Lecture Digital signal processing: Lecture 5 - Zheng-Hua Tan

In this chapter you will learn: Frequency response, system functions, relationship between magnitude and phase, all-pass systems, minimum-phase systems, linear systems with generalized linear phase.

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Digital Signal Processing, V, Zheng-Hua Tan, 2006

1

Digital Signal Processing, Fall 2006

Zheng-Hua Tan

Department of Electronic Systems Aalborg University, Denmark

zt@kom.aau.dk

Lecture 5: System analysis

Course at a glance

Discrete-time signals and systems

Fourier-domain

representation

System analysis

MM1

MM2

MM6

Sampling and reconstruction MM5

System structures

System

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3

System analysis

Time domain: impulse response, convolution sum

Frequency domain: frequency response

z-transform: system function

LTI system is completed characterized by …

) ( ) ( ) (z X z H z

) ( ) ( ) (e jω X e jω H e jω

∑∞

−∞

=

−

=

k

k n h k x n

h n x n

y[ ] [ ] * [ ] [ ] [ ]

4

Part I: Frequency response

System functions

Relationship between magnitude and phase

All-pass systems

Linear systems with generalized linear phase

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5

Frequency response

output

Magnitude Æ magnitude response, gain, distortion

Phase Æ phase response, phase shift, distortion

| ) (

|

| ) (

|

| ) (

|Y e jω = X e jω ⋅ H e jω

) ( ) ( ) (e jω X e jω H e jω

∠

Ideal lowpass filter – an example

Frequency selective filter

⎩

⎨

⎧

<

=

π ω ω

ω ω

ω

|

| , 0

,

|

| , 1 ) (

c

c j

e

H

∞

<

∞

−

n

lp[ ] sin ,

π ω

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7

Make noncausal system causal

Ideal delay

In general, any noncausal FIR system can be made

cause by cascading it with a sufficiently long delay!

But ideal lowpass filter is an IIR system!

] [ ] [n n n d

] [ ] 1 [ ]

[

difference Forward

n n

n

h = δ + − δ

] 1 [ ] [ ] [

difference Backward

−

= n n n

] 1 [ ] [

delay sameple

-One

−

= n

n

]

[n

x

]

[n

]

[n y

8

Phase distortion and delay

1

| ) (

| jω =

id e H

] [ ]

d n j j

id e e

H ( ω) = −ω

π ω ω

∠H id(e j ) n d, | |

Delay distortion

Linear phase distortion

⎩

⎨

⎧

<

= −

π ω ω

ω ω

|

, 0

,

|

| , )

(

c

c n

j j

lp

d

e e

H

∞

<

∞

−

n n

n n n

h

d

d c

lp ,

) (

) ( sin ] [

π ω

Ideal lowpass filter is always noncausal!

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9

Group delay

signal

approximation (though in reality maybe nonlinear)

and the output is approximately

) cos(

] [ ] [n s n 0n

0

) ( ω ≈ − ω − φ

n e

H

) ) ( cos(

] [

| ) (

| ]

n n n

n s e H n

y

)]}

( {arg[

)]

(

ω

d

d e

H

=

An example of group delay

Figure 5.1, 5.2, 5.3

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An example of group delay

12

Part II: System functions

Frequency response

All-pass systems

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13

System function of LCCDE systems

∏

∑

=

−

=

−

=

−

=

−

=

N

k k

M

m m

N

k

k k

M

m

m m

z d

z c a

b

z a

z b z X

z Y

z

H

1

1 1

1

0 0 0 0

) 1 (

) 1 ( ) (

) ( ) ( )

(

∑

=

−

=

− = M

m

m m N

k

k z Y z b z X z

a

0 0

) ( )

(

∑

=

−

=

m m N

k

k y n k b x n m

a

0 0

] [ ]

[

k k

m m

d z z

z d

z c

z

z c

=

−

=

−

at pole a 0

at zero a

r denominato

in the ) 1

(

0

at pole a

at zero a

numerator

in the ) 1

(

1 1

Stability and causality

h[n] absolutely summable

H(z) has a ROC including the unit circle

h[n] right side sequence

H(z) has a ROC being outside the outermost pole

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15

Inverse systems

rational system functions

Poles become zeros and vice versa

ROC: must have overlap btw the two for the sake of

G(z)

] [ ] [

* ] [

]

[

) (

1 )

(

1 ) ( ) ( )

(

n n h n h

n

g

z H z

H

z H z H

z

G

i i

i

δ

=

∏

=

−

=

−

=

−

=

−

=

−

=

M

m m

N

k k i

N

k k

M

m m

z c

z d b

a z H

z d

z c a

b z H

1

1 1

1

0 0

1

1 1

1

0 0

) 1

(

) 1

( ) ( ) (

) 1

(

) 1

( ) ( ) (

16

Example

So,

1 1

1 1 1

5 0 1

9 0 5

0 1

1 5

0 1

9 0 1 )

(

9 0

|

| , 9 0 1

5 0 1 )

(

−

=

−

=

>

−

=

z

z z

z

z z

H

z z

H

i

] 1 [ ) 5 0 ( 9 0 ] [ ) 5 0 ( ]

[

5 0

|

−

=

>

−

n u n

u n

h

z

n n

i

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Part III: Magnitude and phase

Frequency response

System functions

All-pass systems

Relationship btw magnitude and phase

functions, there is constraint btw magnitude and

phase

) (

| ) (

| ) (e jω H e jω e j H e jω

ω

ω ω

ω

j e z j

j j

z H z H e H e H e

∏

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An example

20

An example

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Part VI: All-pass systems

Frequency response

System functions

All-pass systems

1

* 1

1 )

−

=

az

a z z

H ap

ω

ω ω

ω

ω ω

j

j j

j

j j

ap

ae

e a e

ae

a e e

H

−

=

−

= 1

1

1 ) (

*

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An example

P275 Example 5.13,

First-order all-pass system

24

An example

Second-order all-pass

system

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Part V: Minimum-phase systems

Frequency response

System functions

All-pass systems

Minimum-phase systems

system

Stable and causal Æ poles inside unit circle, no

restriction on zeros

Zerosare also inside unit circle Æ inverse system is

also stable and causal (in many situations, we need

inverse systems!)

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Minimum-phase and all-pass decomposition

Any rational system function can be expressed as:

Suppose H(z) has one zero outside the unit circle at

) ( ) ( )

(z Hmin z H z

1

* 1 1 1

* 1 1

1 ) 1 )(

(

) )(

( )

(

−

=

−

=

cz

c z cz z H

c z z H z

H

1

|

,

/

1 * <

= c c

z

28

Frequency response compensation

When the distortion system is not minimum-phase

system:

Frequency response magnitude is compensated

Phase response is the phase of the all-pass

) ( ) ( )

(z H min z H z

) (

1 )

(

min z H z H

d

) ( )

( ) (

)

(z H z H z H z

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Properties of minimum-phase systems

of an all-pass system is negative for

nonminimum-phase (+all-pass nonminimum-phase) always decreases the

continuous phase or increases the negative of the

phase (called the phase-lag function)

Minimum-phase is more precisely called minimum Minimum-phase-lag

system

) ( ) ( )

(z Hmin z H z

π

≤ 0

)]

( arg[

)]

( arg[

)]

( arg[H e jω = Hmin e jω + H ap e jω

Part VI: Linear-phase systems

Frequency response

System functions

All-pass systems

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Design a system with non-zero phase

Constant frequency response magnitude

Zero phase, when not possible

accept phase distortion, in particular linear phase since

it only introduce time shift

Nonlinear phase will change the shape of the input

signal though having constant magnitude response

32

Ideal delay

) (

) ( sin ]

[

α π

−

=

n

n n

h id

1

| ) (

| jω =

id e

H

] [ ]

π ω

ωα

|

| , )

( j j

id e e

H

α

π ω ωα

ω

=

<

−

=

∠

)]

( [

|

| , )

(

j id

e H

grd

e

H

d

n

=

α

when

) (

) ( sin ] [

d

d c lp

n n

n n n

h

−

=

π ω

phase linear with lowpass Ideal

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Generalized linear phase

constants real

are

and

of function real

a is

)

(

) ( )

(

| ) (

| )

(

β

α

ω

β ωα ω ω

ωα ω

ω

j

j j j j

j j j

e

A

e e A e

H

e e H e

H

+

−

=

Summary

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Course at a glance

Discrete-time signals and systems

Fourier-domain

representation

DFT/FFT

System analysis

Filter structures Filter design

Filter

z-transform

MM1

MM2

MM9,MM10

MM3

MM6

MM4

Sampling and reconstruction MM5

System structure

System

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