copyright © 2005. shi ping cuc chapter 2 discrete-time signals and systems content the...

Copyright © 2005. Shi Ping CUC

Chapter 2Discrete-Time Signals and Systems

Content

The Discrete-Time Signal: Sequence

The Discrete-Time System

The Discrete-Time Fourier Transform (DTFT)

The Symmetric Properties of the DTFT

System Function and Frequency Response


The Discrete-Time Signal: Sequences

Elementary sequences

Unit sample sequence

0,0

0,1)(

n

nn

0

00 ,0

,1)(

nn

nnnn



Unit step sequence

0,0

0,1)(

n

nnu

0

00 ,0

,1)(

nn

nnnnu

)1()()( nunun

0

)()(m

mnnu



Rectangular sequence

otherwise

NnnRN ,0

10,1)(

)()()( NnununRN

1

0

)()(N

mN mnnR



Sinusoidal sequence

nnAnx ),cos()( 0

amplitude

digital angular frequency

phase

A

0

)205.0sin(5.1)()205.0cos(5.1)(

2

1

nnxnnx



Real-valued exponential sequence

Rananx n ;,)(

The is convergent when 1|| a)(nx

The is divergent when 1|| a)(nx

n

n

nx

nx

8.02.0)(

2.1001.0)(

2

1



Complex-valued exponential sequence

nenx nj ,)( )( 0

)()(

sincos)( 00

njxnx

njenenx

imre

nn

Attenuation factor

njenx

)85

1(2)(



Classification of sequences

Finite-length sequence

)(nx

21 NnN is defined only for a finite time interval:

where 21 ,NN

examples 88,)( 2 nnnxnny 4.0cos)(

The length of a finite-length sequence can be increased by zero-padding



Right-sided sequence

)(nx1Nn has zero-valued samples for

where 1N

If , a right-sided sequence is called a causal sequence

01 N



Left-sided sequence

)(nx2Nn has zero-valued samples for

where 2N

If , a left-sided sequence is called a anti-causal sequence

02 N



Two-sided sequence

)(nx is defined for any n

a dual-sided sequence can be seen as the sum of a right-sided sequence and a left-sided sequence.



Absolutely summable sequence

n

nx )(

Example:

0,0

0,3.0)(

n

nnx

n

42857.13.01

13.0

0n

n



Square-summable sequence

n

nx2

)(

Example: nnnx 3.0sin)(

It is square-summable but not absolutely summable



Operations on sequence

Time-shifting operation

)()( Nnxny where is an integerN

delaying operation0N

advance operation0N

z-1)(nx )1()( nxnyUnit delay

z)(nx )1()( nxnyUnit advance



Time-reversal (folding) operation

)()( nxny

Addition operation

)(nx )()()( nwnxny Adder

)(nw

Sample-by-sample addition )()()( nwnxny



Scaling operation

)()( nAxny )(nx )()( nAxny Multiplier A

Product (modulation) operation

)(nx )()()( nwnxny modulator

)(nw

Sample-by-sample multiplication )()()( nwnxny



Sample summation

Sample production

2

1

)()()( 21

n

nn

nxnxnx

)()()( 21

2

1

nxnxnxn

nn



Sequence energy

Sequence power

nn

x nxnxnxE 2* |)(|)()(

1

0

2|)(|1

limN

nN

x nxN

P



Decimation by a factor D

)()( Dnxnxd

Every D-th samples of the input sequence are kept and others are removed:



Interpolation by a factor I

otherwise ,0

2 , ,0 ),()( IInI

nxnxp

I -1 equidistant zeros-valued samples are inserted between each two consecutive samples of the input sequence.



The periodicity of sequence

then the is called a periodic sequence,

and the value of N is called the fundamental period.

)(nx

a periodic sequence is usually expressed as

)(~ nx Nnx ))((

)()( kNnxnx if: any integerk: positive integerN



The periodicity of sinusoidal sequence

)cos()( 0 nAnx

)cos()( 00 NnANnx

k

NorkN

00

22

If , : any integerN k

is a periodic sequence and its period is)(nx

0

2

k

N 0

min2N


0min

2

N0

2

If is a integer

0

2

If is a noninteger rational number

QNkP

QkN

P

Q min

00

22

0

2

If is a irrational number

is an aperiodic sequence)(nx

)cos()( 0 nAnx



The periodicity of Complex-valued exponential sequence

when , the periodicity of Complex-valued exponential sequence is the same as the sinusoidal sequence

0

njene

enxnn

nj

00

)(

sincos

)( 0



The periodicity of sinusoidal sequence which is developed by uniformly sampling a continuous-time sinusoidal signal

)cos()( 0 tAtx

)cos(

)2

cos(

)cos()()(

0

0

0

nA

nA

nTAtxnx

T

nTt

Analog angular frequency0

Sampling periodT

Sampling angular frequencyT

Digital angular frequency0



ss f

f

fT 0

000 21

Sampling period : seconds/sample

Analog frequency : hertz (Hz)

Analog angular frequency : radians/second

Digital angular frequency : radians/sample

0

T

0

0f

Units:



The periodicity:

TT

TfTfT0

0000

1 212122

If is a rational number, then 0

2

P

Q

0

2

0PTQT PQ, are positive integers

T0T The period of the continuous-time sinusoidal signal

The sampling period



Sequence synthesis

Any arbitrary sequence can be synthesized in the time-domain as a weighted sum of delayed (advanced) and scaled unit sample sequence.

Unit sample synthesis

k

knkxnx )()()(



Any arbitrary real-valued sequence can be decomposed into its even and odd component:

Even and odd synthesis

return

)()()( nxnxnx oe

)]()([21)(

)]()([21)(

nxnxnx

nxnxnx

o

e

)()( nxnx ee Even (symmetric):

)()( nxnx oo Odd (antisymmetric):



A discrete-time system processes a given input

sequence x(n) to generate an output sequence y(n)

with more desirable properties.

Mathematically, an operation T [ • ] is used.

y(n) = T [ x(n) ]

x(n): excitation, input signal

y(n): response, output signal

Introduction

example

example



Classification

Linear System

Time-Invariant (Shift-Invariant) System

Linear Time-Invariant (LTI) System

Causal System

Stable System



Linear System

A system is called linear if it has two mathematical properties: homogeneity and additivity.

)]([)]([)]()([ 2121 nxTnxTnxnxT

)]([)]([ nxaTnaxT

)]([)]([)]()([ 22112211 nxTanxTanxanxaT

Accumulator



Time-Invariant (Shift-Invariant) System

)()]([ then

)()]([ if

00 nnynnxT

nynxT

Accumulator



Linear Time-Invariant (LTI) System

A accumulator is an LTI system !

A system satisfying both the linearity and the time-

invariance properties is called an LTI system.

LTI systems are mathematically easy to analyze and

characterize, and consequently, easy to design.



The output of an LTI system is called

linear convolution sum

)(*)()()()]([)( nhnxknhkxnxLTInyk

An LTI system is completely characterized in the time domain by the impulse response h(n).

example



Causal System

For a causal system, changes in output samples do not precede changes in the input samples.

In a causal system, the -th output sample

depends only on input samples for and

does not depend on input samples for

0n

)(nx 0nn

0nn

)2()1()()( 321 nxanxanxanye.g.



An LTI system will be a causal system if and only if :

0,0)( nnh

An ideal low-pass filter is not a causal system !

0,0)( nnx

A sequence is called a causal sequence if :)(nx



Stable System

A system is said to be bounded-input bounded-output

(BIBO) stable if every bounded input produces a

bounded output, i.e.

PnyMnx )( then ,)( if

An LTI system will be a stable system if and only if :

n

nhS )(



The M-point moving average filter is BIBO

stable :

1

0

)(1

)(M

k

knxM

ny

A causal LTI discrete-time system:

stablenot is system the 1,|| ifstable is system the 1,|| if

)()( nunh n

prove

prove



Causal and Stable System

)(

)()()(

n

nh

nunhnh

A system is said to be a causal and stable system if

the impulse response is causal and absolutely

summable , i.e.

)(nh

return


The Discrete-time Fourier Transform (DTFT)

The transform-domain representation of discrete-time signal

● Discrete-Time Fourier Transorm (DTFT)

● Discrete-Fourier Transform (DFT)

● z-Transform



The definition of DTFT

dweeXeXIDTFTnx jwnjwjw )(

2

1)]([)(IDTFT:

Existence condition:

|)(| nx

DTFT:

n

jwnjw enxnxDTFTeX )()]([)(



The comparison of vs.)(nx )( jeX

Time domain Frequency domain

discrete continuous

Real valued Complex-valued

Summation integral

)(nx )( jeX



About )( jeX

● It is a periodic function of with a period of 2

The range of

The integral range of

~ ~

● It can be expressed as )()()( jjj eeXeX

)( jeX magnitude function

)( phase function

)(and are all real function of)( jeX example



DTFT vs. z Transform

n

nj

ez

j

enx

zXeX j

)(

)()(



The general properties of DTFT

Linearity

The DTFT is a linear transformation

)()()()( jj ebYeaXnbynax

Time shifting

A shift in the time domain corresponds to the phase shifting

)()( jmj eXemnx



Frequency shifting

Multiplication by a complex exponential corresponds to a shift in the frequency domain

)()( )( 00 jjn eXnxe

Convolution Convolution in time domain corresponds to

multiplication in frequency domain

)()()()( 2121jwjw eXeXnxnx



Energy (Parseval’s Theorem)

deXnx j

n

22)(

2

1)(

Multiplication

deXeXnxnx wjj )()(2

1)()( )(

2121

2)( jeX energy density spectrum



Sequence weighting

)]([)(

jeX

d

djnxn

Multiplied by an exponential sequence

)()( jn ea

Xnxa1



Conjugation Conjugation in the time domain corresponds to the

folding and conjugation in the frequency domain

)()( jeXnx Folding Folding in the time domain corresponds to the folding

in the frequency domain)()( jeXnx

Conjugation and Folding Conjugation and folding in the time domain corresponds

to the conjugation in the frequency domain

)()( jeXnx return


The symmetric properties of the DTFT

Conjugate symmetric sequence: )()( nxnx ee

Conjugate antisymmetric sequence: )()( nxnx oo

For real-valued sequence, it is even symmetric:

)()( nxnx ee

Conjugate symmetry of )(nx

)()( nxnx oo For real-valued sequence, it is odd symmetric:



)()()( nxnxnx oe

Any arbitrary sequence can be expressed as the sum of a conjugate symmetric sequence and a conjugate antisymmetric sequence

)]()([2

1)(

)]()([2

1)(

nxnxnx

nxnxnx

o

e



Conjugate symmetry of )( jeX

)()()( jo

je

j eXeXeX

The can be expressed as the sum of the conjugate symmetric component and the conjugate antisymmetric component

)( jeX

)]()([2

1)(

)]()([2

1)(

jjjo

jjje

eXeXeX

eXeXeX



)()( je

je eXeX －conjugate symmetric

)()( jo

jo eXeX conjugate antisymmetric

)()( je

je eXeX －

For real-valued function, it is even symmetric

)()( jo

jo eXeX

For real-valued function, it is odd symmetric




)](Im[ )(

)](Re[ )(

)( )](Im[

)( )](Re[

jo

je

jo

je

eXjnx

eXnx

eXnxj

eXnx

Implication: If the sequence is real and even, then is also real and even.)( jeX

)(nx



)](arg[)](arg[

)( )(

)](Im[)](Im[

)](Re[ )](Re[

)()(

jj

jj

jj

jj

jj

eXeX

eXeX

eXeX

eXeX

eXeX

If the sequence x(n) is real, then

example



The DTFT of periodic sequences

The DTFT of complex-valued exponential sequences

)(

)2(2)(

)( )(

0

0

0

i

j

nj

ieX

nenx



The DTFT of constant-value sequences

i

j ieX

nnx

)2(2)(

)(- 1)(



The DTFT of unit sample sequences

k

j

i

kNN

eX

iNnnx

)2

(2

)(

)()(



return

The DTFT of general periodic sequences

k

k

kNj

j

ii

kN

kXN

kN

eXN

eX

iNnnxiNnxnx

)2

()(~2

)2

()(2

)(

)()()()(~

2


The representation of a LTI system

Impulse response

Difference equation

System function

)(nh

M

mm

N

kk mnxbknya

00

)()(

)(zH




System function (Transfer function)

The z-transform of the impulse response h(n) of the LTI

system is called system function or transfer function

)()()(

)()]([)(

zXzYzH

znhnhΖzHn

n



The region of convergence (ROC) for H(z)

An LTI system is stable if and only if the unit circle is in

the ROC of H(z)

An LTI system is causal if and only if the ROC of H(z) is

|| zRx

An LTI system is both stable and causal if and only if the H(z) has all its poles inside the unit circle, i.e. the ROC of H(z) is

|| 1 z



System function vs. difference equation

M

mm

N

kk mnxbknya

00

)()(difference equation

take z-transform for both sides

M

m

mm

N

k

kk zXzbzYza

00

)()(

N

kk

M

mm

N

k

kk

M

m

mm

zd

zcK

za

zb

zX

zYzH

1

1

1

1

0

0

)1(

)1(

)(

)()(



Frequency response of an LTI system

n

njj enheH )()(

The DTFT of an impulse response is called the frequency response of an LTI system, i.e.

)](arg[)()( jeHjjj eeHeH

magnitude response function)( jeH

phase response function)](arg[ jeH

example

example



deHd j

g

)](arg[ )(

Group delays

)( jeH In general, the frequency response is a complex function of

is a continuous function of )( jeH

is a periodic function of , the period is 2)( jeH



Response to exponential sequence

)( jeHnjenx 0)( )()( 00 jnj eHeny

The output sequence is the input exponential sequence

modified by the response of the system at frequency 0



Response to sinusoidal sequences

)( jeH)(nx )(ny

)])(arg[cos(|)(|)(

)cos()(00

0

0

jj eHneHAny

nAnx

))](arg[cos(|)(|)(

)cos()(

k

jkk

jk

kkkk

kk eHneHAny

nAnx



Response to arbitrary sequences

)( ),( nheH j)(nx )(ny

)()()( jjj eHeXeY

)()()( nhnxny



Geometric interpretation of frequency response

N

kk

M

mm

MN

N

kk

M

mm

N

k

kk

M

m

mm

dz

czKz

zd

zcK

za

zb

zX

zYzH

1

1)(

1

1

1

1

0

0

)(

)(

)1(

)1(

)(

)()(



)](arg[

1

1)(

)(

)(

)()(

jeHjj

N

kk

j

M

mm

j

MNjj

eeH

de

ceKeeH

K is a real number



N

kk

j

M

mm

j

j

de

ceKeH

1

1

)(

)()(

k

m

jkk

jk

jmm

jm

eldeD

eceC

zero vector

pole vector



)(]arg[

]arg[]arg[)](arg[

1

1

MNde

ceKeH

M

mk

j

M

mm

jj

k

m

jkk

jk

jmm

jm

eldeD

eceC

zero vector

pole vector



)(]arg[)](arg[

)(

11

1

1

MNKeH

lKeH

N

kk

M

mm

j

N

kk

M

mm

j



An approximate plot of the magnitude and phase responses of the system function of an LTI system can be developed by examining the pole and zero locations

To highly attenuate signal components in a specified frequency range, we need to place zeros very close to or on the unit circle in this range

To highly emphasize signal components in a specified frequency range, we need to place poles very close to or on the unit circle in this range



11

2

2

2l

2

1 1l

22

23

je

N

kk

M

mm

j

lKeH

1

1)(



Minimum-Phase and Maximum-Phase system

)()(

arg11

MNKeH N

kk

M

mm

j

the number of zeros inside the unit circleim

the number of zeros outside the unit circleom

the number of poles outside the unit circleopthe number of poles inside the unit circleip

oi ppN oi mmM


A causal stable system with all zeros inside the unit circle is called a minimum-phase delayed system

A causal stable system with all zeros outside the unit circle is called a maximum-phase delayed system

Npp io ,0 A causal stable system

oi

ii

j

mMm

MNpmK

eH

2 2 2

)(222)(

arg2

0)(

arg2

KeH j

MKeH j

2)(

arg2


An anti-causal stable system with all zeros inside the unit circle is called a maximum-phase advanced system

An anti-causal stable system with all zeros outside the unit circle is called a minimum-phase advanced system

o

j

pNKeH

22)(

arg2

)(2)(2)(

arg2

oo

j

mpMNKeH

Npp oi ,0 An anti-causal stable system

)(22)(

arg2

MNmKeH

i

j



Important properties of minimum-phase delayed system

Any nonminimum-phase system can be expressed as the product of a minimum-phase system function and a stable all-pass system

minimum-phase delayed system is often called minimum-phase system for short. It plays an important role in telecommunications

1 ,|)(||)(|

|)(||)(|

0

2min

0

2

1

0

2min

1

0

2

N-mnhnh

nhnh

m

n

m

n

N

n

N

n

|)(| jeH For all systems with the identical



All-pass system Definition

A system that has a constant magnitude response for all frequencies, that is,

0 1|)(| ，jap eH

kzzH )(The simplest example of an all-pass system is a pure delay system with system function

This system passes all signals without modification except for a delay of k samples.



1-th order all-pass system

10 , , 1

)(1

1

rreaazazzH j

ap

1)sin()cos(1)sin()cos(1

11

1)(

)(

)(

jrrjrr

reree

erereeeH

j

jj

jj

jjj

ap



An alternative form of 1-th order all-pass system

10 , , 1

)(1)(

1

11

rreaaz

zazH j

ap

1)( reH jap

a1

a

]Re[z

]Im[zj

Mirror image symmetry with respect to the unit circle



2-th order all-pass system

10 , , 11

)(1

1

1

1

rreaza

az

az

azzH j

ap

10 , , 1

1

1

)(1)(

1

11

1

11

rreaza

za

az

zazH j

ap

1)( jap eH

2)( reH jap

example



example

)()(

cos21cos21

cos21cos2

11)(

12

221

222

221

212

1

1

1

1

zDzD

zzrrzzrrzz

zrrzrrzz

zaaz

azazzHap

10 , rrealet j

221 cos21)( zrrzzD



N-th order all-pass system

)(

)(

1

1)(

1

)1(1

11

)1(1

11

11

1

zD

zDz

zdzdzd

zzdzdd

za

azzH

N

NN

NN

NNNN

N

k k

kap

)()( jj eDeD

1)( jeH



An alternative form for N-th order all-pass system

CR N

k kk

kkN

k k

kap zbzb

bzbz

za

azzH

111

11

11

1

)1)(1(

))((

1)(

RN The number of real poles and zeros

CN The number of complex-conjugate pair of poles and zeros

For causal and stable system 1|| ,1|| kk ba



Application

When placed in cascade with a system that has an undesired phase response, a phase equalizer is designed to compensate for the poor phase characteristics of the system and therefore to produce an overall linear-phase response.

)()()( zHzHzH dap

Phase equalizers



)]()([)()(

)()()(

dapjjd

jap

jd

jap

j

eeHeH

eHeHeH

0)()(

)](arg[ )(

dap

j

d

eHd

Group delays

)()()](arg[ dap

jeH



)()()( min zHzHzH ap

Any causal-stable nonminimum-phase system can be expressed as the product of a minimum-phase delayed system cascaded with a stable all-pass system

)()()()( 111

oo zzzzzHzH

a minimum-phase system )(1 zH

a pair of conjugate zeros outside the unit circle oo zz

1 ,

1

1|| oz

example


)()(

11)1()1()(

1

1

1

1)()()()(

min

1

1

1

111

1

1

1

1

111

1

zHzH

zz

zz

zz

zzzzzzzH

zz

zz

zz

zzzzzzzHzH

ap

o

o

o

ooo

o

o

o

ooo

)1()1()( 111

zzzzzH oo is a minimum-phase system

1

1

1

1

11

zz

zz

zz

zz

o

o

o

ois a 2-th all-pass system



By cascading an all-pass system an unstable system can be made stable without changing its magnitude response

11

)(1

1

1

1

za

az

az

azzHap

example

)()()( zHzHzH ap

unstable system)(zH

stable system)(zH



Relationships between system representations

return

)(zH

)(nhDifference Equation

)( jeH

Inverse ZT

ZT

DTFT

Inverse DTFT

Express H(z) in z-1 cross multiply and take inverse

take ZT solve for Y/X

Take DTFT solve for Y/X

jez substitute


-10 -5 0 5 10 15 200

0.2

0.4

0.6

0.8

1

n

Am

plit

ud

e

unit sample sequence

-10 -5 0 5 10 15 200

0.2

0.4

0.6

0.8

1

n

Am

plit

ud

e

return

)(n

)5( n


-5 0 5 10 15 200

0.2

0.4

0.6

0.8

1

n

Am

plit

ud

e

unit step sequence

-5 0 5 10 15 200

0.2

0.4

0.6

0.8

1

n

Am

plit

ud

e

return

)(nu

)5( nu


-5 0 5 10 15 200

0.2

0.4

0.6

0.8

1

n

Am

plit

ud

e

unit step sequence

-5 0 5 10 15 200

0.2

0.4

0.6

0.8

1

n

Am

plit

ud

e

Rectangular sequence

return

)(10 nR

)5(10 nR


0 10 20 30 40 50 60-2

-1

0

1

2

n

Am

plit

ud

e

Sinusoidal sequence

0 10 20 30 40 50 60-2

-1

0

1

2

n

Am

plit

ud

e

return

) 205.0cos(5.1 n

) 205.0sin(5.1 n


0 5 10 15 20 25 300

0.05

0.1

0.15

0.2

0.25

n

Am

plit

ud

e

Sinusoidal sequence

0 5 10 15 20 25 300

0.05

0.1

0.15

0.2

n

Am

plit

ud

e

Real-valued exponential sequence

return

n2.1001.0

n8.02.0


0 5 10 15 20 25 30 35-0.5

0

0.5

1

1.5

2

n

Am

plit

ud

e

real part

0 5 10 15 20 25 30 35-0.5

0

0.5

1

1.5

n

Am

plit

ud

e

imaginary part

Complex-valued exponential sequence

return

nje

)85

1(2


-20 -15 -10 -5 0 5 10 15 200

20

40

60

80

n

Am

plit

ud

e

finite-length sequence

-20 -15 -10 -5 0 5 10 15 20-1

-0.5

0

0.5

1

n

Am

plit

ud

e

infinite-length sequence

return

)8(172 nRn

)4.0cos( n


-10 -5 0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

n

Am

plit

ud

e

right-sided sequence

-10 -5 0 5 10 15 20 25 300

0.05

0.1

0.15

0.2

n

Am

plit

ud

e

causal sequence

)5(8.02.0 nun

return

)(8.02.0 nun


-30 -25 -20 -15 -10 -5 0 5 100

0.2

0.4

0.6

0.8

n

Am

plit

ud

e

left-sided sequence

-30 -25 -20 -15 -10 -5 0 5 100

0.05

0.1

0.15

0.2

n

Am

plit

ud

e

anti-causal sequence

)5(8.02.0 nun

return

)(8.02.0 nun


-20 -15 -10 -5 0 5 10 15 200

0.05

0.1

0.15

0.2

n

Am

plit

ud

e

two-sided sequence

-20 -15 -10 -5 0 5 10 15 20-0.2

-0.1

0

0.1

0.2

n

Am

plit

ud

e

two-sided sequence

||8.02.0 n

return

)3.0cos(2.0 n


-5 0 5 10 15 20 25 30 350

0.2

0.4

0.6

0.8

1

n

Am

plit

ud

e

absolutely summable sequence

-5 0 5 10 15 20 25 30 350

0.2

0.4

0.6

0.8

1

n

Am

plit

ud

e

absolutely summable sequence

)(3.0 nun

return

)(85.0 nun


-20 -15 -10 -5 0 5 10 15 20-0.05

0

0.05

0.1

0.15

n

Am

plit

ud

e

square-summable sequence

-20 -15 -10 -5 0 5 10 15 20-0.05

0

0.05

0.1

0.15

0.2

n

Am

plit

ud

e

square-summable sequence

nn3.0sin

return

nn6.0sin


-10 -5 0 5 10 15 20 25 300

0.1

0.2

n

Am

plit

ud

e

original sequence

-10 -5 0 5 10 15 20 25 300

0.1

0.2

n

Am

plit

ud

e

delayed sequence

-10 -5 0 5 10 15 20 25 300

0.1

0.2

n

Am

plit

ud

e

advanced sequence

Time-shifting operation )(8.02.0 nun

return

)5(8.02.0 5 nun

)5(8.02.0 5 nun


-20 -15 -10 -5 0 5 10 15 200

0.2

0.4

0.6

0.8

1

n

Am

plit

ud

e

original sequence

-20 -15 -10 -5 0 5 10 15 200

0.2

0.4

0.6

0.8

1

n

Am

plit

ud

e

folding sequence

folding operation

)(8.0 nun

return

)(8.0 nun


0 5 10 15 20 25 30 35 400

0.5

1

n

Am

plit

ud

e

x1(n)

0 5 10 15 20 25 30 35 40-1

0

1

n

Am

plit

ud

e

x2(n)

0 5 10 15 20 25 30 35 40-1

0

1

2

n

Am

plit

ud

e

x1(n)+x2(n)

addition operation

)(8.0 nun

return

)()2.0cos( nun

)()2.0cos()(8.0 nunnun


0 20 40 60 80 100 120 140 160-0.1

0

0.1A

mp

litu

de

x1(n)

0 20 40 60 80 100 120 140 160-1

0

1

Am

plit

ud

e

x2(n)

0 20 40 60 80 100 120 140 160-0.1

0

0.1

Am

plit

ud

e

x1(n)*x2(n)

modulation operation n0125.0sin1.0

return

n125.0sin

)()( 21 nxnx


0 10 20 30 40 50 60 70 80 90-1

-0.5

0

0.5

1

n

Am

plit

ud

e

periodic sequence

0 10 20 30 40 50 60 70 80 90-1

-0.5

0

0.5

1

n

Am

plit

ud

e

periodic sequence

periodic sequence)

8sin( n

return

)16

sin( n


0 10 20 30 40 50 60 70 80 90-1

0

1

Am

plit

ud

e

x1(n)

0 10 20 30 40 50 60 70 80 90-1

0

1

Am

plit

ud

e

x2(n)

0 10 20 30 40 50 60 70 80 90-1

0

1

Am

plit

ud

e

x3(n)

Periodicity of sequence )8

sin( n

return

)103sin( n

)4.0sin( n


0 10 20 30 40 50 60-1

-0.5

0

0.5

1A

mp

litu

de

0 10 20 30 40 50 60-1

-0.5

0

0.5

1

Am

plit

ud

e

Periodicity of sequence 23320

TT

return

3100

TT


0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200

1

2

3

4

5

6

7

8

9

10

11

12

)7(9)6(11)5(10)4(8

)3(6)2(3)1(4)(

nnnnnn

nn

return


-20 -15 -10 -5 0 5 10 15 200

5

10

Am

plit

ud

e

x(n)

-20 -15 -10 -5 0 5 10 15 200

2

4

6

Am

plit

ud

e

xe(n)

-20 -15 -10 -5 0 5 10 15 20-5

0

5

Am

plit

ud

e

xo(n)

return

n9.0


Accumulator

The input-output relation can also be written in the form:

This form is used for a causal input sequence, in which case y(-1) is called the initial condition

)()1()()()()(1

nxnynxlxlxnyn

l

n

l

0 ,)()1()()()(00

1

nlxylxlxny

n

l

n

ll

The output at time instant n is the sum of the input sample at time instant n and the previous output at time instant n-1, which is the sum of all previous input sample values from to n-1

)(ny)(nx )1( ny

return


M-point moving-average system

1

0

)(1)(M

k

knxM

ny

An application: consider )()()( ndnsnx

return

Where is the signal, and is a random noise)(ns )(nd

nnns 9.02)( 8M

7

0

)(81)(k

knxny


0 5 10 15 20 25 30 35 40 45 500

5

10

Am

plit

ud

e

s(n),d(n)

0 5 10 15 20 25 30 35 40 45 500

5

10

Am

plit

ud

e

x(n)

0 5 10 15 20 25 30 35 40 45 500

5

10

Am

plit

ud

e

y(n)

return

nnns 9.02)(


Accumulator

Hence, the above system is linear

)()]([ ,)()]([ 2211

n

l

n

l

lxnxTlxnxT

return

)]([)]([)()(

)]()([)]()([

2121

2121

nxbTnxaTlxblxa

lbxlaxnbxnaxT

n

l

n

l

n

l

)()(

n

l

lxny


Accumulator

Hence, the above system is time-invariant

)]([)()(

)()]([

knxTlxkny

lxknxT

kn

l

kn

l

return

)()(

n

l

lxny


0 5 10 15 20 25 30 35 40 45 500

0.5

1

1.5

Am

plit

ud

e

x(n)

0 5 10 15 20 25 30 35 40 45 500

0.5

1

1.5

Am

plit

ud

e

h(n)

0 5 10 15 20 25 30 35 40 45 500

5

10

Am

plit

ud

e

y(n)

return

)(9.0)( nunh n

)()( 10 nRnx

)()()( nhnxny

Copyright © 2005. Shi Ping CUCreturn

The M-point moving average filter

1

0

)(1)(M

k

knxM

ny

For a bounded input , we havexBnx )(

xx

M

k

M

k

BMBM

knxM

knxM

ny

)(1

)(1)(1)(1

0

1

0

Hence, the M-point moving average filter is BIBO stable


-2 -1 0 1 20

2

4

6

8

pi

Am

plit

ud

e

Amplitude part

-2 -1 0 1 2-0.5

0

0.5

pi

ph

as

e(p

i)

phase part

-2 -1 0 1 20

2

4

6

8

pi

Am

plit

ud

e

real part

-2 -1 0 1 2-5

0

5

pi

Am

plit

ud

e

imaginary part

return

)](9.0[DTFT 113/ nRe jnn


-1 -0.5 0 0.5 10

2

4

6

8

pi

Am

plit

ud

e

Amplitude part

-1 -0.5 0 0.5 1-0.5

0

0.5

pi

ph

as

e(p

i)

phase part

-1 -0.5 0 0.5 10

2

4

6

8

pi

Am

plit

ud

e

real part

-1 -0.5 0 0.5 1-5

0

5

pi

Am

plit

ud

e

imaginary part

return

)](9.0[DTFT 11 nRn


-1 -0.5 0 0.5 1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Real Part

Ima

gin

ary

Pa

rt81.09.0

1)(2

zz

zzH


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

5

10

15

pi

Am

plit

ud

e

Amplitude response

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

-0.5

0

0.5

1

pi

ph

as

e(p

i)

phase response

return

81.09.01)(

2 zz

zzH


-1 -0.5 0 0.5 1 1.5 2

-1

-0.5

0

0.5

1

Real Part

Ima

gin

ary

Pa

rt

21

1

251

2419

127

)(

zz

zzH


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.3

0.35

0.4

0.45

0.5

0.55

pi

Am

plit

ud

e

Amplitude response

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.4

-0.3

-0.2

-0.1

0

pi

ph

as

e(p

i)

phase response

21

1

251

2419

127

)(

zz

zzH

return


-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-1.5

-1

-0.5

0

0.5

1

1.5

Real Part

Ima

gin

ary

Pa

rt

return

)3

,21(

11)(

1

1

1

1

r

zaaz

azazzHap

a1

a

a

a1

Mirror image symmetry


-1 -0.5 0 0.5 1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Real Part

Ima

gin

ary

Pa

rt

2

return

21

4321

48.012.11

83.085.108.361.21)(

zz

zzzzzH

oz

1

oz1


-1 -0.5 0 0.5 1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Real Part

Ima

gin

ary

Pa

rt

2

return

oz

1

oz1

oz

oz


oz

1

-1 -0.5 0 0.5 1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Real Part

Ima

gin

ary

Pa

rt

return

21

21

26.241.21

35.083.01)(

zz

zzzH

a

1

a1

a

a

copyright © 2005. shi ping cuc chapter 2 discrete-time signals and systems content the...

Documents

sequence time

discretetime system

discretetime fourier

discretetime signals

sequences timereversal

sequences twosided sequence

shi ping cuc chapter

input sequence