlcke SUThgYmfu½w`|UTq�b�qTqmUg��h��rYtZ�qT

cos(ωt) � �isµUu`z u`lncke S T�gYmfu½w`| T`���qT`mUg´���~rYtZ�qTRe[H(ω)ejωt] �

cos(jωt) → ² → |H(ω)| cos(ωt+ θ(ω)) =R»e S Thgb��T`S ck| �N��mUg�mfudwqoq}qmfu`lßw`u¬oZoqTqwqoqT`r�g{}`�UcpmfThgb�qc<mfu·�qipm | u·m ��lßr���SU}`|^m �ZrY�Zl$�qcsmkT`�\u`| }`l��hT�g����}hrY�·�qcsmkTZ¼`}`o�oqcsmkT�gb�hTqm }

θ(ω) =Ä�ÅjÆ�Ô�Ç�È0T´Ê%ËÌÅõ®cs³4| u`�Z�qc6mfud�b| T`�q�`g��h�½rY�Zrbm �Z�qT

RCmfu`� ¸ ��t��qT`mku`l�Ø^÷ = ² rY��ikrY�nckgxrY�`z u`�UÚ csµ �`zUu`���hTq¹\u`|Ue{� c±Ú

mfThgYT`w`��m �·zUg{T`�\u`|Ug���t½ckµUexrb³4rY�dy

dt+

1

RCy =

1

RCx(t),

ë�� à ï�hThg\�d��| u`�ZrbmUg��ht½Tqw`�q�h|UgxrY�½cfe S Thg%ë¤m �¬w`uhl®÷qøZï0�

h(t) =1

RCe−t/RC × u(t).

Ü lv��w`uZ¹\isr�u`�Z�qcßm^À4|UT���mUgY�NckexrYu`z u`lncke S T�g��NrY�ZS }`|^m �ZrY�x(t) = ejωt.

í ÷

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² isµ uhz u`lzUe{z csmfThg�Tqw`��mkuZSnm ��w`u

y(t) =∫ +∞

−∞x(λ)h(t− λ)dλ =

∫ t

−∞x(λ)h(t− λ)dλ,

zUg{��mUgh(t− λ) = 0

�Yg{Tλ > t = Q R4��u`�Z�qcPm^À4| Ty(t) = 1

RC

∫ t−∞ ejωλe−(t−λ)/RCu(t− λ)dλ

= 1RC

∫ t−∞ e−(t−λ)/RC+jωλu(t− λ)dλ

= 1RC

∫ t−∞ e−t/RC e[1/RC+jω]λdλ

= 1RC

e−t/RC∫ t−∞ e[1/RC+jω]λdλ

= 1RC

e−t/RC 11/RC+jω

[

e[1/RC+jω]λ]λ= t

λ=−∞

�hThg�m cpo`g��h}y(t) =

1

1 + jRCωejωt.

ë�� í ï¸ �ZS cpwqÀ4l�·r���SU}`|^m �ZrY�d�qcsmkTh��u`| }hlcke SUThg��

H(ω) =1

1 + jRCω.

ë��qôZï

² r���S }h|^m �ZrY���qcpmfT`��uh| }`ld�Zw`u`| cke¶S T�¼h| cp¹\cke\��Thg¶�qc}`o�oqu¾m | �qw`u � Tqw`�¾m �ZS·T`w`�q�h|UgxrY��rbmfTZ¹\ck| tZl�hTqm }hrbmfT`rY�Zl®m �Zl®zUg{T`�\u`|Ug��htZlnckµUe�rY³4rY�Zl@� à �Yg�T�m �ZSncfe�r�u`z u

x(t) = ejωt =² �bcsSUg��htNoq�ZrY�ym �Zl®uh�qu¬��ckS u`�Zlz¬g�Th��u`|Ug���tZlncsµUexrb³4rY�Zl®cfe S Thg\�

y(t) = C e−t/RC ,

�qw`u`�C

T`�¬¹\The{| csm ��rbmfTZ¹\ck| } = ² �bcsSUg��ht oZ�ZrY�Ûm �Zl�wqoqtZ| u`�ZlWz¬g�Th��u`|Ug���t�l¢csµUexrb³4rY�ZlWcke S T�g6mfu}Z¹\|Uuhg�r��qT�m �Zlv�bcsSUg��htZlnoZ�ZrY�Zl®m �Zlu`�qu¬�bcsSUu`�Zln�hThgj�`g{T`ln�qck|Ug��htZloq��r���l®m �ZlvwqoqtZ| u`�ZlckµUexrb³4rY�Zl =î e{T��qck|Ug��ht·oq�ZrY�dcfe S T�gYm �Zl®�qu`|U��tZl

y = D ejωt,�qw`u`���$rYmkTZ¹jcs| }D

��w`u¬oqu¬�be{� csmfThg¬T`w`�Pm �ZSòTqw`T�e m �ZrY�ßS T®cpw`Tqoq�¬¹\cs�Zcfg cp�®mfT`��m �¬m ��mfT`l'm �$zUg{T`�\u`|Ug���tcsµ¬e�rb³4r��è� à = R4�Z|Uexrb�hu`�Z�qc

D =1

1 + jRCω,

uqw`��m c®�·�bcsSUg��htNoq��r��dcke S Thg\�

y(t) = C e−t/RC +1

1 + jRCωejωt.

è �`| u`lCe−t/RC

cke SUThg¬w`T`| u`z¬g´���`l0�h| u`l � zU�¬oqT`zUt$m cke S ckg¬rYmku0�¬mfT`S

t→∞ � �hThg��$oq�ZrY�$rbmfTZ¹jcs| }`l�hTqm }hrbmfT`rY�Zlvcke S Thg��

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y(t) =1

1 + jRCωejωt.

² m cpoqcs��mfThe{T®Th�¬m t�rY��isrY�ßrY�Z�Zwhe�wqm ckg¬�qc'm �ßrY��isr���� í � cpw`u`�qikS^³4l'¼h|Uexrb�hu`�Z�qc4w`}qo`g¬m �ßrY�ZS }`|^m �ZrY��qcpmfT`�\u`| }`l �qô =íe{'Å�Æ�Ë=u�ÊCSõ®TN¼h| u`�Z�qc�m �ZS�isµ u`z u�mfu`�·w`T`| Tqw`}`S ³ÁrY�Zrbm tZ�qTqmfu`l��Yg{T�m ��S�ckexrYu`z u

cos(ωt) = R»e S Thg

cos(jωt) → Re[

H(ω)ejωt]

.

R'w`cfg�zUt���isµ u`zUu`l�rbm �ZSycfe�rYuhz uejωt

cke S Thgj�y = H(ω)ejωt � �½ikµ u`z uhl�rbm �ZS

cos(jωt)¹\T~cke S Thgj�

Re[y] = Q R4��u`�Z�qc®csw`u`�qisS ³4l �y = Re

[

11+jRCω

ejωt]

= Re[

1−jRCω1+R2C2ω2 e

jωt]

= Re[

1√(1+R2C2ω2)

ejθ(ω) ejωt]

�hThg�m cpo`g��h}y =

1√

(1 +R2C2ω2)cos(ωt+ θ(ω)),

�qw`u`�θ(ω) = − tan−1(RCω).× T`| Tqm �Z| u`�Z�qcO�¬mUg$rbm �ZSçisµ u`zUuÙik��u`�Z�qc�mkT`oZ}`S ms³4r����qcWrY�Z��S �¬m ��mfT

ω � Tqo�oq}Ù�qcWzUg{T`��uh| cpmUg��h�wqoq}qmfu`lv�hThg�z¬g�Th��u`| csmUg��ht���}`rY� � w`uh�Ncsµ Th|^m^À4S^mfThg�Tqw`��m �NrY�Z��SU�¬m ��mfT

ω =

J@� î æñ���Mg��ý�����y�ìMç�� ÂËFourier

¸ mfu�w`| u`���bu`�Z�qcsSUu��hcs�\}qoqThg�uW�qcsoZcsm tZrYT`�qcmfu���}hrY�qT�ckS �`ldw`ck|Ug�uhzUg��huh��rYtZ�qTqmfu`ly�hThgÌcfe�zUT`�qcy��mUgmfu��\}`rY�qT�mfu`��Tqw`u�m cpoqcke�mfThgÒTqwh�WisS TWT`|¬g ¹j�qt�r\g{�qu�wqoqtU¹ju`l·r�����S u�m t�ms³»S � m ��¼hT`r�g��ht�rY�Z��S ��m ��mfTω0

�hT�g$mfTÏw`u¬oZoqTqwqoq}`r�g{}Ám ��l � mfu`�Zl"T`S^À'm ck| u`�Zl¢T`| �qu�S¬g´��u`�Zlnω0 = ð ��oZThz t mkuÁ��}`r��qTÙcsSU�`l

w`cs|¬g�u`z¬g´��u`��rYtZ�qTqmfu`l<Tqw`u¬m csoqcke�mkT�ghTqw`�yz¬g�}`�h|Ug m cklß��T`rY�qT`mUg´��isl<��|UT`�q�qisl = ¸ mfu�w`T`| �ZS��hck��}qoqThgxu�¹\Tcpw`cs�`m cke S u`�Z�qcPm �·�qcsoqipm �d�hThgY�Yg{T�mfu���}`rY�`T�csS �hlv�q�·w`cs|Ugxu`zUg��huh��rYtZ�qTqmfu`l =Q R4rYms³.isS TW�q��w`cs|Ugxu`zUg��h�"rYtZ�qT

x(t) = R4}`S~�qT`lNcsS z¬g�Th��is|UckgÒ�q��SUu�mfu�rYtZ�qT��qcsmkThµ ��mk³4S½mUg{�ZÀ4Smfu`�~��| ��SUu`��Tqw`�

t = αip³4l

t = βëα < β

ï � m ��m c��TqmfT`rb�hck��}h� u`�Z�qcyisSUT�}qoZoZu�rYtZ�qTx′(t) � mfuuqw`uhexu�rY�Z�Zwhe´w`m ckgj�qcvmfu�rYtZ�qT

x(t)rbmfu�zUg{}`rbm �Z�qT

[α, β]�hThg�csw`T`S TqoqT`�¬¼h}`SUcpmfThg�w`ck|Ug�uhzUg��h}�cs�`m �`l

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mfu`�çzUg{T`rbm tZ�qTqmfu`l�T`��mfu`� = R»e S T�g0��T`S ck| � �¬mUgñmfu�S isu�r�t��qTx′(t)

cfe S Thgòw`cs|¬g�u`z¬g´��� � �qc�w`cs|Uexu`z uT0 = β − α � �hThg2rY�Z�Zwhe�wqm ckgÒ�qcdmfuOrYtZ�qT

x(t)rYmkuWzUg{}`rYm ���qT

[α, β] =�º u"rYtZ�qTx′(t)

ik��ckg%z �Zuw`cpwhcs| T`r��qisS ckl~ThrY�ZS is��ckg{csl¾rbmfT¢rY�Z�qcfe�T

t = �Thg

t = β � �hThg»mkuç�\}`rY�qT¢mfu`�"is��ckg»zUg{}q�h|Ug�m csl��ThrY�qTqmUg��hikl®�b| T`�`�qisl � �qwq³4lng�r����Zcfgb�bg{T��h}Z¹\cPw`ck|Ug�uhzUg��h��rYtZ�qT =¸ c6w`u¬o�oqikl6whcs|Ug�wqm^À4rYcfg�l$�`�Z³4l$�qThl$csS zUg{T`�\is| cfghmfuy�q�vw`cs|Ugxu`zUg��h�NrYtZ�qT

x(t)r�c0�¬oquymkuyz¬g�}hrbm �Z�qT

mk³4SòmUg��ZÀ»Sòmkuh�$T`w`�t = −∞ ip³4l

t = +∞ = õ®T®z cke{µ uh���qc4��mUg mfuP�\}`rY�qT®T`��mkuh�$mkuh�$�q�$w`cs|Ugxu`zUg��huh�rYtZ�qTqmfu`lvcke S Thg�r���SUcs��isl®�hThg�z¬e�zUcpmfThg�Tqw`��mku¾_ � [pa`V^¦�]`_Ya�[ ¥ V�_ ª Fourier =× |Ug S�w`| u���³4| tZrYu`�Z�qc~rbmfu`�Zl��bcsS¬g´��u`��l¾m ��w`u`�Zl~mkuh�O�qcsmfT`rY���Z�qTqmUg�r��qu`�

Fourier,¹jTçz �`rYuh���qc

isSUT"w`T`| }`z cfg �b�qTÛcsSU�`l�w`cs|Ugxu`zUg��huh�"rYtZ�qTqmfu`l � mfu"uqwhuhe�uÛis��cfg2zUg{}q��|Ug mfuÛ��}`r��qT � ��Thg%¹jT"z cke{µ uh���qcwqÀ4lnmku��\}`rY�qT~mfu`�½m cke S ckgjS T~�Ye S ckg�rY�ZS cs��isl�hTq¹�À4l��½w`cs|Uexu`z uhl�mfu`�½w`cs|¬g�u`zUg��huh�~r�t��qT`mkuhlm cke S ckgrbmfu ∞ � uqw`��m c®mfu�rYtZ�qT��Ye S csmkT�g\�q�·w`cs|¬g�u`z¬g´���bÂ<õ®cs³4| u`�Z�qcPmfu�r���| �`�½mk³4Syu`|�¹\u¬��³4SUe�³4Sdw`Tqoq�ZÀ4Sw`u`�dzUe{z cpmfThgbrbm � ¸ ��isr��� í ë ¸ ��t��qT¾Ø�Zï = ² w`ck|Ue�uhz �`lPmfu`�ycfe S T�gbexrY��w`| u`l

T0 � �hThgb�qwq³4l®��The SUcpmfThgrbmfu ¸ ��t��qT"Ø� � w`T`|Ugxrbm }`SUckgjm �ZS·Tqw`�`rbmfT`r����qcpmfT`µ ��z �Zu�z¬g�Thz u��bg´�hÀ4S½w`Tqoq�ZÀ4S = R4}hS

T0 → +∞ �mfu�rYtZ�qT½m cke S ckg�S T��be S ckg�ikS T`l�q��SUu�u`|�¹\u¬��À4SUgxu`lw`Tqoq�q�`l � �hThg�r���SUcpwqÀ4l®S T��Ye S ckg��q�dw`ck|Ugxu`zUg��h� =º uP��}`r��qT$mfu`�<r�t��qT`mkuhlJ í zUe{z csmkT�g rbmfu ¸ ��tZ�qTPÝ1 =Òè g^��ThrY�qTqmUg��hikl'�b| T`�q�qisl4cfe S ThgUzUg{}q�h|Ug�m csl»�hThg

�ßTqw`�hrbmfT`rY�ß�qcsmfT`µ �$mfu`�Zl»cfe S T�gZe�rY�$w`|Uu`l1/T0 = Q è rYu®�qcs��T`oZ��m ck| u®cke S Thg¬mfu

T0 � m �`rYuPwqoq�Zr�g{isrYm cs| T�n�`e{Tyw`| u`l$m �ZSP}qoZoq�ncke S Thg�uhg���T`r��qTqmUg��hisl$�b| Th�q�qisl = Q è mfT`S

T0 → +∞ � mfud��}`r��qT�m cfe S ckg�S Ty�Ye S ckgrY�ZS ck��ikl = Ü w`�"mku"w`T`| }hz ckg����qTÛT`��m �¢��The S cpmfThg'�hTZ¹\Th| }ç�¬mUg'�`g{T¢�q��w`ck|Ug�uhzUg´��t"rY�ZS }`|^m �ZrY��ik��ckgrY�ZS ck��ikl®�\}`rY�qT =4º u��\}`rY�qT�T`��m ��zUe{z csmfThg�Tqw`��mku��qcpmfT`rY���Z�qTqmUgxrY�q�

Fourier.õ®T�z �hrYu`�Z�qcym^À4| T�mkuh��ly¼`Thr�g��hu`�Zl·m ��w`u`�Zldmk³4S��qcsmkThrY�����qT`mUg�rY�ZÀ4S

Fourier��³4|¬e�lNTqw`�hz ckg{µ � =Ü lv¹\cp³4|Ut�r�u`�Z�qcPmku��q�Nw`ck|Ug�uhzUg��h��rYtZ�qT

x(t) = õ®T~ThS Tqoq�ZrYu`�Z�qc®mfu~r�t��qT~T`��m ��rYcvrbmfuhgx��ckg�À4z ckgxlrY�ZS T`| m t�r�ckgxl

ej2πft � �qw`uh� f �nr�����S ��m ��mkT�ë¤�nu`w`uhe{TNrY�ZS z ikcpmfThgh�qc6m �ZS��h���`o`g��htyrY�Z��S �¬m ��mfTω

�qcm �·rY��ikrY�

f = ω/2πï = Ü w`uhz ckg��hS �ZcsmkT�g\�¬mUg

x(t) =∫ +∞

−∞X(f) ej2πftdf,

ë��Z÷Zï�qw`u`�

X(f) =∫ +∞

−∞x(t) e−j2πftdt.

ë��qø�ïè m ��w`u`le�Z÷Ncke SUThg�T`S mUe�rbmfuhgx��u`lmfu`�·m �¬w`uh� í ô � u�uqw`u�e�u`l�T`S Tqoq�ZckgYmfu�rYtZ�qT

x(t)rYcPr���SUT`|^m tZrYckgxl

z iso�mfT =è m ��w`u`lè�Z÷~cfe S Thg��bcsSUe��hck�ZrY�~mfu`�~m ��w`u`�¾øZô � u�u`w`uhe�uhlNThS Tqoq��cfg¶mfu�w`cs|Ugxu`zUg��h�OrYtZ�qT�rYc�rYcfg{| }

Fourier,rY�ZS T`|^m tZrYcfg»mk³4SOrY�ZS T`|^m tZrYcs³4S

ejnω0t = µ Thg0rbmUg�l�z �Zu�w`cs|Ug�wqm^À4r�ckgxl��ÛT`S }qoq�ZrY�çmfu`�rYtZ�qTqmfu`l

x(t)�Ye S csmkT�gòrYc~w`ck|Ugxu`zUg��hislWrY�ZS T`| m t�r�ckgxl�m �Zl¾�quh| ��tZl

ejnωt � TqoZoZ} rbmfu�S�m ��w`u ø�ô¢�T`S }`oZ�ZrY��Ye S csmfThg�rYc6Th|Ug ¹j�qtZr�g{�qu�wqoqt¬¹\u`l$�`|^³4S � T`S^ÀÞrbmfuZS�m �¬w`uÔ�Z÷v�Ye S csmkT�gh�hTqm }·rY�ZS cs��tvm |U�qw`u =¸ m �rY��isr��@�qø � mfu�r�t��qT

x(t)�qcpmfT`rY���Z�qTqmUe{� csmkT�g`rbmfu�rYtZ�qT

X(f) =4è �qcsmkT`r����Z�qTqmUgxrY�q�hl6T`��m �`luZS u`�q}`� csmfThgò_ � [±ahV^¦�]`_ba�[ ¥ V�_ ª © Fourier. ² r���ikrY�º�Z÷Wu�SUu`�q}`� csmkThg<aq�¬[ � Vj[ £ �E}\� ©�_ � [pa`V^¦�]`_ba1Õ[ ¥ V�_ ª © Fourier.

Ü w`u`zUckg S �ZcsmkT�g���mUg�u½_ � [±ahV^¦�]`_ba�[ ¥ V�_ ª © Fouriercke S T�g��quZS T`zUg��h�hl � �hThg�rY�ZS cpwqÀ4lcp����| }`�Uckg��qu�S uhrYtZ�qT`S^mfTymfuyr�t��qT

x(t) = Ü ��m �drY�Z�qThe S cfgh��mUg�isS Tdr�t��qTy�Zw`uh| cke�S Tyw`Th| T`rbmfTZ¹\cfe�cke�m c

í �

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��w`��m ���qu`| �\tx(t) � t���w`��m ���qu`| �\t

X(f) = ² w`|^À'm ��rY�ZS }h|^m �ZrY�~cp�h�\| }`� cfg�m ��rY�Z�Zw`cs|Ug{�\u`| }mfu`�NrYtZ�qTqmfu`l®³4l®w`| u`lvmfu���| �ZS u � T`S^ÀÙ�Nz cs��m cs| �NzUe S cfgYmku���}`r��qT½mfu`�NrYtZ�qTqmfu`l =× T`| Tqm �Z| tZrYTqm c2��mUg^rbmfuß�qcpmfT`rY���Z�qTqmUgxrY�q�

FourierX(f)cs�q�\T`SUe{� uZS^mfThg �hThg Th| S ��mUg��hisl4r�����S ��m ��m csl ��qwq³4l�ckex��Th�qc½��Thg4rbm �ZS~w`ck|Ue�wqmk³4rY��m ��l�r�ckg{| }`l

Fourierë¤m ��w`u`l~øZôZï = R4}`S�mfu¢rYtZ�qT

x(t)cfe S Thg

w`| Tq�b�qTqmUg��h� � g�rY���Zu`�ZS�u�gYr���ikrYcfg�lX(f) = X∗(−f), ë��1qï

z ��oqT`z t·mfu���}hrY�qT½mfu`�·wqoq}qmfu`�Zlvcke S Thg\rY�Z�q�qcpm |Ug��h� �|X(f)| = |X(−f)| . ë��0�Zï

è �qcpmfT`r����Z�qTqmUgxrY�q�`lFourier,

�qwq³4l®zUe{z cpmfThgb�qc$mfuZSvm ��w`u��qø � cke SUThgbisSUT`l��b| Th�q�`g´���`l®m cpoqcsrYm t�l �F � uÞuqw`uhexu`l�cpwhg{z | }�rbmfuÛr�t��qT

x(t)�hThgñmfuÞ�qcpmfT`rY���Z�qTqmUe{� ckgñrbmfuÛrYtZ�qT

X(f) = ² rY��ikrY����÷cp����| }`�UckgYmfuZS�T`S mUe�rbm |Uu`��u�m csoZckrbm t � F−1 Â

X(f) = F [x(t)], x(t) = F−1[X(f)].ë�Ø� 1 qï

Ä�ÅjÆ�Ô�Ç�È0T´Ê%ËÌÅõ®cs³4| u`�Z�qcvmkuZS·u`|�¹\u¬��À4SUgxu�w`T`oZ�q�

x(t) = Π(

t2

) =�è �qcpmfT`r����Z�qTqmUgxrY�q�`lFourier

¼h|Uexrb�hcpmfThgjT`w`�mfu�S�m ��w`uç�qø �

X(f) =∫ +1

−1e−j 2πftdt = 2

sin(2πf)

2πf.

² rY�ZS }`| m ��r��X(f)

csw`Tqoq�U¹jcs�Zckg%m �ZS�g{zUg{�¬m ��mfTã�0 � cke S Thg4�q}qo`g�rbmfT"w`| Tq�b�qTqmUg��ht � zUg{��mUg2mfu"r�t��qTx(t)

cke S T�g�}`|^mUgxu =

J@� ï Kd��� �d�PýN�� �Pý�ä æ6�ã6åP�Ûæ$� �ä MW�� üÞ��Mç��(energy spectral density)

² csS ik|^�bckg{T~csS �`lrYtZ�qTqmfu`lvu`|Ue{� csmkT�gY³4l

E =∫ +∞

−∞|x(t)|2 dt. ë Ø� �Ø ï

Ü w`u`z ckg��hSU��csmfThgU��mUg^�<csS is| ��cfg{T�cs�h��| }h� cpmfThgUrY�ZS T`|^m tZrYcfgfmfu`�6�qcsmfT`rY���Z�qTqmUg�r��qu`�Fourier

ë ¹\cpÀ4|U���qTParceval),

³4lE =

∫ +∞

−∞|X(f)|2 df. ë Ø� qÝZï

è m ��w`u`lvT`��m �`lcke S Thg�T`S mUe�rbmfuhgx��u`lmkuh�dm ��w`u`��0 d�Yg�T�w`cs|¬g�u`z¬g´��}�rYtZ�qTqmfT =»è �`| uhl

dE = |X(f)|2 df,

ô1

Page 51: simata kai systimata - users.auth.grusers.auth.gr/~hadjidem/simata kai systimata.pdf · @)A B C D E F G H I H A (TIME DOMAIN ANALYSIS) J LKNMO P Q R4SUTWVYXZV\[^]`_badcfe S ThgjikS

cke S Thg��NcsSUis|^�bckg{T�mfu`�·r�t��qT`mkuhlx(t) � rYm ��Sw`ck|Ugxu���tNmk³4S�rY�Z��S u¬m t�mk³4S

f − (f + df) � �hT�g�u��`| u`l

G(f) = |X(f)|2 ,cke S Thg4��w`���hS �¬m ��mfT"m �Zl�csS ik|^�bckg{T`l�rbmfu"�\}`rY�qTÏë

energy spectral density).× T`| Tqm �Z| uh���qc���mUg

mfuÏm csm | }q��³4S uÙmfu`� �qcpmfT`rY���Z�qTqmUgxrY�qu`�Fourier

zUe S cfg$m �ZSçckS is|^�bcfg�T T`SU}Á�quZS }`z T rY�Z��S ��m �¬mfT`lëö�qwq³4l�mku�m csm | }q��³4S u�mfu`��wqoq}qmfu`�Zly�`g�Thl�mkT`oZ}hS^ms³4r���l��qcnrY�Z��S ��m ��mfT

fz¬e S ckg�m ��S·ckS is|^�bcfg�T¾m �Zl

mfTqoq}`S^mk³4rY�Zl�ï =Ä�ÅjÆ�Ô�Ç�È0T´Ê%ËÌÅõ®cs³4| u`�Z�qc�mfu�r�t��qT

x(t) = e−αtu(t).

è �qcsmkT`r����Z�qTqmUgxrY�q�hlFourier

Th�¬mfu`�·mfu`�NrYtZ�qTqmfu`lvcke S Thg�u

X(f) =1

α + j 2πf,

�hT�g��dw`���hSU�¬m ��mfT�m ��lvcsS ik|^�bckg{T`lnrbmfu½�\}`rY�qT�cke S T�g��

G(f) =1

α2 + 4π2f 2.

² csS ik|^�bckg{T~rbm �ZSw`cs|Ugxu���t·mk³4S�rY�Z��S u�m t�ms³4S[−B,+B] cfe S Thg\�

∫ +B

−BG(f)df =

1

πftan−1

2πB

α.

à�� ð æ<þÿåP������� ��æñ���Mg��ý�����y�ìMç�c�ÿ�Fourier

õ®TPz �`rYuh���qc � ��³4|¬e�l»Tqw`�`z cfg{µ � � �qcs|¬g´��}$¼hT`r�g��h}ß¹jcp³4| tZ�qTqmfT�Tqw`��m �0¹jcp³4|Ue{T�ms³4Sò�qcsmkT`r����Z�qTqmUgxrY�ZÀ4SFourier.

î e{Tç¼hT`r�g��htÛg{zUg{�¬m ��mfT�cke S T�gñ��mUgñuÞ�qcsmkThrY�����qT`mUg�rY�q�hlFourier

cfe S ThgòikS T`l¾�b| T`�q�`g��h�hlm cpoqckrbm tZl � ��ThgYmfu�Th�¬m �~gxrY����cfgb�hThg��bg{T�mkuZSyT`S^mUe�rYm | u`��u~�qcpmfT`rY���Z�qTqmUgxrY�q�

Fourier.

ô�Ø

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× ��ñ Ü µ Ü ¸ �¸ tZ�qT î csmfT`rY���Z�qTqmUg�r��q�`l

Fourier

x(t) X(f) = F [x(t)]

1 α1x1 + α2x2 α1X1 + α2X2

2 x(t− t0) X(f)e−j 2πft0

3 x(αt) |α|X( fα)

3a x(−t) X(−f) (= X∗(f)�Yg{T�w`| Tq�b�qTqmUg��h��rYtZ�qT

)

4 x(t) = X(t) X(f) = x(−f)

5 x(t)ej 2πf0t X(f − f0)

5a x(t) cos(2πf0t)12X(f − f0) +

12X(f + f0)

6 dnx(t)dtn

(j 2πf)X(f)

7∫ t−∞ x(t′)dt′ (j 2πf)−1X(f) + 1

2X(0)δ(f)

8 x(t) =∫+∞−∞ x1(t− λ)x2(λ)dλ

= x1(f) ∗ x2(f) X(f) = X1(f)X2(f)

9 x1(t)x2(t)∫+∞−∞ X1(t− λ)X2(λ)dλ = X1(f) ∗X2(f)

¸ �Z�qcke�³4rY�Q R4SUT u¬oquq�`oqtZ|^³4�qT m �Zl��qu`| ��tZl ∫+∞

−∞ x1(t − λ)x2(λ)dλuZS u`�q}`� csmfThg®� ���E+f�j¨ £ °�_Ya � ¥ ¡â�-F4V � °4©

(convolution integral)�hT�g�rY�Z�¬¼hu¬oqe{� csmfThgY�qcPT`rbm ck|Ue�rY�hubÂ

x1(f) ∗ x2(f) =

à�J ï ¾ås�næ�ˤ�Û����� ��æñ���Mã��ý�����y�¯Mç�c�ÿ�Fourier

õ®T�z �`rYuh���qc2�qcs|¬g´��}�Tqwqoq}ßw`Th| T`z cfe �b�qTqmfT®�qcpmfT`rY���Z�qTqmUgxrY�ZÀ4SFourier. º T�w`|^À'mfT$w`ikS^m c%Tqw`uhz ckg��hSU��uZSfÚ

mfThgYck���hu¬oqT�Tqw`��mkuZS�uh|Ug�r��q��mkuh�·�qcsmfT`rY���Z�qTqmUg�r��qu`�Fourier.

• F [Aδ(t)] = A

• F [Aδ(t− t0)] = Ae−j 2πft0

ôZÝ

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• F [A] = Aδ(f)

• F [ej 2πf0t] = Aδ(f − f0)

• F [A cos(2πf0t)] = A2[δ(f − f0) + δ(f + f0)]

• õ®cp³4| uh���qcPm �ZSw`ck|Ug�uhzUg��ht�rY�ZS }`|^m �ZrY�dzUipoZmfT�ëideal sampling waveform)

ys(t) =+∞∑

m=−∞δ(t−mTs).

ë Ø� à ï

è �qcpmfT`rY���Z�qTqmUgxrY�q�`lFourier

cke S Thg�u

Ys(f) = fs+∞∑

n=−∞δ(f − nfs)

ë Ø� í ïP r"Ñ�Ç�È0TòëqÓº u�w`cs|¬g�u`zUg��h�¾rYtZ�qT¾Ø� à T`S Tqoq�ZcpmfThg\rYcPrYckg{| }

Fourier³4l

ys(t) =+∞∑

n=−∞Yne

j 2πnfst,

�qw`u`�Ys =

1

Ts

∫ +Ts/2

−Ts/2δ(t)e−j 2πnfstdt = fs.

Q Ü |UTys(t) = fs

+∞∑

n=−∞ej 2πnfst,

�hThg�ck��T`|U�q�`� uZS^mfT`lmku��qcpmfT`rY���Z�qTqmUgxrY�q�Fourier

�hThg\rbmfT�z �Zu��qisoZ�Ys(f) = fs

∑+∞n=−∞F

[

ej 2πnfst]

+ fs∑+∞

n=−∞ δ(f − nfs).

•F [u(t)] = 1

j 2πf+1

2δ(f).

ë�Ø� qô�ïP r"Ñ�Ç�È0TòëqÓQ R4��u`�Z�qc

u(t) =∫ +∞

−∞δ(λ)dλ,

�hThg�ck��T`|U�q�`� uZS^mfT`lmkuZSnm csoqcsrbm tFourier

�hThg�rYmkT�z �Zu��qipoq� � w`T�e�|US u`�Z�qc

F [u(t)] = 1

j 2πf.1 +

1

2.1.δ(f).

ô Ã

Page 54: simata kai systimata - users.auth.grusers.auth.gr/~hadjidem/simata kai systimata.pdf · @)A B C D E F G H I H A (TIME DOMAIN ANALYSIS) J LKNMO P Q R4SUTWVYXZV\[^]`_badcfe S ThgjikS

• î cpmfT`rY���Z�qTqmUg�r��q�`lFourier

m �ZlvrY�ZS }`|^m �ZrY�Zlsgn(t) =

² r���SU}`|^m �ZrY�sgn(t)

u`|Ue{� csmkT�gY³4l

sgn(t) =

+1 t > 0

−1 t < 0.

ë Ø� Z÷qï

× T`|UTqm �Z| u`�Z�qc���mUg»�sgn(t)

z¬e�zUckg»mfuçw`| �`r����quÛm �Zl��qcsmkTZ¼qoq��m tZlt = ² rY�ZS }`|^m �ZrY�"T`��m t

cp�h�\| }`� csmkT�g�rY�ZS Th|^m tZrYckg�mfu`�N�qu�SUT`zUg{Thexuh�d¼htZ�qTqmfu`l®³4lsgn(t) = 2u(t)− 1. ë Ø� qøZï

R4��T`|U�q�`� uZS^mfT`l�mfuZS�m cpoqcsrYm tFourier

�hThg'rbmfTOz �Zu"�qisoZ��T`��m tZl½m ��l�r���ikrYcp³4l � �hThg2oqT`�¬Ú¼`}hS u�S mkThl��w`�O/��·m �·rY��ikrY��Ø� qô � w`The{| S u`�Z�qc

F [sgn(t)] = 1

j πf.

ë Ø� 0qï

´ | �Zr�g{�quqw`u�g�u`�Z�qc®m^À4| T�m ��Syg{zUg{�¬m ��mfT í m �ZlPw`T`|UTq�b| }`��uh��Ýà ·�hThg�w`The{| S u`�Z�qc®m �·rY��isrY�

F[

1

j πt

]

= sgn(−f)

�hThgYm cso`g´��}F[

1

πt

]

= −j sgn(f). ë�Ø� 1�qï

à·à �a�Oã6M�ý ��æñ��sbed �Pý�ä M¢KN��¾åP�Pý�MWý�ä �¾æñ�ß�üÞ ¾å��ä ������Pý�ä��ån ~KNMO�y���~��ä ¸ ����å��¯MWý�ä

² rY�ZS }`|^m �ZrY�·�qcpmfT`��uh| }`lu`|Uexr�¹\���hc®�qc�mfu�S�m ��w`uç�0 �H(ω) =

∫ +∞

−∞e−j ωλh(λ)dλ,

�hThg�w`T`| T`m ��| uh���qc � rY�����h|Ue S u�S mkT`ld�qcvmfuZSdm ��w`us�qø � u�u`w`uhe�uhldzUe{z cfgjmku¾�qcpmfT`rY���Z�qTqmUgxrY�q�Fourierm �Z����S^mfu`lNrYtZ�qTqmfu`l

x(t) � ë¤oqT`�¬¼h}`S uZS^mfT`lNcpw�e¶wqoqisuZS���w`�O/\�¾�¬mUgÌT`S^mUe�m �Zly�h���`o`g��htZlNrY�Z��S ��m ��mkThlω

is��u`�Z�qc�csz^Àÿm �¢rY�Z��SU�¬m ��mfTf = ω/2π

ï � ��mUgß]�V� �� � £ [^]�V\]Û_ � [paO}\� £ �`© �s� �sa ¥ò� V\]Ù¡ £ � ©�[p�_ � [pa`V^¦P]q_Ya�[ ¥ Vb_ ª Fourier

[^]�©@+ £ �� �V\[ ¥ +�¨h©naZ¡ ª + £j¥ V\]�© �H(f) = F [h(t)]. ë ØZØ� qï

Ü w`��m �·rY��isr��dT`��m t·w`| uq�h��wqm ckgY��Thg��NT`S^mUexrbm | u`�\�NrY��isrY�h(t) = F−1[H(f)].

ë�ØZØ�Ø ïô í

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Ü l�w`}`| u`�Z�qcnm^À4| T¾mfu�SNm �¬w`u í ø � uqw`uhexu`l·z¬e�zUckg�m �ZS·isµUu`z u�rbm �ZSdcfe�r�u`z ux(t) � rY�ZS Th|^m t�r�ckg\m �Zl�h| uh��rYmUg´��t�lTqw`�`�h|UgxrY�Zl �

y(t) =∫ +∞

−∞x(λ)h(t− λ)dλ,

�qw`u`�$cswhe�wqoqiku�S»¹\ikrYT`�qch(t, λ) = h(t−λ) � zUg{�¬mUg^mfu�rY�Zrbm �Z�qT<cfe S T�g rbmfTZ¹jcs| � = �Pw`ckS�¹\�Z�`e{� u`�Z�qc%��mUg

udm ��w`u`l�T`��m �`l®g�rY���Zckg���w`�dm �ZSPw`| u�¡¬w`�q¹\csrY�y�¬mUg���cfe�r�u`z u`lPcs�\T`| �q�`�UcpmfThg�rYc$��|U��S ut = −∞ ��Thg

mfu®rY�Zrbm �Z�qTP��|Ucs�qu`�ZrYc'm ��m c$ë¤cfe���c»�q�Zz ckSUg��ht�isµUu`z uhï = × The{| S u`�Z�qc»mfu®�qcpmfT`r����Z�qTqmUgxrY�q�Fourier

�hT�gmk³4S0z �Zu®�qcpoZÀ4S0T`��m tZl4m �Zl»rY��isrY�Zl'�hThg�ik��uh���qc � rY�Z�q�Y³4SUT��hThg��qc2m �ZS6g{zUg{�¬m ��mfTe<m �Zl4w`T`| T`��|U}`��uh�Ý1 ¾ë¤�bg{T�mkT�u¬oquq�`oq��| À4�qTqmfT�z¬g´wqoZÀ4rYcs³4lfï �

Y (f) = H(f)X(f),�qw`u`�

H(f) = F [h(t)] ��rY�ZS }`|^m �ZrY�½�qcsmkTh��u`| }hl�ëöcp����| T`r��qisS ��rY�ZS T`|^m tZrYcfgYm �ZlnrY�Z��S ��m ��mkT`lfT`S^mUeYm �ZlP�h���`o`g��htZlr�����S ��m ��mkThl

ωï � T`w`��m �ZSnu`w`uhe{T�w`The{| SUu`�Z�qcP�hT�gYm �·rY��ikrY�

H(f) =Y (f)

X(f).

× T`| Tqm �Z| u`�Z�qc��¬mUg'�WrY�ZS }`| m ��r����qcsmkTh��u`| }hl~cke S Thg4u"oq�¬��uhl�mfu`�W�qcsmkT`���Z�qTqmUgxrY�qu`�Fourier

m �ZlcsµU�`z u`�½w`| u`lmfu��qcpmfT`r����Z�qTqmUgxrY�q�

Fourierm �Zlvcfg�rY�hz u`� � ��w`��m ��S�w`| uO¡�w`�Z¹\ckrY�½�¬mUgYmfu~rY�Zrbm �Z�qT

�Z| cs�quh��r�c0m �nrbmUg��b�qtncs�\T`| �qu¬�bt�lßm �Zl$ckgxrY�`zUu`� =»è oq�¬�bu`l�T`��m �`l�cke S Thg�T`SUcsµ }`|^m ��mfu`lßmfu`��rYtZ�qTqmfu`lm �Zl®cfg�rY�hz u`� =² m cpoqcs��mfThe{TßTh�¬m t<rY��isr��0�Zw`u`|Ucke S T���| ��r\g{�quqw`uhg´�¬¹\cke^�hThg ³4l'u`|UgxrY�q�`l'm �Zl2rY�ZS }h|^m �ZrY�Zl%�qcpmfT`�\u`| }`l =õ®T�z u`�Z�qc®�`�Z³4lvrYmkT�csw`�`�qcsS T~�¬mUg�u�u`|UgxrY�q�`lT`��m �`lv�bcsSUg��hck��csmkT�g��qc�m �N��|Ut�r��dms³4S��qcsmfT`rY���Z�qT�Ú

mUgxrY�ZÀ4SLaplace.

íe{'Å�Æ�Ë=u�ÊCSõ®T�¼h| u`�Z�qc6m �nr���SU}`|^m �ZrY��qcsmkTh��u`| }hlßmfu`��b| Th�q�`g��hu`�yr���rYm t��`Tqmkuhl6mfu`� ¸ ��tZ�qTqmfu`lnØ^÷ � ck��T`| �q�ZÚ� uZS^mfT`lvmfu�S�w`T`| Tqw`}hS^³Áu`|UgxrY�q� = ² z¬g�Th��u`|Ug���t½ckµUexrb³4rY�dmfu`�·�h���`oZÀ4�qTqmfu`lT`��mfu`�Ncke S Thg��

LCd2y

dt2+ y = x(t),

�hThg\cs�\T`| �q�`� uZS^mfT`lmfu�S�m cpoqcsrYm tymku`�N�qcsmkThrY�����qT`mUg�rY�quh�Fourier

�hThg\rbmfT�z �Zu��qisoZ� � w`The{| S u`�Z�qcRCj 2πfY (f) + Y (f) = X(f),

�hThg\rY�ZS cpwqÀ4lH(f) =

Y (f)

X(f)=

1

1 + j 2πfRC.

ë ØZØ ÝZïè m ��w`u`lvT`��m �`lrY�Z�Zwhe�wqm ckgY�qc�mfu�S�m �¬w`u��Zô¾ë¤cfe S T�g

ω = 2πfï =

¸ mfu�S�e�z¬g�u¾m ��w`u~��TqmkT`oZt��bu`�Z�qc�T`SNcs�\T`| �q�`r�u`�Z�qcvmfuZSdm csoqcsrbm tNmfu`���qcsmkThrY�����qT`mUg�rY�quh�Fourierrbm �ZS�h| uh��rYmUg´��tNTqw`�`�h|UgxrY�½T`��mfu`�·mfu`�NrY�Zrbm tZ�qTqmfu`l � �Nuqw`uhe{T~zUe{z csmkT�g�Tqw`��mfuZSnm ��w`u�÷qø �

h(t) =1

RCe−t/RC × u(t).

ôZô

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- 1 0 - 8 - 6 - 4 - 2 0 2 4 6 8 1 00 , 0

0 , 2

0 , 4

0 , 6

0 , 8

1 , 0

f r e q u e n c y f

| H ( f ) |

f3-f3

1/ 2

_

¸ ��tZ�qT~ÝZÝh º u~��}`rY�`T½wqoq}qmfu`�Zl®mfu`�NrY�Zrbm tZ�qTqmfu`l�mfu`� ¸ ��tZ�qTqmfu`l·Ø^÷ =è

ð/2

-ð/2

f0

¸ ��tZ�qT~Ý Ã Â º u~��}`rY�qT���}`r�cp³4l®mfu`�NrY�Zrbm tZ�qTqmfu`l�mfu`� ¸ ��tZ�qTqmfu`l·Ø^÷ =Q R4��u`�Z�qc

H(f) = F [h(t)] = 1

RC

11

RC+ j 2πf

=1

1 + j 2πfRC.

àdù ( �Ï�� �Ë

¸ m �ZS0whT`| }q�b| T`�\uNØ øßz ckg{µ T`�qcñ��mUg�isS Tv�b| T`�q�`g��h�nrY�Zrbm �Z�qTPrY�Z�Zw`cs|Ug{�\is| csmkT�gU³4lò�U¹j�q�`l � mfu`��u`w`uhe�uh�mfT���T`| T`�`m �Z|Ug�rYmUg´��}��hTZ¹\u`|¬e�� uZS^mfThgÒTqw`��m �~rY�ZS }h|^m �ZrY���qTqmfT`�\u`| }`l

H(f)ëω = 2πf

ï = Ü S½w = � =cke S Thg |H(f)| = K � �qw`u`� K rbmfTZ¹\ck| } � m ��m cz csS·��w`}`|U��cfg\w`Th| T`�q�`| ��³4rY�~rbmfu�wqoq}qmfu`l�mku`����}hr�Ú�qTqmfu`lrbm �ZSnisµ uhz u =

%��>��� ܨ� ³ 4^Z�8�8�!��D9 �����>�>�h8�ZRC

¿ lOisS T w`T`| }hz ckg����`TÙT`S Th��is|Uu`�Z�qc�mfu �b| Th�q�`g´���ÙrY�Zrbm �Z�qTÞmfu`� ¸ ��t��qT`mkuhlÞØ�÷ = ² rY�ZS }`|^m �ZrY��qcpmfT`�\u`| }`lvmfu`�NzUe{z csmkT�g�Tqw`��m �NrY��isrY�~ØZØ^Ý � m �ZSnu`w`uhe{T��b| }`�\u`�Z�qcP³4l

H(f) =1

1 + j (f/f3),

ô�÷

Page 57: simata kai systimata - users.auth.grusers.auth.gr/~hadjidem/simata kai systimata.pdf · @)A B C D E F G H I H A (TIME DOMAIN ANALYSIS) J LKNMO P Q R4SUTWVYXZV\[^]`_badcfe S ThgjikS

|H(ù)|

f

B-B

(á)

f

(â)öÜóç

¸ ��tZ�qT~Ý í Â$ëöThï º u~��}hrY�qT�wqoZ}`mkuh��lvmfu`�½g{z T`S¬g´��u`���U¹j�qu`� = ë ¼�ï º u��\}`rY�qT���}hrYcp³4l®mfu`��g�zUT`SUg��huh��¬¹\�qu`� =t

H(f) =1

1 + (f/f3)2e−j tan−1(f/f3)

�qw`u`�f3

cfe S Thg\�`g{T�w`T`| }`�qcsm | u`lvw`uh�Ncsµ Th|^m }qmfThg�Tqw`��mfT�rbmfuhgx��cke{T�mfu`�·�h���`oZÀ4�qTqmfu`l �f3 = 1/(2πRC).

R'w`u`�`isS^³4l½mfu�wqoZ}`mkuhlNmfu`���\}`rY�qT`mku`l½m ��l½csµ �`z uh���hT�g%����}hrY�¾mfu`��z¬e�zUu�S^mfThg � ThS^mUe�rYmkuhgx��T � Tqw`�mfu`�Zl®m �¬w`uh��l|H(f)| = 1

1 + (f/f3)2�hThgθ(f) = − tan−1(f/f3).

º T�z¬g�T`��|U}`�q�qTqmfT�wqoq}qmfu`�Zl®�hT�g���}hrYcp³4lvzUe{z uZS^mfThg\rbmfT ¸ ��tZ�qTqmfT~ÝZÝy��Thg\Ý Ã � T`S^mUexrbmfuhg���T =² w`T`| }`�qcsm | u`l

f3cke S Thg��nrY�Z��SU�¬m ��mfTnm �Zl<cfg�rY�hz u`��Yg{Tym �ZSPuqw`uhe{Tdmfu�wqoq}qmfu`l$rYmkTq¹\cs| tZl6�hTqm }hr�Ú

mfT`rY�Zl�m �Zl�ckµ �`z uh�"cke SUThgñe�rYu¢w`|Uu`l~mfu1/√2

mfu`�Wwqoq}qmfu`�Zl~m ��l~cfg�r��`z u`� � zU�¬oqT`zUtO�"g�rY���Zl�m �ZlcsµU�`z u`�½cke S Thg\exrY�·w`| u`lvmfu

1/2m ��lg�rY���Zu`lvm ��lvckgxrY�`z uh� =

%��>��% GIH6Z>AJ!��D���@<Lª²8�����Pw`uZ¹jipmfu`�Z�qcP�¬mUg�is��u`�Z�qc®isS T~g{z T`SUg��h�¾�U¹j�q� � �qcPrY�ZS }`|^m �ZrY�·�qcpmfT`��uh| }`l

H(f) =

Ke−j 2πft0�Yg�T −B < t < B

0T`o�oqu`�

.

º TP��T`|UTq�`m �Z|UgxrbmUg��h}®mfu`�$�U¹j�qu`�<T`��mfu`�$zUe{z uZS^mfThg¬rbmfu ¸ ��t��qTvÝ í = × T`| Tqm �Z| uh���qc4�¬mUgU�<w`cs|Ugxu���t$mk³4SrY�Z��S u�m t¬mk³4S�rbm �ZSw`ck|Ug�u���t

[−B,+B] w`T`| T`�qikS ckg\T`S T`o�oquhe�³'m � � csS ÀÏ�¬oqcsluhg�}`o�oqcklvrY�Z��S ��m �¬m ckl�h��¼huZS^mfThg =

ôZø

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0

tt0

2 K B

¸ ��tZ�qT~ÝZô� ² ��| u`�ZrbmUg��ht½Tqw`�q�h|UgxrY�h(t)

mfu`�Ng{z ThSUg��hu`���¬¹\�quh�

² �h| u`�ZrbmUg��ht½Tqw`�q��|Ug�r��Nmfu`�NrY�Zrbm tZ�qTqmfu`lP�qcPr���SU}`|^m �ZrY�d�qcsmkTh��u`| }`lm ��Snw`T`| T`w`}`S^³Ácke S Thg\�

h(t) = F−1[H(f)] = k∫ +B

−Bej 2πf(t−t0)df = 2KBsinc[2B(t− t0)],

�qw`u`�½�NrY�ZS }`|^m �ZrY�sinc

u`|¬e��UcpmfThg�³4l

sinc =sin(πz)

πz.

² �b| T`�\g��ht�w`T`| }`rYmkThrY��m ��lyrY�ZS }`| m ��r���lh(t)

zUe{z csmkT�g¶rbmfu ¸ ��tZ�qT�ÝZô = × T`| Tqm �Z| u`�Z�qc���mUg¶cke S T�gh(t) 6= 0 �Yg{T

t < 0 � ckS^À¢¹\Tyipw`|Ucpw`c<S TdgxrY���Zckgh(t) = 0

�Yg{Tt < 0 = ¸ ��SUcpwqÀ4l � mfudg{z T`SUg��h�N�je´oZm | u

z ckS��Zw`u`| cfe�S T�cke S T�g�Thg�mUg�u`�h| T`mUg��h� =_ P ÑRQ\Ó�Ñ�ÓQ R4SUT��b| T`�q�`g��h��rY�Zrbm �Z�qTNis��cfgYrY�ZS }h|^m ��r��d�qcpmfT`�\u`| }`l

H(ω) =ω2c

ω2c − ω2 + j√2 ωcω

,

�qw`u`�ωc

w`T`| }`�qcsm | u`l = × uhg{}Ûcfe S Thg»�OTqw`�q��|UgxrY�Owqoq}qmfu`�Zl~�hThg4w`uhg{}Þ�OTqw`�q�h|UgxrY�¢��}`r�cp³4l;× À4l

rY�Z�Zw`ck|Ug��\is| csmfThgb³4lv�¬¹\�q�`l;

à ' î æñ���Mg��ý�����y�¯Mç�s��äHilbert

õ®cs³4| u`�Z�qcPikS T�rYtZ�qTx(t)

�hThg\z �Z�`g�u`�Z|^�bu`�Z�qc®ikS T�S isu�rYtZ�qTx(t)

�qc�mfu��qcpmfT`rY���Z�qTqmUgxrY�q�

x(t) = x(t) ∗ 1πt,

ô1

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z ��oqT`z tx(t) =

∫ +∞

−∞x(λ)

1

π(t− λ)dλ.

ë ØZØ Ã ïè �qcpmfT`r����Z�qTqmUgxrY�q�hl2T`��m �`l4u�S uh�q}`� cpmfThgZ_ � [±ahV^¦�]`_ba�[ ¥ V�_ ª © Hilbert.

× T`| Tqm �Z| u`�Z�qc'�¬mUg uß�qcsmfT`rY���UÚ�qTqmUgxrY�q�`lvTh�¬m �hlvcke S Thg���|UT`�q�`g��h�`ln�qcpmfT`r����Z�qTqmUgxrY�q�`l =õ®T �qcpoqcpm tZrYuh���qc�mfu ��}`r��qT�mfu`�Ûr�t��qT`mkuhl

x(t) = × The{| S uh���qc�mfu �qcsmkT`r����Z�qTqmUgxrY�q�Fourier,

X(f) = F [x(t)] � ��|U��r\g��qu`w`uhg�uh��S m cslvm �ZSyg{zUg{�¬m ��mfT�·mkuh� × e S T`�hT � �F [x(t)] = X(f)F

[

1

πt

]

,

t � oq�¬��³ m �ZlyØ� 0� �X(f) = −j sgn(f)X(f). ë ØZØ í ï

² rY��isr���ØZØ í rY�ZS z ikckgjmfu¾��}hrY�qT�mfu`�~rYtZ�qTqmfu`lx(t)

�qcmfu¾�\}`rY�qT¾mkuh��r�t��qT`mkuhlx(t) = Ü w`��m �

rY��isr��dT`��m tdw`| u`�h��wqm ckg��¬mUg��qc�mku��qcsmfT`rY���Z�qTqmUg�r��q�Hilbert

mfu½wqoq}qmfu`lvw`T`| T`�qikS ckg�Th�qcpm }�¼`oq�¬mfu �∣

∣X(f)∣

∣ = |X(f)| .¿ lvw`| u`lvm �·��}hrY� � oZTh�¬¼`}hS u�S mkThl��w`�O/��N��mUgY�N��T`S mkT`rYmUg´��tN�quZS }`z T

jcs�h��| }h� cpmfThg�³4l

j = ej π/2, −j = e−j π/2,

�hTqmfTqoqt��bu`�Z�qcyrbmfu¾rY�Z�qik| T`rY�qT¾��mUg¶uhg�¹\cpmUg��hiklyrY�Z��S ��m �¬m ckl�w`u¬o�oqTqwqoqT`r�g{}`�Uu�S^mfT�gÌ�qce−j π/2 � �hThguhg2T`| SU�¬mUg��hikl�rY�Z��S �¬m ��m ckl½�qc

j = ej π/2 = Ü ��m �OrY�Z�qThe SUckg%��mUg%�qc·mfu"�qcpmfT`rY���Z�qTqmUg�r��q�Hilbert�¢�\}`rY�¢�qcsmfTqmfuqwhe{� cpmfThgñ�hTqm } −π/2 �Yg{TÛ¹\csmUg´��isl�rY�Z��S �¬m ��m ckl��hThgñ�hT`m }

+π/2�Yg{T�T`|US �¬mUg��hikl

rY�Z��S ��m �¬m ckl =P Ö�Åqå\ÐbÍvTwQCÑ�Ñ=SjËÌÅõ®cs³4| u`�Z�qc�mfu½w`|UTq�b�qTqmUg��h��rYtZ�qT

x(t)�hThg�zU���`gxu`�Z|^�bu`�Z�qcPmfu�rYtZ�qTz = x(t) + j x(t).

ë ØZØ ôZïº unrYtZ�qT

zuZS u`�q}h� cpmfThgbaq�paf�b q[ ¥ + ª V\¨`_ba =%º ur�t��qTvTh�¬m �zUcsS$cke SUThg�w`| T`���`TqmUg��h� � �hThg�¹\Tz cke{µ uh���qc

��mUgYik��cfgY�\}`rY�qT��q�ZS u�rYmUg�lP¹\cpmUg��hiklvrY�Z��S �¬m ��m csl = Q R4��u`�Z�qcF [z] = F [x(t) + j F [x(t)]

= X(f) + sgn(f)X(f),�hThg�m cpo`g��h}

Z(f) = F [z] =

2X(f)�Yg�T

f > 0

0�Yg{T

f < 0.

ë ØZØ^÷qï

ð cfe{µ T`�qc�csw`u`�qisS ³4lP��mUg�mfuNrYtZ�qTzik��ckg��\}`rY�qTN�`��S u½rYmUg�lß¹\cpmUg��hiklPrY�Z��SU�¬m ��m csl � ��Thg�mfuN�\}`rY�qT

T`��m ��rY�Z�Zwhe�wqm ckg¶�qcnmfu�T`S^mUe�rYmku�g���u���}hrY�qT¾mfu`�~w`|UTq�b�qTqmUg��hu`��rYtZ�qTqmfu`lx(t) = Q R4S T�m ismku�g�u�r�t��qT

uZS u`�q}`� csmfThg\rYtZ�qTSSB,

Tqw`��mkT�T`| �bg��h}single sideband.

ô1�

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ó T Å�ËCÑ�Æj{'Õ®Ñ�Óº u�r�t��qT

z(t)�Zw`u`| cfeYSUT�z¬g�Th�qu`| �Y³2¹jcke��qc�mfu���is|Uu�SyrYtZ�qT

ej ω0t ='º u~zUg{T`�qu`| ��³4�qisS u~rYtZ�qT�cke S Thgmfu

z ej ω0t

�hThg\is��ckg��q�ZS u�m �ZSn}`S ³Ï�\T`rY�qTqmUg��htNw`cs|Ugxu���t = × The{| S u`�Z�qc®m^À4| T�mku��qcpmfT`rY���Z�qTqmUgxrY�q�Fourier,

F[

z ej ω0t]

= Z(f − f0).ë ØZØ øZï

Ü w`��m �ZSNm cpoqcs��mfThe{T�Th�¬m t�rY��ikrY�~��The SUcpmfThgÌ�¬mUg¶mfu��\}`rY�qT�mfu`��zUg{T`�qu`| ��³4�qisS u`��rYtZ�qTqmfu`ldcke S ThgmfuWe{zUg�uO�qc�mfu��\}`rY�qT�mfu`�¾T`|U��g��huh��rYtZ�qTqmfu`l

z(t) � �qcpmfTqmfuqwhgxrY�qikS u��hTqm }�m �¾rY�Z��S ��m ��mkTf0

mfu`���ik| uZS^mfu`lrYtZ�qTqmfu`l = Ü wh��mku�rYtZ�qT½Th�¬m ��w`The{| S uh���qc®mfu�w`| Tq�b�qTqmUg��h�

SSBrYtZ�qT

Re[

z ej ω0t]

.

Ä�ÅjÆ�Ô�Ç�È0T´Ê%ËÌÅ�Ñ=SjË�Å\ÍzuNUSSB

ð e{z csmkT�g<mfuÙrYtZ�qTx(t) = cosω0t

�hThgß� ��mfu`���`c�mfuÏr�t��qTx(t) = º uÙr�t��`T

x(t) = cos(ω0t)cp����| }`�UcpmfThg\�¬w`���`g���ThzUg��ht½�quh| ��t·³4l

x(t) = cosω0t) =1

2

(

ej ω0t + e−j ω0t)

.

º u·rYtZ�qTx(t)

w`| uq�h��wqm ckg�Tqw`�·mkudr�t��qTx(t)

T`S®w`u¬oZoZT`wqoZThr�g{}`rYu`�Z�qc<mUgxl<¹\cpmUg��hikl$r�����S ��m ��m csl$cpwhe−j �hThg�mUg�lT`| S ��mUg��hislcswhe

+j = Q R4��u`�Z�qcx(t) = 1

2(−j ej ω0t + j e−j ω0t)

= 12 j(−j2 ej ω0t + j2 e−j ω0t)

= 12 j(ej ω0t + e−j ω0t) ,

�hThg�m cpo`g��h}x(t) = sin(ω0t).¸ mfu~e�z¬g�u~Tqw`u�m ipoqcsr��qT½�hT`mkT`oZt��bu`�Z�qc®ThS���| �Zr�g{�quqwhuhg{t�r�u`���`c�mfuZSn�bcsS¬g´����m ��w`u�ØZØ Ã Â

x(t) =∫+∞−∞ x(λ) 1

π(t−λ)dλ

= 1π

∫+∞−∞

cos(ω0λ)t−λ

= sin(ω0t).

) g{T�m �ZS�Tqw`�`z cfg�µU�Nmk³4Snw`T`| Tqw`}hS^³Á��| �Zr�g{�quqw`uhg{tZrYT`�qcPmfuZSnm ��w`u∫ +∞

0

sin(ω0x)

xdx =

1

π.

÷0

Page 61: simata kai systimata - users.auth.grusers.auth.gr/~hadjidem/simata kai systimata.pdf · @)A B C D E F G H I H A (TIME DOMAIN ANALYSIS) J LKNMO P Q R4SUTWVYXZV\[^]`_badcfe S ThgjikS

àèW î æñ���Mg��ý�����y�¯Mç�s��äLaplace

è �qcsmfT`rY���Z�qTqmUg�r��q�`lLaplace

cfe S T�gjT`S mUe�rbmfuhgx��u`l�mfu`���qcpmfT`rY���Z�qTqmUgxrY�qu`�Fourier

ckS �`l�rYtZ�qTqmfu`lx(t) � �hThg�ik��ckgbmfu½wqoqcsuZS ip�`m �Z�qT��¬mUg��¬w`}h| ��ckgY�Yg�T�ck�Z| ��m cs| u�rY�ZS u¬oqu�rY�ZS T`|^m tZrYcs³4S � rYcPrY�����h|Ug�r���qc�mfu`�Zlv�qcpmfT`r����Z�qTqmUgxrY�qu`�Zl

Fourier. è �qcsmkThrY�����qT`mUg�rY�q�hlFourier

uh|Ue{� cpmfThg\�qc�mfu�S�m ��w`uç�qø �X(f) =

∫ +∞

−∞x(t) e−j2πftdt,

ë ØZØ�qï�hThg�w`T`| Tqm �Z| uh���qcv�¬mUg�cke SUThg

|X(f)| ≤∫ +∞

−∞|x(t)| dt. ë�Ø�Ø��qï

Ü w`�Om �ZS�m csoZck��mkThe{T"T`��m t�r���ikrY��¼qoqisw`u`�Z�qcN�¬mUg%�Yg�T¢S T"��w`}`| ��ckg2u"�qcsmkThrY�����`TqmUg�r��q�`lFourier,w`| isw`ckgYmku�u¬oquq�`oqt�| ³4�qT

∫ +∞

−∞|x(t)| dt

S Tcfe S T�g�w`ck| Tqmk³4�qisSUu = Ü �¬m �vw`ck|Ug�uh|Ue{� ckgqmkurY�ZS u¬oquvmk³4S<rY�ZS T`|^m tZrYcs³4Sx(t)

�Yg{TmUg�l0uqw`uhe{csl<�Zw`u`| ckeS T�u`|Ugxr�¹\cfeYu��qcpmfT`rY���Z�qTqmUgxrY�q�`l

Fourier.¸ mfT�csw`�`�qcsSUTn¹\T�z �`rYu`�Z�qcòmfu`�Zlò¼`Thr�g��hu`�Zl<m �¬w`uh��l6mfu`�v�qcsmkThrY�����qT`mUg�rY�quh�

Laplace��Thg`mfu`�vT`S�Ú

mUexrbm | u`�\u`�d�qcsmkThrY�����qT`mUg�rY�quh�Laplace,

��³4|Uexl®Tqw`�hz ckg{µ � = × |Ue Sn�`�Z³4l®T`w`�½mfu`�Zl�m ��w`u`�ZlPuh|Ug�r��qu`�mfu`���qcsmkT`r����Z�qTqmUgxrY�quh�

Laplace�hThg�mfu`��T`S^mUgxrbm | �`�\u`��mkuh� � ¹\T·z cke{µ u`�Z�qc$wqÀ4lß�qcsmkTZ¼`T�e S u`�Z�qcßTqw`�

mfu`�Zld�qcpmfT`rY���Z�qTqmUgxrY�qu`�ZlFourier

rbmfu`�Zly�qcsmfT`rY���Z�qTqmUg�r��qu`�ZlLaplace,

À4rYm cnS T�r����Z�qTqmUexrYu`�Z�qc�`g{T��hTqoq��m cs| �Ncfg´����S T~�Yg�T�mfu`�Zlv�qcpmfT`rY���Z�qTqmUgxrY�qu`�Zl

Laplace.Ü l$�¬whuZ¹\isr�u`�Z�qc0��mUg`mfuyu¬oquq�hoZtZ|^³4�qT·rbmfu�zUcsµUg{�d�qisoqu`l<m ��l$rY��isr���l®ØZØ��z csSPcfe S Thg�w`cs| T`ms³4�qikS u �uqw`��m c�z ckS¾��w`}`|U��cfg'u¢�qcpmfT`r����Z�qTqmUgxrY�q�`l

Fourier�bg{T¢mfu"r�t��qT

x(t) = Ü S mUe%mfu`�WrYtZ�qTqmfu`lx(t)¹\cs³4| u`�Z�qc�m^À4| T�mku�rYtZ�qT

x(t) e−σt, σ > 0.ë�Ø Ý1 Zï

è �qcpmfT`r����Z�qTqmUgxrY�q�`lFourier

Th�¬mfu`�·mfu`�NrYtZ�qTqmfu`l®w`| uq�h��wqm cfg�Tqw`��mfuZSn�bcsS¬g´����m �¬whu�ØZØ� � ³4l

F[

x(t)eσt]

=∫ +∞

o−x(t) e−σt e−j2πftdt,

ë Ø Ý�Ø ï�qc2m ��S0cpwhg�wqoqisuZS0��w`�Z¹\ckrY�<��mUg

x(t) = 0�bg{T

t < 0 � u`w`�¬m c'mfuß��}qmk³��`|UgxuPmfu`�$u¬oquq�`oq�Z|^À4�qTqmfu`lñcfe S Thgmfu0−

T`S^mUe�mfu`� −∞ = Ü ��m �½z csSv¹\ism ckg�rYu�¼hT`|U�Nw`ck|Ug�uh|Ug�r��q� � zUg{�¬mUgY�¬oZT½mfT½rYtZ�qTqmfTNcs�\T`| �q�`� uZS^mfThgrYc�ikS T�rY�Zrbm �Z�qT��h}qw`uhg{TW��| uZSUg��ht�rbmUg����qt � m �ZS½uqw`uhe{T�¹\cs³4| u`�Z�qc

t = 0 = ² rY��isrY��Ø Ý�Ødw`The{| S cfgm �·�qu`|U��t �

F[

x(t)eσt]

=∫ +∞

o−x(t) e−(σ+jω)tdt = X(σ + jω),

ë�Ø^ÝZÝZï�qw`u`�·¹\ikrYT`�qc

ω = 2πf = µ }`S u`�Z�qcPm^À4| T½m �ZS�TqoZoqTq�btN�qcpmfT�¼`oq��m t�l

s = σ + j ω,

÷�Ø

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�hThg�w`The{| S u`�Z�qc � �bg{T�mku��qcpmfT`rY���Z�qTqmUgxrY�q�Fourier

mfu`�NrYtZ�qTqmfu`lyØ Ý1 �F[

x(t)eσt]

=∫ +∞

o−x(t) e−stdt,

�hThg\rY�����h|Ue SUu�S^mfT`ln�qc�mfu�S�m �¬w`u�Ø^ÝZÝdik��uh���qcPm csoqg��h}

X(s) =∫ +∞

o−x(t) e−stdt.

ë Ø Ý Ã ï

è T`S^mUexrbm | u`�\u`l<�qcsmfT`rY���Z�qTqmUg�r��q�`lFourier

csS �hl<rYtZ�qTqmfu`lx(t)

z¬e�zUcpmfThghTqw`��mfu�Sß�bckSUg��h��m ��w`u~�Z÷ �x(t) =

∫ +∞

−∞X(f) ej2πftdf,

�hThg\cpw`u`�`isS^³4lu�ThS^mUe�rYm | u`��u`l�qcsmfT`rY���Z�qTqmUg�r��q�`lFourier

mkuh�·r�t��qT`mkuhlyØ Ý1 ·cke SUThg�u

x(t)e−σt =∫ +∞

−∞X(σ + j ω)ej ωtd(

ω

2π),

tx(t)e−σt =

1

∫ ω=+∞

ω=−∞X(σ + j ω)ej ωtdω,

�hThg�m cpo`g��h} � �bg{T�m �·rY�ZS }`|^m �ZrY�x(t) �x(t) =

1

2π j

∫ s=σ+j ∞

s=σ−j ∞X(s)estds,

ë Ø Ý í ï�qw`u`��cpwhg�wqoqisuZSy�h}`SUT`�qcvm ��SyTqoZoqTq�bt½�qcpmfT�¼`oq�¬m tZl

s = σ + j ω = ¸ mfu�Sym ��w`u�Ø Ý í mfuσ

¹\cs³4| cke�mkT�grbmfTZ¹\ck| � � z �¬oqT`zUt���u¬oquq�`oqtZ|^³4rY���Ye SUcpmfThgÒ�hTqm }"�qt��hu`l��`g{T`l�cs�¬¹\cfe{T`lNw`T`| }`o�oq��oq��l�rbmfu�S~}`µUu�S Tmk³4S���T`S^mfT`rYmUg´�hÀ4S � rYmku��`g��bT`zUg��h��cpwhe�w`cszUu =× T`| Tqm �Z| tZrYTqm c�¬mUgju�m ��w`u`lNØ Ý Ã �qcsmkThrY�����`TqmUe{� ckg�mfu~rYtZ�qT

x(t)rYmku~rYtZ�qT

X(s)�hThgju~m ��w`u`l

Ø Ý í zUe{z ckgYmfu�SyT`S^mUexrbm | u`�\u�T`��mfu`�·mfu`�N�qcpmfT`rY���Z�qTqmUgxrY�qu`� =× | u���³4| u`�Z�qc<m^À4|UTdrbmfuZS®u`|Ug�r��q�dmfu`�·_ � [pa`V^¦�]`_ba�[ ¥ V�_j� X Laplace, X(s) � csS �hlßr�t��qT`mku`l

x(t)Â

X(s) =∫ +∞

o−x(t) e−stdt,

ë Ø ÝZôZï�qw`u`�

s�`g��bT`zUg��ht��qcpmfT�¼`oq�¬m t =4è aq�¬[ � V\[ £ �E}\� ©P_ � [±ahV^¦�]`_ba�[ ¥ V�_ ª © Laplace

cke S Thg�u

x(t) =1

2π j

∫ s=σ+j ∞

s=σ−j ∞X(s)estds.

ë Ø Ý�÷qï

è g�m �¬w`u�g�T`��mfuhe�cfe S Thgb�`�quhgxuhgY�qc<mfu`�Zlßm ��w`u`�ZlnØ Ý Ã �hThg�Ø Ý í � T`S^mUexrbmfuhgx��T � T`o�oq}NrYmkuh��lßm ��w`u`�ZlnØ Ý�ô�hThgÌØ Ý¬÷·�N�qcpmfT�¼`oq��m t

scke S Thg��`g��bT`zUg��ht��qcpmfT�¼`oq��m t � zU�¬oqT`z t·mfu

σcfe S Thg\csSn�bikS ckg��qcpmfT�¼`oq�¬m � =

è �qcpmfT`r����Z�qTqmUgxrY�q�`l¾Ø Ý�ô¾uZS u`�q}`�UcpmfThg»_j�^� ª ¡ � �   £ � ©·_ � [±ahV^¦�]`_ba�[ ¥ V�_ ª © Laplaceëone sided

Laplace transform) =÷qÝ

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è �qcpmfT`rY���Z�qTqmUgxrY�q�`lLaplace

cfe S ThgÒ�b| T`�q�`g��h�hl½�hThg%cfe S Thg%�quZS T`z¬g´���`l � zU�¬oqT`zUt�rbmfu�rYtZ�qTx(t)T`S^mUgxrbmfuhgx��cke�ikS T`lv�hThg��q�ZS uZSyikS T`l�qcpmfT`rY���Z�qTqmUg�r��q�`l

Laplace.

è �`cpmfT`rY���Z�qTqmUg�r��q�`lLaplace

cke SUThg¶ikS T`ly�b| T`�q�hg´���`ldm csoZckrbm tZl � L � �hT�g¶u�g�m ��w`uhg%Ø ÝZô½��Thg'Ø^Ý�÷cp����| }`�Uu�S^mfThg\�hThgY³4l

X(s) = L[x(t)], x(t) = L−1[X(s)].ë�Ø Ý�øZï

º u�u¬oquq�`oqtZ|^³4�qT�Ø Ý�ôymfu`�N�qcpmfT`r����Z�qTqmUgxrY�qu`�Laplace

rY�����`o`e S cfgb�Yg{T��h}q¹\c

σ = Re(s) > c,

T`S∫ L

o|x(t)| dt ≤ K <∞,

�Yg{T��h}q¹\cL > 0

�hThg\T`Sy�·r���SU}`|^m �ZrY� |x(t)| cpw`Tqoq�¬¹\ck��cfg�m �NrY��isrY�

|x(t)| ≤ Aect, t > 0.

Q R4SUT·rYtZ�qTx(t)

w`u`��is��ckg�T`��m tm �ZS®g{zUg{�¬m ��mfT·u�SUu`�q}`� csmkThgjV\¨`_ba � +O� � [ ¥ +�¨�©P[f�q� � °4© = ² �`g��h| ��m cs| �mUg{�qt�mkuh�

cu�SUu`�q}`� csmkThgß[ � [s_Ò]`_��^�¬]çaZ¡ ª �� `[^]�©�VYX i +f� ¥ V\]�© =ç¿ l~isSUT"w`T`| }hz ckg����`TÛrYtZ�qTqmfu`l�w`u`�

is��cfgYT`��m t·m ��Syg{zUg{�¬m ��mfT�T`SUT`��ik| u`�Z�qcPmfu�rYtZ�qTu(t) � �Yg{T�mfu�uqw`uhexu~cke S Thg

A = 1�hThg

c = 0 =

àÔ` ·¹¸ y�� ����¯Mç�s��ä ��æñ���Mg��ý�����y�ìMç�� édLaplace

%�­>��� Y[Z>46B>H6�N! ³ 8�Z½�õ®cs³4| u`�Z�qc�mfu�r�t��qT

x(t) = u(t).

è �qcpmfT`r����Z�qTqmUgxrY�q�`lLaplace

cke S T�g�u

X(s) =∫∞0 e−stdt

=[

e−st

−s

]t=∞

t=0

=[

e−σte−j ωt

−s

]t=∞

t=0

= −[

e−σt

s(cosωt− j sinωt)

]t=∞

t=0.

) g{T�S T�rY�����`o`e SUckgbmfu�z csµUg{���`ipoqu`l®w`| isw`ckg�S T�cke SUThg

σ = Re(s) > 0.

÷ Ã

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º ��m c®u��qcpmfT`rY���Z�qTqmUg�r��q�`lLaplace

mfu`�N�qu�S ThzUg{Thexu`�·¼htZ�qTqmfu`lu(t)

cke S T�g�u

X(s) =1

s, Re(s) > 0.

ð ��oqT`z tNu��qcsmkThrY�����qT`mUg�rY�q�hlLaplace

m �Zlu(t)

g�r�����cfgY��mkT`Sd�·�qcsmfT�¼qoq��m tscke SUThg�rbmfu��`g��bT`zUg��h�

cpwhe�w`ckz uRe(s) > 0 = Q Ü | T � m cpo`g��h}

L[u(t)] = 1

s, σ = 0.

ë�Ø Ý1Zï

%�­>��% Y[Z>46B>H6�N! ³ 8�Zô%õ®cs³4| u`�Z�qc�mfu�r�t��qT

x(t) = e−αt × u(t).Q R4��u`�Z�qcL[e−αt × u(t)] =

∫∞0 e−(s+α)tdt

= −[

e−(α+σ)t e−j ωt

s+α

]t=∞

t=0.

º u�zUcsµUg{���qisoZuhlvrY�����`o`e S ckgY�Yg�T

Re(s) = Re(α) + σ > 0.

Q Ü | TL[e−αt × u(t)] =

1

s+ α, σ > −Re(α). ë�Ø^Ý1�qï

%�­>�&� Y[Z>46B>H6�N! ³ 8�Za�õ®cs³4| u`�Z�qc�mfu�r�t��qT

x(t) = δ(t).Q R4��u`�Z�qcX(s) = L[δ(t)] =

∫ ∞

0−δ(t)e−stdt = 1.

ë�Ø Ã Zïº u�u¬oquq�hoZtZ|^³4�qT~T`��m ����w`}`|U��cfgY�Yg�T��h}Z¹jc

σ = Re(s)�hThg�r���SUcpwqÀ4l

σ = −∞ =

%�­>� 2 Y[Z>46B>H6�N! ³ 8�Z 2õ®cs³4| u`�Z�qc�mfu�r�t��qT

cosω0t× u(t).

÷ í

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� rY���ZckgY�NrY��isrY�cosω0t =

1

2

(

ej ω0t + e−j ω0t)

,

�hThg\rY�ZS cpwqÀ4l �L[cosω0t× u(t)] =

1

2L[ej ω0t × u(t)] +

1

2L[e−j ω0t × u(t)],

�hThg\�qcPcs�\T`| �qu¬�bt·mfu`�·m ��w`u`��Ø Ý�ô �L[cosω0t× u(t)] =

1

2

(

1

s− j ω0+

1

s+ j ω0

)

,

�hThg�m cpo`g��h}L[cosω0t× u(t)] =

s

s2 + ω20,

�Yg{Tσ > 0.

ë�Ø Ã Ø^ï

%�­>��X Y[Z>46B>H6�N! ³ 8�ZôXî c®�`�quhgxu�m | �qw`u�T`w`u`z ckg��hSU��csmfThg\��mUg

L[sinω0t× u(t)] =ω0

s2 + ω20,

�bg{Tσ > 0.

ë�Ø Ã Ý�ï

÷qô

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¸ mfu�S"whe S Tq�hT w`u`�ÞTq�hu¬oqu`�¬¹\ckeò¹\T�z �hrYu`�Z�qc�mfu �qcpmfT`rY���Z�qTqmUgxrY�q�Laplace

�qck|Ug��`À4S�¼hT`r�g��`À4SrY�ZS T`| m t�r�cp³4S =

× ��ñ Ü µ Ü ¸ �&�

¸ tZ�qT î cpmfT`rY���Z�qTqmUgxrY�q�`lLaplace

whT`| }`�qcsm | u`lc

x(t) X(s) = L[x(t)] c

1 δ(t) 1 −∞

2 u(t) 1s

0

3 e−αt u(t) 1s+α

−Re(α)

4 cosω0t u(t)s

s2+ω20

0

5 sinω0t u(t)ω0

s2+ω20

0

6 tn−1

(n−1)! u(t), n > 0 1sn

0

7 tn e−αt

n!u(t), n > 0 1

(s+α)n+1 0

8 e−αt cosω0t u(t)s+α

(s+α)2+ω20

0

9 e−αt sinω0t u(t)ω0

(s+α)2+ω20

0

÷Z÷

Page 67: simata kai systimata - users.auth.grusers.auth.gr/~hadjidem/simata kai systimata.pdf · @)A B C D E F G H I H A (TIME DOMAIN ANALYSIS) J LKNMO P Q R4SUTWVYXZV\[^]`_badcfe S ThgjikS

¸ mfu�S½whe SUTq�hT�w`u`��Tq��u¬oZuh�U¹jcke�¹\T�z �`rYuh���qcy�qcs|Ug��hikln¼hT`r�g��hikl·g{zUg{�¬m ��m cklymfu`�~�qcsmkThrY�����qT`mUg�rY�quh�Laplace.

× ��ñ Ü µ Ü ¸ �&�&�

¸ tZ�qT î csmkThrY�����qT`mUg�rY�q�hlLaplace

x(t) (x(t) = 0, t < 0) X(s) = L[x(t)]

�b| T`�q�`g��h��m �¬mfTα1x1 + α2x2 α1X1 + α2X2w`T`| Tq��À'�YgxrY� dx

dtsX(s)− x(0−)

dnxdtn

snX(s)− sn−1x(0−)− sn−2x(1)(0−) . . .− x(n−1)(0−)

u¬oquq�`oqtZ|^³4rY�y(t) =

∫ t−∞ x(λ)dλ X(s)

s+ y(0−)

s

�qw`u`�y(0−) =

[

∫ t−∞ x(λ)dλ

]

t=0−

x(t)e−αt X(s+ α)

x(t− t0)× u(t− t0) e−st0X(s)

tx(t) −dX(s)ds

x1 ∗ x2 ≡∫∞0 x1(λ)x2(λ)dλ X1(s)X2(s) (x1(t) = x2(t) = 0, t < 0)

x1(t)x2(t)12π j

∫ c+j ∞c−j ∞ X1(s− λ)X2(λ)dλ

lim t→0+ x(t) lim s→+∞ sX(s)

lim t→+∞ x(t) lim s→0 sX(s)

à f û dNåæ6MWý �Pý�ä æ�b �s�n ~K �Ûå¾�������� éd M¢KNMO�P������� �ä

² isµ u`zUu`lWckS �`l���|UT`�q�`g��hu`��rYmkTq¹\cs| uh�ÛrY�Zrbm tZ�qTqmfu`l��Zw`uh| cke6S T �qcpoqcpm �¬¹\cfeñ�qc¾m �¢¹jcp³4|Ue{T ms³»S�qcpmfT`r����Z�qTqmUgxrY�ZÀ4S

Laplace.É |Ue�rY�hu`�Z�qc�w`|^À'mfT¾mfu��qcsmkThrY�����qT`mUg�rY�q�

Laplace, Y (s) � �hT�g��qcpm }¼h|Uexrb�hu`�Z�qc4m �ZS6isµ uhz uvw`The{| S uZS^mkThlòmfuZS6ThS^mUe�rYm | u`��u�qcpmfT`rY���Z�qTqmUgxrY�q�Laplace, y(t) = L−1[Y (s) =

õ®Tn¼h| u`�Z�qc<rbm �ZSßw`T`|U}q�b| T`��u½T`��m tvmfu`�Zl<m ��w`u`�Zl<�Yg{Tdm �v�bckSUg��htnw`cs|¬e´wqmk³4rY� � ��Thg�rbmUg�l$csw`�`�qcsSUcslw`T`| T`��|U}`��uh��l¹\T½w`T`| uh��r\g�}hrYu`�Z�qc®zUg{}`�\u`| T~w`T`| T`z cfe �b�qTqmfT =õ®cs³4| u`�Z�qc��b| T`�q�`g��h��rY�Zrbm �Z�qT � mku�uqw`uhexu~��T`| T`�`m �Z|Ue{� cpmfThgjTqw`��m �·z¬g�Th��u`|Ug���t�csµUexrb³4rY�

any(n) + an−1y

(n−1) + . . .+ a0y = bmx(m)(t) + bm−1x

(m−1)(t) + . . .+ b0x(t).ë�Ø ÃZà ï

÷qø

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² ckexrYu`z uhlcke SUThg��x(t)

�hThg\�·ikµ u`z uhl�y(t) =

�Pw`uq¹\ipmfu`�Z�qc���mUg4�OckexrYu`zUu`lx(t)

��Thg»�¬oZckl�uhg'w`T`| }`��³4��u�gñ�qis��|Ug'm }`µ cs³4lm

cfe S ThgñexrYcsl~w`| u`l�q�Zz isSn�Yg{T

t = 0− �x(i)(0−) = 0, i = 0 . . .m,�hThg\�¬mUgYmfu�rY�Zrbm �Z�qT½is��ckg�Tqw`uZ¹\���hck���qikS �NckS is|^�bckg{T�m �·rbmUg��b�qt

t = 0− �y(j)(0−) 6= 0, j = 0 . . . n.

Ì ��mfu`�Z�qcßm �ZSnikµ u`z u�mfu`�NrY�Zrbm tZ�qTqmkuhl =× The{| S u`�Z�qc®mfu��qcpmfT`r����Z�qTqmUgxrY�q�

Laplace��ThgYmk³4S�z �Zu��qcsoZÀ4Sm �ZlvcsµUexrb³4rY�ZlyØ ÃZà �hThg�ik��u`�Z�qc

an[

snY (s)− sn−1y(0)− . . .− y(n−1)(0)]

+an−1 [sn−1Y (s)− sn−2y(0)− . . .]

. . . . . . . . .

+a1 [sY (s)− y(0)]

+a0Y (s) = [mmsm + bm−1S

m−1 + . . .+ b0]X(s),ë Ø Ã í ï�qw`u`�½csoZ}Z¼`Th�qcP��w`�O/\�N�¬mUg

x(i)(0−) = 0, i = 0 . . .m =² rY��isrY�~Ø Ã í ��|U}`��csmkT�gY³4l

Y (s) = H(s)X(s) +C(s)

D(s),

ë Ø Ã ôZï�qw`u`�Nmfu½w`u¬oq��À4S �Z�qu

C(s)cke S T�gYm �Zl®�qu`|U��tZl

C(s) = [. . .]y(0) + [. . .]y(1)(0) = . . .+ [. . .]y(n−1)(0),

�hThgòuhg»r���S m cpoqcsrbm ikl�mk³4Syi(0)

csµ Th|^m^À4S^mfThgñ�q��S uÞTqwh�ÛmkTai = º uÛw`u¬oq�¬À4SU���qu

D(s)cke S Thg»mfu

��T`| Tq�hm ��|¬g�rbmUg��h�~w`u¬oq��À4S �Z�qu½m �Zl®z¬g�Th��u`|Ug���t�lnckµUexrb³4rY�ZlyØ ÃZà �hThg�z¬e�zUcpmfThg�Tqw`��m �·rY��isr��D(s) = ans

n + an−1sn−1 + . . .+ a0.¸ m �·rY��isrY��Ø Ã ôyw`T`| Tqm �Z| u`�Z�qcP��mUgb�¬oqckl®uhgbwqoq�Z| u`��uh|Ue�ckl®�Yg{T½mUgxl®T`| �bg��hislr���S^¹\t��hcsl�w`ck|Ug{is��u�S mkThg

�q�ZS uWrYmkuZS��h| uC(s) � zU�¬oqT`zUt�rbmfu�S~z cs��m cs| uW�q�ZS uO�`| uWmfu`��z ckµUg�u`���qipoqu`�Zl·m �Zl�Ø Ã ô =~è �`|Uu`l

H(s)rbm �WrY��ikrY�çØ Ã ô�cfe S Thg»�Wr���S }h|^m �ZrY���qcsmkT`�\u`| }`l � �qwq³4l�¹\T¢zUu`�Z�qc�rYm ��S�whT`| }q�b| T`�\u à =¸ m �NrY��isrY�~Ø Ã ôdu��`|Uu`l

H(s)cfe S Thgje�rYuhl®w`| u`l

H(s) =bms

m +mm−1sm−1 + . . .+ b0

ansn + an−1sn−1 + . . .+ a0≡ N(s)

D(s),

�qw`u`�Nmfu½w`u¬oq��À4S �Z�quN(s)

cfe S T�gYmfuN(s) = bms

m +mm−1sm−1 + . . .+ b0.

÷0

Page 69: simata kai systimata - users.auth.grusers.auth.gr/~hadjidem/simata kai systimata.pdf · @)A B C D E F G H I H A (TIME DOMAIN ANALYSIS) J LKNMO P Q R4SUTWVYXZV\[^]`_badcfe S ThgjikS

R=2Ù

L=1H

4Vi(t)

1

2

¸ ��tZ�qT~Ý�÷�Â

² rY�ZS }`|^m �ZrY�H(s)

csµ T`| m }qmfThg��q�ZS u�Tqw`��mUg�lvw`T`| T`�qism | u`�Zlvmfu`�·r���rYm t��`Tqmkuhl =è T`S^mUe�rYm | u`��uhl<�qcpmfT`r����Z�qTqmUgxrY�q�`l

Laplacem �Zl

Y (s) � �qwq³4l<zUe{z csmkT�g`rbm �®r���ikrY�yØ Ã ô � ¹\T�z �`rYcfgm �ZSnikµ u`z u

y(t) �y(t) = L−1[H(s)X(s)] + L−1

[

C(s)

D(s)

]

.ë Ø Ã ÷qï

× T`| Tqm �Z| u`�Z�qc$��mUghmfuyw`u¬oq�¬À4SU���quC(s)

g�rYuh�¬mfThg�w`| u`l��q��zUisS®�¬mfT`S®mfudrY�Zrbm �Z�qT��Z| cs�`cke`�Yg{Tt = 0 =¸ m �ZSnw`cs|¬e´wqmk³4rY�NT`��m t·�NrY��isrY�~Ø Ã ôyw`The{| S ckg�m �·�qu`| �\t

Y (s) = H(s)X(s).¸ m �NrY��isrY�~Ø Ã ÷·u��`|Uu`lL−1[H(s)X(s)]zUe S ckg<m �ZSçTqw`�q��|Ug�r�� �q�Zz csS¬g´��t�lO�hT`mkT`rYm }`rYcs³4l ë

zero state response),z ��oqT`z t�m �ZSçTqw`�q�h|UgxrY�

mfu`�çrY�Zrbm tZ�qTqmfu`l���mfT`SWm �¢rbmUg��b�qt¢cs��Th| �qu¬��tZl�m �Zl¾cfg�rY�hz u`�çmku�rY�Zrbm �Z�qTç�Z| cs�quh��r�cOëöz csSOckex��cTqw`uZ¹j�¬��cs�Z�qisS �nckS is|^�bcfg�T�ï � z ��oqT`z t y(j)(0−) = 0, j = 0 . . . n

ëöuqw`��m c<cke SUThgC(s) = 0

ï ='è �h| u`lT`��m �`lz csSycsµ T`| m }qmfThg�Tqw`��mUg�lT`| �bg��hislnrY�ZS�¹\t��hckl = R'w�e�rY�Zlvu��`| uhl

L−1[

C(s)

D(s)

]

,

cke S Thg�ud�`| u`lßw`u`�nw`cs|Ug{is��cfghmUgxl<wqoq�Z| u`��uh|Ue{csl�Tqw`�ymUgxl$Th| �bg´��isl�r���S�¹jt¬��csl = ¸ c6w`u¬oZoqisl$w`cs|Ug�wqm^À4rYcfg�lu¾�`| u`lyTh�¬m �hlnm cke S ckg�rbmku��q�Zz ikSd��mfT`SNu���| �ZS u`lym cke S ckgjrYmku�}qw`cfg�|Uu � �hThg�r���SUcpwqÀ4ln��oZ�ZrY�½m cke S ckgw`| u`lPm �yoZ�ZrY�yrbmfTZ¹\ck| }`l��hTqm }hrbmfT`rY�Zl L−1[H(s)X(s)] = ¸ m ��Sw`cs|Ue�wqmk³4rY�dT`��m ty�`|Uu`l L−1[C(s)

D(s)]uZS u`�q}`� csmfThg�w`T`| u`z¬g´���`l��`| u`l =

àÔ� ï ¾ås�næ�ˤ�Û����� ��æñ���Mã��ý�����y�¯Mç�c�ÿ�Laplace

%�õ>��� Y[Z>46B>H6�N! ³ 8�Z½�õ®cs³4| u`�Z�qc�mku��h���`oZ³4�qT�mkuh� ¸ ��tZ�qTqmfu`l½Ý�÷ =·è zUg{Tq�h�`wqm ��lNcfe S T�g�rYm ��¹\isrY��Ø·�hThg�w`���bThe S cfgÌrbm �¹\ikrY�~Ý�m �~rbmUg �b�qt

t = 0 = Ì ��mkuh���qcnm ��S½isµUu`z ui(t) =nº u�rY�Zrbm �Z�qT�z ckS½is��ckg�ckexrYu`z u��bg{T

t > 0 �is��cfgY�`�Z³4lTqw`uZ¹\���hck���`isS �NcsSUis|^�bckg{T�m �·rbmUg��b�qtt = 0 � z ��oqT`z t i(0−) 6= 0 =÷0�

Page 70: simata kai systimata - users.auth.grusers.auth.gr/~hadjidem/simata kai systimata.pdf · @)A B C D E F G H I H A (TIME DOMAIN ANALYSIS) J LKNMO P Q R4SUTWVYXZV\[^]`_badcfe S ThgjikS

R

L

i(t)x(t)+

-

C

¸ ��tZ�qT~ÝZø�Â

² cfe�r�u`z u`l�cke SUThgj�Nw`����t4V

�½uqw`uhe{T¢ë¤�bg{T~mfu��h���`oZ³4�qThï$�q�Zz csSUe{� csmkT�g��Yg�Tt > 0

ëözUg{�¬mUg\�`oqcke SUckgu�zUg{Tq�h�qwqm �Zl�ï = ² zUg{T`��uh|Ug´��t�csµUexrb³4rY�·cfe S Thg\�

di

dt+ 2i =

4�Yg�T

t ≤ 00

�Yg�Tt > 0.

ë Ø Ã øZïõ®Téoq�ZrYu`�Z�qcÛT`�¬m t m � zUg{T`��uh|Ug´��téckµUe�rY³4rY� �qcçm �Á¹\cp³»|Ue�Témk³4SÏ�qcsmfT`rY���Z�qTqmUg�r��ZÀ4S

Laplace.× The{| S u`�Z�qc®mfu��qcpmfT`r����Z�qTqmUgxrY�q�Laplace

��ThgYmk³4S�z �Zu��qcsoZÀ4S��Thg�ik��uh���qcP�Yg{Tt > 0 �

L[

di

dt+ 2i

]

= 0�Yg{T

t > 0,

tsI(s)− i(0−) + 2I(s) = 0.

ë�Ø Ã qï�Pw`uZ¹jipmfu`�Z�qcN�¬mUg%mfuW�h���`oZ³4�qTO�Yg{T

t < 0cke S Thg2rbm ��rbmfTZ¹\ck| }W�hTqm }`rYmkThrY�Þë

steady state),�hThg

rY�ZS cswqÀ4li(0−) =

4

2= 2A,

uqw`��m c®Tqw`��m �ZSNØ Ã ·is��u`�Z�qcI(s)[s+ 2]− 2 = 0 � t

I(s) =2

s+ 2.

² m csoqcs��mkT�e�T"T`��m t�rY��isrY��z¬e S cfgÒmfuO�qcpmfT`rY���Z�qTqmUgxrY�q�Laplace

m �Zl½ckµ �`z u`� = ² isµ uhz u`li(t)

¹\T¼h| c±¹jckeÒTqw`�OmfuZS�T`S^mUe�rYm | u`��u"�qcpmfT`r����Z�qTqmUgxrY�q�

Laplace.Q R4��u`�Z�qcNm^À4| T � Tqw`�Omfu�S~whe S Tq�hT¢ms³4S

�qcpmfT`r����Z�qTqmUgxrY�ZÀ4SLaplace,

i(t) = 2L−1[

1

s+ 2

]

= 2e−2t,�Yg{T

t > 0.

%�õ>��% Y[Z>46B>H6�N! ³ 8�Zô%õ®cs³4| u`�Z�qc<mfud�h���`oZ³4�qT·mkuh� ¸ ��tZ�qTqmfu`l®Ý�ø =4è zUg{Tq�h�qwqm �ZlP�`oqcke SUckg�m ��rbmUg �b�qt

t = 0 = Ì ��mfu`�Z�qc6mfu�qcpmfT`r����Z�qTqmUgxrY�q�

Laplacem �Zlvcsµ �`zUu`� =4º u�r���rYm ���qT½is��cfgYckexrYu`zUu��Yg�T

t > 0 =ø1

Page 71: simata kai systimata - users.auth.grusers.auth.gr/~hadjidem/simata kai systimata.pdf · @)A B C D E F G H I H A (TIME DOMAIN ANALYSIS) J LKNMO P Q R4SUTWVYXZV\[^]`_badcfe S ThgjikS

² zUg{T`��uh|Ug��ht�csµ¬e�rb³4r��dcfe S T�g��

Ldi

dt+Ri+

1

c

∫ t

−∞i(λ)dλ = x(t).

ë Ø Ã �qï× The{| S u`�Z�qc®mfu��qcpmfT`r����Z�qTqmUgxrY�q�

Laplace��ThgYmk³4S�z �Zu��qcsoZÀ4S��Thg�ik��uh���qc �

LsI(s)− Li(0−) +RI(s) +1

c

I(s)

s+1

c

1

s

[∫ t

−∞i(λ)dλ

]

t=0−= X(s).

ë�Ø í ZïR»e S Thg�m^À4| T

i(0−) = 0ë¤zUg{��mUgYmku��h���`oZ³4�qT�cke S Thg�ThS uhg��`m �~�bg{T

t ≤ 0 ï = R'whexrY�Zlvcke SUThg1

c

∫ 0−

−∞i(λ)dλ =

qc(0−)

c= vc(0

−),

�hThg\�·ckµUe�rY³4rY�~Ø í ·z¬e�zUckgI(s) =

sX(s)− vc(0−)

L[

s2 +(

RL

)

s+ 1LC

] .

² isµ uhz u`li(t)

ck��|Uexrb�hcsmfThg�Tqw`��mfuZS�T`S^mUexrbm | uh��u��qcsmkT`r������qT`mUg�rY�q�Laplace,

i(t) = L−1 [I(s)] .

%�õ>�&� Y[Z>46B>H6�N! ³ 8�Za�

è �qcpmfT`r����Z�qTqmUgxrY�q�`lLaplace

m ��ly(t)

cke S Thg�u

Y (s) =s+ 8

s2 + 6s+ 13.

ë Ø í Ø ïÌ ��mfu`�Z�qc2m �$r���SU}`|^m �ZrY�

y(t) = L−1 [Y (s)] =Òº u®�hoZ}hrY�qT®rYmku®z csµ¬g���qipoqu`l»m ��l$Ø í Øñmfu®T`S Tqoq�Zu`�Z�qcrYcPTqwqoq}��`oq}`r��qTqmfTbÂ

s+ 8

s2 + 6s+ 13=

s+ 3

(s+ 3)2 + 22+

5

(s+ 3)2 + 22.

�hThg\rY�ZS cpwqÀ4ly(t) = L−1

[

s+ 3

(s+ 3)2 + 22

]

+ L−1[

5

(s+ 3)2 + 22

]

.

î cPm ��¼hu`t¬¹\ckg{T�ms³4Syg{zUgxu¬m t�mk³4S�d��Thgq�·mfu`�·whe S Tq�hT �&�&� cs�Z|Uexrb�hu`�Z�qc

y(t) =(

e−3t cos 2t+ 52e−3t sin 2t

)

× u(t)

= e−3t(

cos 2t+ 52sin 2t

)

× u(t).

ø�Ø

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%�õ>� 2 Y[Z>46B>H6�N! ³ 8�Z 2

è �qcpmfT`r����Z�qTqmUgxrY�q�`lLaplace

m ��ly(t)

cke S Thg�u

Y (s) =10

s2 + 10s+ 16.

ë Ø í ÝZïÌ ��mfu`�Z�qc2m �$r���SU}`|^m �ZrY�

y(t) = L−1 [Y (s)] =Òº u®�hoZ}hrY�qT®rYmku®z csµ¬g���qipoqu`l»m ��l$Ø í Ý<mfu®T`S Tqoq�Zu`�Z�qcrYcPTqwqoq}��`oq}`r��qTqmfTbÂ

10

s2 + 10s+ 16=

10

(s+ 2)(s+ 8)=

A

s+ 2+

B

s+ 8.

É |Uexrb�hcpmfThg���mUgYcfe S T�gA = 5/3 � B = −5/3 �hThg�r���S cswqÀ4l

Y (s) =5

3

(

1

s+ 2− 1

s+ 8

)

,

uqw`��m c � rY�Z�q�Y³4S T��qc�m �ZS�g{zUg{��m ��mkT à mfu`�·whe SUTq�hT �&�&� �y(t) =

5

3

(

e−2t − e−8t)

× u(t).

%�õ>��X Y[Z>46B>H6�N! ³ 8�ZôX

è �qcpmfT`r����Z�qTqmUgxrY�q�`lLaplace

m ��ly(t)

cke S Thg�u

Y (s) =2s2 + 6s+ 6

(s+ 2)([s+ 1]2 + 1).

Ì ��mfu`�Z�qc4m �PrY�ZS }`| m ��r��y(t) = L−1 [Y (s)] = Ü S Tqoq�Zu`�Z�qc»mfu�`oq}`rY�qTmfu`�®z csµUgxu`�v�qipoqu`�Zl0rYcñTqwqoq}

�`oq}`rY�qT`mkTYÂY (s) =

A

s+ 2+

Bs+ C

s2 + 2s+ 2R4�Z|Uexrb�hcsmkT�g��¬mUg�uhgYw`T`| }`�qcsm | uhgA � B � C cke S T�g\exrYuhgYw`| u`l

A = 1 � B = 1 � C = 2 � uqw`��m cY (s) = 1

s+2+ s+2(s+1)2+1

= 1s+2

+ s+1(s+1)2+1

+ 1(s+1)2+1

,

�hThg\rY�ZS cpwqÀ4l¬Â

y(t) = L−1[

1

s+ 2

]

+ L−1[

s+ 1

(s+ 1)2 + 1

]

+ L−1[

1

(s+ 1)2 + 1

]

.

î cPm ��¼hu`t¬¹\ckg{T�ms³4Syg{zUgxu¬m t�mk³4S à � ·�hThgq�·mfu`�dw�e S Tq�hT �&�&� is��u`�Z�qcPm cpo`g´��}bÂy(t) =

(

e−2t + e−t cos t+ e−t sin t)

× u(t).

øZÝ

Page 73: simata kai systimata - users.auth.grusers.auth.gr/~hadjidem/simata kai systimata.pdf · @)A B C D E F G H I H A (TIME DOMAIN ANALYSIS) J LKNMO P Q R4SUTWVYXZV\[^]`_badcfe S ThgjikS

v(t)

Ci(t)+ -

(a)

V(s)

I(s)+

Z(s)=1/cs

+ - -

v(0 )/s-(b)

¸ ��tZ�qT~Ý1�ÂàÔ� � Mç s�ed½���� MO�ß ��&�Oæ�˱ ��Kd�����ÿ����� �ä þ ä ¸ å ¾ä

��æñ���Mg��ý�����y�ìMç�s�Laplace

õ®T�z �`r�u`�Z�qc�mku�gxrYu`z �ZS T`�qu��h���`oZ³4�qT � ³4ldw`| u`ldmfu��qcsmfT`rY���Z�qTqmUg�r��q�Laplace (Laplace trans-

form equivalent circuit),�bg{T~�qcs|Ug��h}½¼hT`r�g��h}~rbmfuhg���cke{T~csS �hl®�h���`oZÀ4�qTqmfu`l =

%�ö>��� Y°?��DA6�Â:=�>#õ®cs³4| u`�Z�qc�mkuÏrbmfuhgx��ckexuÙcsS �`lW���¬�`oZÀ4�qT`mku`lOmfuÏuqw`uhexuÙT`w`u¬m csoZcfe mfThg$Tqwh�ÏikS T`S¢w`���hS ³'m t � �qwq³4l��T�e S csmkT�g�rbmfu ¸ ��tZ�qT~Ý1ZT =L� rY����cfgY�·rY��ikrY�

v(t) =1

c

∫ t

−∞i(λ)dλ,

Tqw`��m ��S�uqw`u�e�T~w`The{| S u`�Z�qc � �qcP�qcpmfT`rY���Z�qTqmUgxrY�q�Laplace,

V (s) = 1c

[

I(s)s+ 1

s

∫ 0−

−∞ i(λ)dλ]

,

= 1csI(s) + 1

s

[

1c

∫ 0−

−∞ i(λ)dλ]

,

�hThg\cpw`cfg�zUt1

c

∫ 0−

−∞i(λ)dλ = v(0−),

is��u`�Z�qcPm cpo`g��h}V (s) =

1

csI(s) +

v(0−)

s.

ë Ø í à ïÜ w`�vm ��r���ikrY��T`��m tßw`T�e�|US u`�Z�qcñmfuv�qcsmkThrY�����qT`mUg�rY�qikS u®�hTqm }

LaplacegxrYu`z �ZS T`�qu�h���`oZ³4�qT

KVLw`u`���\The S csmkT�g2rbmfu ¸ ��tZ�qT¢ÝÃU¼ =�è w`���hS ³'m t�l�T`S^mUg�w`| uhrb³'w`cs�ZcpmfThg2Tqw`�"m ��S�T`S^mUexrbmfT`rY��ckgxrY�`zUu`�ëimpetance) 1/(cs)

��Thg�uhg�T`|U��g��hiklnrY�ZS�¹\t��hcsl®ThS^mUg�w`| u`rb³'w`ck��uZS^mfThg\Tqw`��m �ZSw`���btv(0−)/s =

ø Ã

Page 74: simata kai systimata - users.auth.grusers.auth.gr/~hadjidem/simata kai systimata.pdf · @)A B C D E F G H I H A (TIME DOMAIN ANALYSIS) J LKNMO P Q R4SUTWVYXZV\[^]`_badcfe S ThgjikS

+_

i(t)L

v(t)

(a)

+_

I(s)sL

V(s)

+ -

(b)

¸ ��tZ�qT~Ý1��Â

v(t)

i(t) R+ -

(a)

R+ -

V(s)

I(s)

(b)

¸ ��tZ�qT à �Â%�ö>��% Y*<>A6�N�7�õ®cs³4| u`�Z�qcòmfu�rbmfuhgx��cfe�uyckS �`l<�h���`oZÀ4�qTqmfu`l<mkuyuqw`uhexudTqw`u¬m csoqcke�mkT�ghTqw`�yikS Tnw`�ZS cfe�u � �qwq³4l$��The SUcpmfThgrbmfu ¸ ��t��qT~Ý1�ZT =L� rY���ZckgY�·r���ikrY�

v(t) = Ldi

dt,

Tqw`��m ��S�uqw`u�e�T~w`The{| S u`�Z�qc � �qcP�qcpmfT`rY���Z�qTqmUgxrY�q�Laplace,

V (s) = L[sI(s)− i(0−)]

tI(s) =

1

sLV (s) +

1

si(0−).

º u~gxrYu`zU��S Th�qu��h���`oZ³4�qTKVL

zUe{z csmkT�g�rbmfu ¸ ��tZ�qT~Ý1�U¼ � �hThgjg�r�����cfgY�·r���ikrY�

V (s) = LsI(s)− Li(0−).ë Ø íZí ï

%�ö>�&� ��A=:J�$�@:CZ>�D<õ®cs³4| u`�Z�qc�mfuÏrbmfuhg���cfe�uÙcsS �hl����¬�`oZÀ4�qT`mku`lOmkuÏuqw`uhexuÁTqw`u¬m csoqcke�mkT�g<Tqw`�Ù�`g{TÏT`S^mUexrbmfT`rY� � �qwq³4l��T�e S csmkT�g�rbmfu ¸ ��tZ�qT à ZT =L� rY����cfgY�·rY��ikrY�

v(t) = Ri(t),

Tqw`��m ��S�uqw`u�e�T~w`The{| S u`�Z�qc � �qcP�qcpmfT`rY���Z�qTqmUgxrY�q�Laplace,

V (s) = RI(s).ë�Ø í ôZï

ø í

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R

L

i(t)x(t)+

-

C(a)

+

-

+ +- -

V(s) I(s)

s/101_

10i (0)L

8/5s

1

v (0 )/s-

c(b)

¸ ��tZ�qT à ØZÂ

º u~gxrYu`zU��S Th�qu��h���`oZ³4�qTKVL

zUe{z csmkT�g�rbmfu ¸ ��tZ�qT à U¼ =íe{'Å�Æ�Ë=u�ÊCSî c4m �<¼hu`t¬¹\cfg�Tvms³»S<gxrYu`z �ZS T`�Z³4S<�h���`oZ³4�q}qmk³4S6w`uh�Pcke{z T`�qc»w`| u`���bu`�Z�qisS ³4l � �Zw`u`| u`�Z�qcñS T�¼h| u`�Z�qcmfuég�r�u`z �ZS T`�qu �h���`oZ³4�qTécsSU�`lÞrY�ZS�¹jcpmfu`�Ï�h���`oZÀ4�qTqmfu`l = î cpm } cfe S T�gvcs���hu¬oqTéS TÁ¼h| uh���qc¢mfu�qcpmfT`r����Z�qTqmUgxrY�q�

LaplaceTh�¬mfu`�·mfu`�·�h���`oZÀ4�qTqmfu`l =

Ü lP¹\cp³»| t�r�u`�Z�qcßmfu½�h���`oZ³4�qT�mfu`� ¸ ��t��qT`mku`l à Ø^T��qwhu`�½�dw`���bt·cfe S Thg\�x(t) = (−0.5 cos t+ 2.5 sin t)× u(t).

è gYTh| �bg´��islnrY�ZS�¹\t��hckl®cke S T�giL(0

−) = 0 A , vc(0−) = −2 V ,�hThg\uhgYmUg{�qisl®mk³4Snw`T`| T`�qism |^³4Sycke S Thg

R = 1 ¿ , L =1

10H , c =

5

8F .

º uÁgxrYu`z �ZS T`�`uÏ�h���`oZ³4�qTÙzUe{z cpmfThg$rYmku ¸ ��t��qT à Øk¼ = © T`�¬¼h}`S uZS^mfT`lOm^À4| TÏ��w`�O/\�ÞmUgxlOT`| �bg��hiklrY�ZS�¹\t��hcklP�Yg�T�mfT

iL(0−)

�hThgvc(0

−) � is��u`�Z�qcV (s) =

s

10T (s) +

8

5sI(s)− 2

s+ I(s).

Ü w`��m �·rY��isr��dT`��m t·w`The{| S u`�Z�qc®m cpo`g��h}

I(s) =V (s) + 2

ss10+ 85s+ 1

.

² V (s)w`| uq�h��wqm cfg�Tqw`��m �ZS�ckexrYu`z u�³4l

x(t) �V (s) =

−0.5s+ 2.5s2 + 1

,

�hThg�m cpo`g��h} �I(s) =

15s2 + 2.5s+ 20

(s2 + 1)(s2 + 10s+ 16).

Ü w`�¾m �~rY��isr���T`��m t½¼h|Ue�rY�hu`�Z�qcnm �ZSNisµ uhz ui(t)

w`T�e�|US u�S mkThldmfuZS½T`S^mUexrbm | u`�\u��qcsmkThrY�����qT`mUg�rY�`�Laplace �

i(t) = L−1[I(s)].

øZô

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Fouri-er

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C(s) = 0ï �

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Laplacem �Zl�csµU�`z u`� ��Thg

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Laplacem �Zlvcfg�rY�hz u`� = ² rY�ZS }`|^m �ZrY�

H(s)cfe S ThgjT`S ckµ }`|^m ��m �Nm �ZlckgxrY�`z uh�

x(t) � �hThgjcsµUT`|^m }qmfThg�q�ZS u�SOTqw`�ÛmUgxl�w`T`| T`�qism | u`�Zl¾mfu`�"r���rYm t��qT`mkuhl = ² rY�ZS }`|^m �ZrY�OTh�¬m t¢u�S uh�q}`� cpmfThgßV¶ ¬� � £ [^]�V\]_ � [paO}\� £ �`© = R'w`u`�qikS^³4l � ��ckSUg��hcs�Zu`�Z�qcdmfuZS�u`|UgxrY�q��m �Zl·rY�ZS }`| m ��r���ld�qcsmfT`��u`|U}`l � �qwq³4lNz �Z¹j�¬��crbm �ZSw`T`|U}q�b| T`��u¾ÝZÝ � ³4lÐ ÆvT{Ñ2ËCÑ"U Â�÷ V� �� � £ [^]�V\] _ � [paO}�� £ �`©s« H(s) « � � ª © i�£ a�_�_ ¥ +`�UXÞV\[paO� �ö£ � XÛV� ZV\[^¨`_ba�[p� © �k� �pa ¥�·� ªØi � ©�[p�� �_ � [pa`V^¦�]`_ba�[ ¥ V�_j� X Laplace

[^]�© � � ª � �� Û¡ £ � ©¾[p�¾_ � [pa`V^¦�]`_Ya�[ ¥ V�_ ª Laplace[^]�©

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Y (s)

X(s),

ë Ø í ÷qï

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ckgxrY�`zUu`�x(t) =

Ä�ÅjÆ�Å\ÍzSjÆ`ÓjÑÌÓQ è mfT`Sçu �h| u`l L−1[C(s)

D(s)]cke SUThg6w`T`|Uu`zUg��h�hl"�`|Uu`l � z ��oZThz tÞm cke SUckg6rYmku �q�Zz ikS¢�¬mfT`S

t → ∞ � �rY�ZS }`| m ��r��v�qcpmfT`�\u`| }`l<¼`|¬e�rb��cpmfThg�Tqw`�dmfu�S®�`| u·rbmfTZ¹\ck| }`l<�hT`m }`rbmfT`rY�Zl L−1[H(s)X(s)] = ë ¸ �¬��Ú�h|Ue S Tqm cnmfuZSdm ��w`uOØ Ã ÷��qcmfu�SNu`|UgxrY�q�"Ø í ÷qï = Ü ��m �¾��The S cpmfThg¶rbmfu~w`T`| }hz ckg��b�qT¾m �Zlnw`T`| T`��|U}`��uh�Ø ø =������% ®n¯@�N�D<��@?�A6B>4=:C<��D<�#�8��-:CZ>¥��>46B>#��DZ�! �D46�M?M�@:J!�����#

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Laplacem �ZlrY�ZS T`| m t�r�cp³4lvz isoZmkT~cfe S Thgj�½�quZS }`zUT � L[δ(t)] = 1 � �hT�gjrY�ZS cpwqÀ4l X(s) = 1 =0º ��m c

uÞm �¬w`uhl�u`|UgxrY�qu`� Ø í ÷"m ��l�rY�ZS }`|^m �ZrY�Zl��qcsmkTh��u`| }hl�z¬e�z cfgY (s) = H(s) � �qw`u`�

Y (s)cfe S Thg0u

�qcpmfT`r����Z�qTqmUgxrY�q�hlLaplace

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z ��oqT`z t¾�OV� ��^� £ [^]�V\]�_ � [paO}\� £ �`© �s� �pa ¥ �N_ � [pa`V ¦�]q_Ya�[ ¥ Vb_ ª © Laplace[^]�©è+ £ �� �V\[ ¥ +�¨�©½aq¡ ª + £ Õ

¥ V\]�© =

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any(n) + an−1y

(n−1) + . . .+ a0y = bmx(m)(t) + bm−1x

(m−1)(t) + . . .+ b0x(t),ë�Ø í Zï

�qw`u`�n � m Tq�his|UThg�u�g��hT�g

n 6= m = ² ckµUe�rY³4rY�½Th�¬m t�cfe S T�g��`g�T¾�q��u`�qu¬�bcsS tZlyzUg{T`��uh|Ug��ht�csµUexrb³4rY�n

m }`µ cs³4l � �qc�rbmfTZ¹\ck| u`�Zl~rY�ZS^m csoqcsrbm ikl =çº uÛzUcsµUg{�Û�qisoZuhl~uh|Ue{� cpmfThgñTqw`�çm �ZS�cfe�r�u`z ux(t)

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�Z| cs�quh��r�cß�Yg{Tt ≤ 0 �

x(i)(0−) = 0, y(j)(0−) = 0, i = 0 . . .m, j = 0 . . . n.

Q R4��u`�Z�qc[

ansn + an−1s

n−1 + . . .+ a0]

Y (s) =[

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m−1 + . . .+ b0]

X(s),

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H(s) =bms

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D(s),

ë Ø í �qï�qw`u`�½u`|¬e�rYTh�qcPmfT½w`u¬oq��À4S ���`T

N(s) � D(s) � ³4lN(s) = bms

m +mm−1sm−1 + . . .+ b0,

D(s) = ansn + an−1s

n−1 + . . .+ a0.

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m��Thg

n � ThS^mUe�rYmku�g���T = × T`| Tqm �Z| uh���qc�Tq�h�h�qTÏ��mUg0mku w`u¬oq��À4S �Z�quD(s)

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Page 78: simata kai systimata - users.auth.grusers.auth.gr/~hadjidem/simata kai systimata.pdf · @)A B C D E F G H I H A (TIME DOMAIN ANALYSIS) J LKNMO P Q R4SUTWVYXZV\[^]`_badcfe S ThgjikS

++_

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v (t)i(t)

R

Ci

v (t) o

(a)

++_

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R

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vi(t)�hThg��nisµUu`z u`l��

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Vi(s) = RI(s) + Vo(s),

�qw`u`�Vi(s)

u��qcpmfT`rY���Z�qTqmUgxrY�q�`lLaplace

m �Zl�cfg�rY�hz u`�~�hThgVo(s)

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sCI(s).

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H(s) =Vo(s)

Vi(s)=

1

sRC + 1.

ë Ø ô1 qïR»e S Thg4�\T`S cs| �ç��mUg'�WrY�ZS }`|^m �ZrY���qcpmfT`�\u`| }`l

H(s)ckµ T`|^m }qmfThg4�q�ZS uçTqw`�¢mUg�l�w`T`| T`�qism | u`�Zl�mfu`�

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scfe S Thg��`g �bT`zUg��ht �

s = σ + j ω = R'w`u`�`isS^³4lnu�m ��w`u`lNØ^ô1 Nr����qwhe´wqm cfgY�qc®mfu�Sym ��w`u�Ø ô�ØnT`Sd�½�`g���ThzUg��ht��qcpmfT�¼`oq�¬m ts��g S cke�mkT�g\rbmfu�S��\T`S^mfT`rbmUg��h�~}`µ uZS T � s = j ω =

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× T`| Tqm �Z| u`�Z�qc���mUg2u"�qcsmfT`rY���Z�qTqmUg�r��q�`lLaplace

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�q�dw`cs| T`ms³4�qikS u �y(t) = L−1[c1] = c1δ(t),

ø1�

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D(s),

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è �qcpmfT`r����Z�qTqmUgxrY�q�`lLaplace

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w`cs|Ug{is��cfg�isS ThSy�`| u�m �Zlv�qu`| ��tZle−αt cos(ω0t)

y(t) = L−1[Y (s)] = . . .+

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× u(t) + . . .

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