digital signal processing lecture 10

Remote Sensing LaboratoryDept. of Information Engineering and Computer Science

University of TrentoVia Sommarive, 14, I-38123 Povo, Trento, Italy

Remote Sensing LaboratoryDept. of Information Engineering and Computer Science

University of TrentoVia Sommarive, 14, I-38123 Povo, Trento, Italy

Digital Signal ProcessingLecture 10

Digital Signal ProcessingLecture 10

E-mail: [email protected] page: http://rslab.disi.unitn.it

Quote of the Day

When you look at yourself from a universal standpoint, something inside always reminds or informs you that there are

bigger and better things to worry about.

Albert Einstein

University of Trento, Italy

University of Trento, Italy B. Demir

Reminder

Pole-zero plot and region of convergence of a causal exponential sequence x[n]=anu[n] with a) decaying exponential, b) unit step sequence and c) growing sequence


Reminder

Pole-zero plot and region of convergence for the a) causal, b) anticausal and c) two sidedexponential sequences.


Example

Pole-zero plot and region of possible convergence for the z transform in

1

1 1

11 (1 0.5 )

zX zz z


System function of LTI systems

Procedure for the analytical computation of the output of an LTI system using the convolution theorem of the z-transform:


System function of LTI systems

Equivalent system function of linear time-invariant systems combined in a) parallel connection and b) cascade connection


Stability-Causality


Linear constant-coefficient difference equations

0 0[ ] [ ]

N M

k kk k

a y n k b x n k

0

0

Mk

kkN

kk

k

b zH z

a z

A class of LTI systems whose input and output sequences satisfy a linear constant-coefficient difference equation is written as:

Using the properties of linearity and shifting we obtain:

Then the system function (transfer function) is obtained as:

0 0( ) ( )

N Mk k

k kk k

a z Y z b z X z


Example


Example

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Example-Cont



1

0 0 1 0 1

10 0

0 1 1

1

1

M MMk

k kkN Mk k k

N N Nk

k k kk k k

z z z zb zb bH z za aa z p z z p

H(z) is a rational function, that is, the ratio of two polynomials in z-1.

We recall that any rational function of z-1 can be expressed in the form of:

Where the first summation is included only if M≥N.

10 1 1

M N Nk k

kk k k

AH z C zp z



0 1 0[ ]

M N Nn

k k kn k k n

h n C A p

If we assume that the system is causal, then the ROC is the exterior of a circle starting at the outermost pole, and the impulse response is:

For a causal system to be stable, the impulse response must be absolutely summable.

This is possible if and only if |pk|<1. Hence a causal LTI system with a rational system function is stable if and only if all poles of H(z) are inside the unit circle in the z plane. The zeros can be anywhere.

0 1[ ] [ ] ( ) [ ]

M N Nn

k k kk k

h n C n k A p u n


Example

Pole-zero plot for the system function given by:

This causal system is stable because its poles are inside the unit circle.

2

1 2 3

11 0.9 0.6 0.05

zH zz z z


Example

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One-sided z-Transform


Example


Example-Cont


Systems that are designed to pass some frequency components withoutsignificant distortion while severely or completely eliminating others areknown as frequency selective filters.

By definition, an ideal frequency selective filter satisfies the requirementsfor distortionless response over one or more frequency bands.

Ideal filters are used in the early stages of a design process to specifymodules in a signal processing systems. However they are not realizablein practice, they must be approximated by practical and non-ideal filters.

This is usually done by minimizing some approximation error between thenon-ideal filter and a prototype ideal filter.

Ideal and Practical filters

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Filters-Reminder

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Ideal frequency selective filters either pass or eliminate a region of inputspectrum perfectly and they have abrupt transitions between passbands andstopbands.

However ideal filters can not be implemented in practice; therefore they have tobe approximated by practically realizable filters. Practical filters differ from idealfilters in several respect:

The bandpass responses are not perfectly flat; The stopband responses can not completely eliminate bands of

frequencies; The transition between passband and stopband regions takes place over

a finite transition band.


Filters-Reminder


Frequency responses of four types of ideal filter

a) Lowpass filter; b) bandpass filter; c) highpass filter; bandstop filter.



As the impulse response corresponding to each of these ideal filters isnoncausal and of infinite length, these filters are not realizable.

In practice, the magnitude response specifications of a digital filter in thepassband and in the stopband are given with some acceptabletolerances.

In addition, a transition band is specified between the passband andstopband.


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Filter design

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Ideal frequency selective filters either pass or eliminate a region of inputspectrum perfectly and they have abrupt transitions between passbands andstopbands.

However ideal filters can not be implemented in practice; therefore they have tobe approximated by practically realizable filters. Practical filters differ from idealfilters in several respect:

The bandpass responses are not perfectly flat; The stopband responses can not completely eliminate bands of

frequencies; The transition between passband and stopband regions takes place over

a finite transition band.


Filters-Reminder


1 , 0 ,

kj

k

H e

?][ nh

Ex Example from DTFT:

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Ideal Low Pass Filter:


1 1[ ]2 2

kk k

k

j n j nj ne e eh njn jn

1 1 2 sin sin[ ] ,2 2

k kj n j nk ke e j n nh n n

jn jn n

sin

j j nk

n

nH e en

Example-Cont:

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1 1 1[ ]2 2 2

kk

k k

j nj j n j n eh n H e e d e d

jn


sin

Mj j nk

Mn M

nH e en

Example-Cont:

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Filters-Reminder



Typical characteristics of a practical bandpass filter

A good filter should have only a small ripple in the passband, highattenuation in the stopband, and very narrow transition bands.


Frequency Response of Rational System Functions

• All LTI systems of practical interest are described by a difference equation of the form:

and have a rational system function:

• For a stable system the system function convergences on the unit circle. Therefore, we obtain:

which expresses H(ejω) as a ratio of two polynomials in the variable ejω

0 0[ ] [ ]

N M

k kk k

a y n k b x n k

0

0

Mk

kkN

kk

k

b zH z

a z

0

0

Mj k

kj k

Nj k

kk

b eH e

a e

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Frequency Response of Rational System Functions

• DTFT of a stable and LTI rational system function:

• Magnitude Response:

• Phase response:

0 1

0

1

1

1

Mj

kj k

Nj

kk

z ebH ea p e

0 0 1

0

0 1

1

1

MMjj k

kkj k k

N Nj k j

k kk k

z eb ebH eaa e p e

0

1 10

( ) (1 ) (1 )M N

j j jk k

k k

bH e z e p ea

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Dependence of frequency response on poles and zeros

• This equation consists of factors of the form and

• The factor is a complex number represented by a vector drawn from the point zk (zero) to the point z=ejω in the complex plane and can be written as:

• is the distance from the zk to the point z=ejω , and is the angle of the vector with the positive real axis.

( )0 1 0 1

0 0

1 1

1

1

M Mj j

k kj j N Mk k

N Nj j

k kk k

z e e zb bH e ea ap e e p

jke z j

ke p

jke z

kjjk ke z Q e

kQ k



The factor is a complex number represented by a vector drawn from the point pk (pole) to the point z=ejω in the complex plane.

is the distance from the pole to the point, and is the angle of the vector with the positive real axis.

Then, we can write:

jke p

kjjk ke p R e

kRk

0 1 0

1 10 0

1

exp ( ) ( ) ( )

M

k M Mj k

i kNk k

kk

Qb bH e N Ma aR

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• We note that if because , and

if because

0 1 0

1 10 0

1

exp ( ) ( ) ( )

M

k M Mj k

i kNk k

kk

Qb b

H e N Ma aR

0

0

ba

0

0

0ba

0

0

0ba

0

0

0ba

1je

0 1je

0 1

0

1

M

kj k

N

kk

Qb

H ea R

0

1 10

( ) ( ) ( )M M

ji k

k k

bH e N M

a

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0 1

0

1

M

kj k

N

kk

Qb

H ea R

0

1 10

( ) ( ) ( )M M

ji k

k k

bH e N M

a

distance of the - zero from

distance of the - pole from

angle of the - zero with the real axis

angle of the - pole with the real axis

jk

jk

k

k

Q k th e

R k th e

k th

k th

where ω is the angle of the point z=ejω with the positive real axis and

Therefore the magnitude response at a certain frequency ω is given by theproduct of the lengths of the vectors from the zeros to z=ejω divided by theproduct of the lengths of vectors drawn from the poles to z=ejω.

Similarly the phase response is obtained by subtracting from the sum ofangles of zeros the sum of angles of poles. All angles are determined withrespect to the positive real axis. We can determine H(ejω ) for each value ofω or equivalently any location of the point ejω on the unit circle.

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To suppress a frequency component at ω=ω0, we should place a zero atangle ω0 on the unit circle.

To enhance or amplify a frequency component at ω=ω0 we should place apole at angle ω0 close but inside the unit circle.

Complex poles or zeros should appear in complex conjugate pairs to assurethat the system has real coefficients. This stems from the fact that thefrequency ω is defined as the angle with respect to positive real axis.Therefore all symmetries should be defined with respect to the real axis.

Poles and zeros at the origin do not influence the magnitude responsebecause their distance from any point on the unit circle is unity. However apole at the origin adds (or subtracts) a linear phase of ω rads to the phaseresponse. We often introduce poles and zeros at z=0 to assure that N=M

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Significance of poles and zeros


Significance of poles and zeros

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Allpass systems

The frequency response of an allpass system has constant magnitude at all frequencies.

If we assume that G=1, allpass systems preserve that the power andenergy of their input signals.

In this sense allpass systems are said to be lossless systems. Thesimplest allpass system simply scale and delay the input signal:

A more interesting all pass system is:

jH e G

kapH z Gz

1 *

11k

apk

z pH zp z

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Allpass systems

We note that each pole pk of an allpass system should be accompanied by a complex reciprocal zero 1/pk

*.

Since causal and stable allpass systems must have all poles inside the unit circle, all zeros are outside the unit circle.

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Invertibility and minumum-phase systems

The zeros of H(z) become the poles of its inverse system and vice versa.A causal and stable H(z) should have its poles inside the unit circle; itszeros can be anywhere.

In inverse filtering applications the system Hinv(z) should be causal andstable as well.

A causal and stable LTI system with a causal and stable inverse isknown as a minumum-phase system.

A LTI system is stable and causal and also has a stable and causalinverse if and only if both the poles and the zeros of H(z) are inside theunit circle.

invh n h n n

1invH z H z

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1

invH zH z


Minimum phase and allpass decomposition

Any system with a rational system function can be decomposed into a minimum phase system and an allpass system.

Suppose that H(z) has one zero z=1/a* where |a|<1 outside the unit circle, and all the other poles and zeros are inside the unit circle. Then we can factor H(z) as:

If we repeat this process for every zero outside the unit circle, we obtain:

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1 *1( ) ( )( )H z H z z a

1 *1

1 1

( )( ) ( )(1 )(1 )z aH z H z az

az

min( ) ( ) ( )apH z H z H z

minimum phase system allpass system


Relationship between time, frequency and z-transform domains

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digital signal processing lecture 10

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