5장 Transform Analysis of LTI Systems
5.1 The Frequency Response of LTI Systems
thesystem of
shift phase
or response phase
: ) e ( H
system the
of gain or
resonse maginitude
: ) e ( H
) e ( X )
e ( H )
e ( Y
) e ( X ) e ( H )
e ( Y
) e ( X ) e ( H ) e ( Y
) z ( X ) z ( H ) z ( Y
) n ( h ) n ( x ) k n ( h ) k ( x )
n ( y
j j
j j
j
j j
j
j j
j k
w w
w w
w
w w
w
w w
w
¥
-¥
=
Ð
×
Ð + Ð
= Ð
=
=
×
=
×
*
= -
=
×
å
5.1.1 Ideal Frequency-Selected Filter
(1) LPF
¥
<
<
¥ p -
= w
×
çç è æ
p
£ w
<
w
w
<
= w
× w
n n ,
n ) sin
n ( h
, 0
, ) 1
e ( H
c lp
c j c
lp
(2) HPF
çç è æ
p
<
w
<
w
w
<
= w
× w
c j c
hp 1,
, ) 0
e (
H 1
(3) • The ideal filter are noncausal, and their impulse response extend from -¥ to + ¥, and the system are not computationally realizable.
• The phase response of the ideal filter is specified to be zero.
5.1.2 Phase Distortion and Delay
(1) Ideal delay system
p
<
w w
-
= Ð
=
=
×
- d
=
×
w - w
, n H
1 H
e ) e ( H
) n n ( ) n ( h
d id
id
n j j
id
d id
d
(2) Ideal LPF with liner phase
· time domain shift (delay distortion) : frequency domain linear phase (phase distortion)
¥
<
<
¥ - -
p
-
= w
×
çç è æ
p
£ w
<
w
w
<
= w
×
w - w
n ) ,
n n (
) n n ( ) sin
n ( h
, 0
, ) e
e ( H
d d c
lp
c
c n
j j
lp
d
• 차단 주파수가 LPI와 같을 경우
) n ( h ) n ( )
n ( h
) e ( H 1 ) e ( H
hp hp
j lp j
hp
- d
=
-
= w
w
2
(3) Group delay
· The deviation of the group delay from a constant indicates the degree of nonlinearity of the phase.
)]
e ( H [ grd )])
e ( H (arg[
d ) d
(
Z jw = jw
- w
= w
×
(4) Example (effect of attenuation and group delay)
· 입력신호 : 주파수가 0.85p, 0.25 p, 0.5 p인 3개의 narrowband phase 구성.
· 필터 : 저역통과 특성을 가지며 group delay 가 상수가 아님.
3
4
· 출력신호 : 입력신호중 0.85 p 주파수 성분은 시스템에 의해 감쇄 되었으며, 0.25 p, 0.5 p 주파수 성분은 각 주파수에서 시스템의 group delay가 달라서 출력에서의 위치가 바뀌게 된다.
· 그러므로 필터 설계에 있어서 선형위상(constant group delay) 특성은 중요하다.
5.2 System Functions for Systems Characterized by Linear Constant-Coefficient Difference Equations.
( )
( k 1)
N
1 k
1 k M
1 k k
N
0 k
k M
0 k
k k
Z d 1
Z c 1
0 0 Z
Z b M
0 k
k k k
N
0 k
M
0 k
k M
0 k
k
b )
z ( X
) z ( ) Y z ( H
) z ( X Z b )
z ( Y Z
) k n ( x b )
k n ( y
-
=
-
= -
=
= -
- -
=
- -
=
=
=
P
= P å
= å
=
×
=
×
-
= -
×
å å
å å
a a
a
a k
5.2.1 Stability and Causality
(1) Stability : the impulse response must be absolutely summable.
å
¥
-¥
=
¥
<
n
) n (
h 5
(2) Causality : the h(n) must be right-sided sequence, and therefore, the ROC of H(z) must be outside the outermost pole.
(3)
The condition for stability is equivalent to the condition that the ROC of H(z) include the unit circle.
1 z for , Z
) n ( h )
n ( h
n n
n < ¥ =
=
¥
å
<å
¥
-¥
=
¥
-¥
=
-
(4) In order for a LTI system to be both causal and stable, the ROC of the
corresponding system function must be outside the outermost pole and include the unit circle. Clearly, this requires that the poles of the system function be inside the unit circle.
5.2.2 Inverse Systems
(1) LTI system H(z)의 inverse system Hi(z)는
) n ( ) n ( h ) n ( h
) z ( H ) 1 z ( H 1
) z ( H ) z ( H
i
i i
d
=
*
×
=
®
=
×
6
(2) Hi(z)의 pole 은 H(z)의 zero 이다.
( )
( )
å
= å P
= P
=
-
=
- -
=
-
=
- - - -
M
1 k
1 k N
1 k
1 k
1 k N
1 k
1 k M
1 k
) Z c 1 (
) Z d 1 (
0 0 i
Z d 1
Z c 1
0 0
) b z ( H
) b z ( H
a a
(3) 일반적으로 H(z)와 가 모두 causal, stable 해야 하므로 두
시스템에서 pole과 zero는 모두 단위원내에 존재하여야 한다.
(4) Minimum-phase system : pole과 zero가 모두 단위원내에 존재 )
z ( Hi
7
5.2.3 Impulse Response for Rational System Function
(1) IIR system (N>M)
) n ( u d A )
n ( h
Z d 1 ) A
Z ( H
n k k N
1 k
1 k
k N
1 k
å å
=
-
=
=
= -
(2) FIR system(N=0)
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æ £ £
= -
d
=
=
å å
=
-
=
otherwise ,
0
M n
0 , ) b
k n ( b )
n ( h
Z b )
Z ( H
n k
N
1 k
k k M
1 k
8
