bgl/snu 5.linear time-invariant system 5.1 the frequency response of lti systems 5.2 system...
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BGL/SNU
5.Linear Time-Invariant System
5.1 The Frequency Response of LTI systems
5.2 System Functions for Systems Characterized by Linear
Constant-Coefficient Difference
5.3 Frequency Response for Rational System Functions
5.4 Relationship between Magnitude and Phase
5.5 All-Pass Systems
5.6 Minimum-Phase Systems
5.7 Linear Systems with Generalized Linear Phase
5.8 Summary
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5.1 Frequency Response
][*][][ nhnxny )()()( zHzXzY )(|)(|)(
jeHjjj eeHeH
(e.q) Frequency selective filter - ideal
|)(| jeH
cc
)( jeH
cjeH ||,1|)(|
elsewhere,0
)( jeH
)(
)(sin][
n
nnh c
( : delay,centerpoint of sync function)
1
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Group dalay
)(arg)(grd)(
jj eH
d
deH
)(arg).( jeHqe )(
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5.2 System Functions for Constant Coefficient Systems
M
kk
N
kk knxbknya
00
][][
N
kk
M
kk
N
k
kk
M
k
kk
zd
zc
a
b
za
zb
zX
zYzH
1
1
1
1
0
0
0
0
)1(
)1(
)(
)()(
Stable if Causal if Roc includes
kdk 1,|| z
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-Inverse System
1)()( zHzH i
N
kk
M
kk
i
zc
zd
a
b
zHzH
1
1
1
1
0
0
)1(
)1(
)(
1)(
][][*][ nnhnh i
Inverse system, stable if all poles and zeros, inside the uc minimum-phase system
)1()9.0(5.0)()9.0()( , 9.01
5.01)( (e.q) 1
1
1
nununhz
zzH nn
)1()5.0(9.0)()5.0()( , 5.01
9.01)( 1
1
1
nununhz
zzH nn
ii
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-FIR vs IIR
N
k k
kNM
r
rr zd
AzBzH
11
0 1)(
N
k
nkk
NM
rr nudArnBnh
10
][)(][][
FIR part IIR part
BGL/SNU
5.3 Frequency Response of Rational System Functions
)1)1) : phase0
0 N
jωk
Mjω
kj ed(ec(
a
bH(e
)1arg)1arg) grd :delay N
jωk
Mjω
kj ed(
d
dec(
d
dH(e
(note) arg : continuous phase ARG : its principal value in
),( )(2)( ARG) arg reHH(e jj
N
k
jk
M
k
jk
j
ed
ec
a
beH
1
1
0
0
|)1(|
|)1(||)(| : Magnitude
BGL/SNU
(example)
11) zreH(z j
jjj ereH(e 1)
)]cos21log[10 2 θ(wrr)||H(e jω
])cos1
)sinarctan
θ(wr
θ(wr)H(e jω
)cos21
)cos grd
2
2
θ(wrr
θ(wrr)H(e jω
Check how they change as r and vary.
BGL/SNU
5.4 Relationship between Magnitude and Phase
)(ˆ][ˆ
)()(][ )](arg[
j
eXjjj
eXnx
eeXeXnxj
F
F
(complex cepstrum of x[n])
from Eqs. (11.28) and (11.29) (pp. 781)
deXx
deXxeX
deXeX
j
jj
jj
)(log2
1]0[ˆ,
2cot)](arg[
2
1]0[ˆ)(log
2cot)(log
2
1)](arg[
where
-
-
P
P
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※Relationship between real part and imaginary part of complex sequence (single-side band)
single-side band sequence 0,0)( jeX
)()()(
][][][ j
jj
rj
ir
ejXeXeX
njxnxnx
(complex sequence)F
)()(2
1)(][][
2
1][
)()(2
1)(][][
2
1][
**
**
jjjii
jjjrr
eXeXejXnxnxnjx
eXeXeXnxnxnx
0),(
0),()(
jr
jrj
iejX
ejXeX or
0,
0,)(
)()()(
where
j
jeH
eXeHeX
j
jr
jji
Hilbert Transform
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Illustration of decomposition of a one-sided Fourier transform
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Inverse Hilbert transform
0,0
0,)2/(sin2
2
1
2
1][
)()()()(
1)(
2
0
0
n
nn
n
djedjenh
eXeHeXeH
eX
njnj
ji
jjij
jr
impulse response of an ideal Hilbert transformer
Hilbert transformer
xr[n] xr[n]
xi[n]
x[n]
BGL/SNU
5.5 Allpass System
)cos(21
1)(
)cos(1
)sin(arctan2)(
1)(
)(
)(
1
1
1)(
)(1
)(
2
2
***
1
*1
rr
reHgrd
r
reH
foreH
eA
eAe
ae
eae
ae
aeeH
reaaz
azzH
jap
jap
jap
j
jj
j
jj
j
jj
ap
jap
1a
*
1
a1
)( jap eH
2
)( jap eHgrd
2
1r
r
r
1
1
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- 2nd order allpass function
- Nth order allpass function ( real-coeff)
)(
)()(
1
zA
zAzzH
N
ap
22
11
21
12
1*1
1*1
1)1()1(
)()()(
zaza
azaz
zaaz
azazzH ap
coeffs.-real
1
2
)( jap eH )( j
ap eHgrd
2 2
1a
a
1
*
1
a
*a
BGL/SNU
5.6 Minimum phase System
)( and )( ofn combinatioin drepresente becan functions system All -
Circle Unit theinside zeros all and poles all ),( -
min
min
zHzH
zH
ap
).( ge
)()(
11)1)(1)(1)(1(
1,1),)()(1)(1()(
min
1
*1
1*
11*11*1
*111*1
zHzH
bz
bz
zb
bzzbbzzaaz
babzbzzaazzH
ap
ab
1 )(zH )(min zH )(zH ap
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)()(
? " phase-minimum "Why
min jj eHeH
1
2
3
) phase negative ( lag-phase minimum
)(arg)(arg)(arg min
jap
jj eHeHeH) 0grd ( slope negative
delay group minimum
)()()( min
jap
jj eHgrdeHgrdeHgrd
) 0grd (
) 5.32 figure see (delay energy minimum
)nwhen (,)()(0
2
min0
2
n
m
n
m
mhmh
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Sequences all having the same frequency response magnitude
( zeros are at all combinations of 0.9ej0.6 and 0.8ej0.8 and their reciprocals)
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)0()0(
))(0()()(lim)(lim0 (note)
5.66) prob see (
min
minmin
hh
bhzHzHzH)h( apzz
1|| bsince
BGL/SNU
- Frequency response Compensation
Distoring System
C ompensating System
][ns ][nsd ][nsc
)(zH d )(zH c
)(zG
)()()(such that )( Choose zHzHzHzH apcdc
BGL/SNU
5.7 Generalized Linear Phase System jj
Rj eeHeH )()(
Part Real
delayconstant )(
phaselinear )(arg
j
j
eHgrd
eH
)order highest thehalf (2
M
1 case
phase!linear )(
...})12
(cos22
cos2{
...)()(
...)(
)Cond. Sufficient(,...)(
102
)12
()12
(2
1222
0
0)1(
110
110
jR
j
Mj
Mj
Mj
Mj
Mj
Mj
Mj
MjMjjj
nMnM
M
eHe
Ma
Mae
eeeaeeea
eaeaeaaeH
aazazaazH
BGL/SNU
quadin zeros 0)(0)(
)()...(
)...()(
1
10
11
110
ii
MMM
MM
M
MMMM
zHzH
zHzazazaz
azazazzH
1
iMi aa
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location zeroon restrict no
)1()1(
)1()1(
)()(
I Type
11
11
111
HH
HH
zHzzH
ordereven
aa
M
nMn
0)1(
)1()1(
)1()1(
)()(
II Type
2
22
22
122
H
HH
HH
zHzzH
orderodd
aa
M
nMn
11
BGL/SNU
) example (
0 1 2 3 4 5 6 n0 1 2 3 4 5 6 n
)(1 nh )(2 nh
2sin
25
sin)(
...1)(
21
411
jj eeH
zzzH
2sin
2sin)(
...1)(
2
3
2
312
jj eeH
zzzH
0 2 0 2
2delay 3/2delay
BGL/SNU
2 case
2,
2phase!linear dgeneralize)(
...})12
(sin22
sin2{
...)()()(
,...)(
102
)12
()12
(2
1222
0
)1(1
110
MeHe
Ma
Maje
eeeaeeeaeH
aazazazaazH
jR
jj
Mj
Mj
Mj
Mj
Mj
Mj
Mjj
nMnM
MM
M
quadin zeros 0)(0)(
)()...(
)...()(
1
10
11
110
ii
MMM
MM
M
MMMM
zHzH
zHzazazaz
azazazzH
BGL/SNU
0)1(,0)1(
)1()1(
)1()1(
)()(
III Type
33
33
33
133
HH
HH
HH
zHzzH
ordereven
aa
M
nMn
0)1(
)1()1(
)1()1(
)()(
IV Type
4
44
44
144
H
HH
HH
zHzzH
orderodd
aa
M
nMn
1 1
BGL/SNU
) example (
)(3 nh )(4 nh
)sin22sin2()(
01)(2
3
4313
jeeH
zzzzHjj )
2sin2
2
3sin2()(
1)(
2
3
4
3214
jeeH
zzzzH
jj
0 2 0 2
2delay 3/2delay
0 1 2 3 4 5 6 n0 1 2 3 4 5 6 n
) zero be tohappened (
BGL/SNU
H.W. of Chapter 5
Ref : [1] Project 4.1 Transfer Function Analysis
Text : [2] 5.10 [3] 5.21 [4] 5.38 [5] 5.45