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1Digital Signal Processing A.S.Kayhan

DIGITAL SIGNAL

PROCESSING

Textbook: Discrete-Time Signal Processing, Oppenheim and Schafer, Prentice-Hall.

Digital Signal Processing A.S.Kayhan

Course Outline

Review of Discrete-time signals, systems, Fourier tr.

Review of Z-transform

Sampling, Decimation, Interpolation

Time and frequency response of systems

Flow Graph Realization

Filter Design

Discrete-time Fourier Series, Discrete Fourier Transform

Fast Fourier Transform

2-D Signal Processing


SIGNALS:A signal is a function of time representing a physical variable, e.g. voltage, current, spring displacement, share market prices, number of students asleep in the class, cash in the bank account.

Continuous time signals : x(t), x(t1,t2,...)Speech signals, Image signals,Video signals, Seismic signals, Biomedical signals (ECG,EEG,...)

Discrete-time signals: x[n], x[n1,n2,...]Inherently discrete-time (Stock market)Discretized continuous-time(Most signals)


SIGNALS:Deterministic or Random


Speech data (8kHz):


Discrete-time signal example:Stock Market daily closing values


Annual max. temperature and precipitation for Ankara (data: MGM)


Graphical Representation of Signals

Continuous-timeSignal

Discrete-timeSignal


Reflection: x[-n]


Time-scaling: x(at)

Time-scaling in discrete-time is not straightforward.Discussed in Decimation and Interpolation.


Time-shift: x[n-no]


Basic Discrete-time Signals:Unit Step Function :

0,1

0,0][

n

nnu


Unit Impulse Function:

0,1

0,0][

n

nn

n

k

knu

nunun

][][

]1[][][

][]0[][][ nxnnx


Real Exponential: nn aCeCnx ][


Complex Exponential: nj oeAnx ][Sinusoidal Signal: )cos(][ nAnx o

}{][ )( nj oAenx Re


Periodicity Properties:

njnj oo ee )2(

Therefore, consider only 20 o

njNnj oo ee )(If periodic, then

N

m

e Nj o

2

1

0

m and N integer.


If periodic with fundamental period N, then fundamental frequency is N/2

If there are a few exponentials in the form:

nN

jk

k enx2

][

they are harmonically relatedThere are only N such distinct signals.


SYSTEM:Any process that transforms signals

10



Interconnection of Systems:Series(Cascade), Parallel, Series/Parallel.

11


Properties:Memoryless if output depends only on input at the same time.

Example: Discrete-time systems with memory:

n

k

kxny ][][

]1[][ nxny


Causality: Causal if output depends only on inputs at the present time and in the past.

12


M

Mk

knxM

ny ][12

1][

]1[][][ nxnxny

Example: Causal system:

Example: Noncausal systems:

][]1[][ nxnxny

t

dxC

ty )(1

)(


Stability: Stable if small inputs do not lead to diverging outputs.

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]1[][][ nxnxnyExample: Stable system:

Example: Unstable systems:

].[][ if ],[)1(][][ nunxnunkxnyn

k


Time Invariance: Time-invariant if a time shift in input causes same time shift in output signal.

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Example:Time-varying: ][][ nnxny

Consider two signals

][][],[ 121 onnxnxnx

][][ 11 nnxny

][][

][][

12

22

onnnxny

nnxny

shift ][1 ny

][][)(][ 211 nynnxnnnny ooo


Linearity: Linear if posses superposition property:Additivity and scaling (homogeneity):

1. Response to is 2. Response to is

][][ 21 nbxnax

][1 nax ][1 nay

Combining these two, we get:

][][ 21 nxnx

][][ 21 nbynay

][][ 21 nyny

Example: Linear: ][][ nnxny

Example: Nonlinear: 2])[(][ nxny

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Linear Time-Invariant Systems:LTI systems can be analyzed in great detail.Many physical processess can be modeled by LTI systems.Unit impulse function will be used as building block and response of LTI systems to a unit impulse will be used to characterize such systems.


Input/output relationship for a LTI system is given as:

k

knhkxny ][][][

Convolution of x[n] and h[n]. h[n] is the impulse response.

][*][][ nhnxny

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Example:


Properties:Commutative:

][*][][*][ nxnhnhnx

r

knr

k

rhrnxknhkx ][][][][

Use of this property:If it is easier; reflect and shift x[k] to obtain x[n-k] first, then multiply with h[k] to find convolution result at time n.

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Associative:

][*])[*][(])[*][(*][ 2121 nhnhnxnhnhnx


Distributive:

][*][][*][])[][(*][ 2121 nhnxnhnxnhnhnx

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Memory: System is memoryless if output depends only on input at same time.Then, system is memoryless if:

][][ nKnh then ].[][ nKxny

Similarly for continuous-time, memoryless if

).()( tKxty

)()( tKth hence


Causal: Consider convolution sum :

k

knxkhny ][][][

For a LTI system to be casual y[n] must not depend on input x[k] for k > n.If system is causal :

.0for ,0][ nnh

Then,

n

kk

knhkxknxkhny ][][][][][0

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Stability: Stable if bounded inputs do not lead to diverging outputs. Assume

. allfor ,][ nBnx

Then,

k

knxkhny ][][][

k

knxkhny ][][][

with, Bknx ][

k

khBny ][][

System is stable iff,

k

kh ][


Example: A system with impulse response h[n]=[n-no] shifts the input

n n

onnnh 1][][

Example: Consider the fallowing accumulator

n

k

kxny ][][

Unstable][][0

n n

nunu

this system has unit step as the impulse response

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Discrete-Time Fourier Transform (DTFT):For a discrete-time signal x[n], DTFT is defined as

n

njj enxeX ][)(

the inverse DTFT (synthesis equation) is defined as

2

)(2

1][ deeXnx njj

Differences between continuous-time FT and DTFT:X(ej) is periodic with 2. Integral in synthesis eq. is over 2 interval (0 to 2 or - to ).


Example:Consider 1],[][ anuanx n

n

njnj nueaeX ][)(

jn

njj

aeaeeX

1

1)()(

0

)cos(21

1)(

2

2

aaeX j

)cos(1

)sin(tan)( 1

a

aeX j

)sin())cos(1(

1)(

jaaeX j

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Common DTFT Pairs:

)2(2)( allfor ,1][ leXnnx j

1)(][][ jeXnnx

)cos(][ nnx o

)]2()2([)( lleX ol

oj

nj oenx ][

l

oj leX )2(2)(

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Convergence: Analysis equation will converge if x[n] is absolutely summable or it has finite energy

n

nx ][

n

nx2

][

Example:Consider

Nn

Nnnx

,0

,1][

)2/sin(

))2/1(sin(

1

1)(

)12(

N

e

eeeeX

j

NjNj

N

Nn

njj


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Properties of DTFT:Linearity: DTFT is linear operator:

)()(][][

)(][

)(][

jj

j

j

eYbeXanbynax

eYny

eXnx

Time and frequency shifting:

)(][

)(][

)(][

)( oo

o

jnj

jnjo

j

eXnxe

eXennx

eXnx


Symmetry: If x[n] is real, then

)()( * jj eXeX

Real part is even, imaginary part is odd function of .Similarly, magnitude is even, phase is odd funtion.

Time and frequency scaling: Time scaling is not straightforward for discrete-time signals. Consider special case; x[an], a = -1

)(][ jeXnx

)()(

k of multiplenot isn if 0,

k of multiple isn if],/[][

1

1

jkj eXeX

knxnx

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Parsevals relation: Energy is equal on both time and frequency domains:

2

22

)(2

1][ deXnx j

n

Since the right hand term is the total energy in frequency domain, is called energy spectral density function (or spectrum).

2)( jeX

Convolution: Convolution in time corresponds to multiplication in frequency domain.

)()()(][*][][ jjj eHeXeYnhnxny


nj

n

j enheH

][)(

where, is the frequency response defined as)( jeH

Example:Consider a LTI sytem withThen the frequency response is

][][ onnnh

onjj eeH )(

For any input x[n] with Fourier transform Fourier transform of output

)( jeX

)()( jnjj eXeeY o

The output is equal to the input with a time shift

].[][ onnxny

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Example:Consider a LTI sytem with |a|,|b|

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Z- Transform :For a discrete-time signal x[n], z-transform is defined as

n

nznxzX ][)(

z is a general complex variable shown in polar form as jerz

We get Fourier transform as a special case when r=1


Convergence: Z-transform exists if

n

nrnx |][|

Poles-Zeros: consider X(z) as a rational function

)(

)()(

zQ

zPzX

where P(z) and Q(z) are polynomials in z. Values of z making X(z)=0 are called zeros of X(z) (roots of P(z)).Values of z making X(z)= are called poles of X(z) (roots of Q(z)).

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28


||||,1

1)( 1 azaz

z

azzX

n

n

az || 1For convergence

0

1 )(][)(n

n

n

nn azznuazX

][][ nuanx nConsiderExample:


||||,1

11)( 1 azaz

z

zazX

n

n

za || 1For convergence

0

1 )(1)(n

nzazX

]1[][ nuanx nConsiderExample:

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||||||,)( bzabz

z

az

zzX

Then

abnubnuanx nn ],1[][][

ConsiderExample:

||||,][ azaz

znua n

||||,]1[ bzbz

znub n


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Properties of convergence:ROC is a ring or disk centered at origin.DTFourier transform exists if ROC includes Unit CircleROC can not contain any polesIf x[n] has finite duration ROC is entire plane.If x[n] is right-sided, ROC is outward from outermost poleIf x[n] is left-sided, ROC is inward from innermost poleIf x[n] is two-sided, ROC is a ring not containing any pole.


Example:

],1[)2(][)4.0(][ nununx nn

|2||||4.0|,24.0

)(

zz

z

z

zzX

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Transfer Function: Z-transform of impulse response h[n]

n

nznhzH ][)(

If impulse response h[n] is causal then ROC will be outward.If the system is causal and stable, all the poles will be inside the unit circle and ROC will include the unit circle.


Inverse Z-Transform:Power Series Expansion: Consider

12

2

11

2

1)( zzzzX

remember

1012 ]1[]0[]1[]2[

][)(

zxzxzxzx

znxzXn

n

therefore]1[

2

1][]1[

2

1]2[][ nnnnnx

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Example (Long Division): |z| > |a|

33221

11

1

1)( zazaaz

azzX

therefore ][][ nuanx n

Example (Long Division): |z| < |a|

33221)( zazazaza

zzX

therefore ]1[][ nuanx n


Partial fraction expansion: Consider

kdzkk

N

k k

k

k

N

k

k

M

k

o

oN

k

kk

M

k

kk

zXzdAzd

A

zd

zc

a

b

za

zbzX

|)()1(,1

)1(

)1()(

1

11

1

1

1

1

0

0

then

N

k

nk nudAnx k

1

][][

33


Example : Consider

.2

1||,

)21

1)(41

1(

1)(

11

zzz

zX

)21

1()41

1()(

1

2

1

1

z

A

z

AzX

2|)()2

11(

1|)()4

11(

2/11

2

4/11

1

z

z

zXzA

zXzA

][4

1][

2

12][ nununx

nn


Properties of Z-Transform:Linearity:

)()(][][

)(][

)(][

zYbzXanbynax

zYny

zXnx

Time shift:

x

n

RROC

zXzzYnnxny

zXnxo

)()(][][

)(][

0

yx RRROC

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Multiplication by Exponential:

xo

ono

RzROC

zzXnxz

zXnx

||

)/(][

)(][

Conjugation:

xRROC

zXnx

)(][ ***


Time reversal:

xRROC

zXnx

/1

)/1(][ ***

xRROC

zXnx

/1

)/1(][

Convolution:

yx RRROC

zYzXnxny

)()(][*][

35


Example: ][][ nuanx n ][][ nunh

||||,)( azaz

zzX

|1|||,

1)(

z

z

zzH

][*][][ nhnxny

11

2

11

1

1

1)(

|1|||,)1)((

)()()(

az

a

zazY

zzaz

zzHzXzY

y[n] is obtained as:

])[][(1

1][ 1 nuanu

any n


Sampling: Consider a signal x[n] which is obtained by taking samples of a continuous time signal xc(t):

)(][ nTxnx cwhere T is the sampling period, its reciprocal, is the sampling frequency.

Tf s1

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37


where has the quantized samples and xB[n] has the coded samples (such as 2s complement).

])[(][ nxQnx


38



Frequency Domain Representation of Sampling: Consider the sampled signal xs(t) obtained from xc(t) by multiplying with periodic impulse train:

n

nTtts )()(

then

n

ccs nTttxtstxtx )()()()()(

The discrete time signal is x[n]= xc(nT). The Fourier transform of s(t) is

TkTS s

ks

2,)(

2)(

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Then, Fourier transform of xs(t) is

k

scs kXTSXX )(

1)(*)(

2

1)(

Ns 2


Ns 2

We observe that when the sampling frequency is not chosen high enough, copies of spectrum overlap; this is called aliasing (distortion). The minimum sampling frequency (rate) to avoid aliasing is 2N (Nyquist rate) :

Ns T 2

2

40


If sampling rate is greater than Nyquist rate, then the original signal xc(t) may be obtained without distortion using a low-pass filter.


Fourier Tr. of xs(t) may be written as

n

njj enxeX ][)(

n

Tnjcs enTxX )()(

On the other hand DTFT of x[n] is

Comparing these equations we observe that )(|)()( TjT

js eXeXX

k

cj

T

k

TX

TeX )

2(

1)(

n

cs nTttxtx )()()(

k

scTj kX

TeX )(

1)(

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Example:Consider

ttx oc cos)(


)cos()( ttx or

))cos(()( ttx osr

42


Example: Consider

4000,cos)( ooc ttxwith sampling period T=1/6000, we obtain

nnTnx3

2cos4000cos][

Nyquist sampling condition is satisfied , there is no aliasing.

80002120002

os T


Example: Consider

16000,cos)( ooc ttxwith sampling period T=1/6000, we obtain

nnn

nnTnx

3

2cos6000/40002cos

6000/16000cos16000cos][

160002120002

os T

Nyquist sampling condition is not satisfied , there is aliasing.

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Reconstruction from Samples:Use ideal LPF to recover original signal


With

n

r nTthnxtx )(][)(

Ttth sinc)(

n

r T

nTtnxtx sinc][)(

n

ncs

nTtnx

nTtnTxtx

)(][

)()()(

44



Discrete-time Processing of Cont.-time Signals:

)()(

)(][

THeH

nThTnh

cj

c

with h(t) is impulse response of continuous-time system

][ nh

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Changing the Sampling Rate:Downsampling and Decimation:Sampling rate can be reduced by further sampling in discrete-time:

)(][][ nMTxnMxnx cd

Let MTT '

Then )'(][ nTxnx cd


If NcX for ,0)( NMTT 2

2

'

2 and

orNM

22

(N is BW of x[n])

Then xc(t) can be recovered without distortion.

Remember that

k

cj

T

k

TX

TeX )

2(

1)(

Similarly

k

cj

d T

k

TX

TeX )

'

2

'(

'

1)(

k

cj

d MT

k

MTX

MTeX )

2(

1)(

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Let

otherwise,0

,2,,0],[][1

MMnnxnx

neM

nxnxM

l

Mnlj ,1

][][1

0

/21

( [ ]=Discrete Fourier Series rep. of impulse train with period M )

][][][ 1 MnxMnxnx d

nj

n

nj

nd

jd eMnxenxeX

][][)( 1

MknMnk /

Mkj

k

jd ekxeX

/1 ][)(


1

0

//2

/1

0

/2

][1

1][)(

M

l k

MkjMlkj

Mkj

k

M

l

Mlkjjd

eekxM

eeM

nxeX

1

0

)2

()(

1)(

M

l

M

l

Mjj

d eXMeX

Combining the exponential terms

First obtain then shift by increments to get)( Mj

eX

2

)( )/2'( MljeX

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48


IfNM

22


A general system for downsampling called decimator is

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Upsampling and Interpolation:Sampling rate can be increased by obtaining intermediate values:

2,/'),'(][ LLTTnTxnx cigiven ).(][ nTxnx c

,2,,0),/(]/[][ LLnLnTxLnxnx ci Consider the following system


The block with inserts (L-1) zeros between samples:L

otherwise,0

2,,0,/][

LLnLnxnx e

or

.][ kLnkxnxk

e

In frequency domain:

n

nj

k

je ekLnkxeX

)(

)()( Ljk

kLjje eXekxeX

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L=2


k

i LkLn

LkLnkxnx

/)(

/)(sin][][

Ln

Lnnh

/

/sin

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Changing the sampling rate by noninteger factor:Following system may be used


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