임펄스 응답으로 표현된 시스템의 응답

 임펄스 함수의 Shifting property :

임펄스 응답으로 표현된 시스템의 응답

0 0 0 0

( ) ( ) (0) ( ) (0)

( ) ( ) ( ) ( ) ( )

x t t dt x t dt x

x t t t dt x t t t dt x t

 

 

 

 

 

 

 

   

 

 

 

0 0 0

( ) ( ) ( ) ( ) ( t) ( ) ( )

x t

x t  t t dt

x t  t dt  t  t

 

      ^  

( ) ( ) ( )

x t

x   t   d

  



 임펄스 응답(Impulse Response)

• 정의 :

• 선형 시불변 시스템(LTI System) : 1) :

2) 3)

( ) [ ( )]

h t  T  t

[ ] T  ( )t

 ^{h t( )}

t ( ) ( )

x t  t

0 t

( ) ( ) y t h t

0 (1)

[ ( )] ( )

T  t   h t  [3 ( )] 3 ( )

T  t   h t 

1 2 1 2 1 2

[3 ( ) 5 ( )] 3 [ ( )] 5 [ ( )] 3 ( ) 5 ( ) T  t    t   T  t  T  t   h t  h t

1 1

[3 ( )] 3 ( ),

T  t   h t T[5 ( t ₂)]5 (h t₂)

 

( ) ( ) ( ) ( ) ( ) ( )

T ^ x   t  d ^ x  T  t  d ^ x  h t  d

  

      

 





 

 컨볼루션 적분(Convolution integral)

•

•

 

( ) ( ) ( ) ( ) ( ) ( )

T ^ x   t  d ^ x  T  t  d ^ x  h t  d

  

      

 





 

 

T x t y t ^ x  h t  d x t h t

 



[ ] T 

( ) ( ) ( )

x t ^ x   t  d





( ) ( ) x t  t

( ) ( ) ( ) ( ) ( ) y t x t h t ^ x  h t  d

  



( ) ( ) ( ) ( ) ( ) ( )

y t  t h t ^   h t  d h t

  



 [예제3.5]

Convolution 적분의 계산

( ) ( ), 0

( ) ( )

x t e

u t a h t u t







( ) x t

0 t

1

( ) h t

0 t 1

0 

1 x( )

( )

h t

shift

t

( ) ( ) ( )

( ) ( ) y t x t y t

x  h t   d





 

  

0  1

( )

h t

t 0

1 ( ) x

i) t0



0 

1

( )

h t

0 t 1 ( ) x

 t

ii) t 0

no overlap ( ) 0

 y t  ⁰

2

( ) ( ) ( ) 1(1 )

If 2, then ( ) 1(1 ) ( ) 2

t a at

t

y t x h t d e d e

a

a y t e u t

    

  





    

    

 

 

 

( ) 1 (1 ^at) ( )

y t e u t

a

  

( ) y t

0 t

1

a

 [예제3.6]

( ) x t

0 t

1

a

( ) h t

0 t

1

b

( ) ( ) ( )

( ) ( ) ( ),

x t u t u t a

h t u t u t b b a

  

   

( ) x

0 

1

a

( )

h t

0 1

tb 

t

(a) t0

( ) x

0 

1

a

( )

h t

0 1

tb 

t (b) 0 t a

no overlap  y t( )  0 y t( )



( ) x

0 

1

a

( )

h t

0 1

tb 

t (c) a t b

( ) x

0 

1

a

( )

h t

0 1

tb 

t (d) b  t a b

( ) 0^a(1)

y t 



_

( ) x

0 

1

a

( )

h t

0 1

tb 

t (e) a b t

( ) y t

0 t

a

a b a b

( ) 0 y t 

 교환성

 결합성

 분배성

Convolution의 성질

( ) ( ) ( ) ( )

x t

h t

h t

x t

1 2 1 2

{ ( )

x t

h t

h t

x t

h t

h t

1 2 1 2

( ) { ( ) ( )} ( ) ( ) ( ) ( ) x t  h t  h t  x t  h t  x t  h t

( ) h t ( )

x t y t( )



1( ) ( ) h t

x t h t₂( ) y t( )

1 2

( ) ( ) ( )

h t h t h t ( )

x t y t( )



1( ) h t ( )

x t

2( ) h t

( )

^ y t





[예제 3.3]

1) 균일해

2) 특수해

3) 완전해

( ) 2 ( ) ( ), (0) 4

dy t y t u t y

dt   

2 0 2

s   s  

yp  A

2 1

( ) 2

t

h p

y t  y  y  Ke^ 

7 2 1

( ) , 0

2 2

y t e^ t t

   

0 2 1 1

A A 2

   

st

yh  Ke

Initial condition: (0) 1 4 y   K 2

( )

y t

 Ke

미분방정식과 임펄스 응답의 관계

[예제 3.3]

1) 균일해

2) 특수해

3) 완전해

(0) 0 ( )

y no initial stored energy

 

2 0 2

s   s  

yp  A

2 1

( ) 2

t

h p

y t  y  y  Ke^ 

1 2 1

( ) , 0

2 2

y t e^ t t

    

0 2 1 1

A A 2

   

st

yh  Ke

Initial condition: (0) 1 0 y   K 2

( )

y t

 Ke

( ) 2 ( ) ( ), (0) 4

dy t y t u t y

dt   

Let

1) 균일해

2) 특수해

3) 완전해

( ) 2 ( ) ( ), (0) 0

dy t y t u t y

dt   

2 0 2

s   s  

yp  A

2 1

( ) 2

t

h p

y t  y  y  Ke^ 

1 2 1

( ) , 0

2 2

y t e^ t t

    

0 2 1 1

A A 2

   

st

yh  Ke

Initial condition: (0) 1 0 y   K 2

( )

y t

 Ke

Let

1) 균일해

2) 특수해

3) 완전해

( ) 2 ( ) ( ), (0) 0 ?

dy t y t t y

dt   

2 0 2

s   s  

yp  A

( ) _{h p} 2^t

y t  y  y  Ke^

( ) 0 ?, 0

y t t

  

0 2 A 0  A 0

st

yh  Ke

Initial condition: (0)y  K  0 ?

( )

y t

 Ke

Let

1) 균일해

2) 완전해 3) 초기조건

좌변 우변

4) 임펄스 응답

2 0 2

s   s  

( ) _h( ) 2^t

h t  h t  ICs  Ke^ ICs ( ) ^st

h th  Ke

( )

h t

 Ke

( ) 2 ( ) ( ), (0) ?

dh t h t t h

dt   

Initial condition: (0 )h ^  K 1 (0) :

h

0 0

0 0

0 0

( ) ( ) (0 )

( ) 1 (0 ) 1

dh d h d h

dt

d h

   

  

 

 









  

   

 



2 2

( ) ^t, 0 ( ) ^t ( )

h t e^ t or h t e u t^

   

5) Convolution 적분 :

라고 가정하면 [예제 3.5]의 결과로부터

( ) ( ) x t  u t

2 2

0

2

( ) ( ) ( )

( ) ( )

1 (1 ), 0

2 ( ) 1 (1 ) ( )

2

t t

t

y t x t y t

x h t d

e d e t

y t e u t



  







 



 

 

   

  





 [예제3.8] : Overall impulse response

1( ) h t ( )

x t

3( ) h t

( )

  y t



2( ) h t

4( ) h t

3 1

2

2 3

4

( ) ( )

( ) ( )

( ) ( )

( ) ( 1)

t

t

h t e u t h t u t h t e u t h t  t











 

 

1 2 3 4

3 2( 1)

( ) ( ) ( ) ( ) ( )

1 1 ( ) ( 1)

3

t t

h t h t h t h t h t e

u t e

u t

   

   

 Identity 시스템과 delay 시스템의 임펄스 응답

 적분기의 임펄스 응답

 Convolution is a smoothing operator

Convolution의 성질

0 0

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

h t t y t x t

h t t t y t x t t





  

    

( ) ( ) ( ) ( )

( )

x t u t

x  u t   d x   d

 

     

( ) ( ) ( )

( )

h t u t y t x   d

    

 미적분 특성

( ) h t ( )

x t y t( )

( )

1 h t

( ) dx t( ) x t  dt

( )

2( ) ^t ( ) h t x t x  d





1

( ) dy t( ) y t  dt

2( ) ^t ( )

y t y  d





d dt

d dt

t



t



( ) h t ( )t

 ^{h t( ) T}

 

( ) h t ( )

u t ^{( )}

 

t ( ) s t T u t

h  d





t



t



impulse response

unit step response

 Memoryless system

 Causal system

• LTI system is causal if

• Output of causal LTI system

 Stable system

• LTI system is BIBO stable iff impulse response is absolutely integrable.

LTI System의 임펄스 응답

( ) ( ), ( ) ( )

y t

Kx t K

h t

K  t

( ) 0, 0 h t  t 

0

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

t

y t x h t d x h t d

h x t d h x t d

     

     



 

 



   

   

 

 

( ) h t dt



  

