Digital Signal Processing, Fall 2006

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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

[email protected]

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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System analysis

Three domains

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

Y =

) ( ) ( )

(e^j^ω X e^j^ω H e^j^ω

Y =

∑

^∞

−∞

=

−

=

k

k n h k x n

h n x n

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

Part I: Frequency response

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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Frequency response

Relationship btw Fourier transforms of input and output

In polar form

Magnitude Æmagnitude response, gain, distortion

Phase Æphase response, phase shift, distortion

| ) (

|

| ) (

|

| ) (

|Y e^j^ω = X e^j^ω ⋅ H e^j^ω ) ( ) ( )

Y =

) ( ) ( )

Y =∠ +∠

∠

Ideal lowpass filter – an example

Frequency response

Frequency selective filter

Impulse response

⎩⎨

⎧

<

= <

π ω ω

ω

ω ω

|

| , 0

,

|

| , ) 1 (

c j c

e H

∞

<

∞

−

= n n

n

h sin ^c , ]

[ ω

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Make noncausal system causal

Cascading systems

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

h =δ −

] [ ] 1 [ ] [

difference Forward

n n

n

h =δ + −δ

] 1 [ ] [ ] [

difference Backward

−

= n n

n

h δ δ

] 1 [ ] [

delay sameple -

One

−

= n n

h δ

] [n x

] [n

x y[n]

] [n y

Phase distortion and delay

Ideal delay system

Ideal lowpass filter with linear phase

1

| ) (

|H_id e^j^ω =

] [ ]

[ _d

id n n n

h =δ −

nd j j

id e e

H ( ^ω)= ⁻^ω

π ω ω

ω =− <

∠H_id(e^j ) n_d,| |

Delay distortion

Linear phase distortion

⎩⎨

⎧

<

= ⁻ <

π ω ω

ω

ω ω

ω

|

, 0

,

|

| ) ,

(

c

c n

j j

lp

e d

e H

∞

<

∞

− −

= − n

n n

n n n

h

d d c

lp ,

) (

) ( ] sin

[ π

ω

Ideal lowpass filter is always noncausal!

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Group delay

A measure of the linearity of the phase

Concerning the phase distortion on a narrowband signal

For this input with spectrum only around w₀, phase effect can be approximated around w₀ as the linear approximation (though in reality maybe nonlinear)

and the output is approximately

Group delay

) cos(

] [ ]

[n s n ₀n

x = ω

) 0

( ^ω ≈−ω −φ

∠H e^j n_d

0 w₀

) ) ( cos(

] [

| ) (

| ]

[n ≈ H e^j^ω⁰ s n−n_d ω₀ n−n_d −φ₀ y

)]}

( {arg[

)]

(

[ ^ω ^ω

τ ^j ω H e^j

d e d

H

grd =−

=

An example of group delay

Figure 5.1, 5.2, 5.3

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An example of group delay

Part II: System functions

Frequency response

System functions

All-pass systems

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System function of LCCDE systems

Linear constant-coefficient difference equation

z-transform format

∏

∑

=

−

=

−

=

−

=

−

=

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 z Y H

1

1 1

1

0 0 0 0

) 1 (

) 1 ( ) (

) (

) ) ( (

∑

=

−

=

− = ^M

m m m N

k k

kz Y z b z X z

a

0 0

) ( )

(

∑

= =

−

=

− ^M

m m N

k

kyn k b xn 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

Stable

h[n] absolutely summable

H(z) has a ROC including the unit circle

Causal

h[n] right side sequence

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

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Inverse systems

Many systems have inverses, specially systems with 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 z H 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

z a H

z d

z c a

z b H

1

1 1

1

0 0

1

1 1

1

0 0

) 1

(

) 1

( ) ( ) (

) 1

(

) 1

( ) ( ) (

Example

So,

1 1 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

H

z z z z

H

i

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

5 . 0

|

1 −

−

=

>

− u n 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

In particular, for systems with rational system functions, there is constraint btw magnitude and phase

Consider the square of the magnitude

)

| (

) (

| )

(e^j^ω H e^j^ω e^j ^H ^e^j^ω

H = ^∠

ω ω

ej z j

j

j H e H e H z H z

e

H( )| = ( ) ( )= ( ) (1/ )| ₌

| ² ^* ^* ^*

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An example

P271, Example 5.11

An example

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Part VI: All-pass systems

Frequency response

System functions

All-pass systems

Consider the following stable system function

1

* 1

) 1

( ₋

−

= − az

a z z

H_ap

ω ω ω

j j j

ap

e e a

ae a e e

H

−

= −

−

= −

1

) 1 (

*

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An example

P275 Example 5.13, First- order all-pass system

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

Magnitude does not uniquely characterize the system

Stable and causal Æpoles inside unit circle, no restriction on zeros

Zerosare also inside unit circle Æinverse 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

minimum-phase all-pass ) ( ) ( )

(z H_min z H z

H = _ap

1

* 1 1 1

* 1 1

)1 1

)(

(

) )(

( ) (

−

− −

−

− −

=

−

=

cz c cz z

z H

c z z H z H

1

|

| , /

1 ^* <

= c c z

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

H_d = _d _ap

) ( ) 1

(

min z z H

H

d

c =

) ( )

( ) ( )

(z H z H z H z

G = _d _c = _ap

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Properties of minimum-phase systems

From minimum-phase and all-pass decomposition

From previous figures, the continuous-phase curve of an all-pass system is negative for

So change from minimum-phase to nonminimum- phase (+all-pass 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 phase-lag system

) ( ) ( )

(z H_min z H z

H = _ap

π ω≤

≤ 0

)]

( arg[

)]

( arg[

)]

(

arg[H e^j^ω = H_min e^j^ω + H_ap e^j^ω

Part VI: Linear-phase systems

Frequency response

System functions

All-pass systems

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Design a system with non-zero phase

System design sometimes desires

Constant frequency response magnitude

Zero phase, when not possible

accept phase distortion, in particular linear phasesince it only introduce time shift

Nonlinear phase will change the shape of the input signal though having constant magnitude response

Ideal delay

) (

) ( ] sin

[ π α

α π

−

= − n n n

h_id

1

| ) (

|H_id e^j^ω =

] [ ]

[ _d

id n n n

h =δ − π ω

ωα ω)= ⁻ ,| |<

( ^j ^j

id e e

H

α π ω ωα

ω ω

=

<

−

=

∠

)]

( [

|

| , )

(

j id

e H grd

e H

nd

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

Linear phase filters

Generalized linear phase filters

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

Frequency response

System functions

All-pass systems

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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

MM7 MM8

Sampling and

reconstruction MM5

System structure System