Post on 31-Jan-2021
TA for LTI1
Transform Analysis of Linear Time-Invariant SystemspIntroductionpThe Frequency Response of LTI SystemspSystem Functions for System Characterized by Linear
Constant-Coefficient Difference EquationspFrequency Response for Rational System FunctionspRelationship between Magnitude and Phase
TA for LTI20. IntroductionpFour System Descriptions
l Impulse Responsel Difference Equationsl Frequency Responsel Z-Transform ( Transfer Function
and System Function)
pFocus of the Chapterl Frequency Responsel System Function
LTI SystemsLTI Systemsx[n] y[n]
y n x n h n x k h n kk
[ ] [ ]* [ ] [ ] [ ]= = −=−∞
∞
∑
a y n k b x n kkk
N
kk
M
[ ] [ ]− = −= =∑ ∑
0 0
Y(ejω)=H(ejω)X(ejω)
Y(z)=H(z)X(z)
TA for LTI31. The Frequency Response of LTI SystemspMagnitude and Phase Response
pIdeal Frequency-Selective Filtersl Lowpass
l Highpass
Y e H e X ej j j( ) ( ) ( )ω ω ω=
Y e H e X ej j j( ) ( ) ( )ω ω ω= ∠ = ∠ + ∠Y e H e X ej j j( ) ( ) ( )ω ω ω
H elpj c
c
( ),
,ω ω ω
ω ω π=
<< ≤
1
0
H elpj c
c
( ),
,ω ω ω
ω ω π=
<< ≤
0
1
h nn
nnlp
c[ ]sin
,= − ∞ < < ∞ωπ
h n nn
nnhp
c[ ] [ ]sin
,= − − ∞ < < ∞δωπ
Phase Response ?Causalilty ?
Phase Response ?Causalilty ?
TA for LTI41. The Frequency Response of LTI Systems (c.1)pPhase Distortion and Delay
l Observation 1
l The Ideal Lowpass Filter with linear phase
l Observation 2-- A narrow band signal s[n]cos(ω0n)The phase for the ω0 can be approximated as
l Group Delay-- A measure for the nonlinearity of the phase
h n n nid d[ ] [ ]= −δ H e eidj j nd( )ω
ω= −Delay Linear Phase
H ee
lpj
j nc
c
d
( ),
,ω
ω ω ωω ω π
=<
< ≤
−
0h n
n n
n nnlp
c d
d
[ ]sin ( )
( ),=
−−
− ∞ < < ∞ωπ
Ideal Filters with Causality ?Ideal Filters with Causality ?
{ }τ ω ωω ω( ) [ ( ) ] ( )= = − ∠g rd H e
d
dH ej j
∠ ≈ − −H e nj d( )ω φ ω0
y n s n n n nd d[ ] [ ] c o s ( )= − − −ω φ ω0 0 0
TA for LTI5
2. System Functions for System Characterized by Linear Constant-Coefficient Difference Equations
a y n k b x n kkk
N
kk
M
[ ] [ ]− = −= =∑ ∑
0 0
pDifference Equations
pZ-Transform
pEx.
H zY z
X z
b z
a z
kk
k
M
kk
k
N( )
( )
( )= =
−
=
−
=
∑
∑0
0
Equivalent Information
a z Y z b z X zkk
k
N
kk
k
M−
=
−
=∑ ∑=( ) ( )
0 0
H zz z
z z
( )
( )( )
=+ +
− +
− −
− −
1 2
11
21
3
4
1 2
1 1
TA for LTI6
2. System Functions for System Characterized by Linear Constant-Coefficient Difference Equations (c.1)
pStability l The condition for system stabillity is equivalent to the condition that
the ROC of H(z) include the unit circle.System stableROC
l Example
pRequirements for both Causal and Stablel The poles of the system function must be inside the unit circle.
h nn
[ ]= − ∞
∞
∑ < ∞
h n z n
n
[ ] −
= − ∞
∞
∑ < ∞
y n y n y n x n[ ] [ ] [ ] [ ]− − + − =5
21 2
TA for LTI7
2. System Functions for System Characterized by Linear Constant-Coefficient Difference Equations (c.2)
pInverse Systeml For a given LTI system with system
function H(z), the corresponding inverse system is defined to be the system with system function Hi(z) such that
H(z)Hi(z) = 1l Existence for the Inverse System
A causal and stable system has a causal and stable inverse system if and only if both the poles and the zeros are inside the unit circle.
l Ex. H z zz
( ).
.=
−−
−
−
1 0 5
1 0 9
1
1
H zH z
h n h n n
H eH e
i
i
ij
j
( )( )
;
[ ]* [ ] [ ] ;
( )( )
=
=
=
1
1
δ
ωω
H zb
a
c z
d z
kk
M
kk
N( )
( )
( )
=
−
−
−
=
−
=
∏
∏0
0
1
1
1
1
1
1
H za
b
d z
c z
kk
N
kk
M( )
( )
( )
=
−
−
−
=
−
=
∏
∏0
0
1
1
1
1
1
1
Poles
Zeros
TA for LTI8
2. System Functions for System Characterized by Linear Constant-Coefficient Difference Equations (c.3)
pImpulse Response for Rational Functions
l Infinite Impulse Reponse (IIR) SystemsThe length of the impulse response is infinite.
l Finite Impulse Response (FIR) SystemsThe length of the impulse reponse is finite.
pExamples
H z B zA
d zr
r
r
M Nk
kk
N
( ) = +−
−
=
−
−=
∑ ∑0
11 1
h n B n r A d u nrr
M N
k kn
k
N
[ ] [ ] [ ]= − +=
−
=∑ ∑δ
0 1
How to check ?
y n a x n kk
k
M
[ ] [ ]= −=∑
0
y n ay n x n a x n MM[ ] [ ] [ ] [ ]− − = − − −+1 11
TA for LTI9
3. Frequency Response for Rational System Functions-- Magnitude ResponsepA stable linear time-invariant system
l Rational Function
l Magnitude Response
l Gain (dB)
is approximately 6m dB, while is approximately 20m dB
H e
b e
a e
b
a
c e
d e
jk
j k
k
M
kj k
k
N
kj
k
M
kj
k
N( )
( )
( )
ω
ω
ω
ω
ω
= =
−
−
−
=
−
=
−
=
−
=
∑
∑
∏
∏0
0
0
0
1
1
1
1
H eb
a
c e
d e
jk
j
k
M
kj
k
N( )ω
ω
ω
=−
−
−
=
−
=
∏
∏0
0
1
1
1
1
( )( )
( )( )H e
b
a
c e c e
d e d e
jk
jk
j
k
M
kj
kj
k
N( )
*
*
ω
ω ω
ω ω
20
0
2
1
1
1 1
1 1
=− −
− −
−
=
−
=
∏
∏
20 20 20 1 20 110 100
0
101
101
log ( ) log log logH eb
ac e d ej
k
M
kj
k
N
kjω ω ω= + − − −
=
−
=
−∑ ∑H e j m( )ω = 2 H e j m( )ω = 1 0
20 20 2010 10 10log ( ) log ( ) log ( )Y e H e X ej j jω ω ω= +
TA for LTI10
3. Frequency Response for Rational System Functions-- Phase ResponsepPhase Response (c.1)
l The principal value of the phase is denoted as ARG[H(ejw)]
l Principal Values = Sum of Individual PVs
∠ =
∠
+ ∠ − − ∠ −−
=
−
=∑ ∑
H e
b
ac e d e
j
kj
k
M
kj
k
N
( )
[ ) [ )
ω
ω ω0
0 1 1
1 1
− < ≤
∠ = +
π π
π ω
ω
ω ω
ARG H e
H e ARG H e r
j
j j
[ ( )]
( ) [ ( )] ( )2
ARG H e ARGb
aARG c e
ARG d e r
jk
j
k
M
kj
k
N
[ ( )] [ )
[ ) ( )
ω ω
ω π ω
=
+ −
− − +
−
=
−
=
∑
∑
0
0 1
1
1
1 2
TA for LTI11
3. Frequency Response for Rational System Functions-- Phase Response (c.1)pPhase Response
l Alternative relation
pGroup Delayl Derivative of the continuous phase function
l That is
l Can be obtained from the principle values except at discontinuities.
grd H ed
dH e
d
dd e
d
dc ej j k
j
k
N
kj
k
M
[ ( )] {arg[ ( )]} ( arg[ ] ( arg[ ])ω ω ω ωω ω ω
= − = − − −−=
−
=∑ ∑1 1
1 1
grd H ed d e
d d e
c c e
c c e
j k kj
k kj
k
Nk k
j
k kj
k
M
[ ( )]Re{ }
Re{ }
Re{ }
Re{ }
ωω
ω
ω
ω=
−
+ −−
−
+ −
−
−=
−
−=
∑ ∑2
21
2
211 2 1 2
A RG H eH e
H ej I
j
Rj
[ ( )] a rc tan( )
( )
ωω
ω=
TA for LTI12
3. Frequency Response for Rational System Functions--Frequency Response of a Single Zero or Pole
pSingle Pole or Zerol The form
l The magnitude squared
l The log magnitude in dB is
l The phase
l Group Delay
1 1 1 1 22
2− = − − = + − −− − −re e re e re e r rj j j j j jθ ω θ ω θ ω ω θ( )( ) cos( )
( )1 1− −p z
20 1 10 1 210 102log log [ cos( )]− = + − −−re e r rj jθ ω ω θ
[ ]ARG re e rr
j j11
− =−
− −
−θ ω ω θω θ
arctansin( )
cos( )
[ ]grd re e r rr r
r rj jj j
11 2
2
2
2
2− =
− −+ − −
=− −−
−
θ ω
θ ω
ω θω θ
ω θcos( )cos( )
cos( )
re e1 −
TA for LTI13
3. Frequency Response for Rational System Functions--Frequency Response of a Single Zero or Pole (c.1)
pEx. ( )1 1− −p zv
v3
1
φ φ φ ω3 1 3− = −
TA for LTI14
3. Frequency Response for Rational System Functions-- Frequency Response of a Single Zero or Pole (c.2)
pFrequency Response for a Single Zero at π
TA for LTI15
3. Frequency Response for Rational System Functions-- Frequency Response of a Single Zero or Pole (c.3)
pFrequency Response for a Single Zero near π
TA for LTI164. Relationship between Magnitude and PhasepMagnitude Response ? ==> Phase Response
l Magnitude Squared of the System Frequency Response
l System Function
l Reciprocal Pairs of Poles and Zerosl Pole Selections ?l Zero Selections ?
pEx.
H e H e H e H z H zj j jz e j
( ) ( ) *( ) ( ) ( / *)ω ω ω ω2
1= ==
H z Hz
b
a
c z
d z
c z
d z
kk
M
kk
N
kk
M
kk
N( ) *(
*)
( )
( )
( )
( )
*
*
11
1
1
1
0
0
1
1
1
1
1
1
=
−
−
−
−
−
=
−
=
=
=
∏
∏
∏
∏
H z H zz z z z
e z e z e z e zj j j j( ) *( / *)
( )( . ) . ( )( . )
( . )( . )( . )( ./ / /1
2 1 1 0 5 0 5 1 1 0 5
1 0 8 1 0 8 1 0 8 1 0 8
1 1
4 1 4 1 4=
− + − +− − − −
− −
− − − −π π π π )/ 4
Transform Analysis of Linear Time-Invariant Systems0. Introduction1. The Frequency Response of LTI Systems1. The Frequency Response of LTI Systems (c.1)2. System Functions for System Characterized by Linear Constant-Coefficient Difference Equations2. System Functions for System Characterized by Linear Constant-Coefficient Difference Equations (c.1)2. System Functions for System Characterized by Linear Constant-Coefficient Difference Equations (c.2)2. System Functions for System Characterized by Linear Constant-Coefficient Difference Equations (c.3)3. Frequency Response for Rational System Functions-- Magnitude Response3. Frequency Response for Rational System Functions-- Phase Response3. Frequency Response for Rational System Functions-- Phase Response (c.1)3. Frequency Response for Rational System Functions-- Frequency Response of a Single Zero or Pole3. Frequency Response for Rational System Functions-- Frequency Response of a Single Zero or Pole (c.1)3. Frequency Response for Rational System Functions-- Frequency Response of a Single Zero or Pole (c.2)3. Frequency Response for Rational System Functions-- Frequency Response of a Single Zero or Pole (c.3)4. Relationship between Magnitude and Phase