5.7 Linear System with Generalized Linear Phase.
• For causal system, zero phase is not attainable, and consequently, some phase distortion must be allowed.
• The effect of linear phase with integer slope is a simple time shift.
A nonlinear phase, on the other hand, can have a major effect on the shape of signal, even when the frequency-response magnitude is constant.
• Thus, in many situation it is particularly desirable to design systems to have exactly or approximately linear phase.
5.7.1 System with Linear Phase
(1) Ideal delay with linear phase
) k n (
) k n ( ) sin k ( x )
n ( h
* ) n ( x ) n ( y
) n (
) n ( ) sin
n ( h
, e ) e ( H
k
k id
id
j j
id
a - - p
a - -
= p
=
a - p
a -
= p
p
<
w
=
×
å
¥
=
-¥
= wa
- w
1
) n n ( x ) n n (
* ) n ( x ) n ( y
) n n ( ) n ( h
) eger (int
n
d d
d id
d
-
= -
d
=
- d
=
= a
×
(2) Ideal LPF with linear phase
) n ( ) h
n n (
) n n ( sin )
n n n 2 (
) n n n 2 ( ) sin
n n 2 ( h
) eger (int n
) n (
) n ( ) sin
n ( h
, 0
, ) e
e ( H
lp d
d c d
d
d d
c d
lp d
c lp
c c j
lp
j
- = p
-
= w -
- p
- -
= w -
= a
×
a - p
a -
= w ç ç è æ
p
£ w
<
w w
<
= w
×
wa -
w
Ü the impulse response is symmetric about
n=n
d• Zero-phase and
a= n
d) n ( n h
n ) sin
e ( h
) e ( H e
) e ( H ) e ( H
lp
^ j c
lp
^
j lp n
j j lp lp j
^
d
- p =
= w
=
=
w
w w
w w
• 2a is an integer
h
lp(2a-n)=h
lp(n)Ü the point of symmetry a is an integer plus one-half.
• If the a or 2a is not an integer, there is no symmetry at all.
• Impulse response가 대칭이면 선형위상 특성을 가지며 그 역은 성립하지 않는다.
3
5.7.2 Generalized Linear Phase
• A system is referred to as a generalized linear-phase system if it’s frequency response can be expressed in the form
.
b + aw - w
w
=
j j jj
) A ( e ) e e
( H
where a and b are constants and is a real ( possibly bipolar ) function of w.
(1) Derivation of an equation that must be satisfied by h(n), a and b for constant group-delay systems.
) e ( A
jwn cos ) n ( h
n sin ) n ( h )
( cos
) (
) sin tan(
. Re]
[Im/
n sin ) n ( h j
n cos ) n ( h e
) n ( h )
e ( H
) sin(
) e ( jA )
cos(
) e ( A e
) e ( A ) e ( H
n n
n n
n j n
j
j j
) ( j j j
w w -
wa = - b
wa -
= b wa - b
×
w -
w
=
=
×
wa - b +
wa - b
=
=
×
å å
å å
å
¥
-¥
=
¥
-¥
=
¥
-¥
=
¥
-¥
= w
-
¥
-¥
= w
w w
aw - b w w
된다 같게
도 같으므로
식이 두
위 시스템이면 같은
5
· 위의 두식에서 오른쪽 두항을 cross multiply 해서 조합하면
[ ]
å å
¥
-¥
=
¥
-¥
=
= b + a - w
=
wa - b w
+ wa - b w
n n
. 0
) n ( sin ) n ( h
)]
cos(
n sin )
sin(
n )[cos n
( h
된다 이
이 방정식은 시스템이 선형위상 (상수의 group delay) 특성을 가지기 위한 h(n), a, b 에 대한 필요조건이다.
(2) 일반적으로 b값에 대해서는 H(e
jw)의 크기가 양 혹은 음인 경우 (b=0 or p) 와
H(e
jw) 가 순허수인 경우 에 대해 고려할 수 있으며, 그리고
5.7.1절에서의 대칭조건 h(2a-n)=h(n)을 고려하면,
÷ø ç ö
è
æ p p
=
b 2
or 3 2
[ ]
[ ]
ç ç ç ç çæ
-
= - a
= a
÷ ø ç ö
è
æ w -a =
p
= p b
× ç ç ç ç ç ç
è æ
= - a
= a
÷ ø ç ö
è
æ w -a =
p
= b
×
å å
¥
-¥
=
¥
-¥
=
) n ( h ) n 2 ( h
) eger int an ( M 2
0 ) n ( cos ) n ( 2 h
or 3 2
) n ( h ) n 2 ( h
) eger int an ( M 2
0 ) n ( sin ) n ( h or
0
n n
