제2강 기본적인 시스템의 성질 및 선형시불변 시스템의 정의
1
Basic system properties
A system is described by
System
[n ] ] y
[n x
(t )
x y (t )
]) [ ( ]
[
)) ( ( )
(
n x f n
y
t x f t
y
Continuous-time system
Discrete-time system
Basic system properties
Systems with memory vs without memory
] [ ]
1 [
2 ] 1 [
] [
]) [ ]
[ 2 ( ]
[
2 2n x n
x n
y n
y
n x n
x n
y
Accumulator
Delay
Capacitor
nk
k x n
y [ ] [ ] ] 1 [
]
[ n n x y
tx d
t C
y 1 ( ) )
(
System without memory
System with memory
Basic system properties
Causality (인과성)
• A system is causal if the output at any time depends on values of the input at only the present and past times
] 1 [
] [ ]
[ n x n x n y
MM k
k n M x
n
y [ ]
1 2
] 1
Moving average
[
x [ n 1 ], x [ n ], x [ n 1 ],
Present Future
Past
Basic system properties
Inverse system
System Inverse
system
] x [n ]
[n ] y
[n x
nk
k x n
y [ ] [ ] w [ n ] y [ n ] y [ n 1 ] )
( 2 )
( t x t
y ( )
2 ) 1
( t y t
w
Examples:
Basic system properties
Stability in bounded input and bounded output (BIBO) sense
• Examples
) ( )
( t tx t
y
' )
( )
( t B y t B
x
' ]
[ ]
[ n B y n B
x
)
)
(( t e
x ty
Basic system properties
Time invariance (시불변)
• Are the following systems time-invariant?
] [ ]
[ n y n
x x [ n n
0] y [ n n
0]
) 2 ( )
(
] [ ]
[
)]
( sin[
) (
t x t
y
n nx n
y
t x t
y
) ( )
( t y t
x x ( t t
0) y ( t t
0)
Basic system properties
Linearity (선형성)
] [ ]
[
] [ ]
[
2 2
1 1
n y n
x
n y n
x
ax
1[ n ] bx
2[ n ] ay
1[ n ] by
2[ n ]
]) [ ( ]
[
) ( )
(
) ( )
(
2
n x RE n
y
t x t
y
t tx t
y
) ( )
(
) ( )
(
2 2
1 1
t y t
x
t y t
x
ax
1( t ) bx
2( t ) ay
1( t ) by
2( t )
Discrete-time LTI systems
Any discrete-time signal can be represented by
k
k n k
x n
x [ ] [ ] [ ]
Discrete-time LTI systems
Impulse response (임펄스 응답)
] [n
x y[n]
Linear system
]
[n h[n]
Linear system
Impulse response
Discrete-time LTI systems
Linear system (선형 시스템)
] [n Linear system y
k
k n k
x n
x [ ] [ ] [ ]
k
k
n h k x n
y [ ] [ ] [ ]
] [ ]
[ n k h
kn
Discrete-time LTI systems
Time-invariant system (시불변 시스템)
] [nk
h[n k]
LTI system
Impulse response
] [
]
[ n h n k h
k
For time-invariant system ]
[n h[n]
LTI system
]
[n
h
Discrete-time LTI systems
Linear time-invariant (LTI) system (선형 시불변 시스템)
] [n LTI system y
k
k n k
x n
x [ ] [ ] [ ]
k
k n h k x n
y [ ] [ ] [ ]
Linear convolution
k
k n
h k x
n h n
x n
y
] [
] [
] [
* ] [ ]
[
Discrete-time LTI systems
How to calculate the linear convolution.
1. Obtain by flipping the signal ] with respect to 0.
2. Obtain by shifting by .
3. Take a product of ] and for all ∈ ∞, ∞ 4. Sum the product over ∈ ∞, ∞
Discrete-time LTI systems
Example. Consider an LTI system whose impulse response is h[n]. Find the output when the input is given by x[n].
1
0.5
2 ]
[n h
] [n x
0 1 2
0 1
k
k n h k x n
y[ ] [ ] [ ]
Discrete-time LTI systems
Example
Example
] [ ]
[
], [ ]
[
n u n
h
n u n
x
n
otherwise n n
h
otherwise n n
x
n
, 0
6 0
] ,
[
, 0
4 0
, ] 1
[
Continuous-time LTI systems
Continuous LTI systems
• Impulse response (임펄스 응답)
)
(t h(t)
LTI system
Impulse response
Continuous-time LTI systems
Continuous-time LTI systems (연속시간 선형 시불변 시스템)
h t d x
t x t
h t
y
) (
) (
) (
* ) ( )
(
) (t
x h(t)
LTI system with impulse response
x h t d t
y ( ) ( ) ( )
Linear convolution
Continuous-time LTI systems
Example
