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