digital signal processing, mm3.ppt - aalborg...

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Digital Signal Processing, Fall 2010

Lecture 3: Sampling and reconstruction, transform anal sis of LTI s stems

Zheng-Hua Tan

transform analysis of LTI systems

Digital Signal Processing, III, Zheng-Hua Tan1

Department of Electronic Systems Aalborg University, Denmark

[email protected]

Course at a glance

Discrete-time signals and systems

MM1 System

Fourier transform and Z-transform

Filt d i

MM2

Sampling andreconstruction

MM3

Systemanalysis


DFT/FFT

Filter design

MM5

MM4

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Part I-A: Periodic sampling

Part I: sampling and reconstruction Periodic sampling

F d i i f h li Frequency domain representation of the sampling

Reconstruction

Part II: system analysis


Periodic sampling

From continuous-time to discrete-time)(txc ][nx

nnTxnx c ),(][

Sampling period

Sampling frequencyT

Tf

T

s

s

/2

/1


3

Two stages

Mathematically Impulse train modulator

Conversion of the impulse

In practice?

Conversion of the impulse train to a sequence

nc

cs

n

nTttx

tstxtx

nTtts

)()(

)()()(

)( )(


n

c nTtnTx )()(

) )()()( ( dtxtx cc

nnTxnx c ),(][

Periodic sampling

Tow-stage representation Strictly a mathematical representation that is

convenient for gaining insight into sampling inconvenient for gaining insight into sampling in both the time and frequency domains.

Physical implementation is different.

a continuous-time signal, an impulse train, zero except at nT

a discrete-time sequence time normalization

)(txs

][nx


a discrete-time sequence, time normalization, no explicit information about sampling rate

Many-to-many in general not invertible

][nx

4

Part I-B: Freq. domain represent.



Reconstruction



Frequency-domain representation

From xc(t) to xs(t)

skT

jSnTtts )(2

)( )( )(

The Fourier transform of a periodic impulse train is a periodic impulse train.

nc

ksc

nc

ccs

ks

n

nTtnTx

kjXT

nTttx

jSjXjXtstxtx

Tj

)()(

))((1

)()(

)(*)(2

1)( )()()(

)()()()(

s

?


The Fourier transform of xs(t) consists of periodic repetition of the Fourier transform of xc(t).

5

Frequency-domain

ksc

ks

kjXT

jX

kT

jS

))((1

)(

)(2

)(

s


NsNNs 2or

Recovery

)()()( srr jXjHjX

Ideal lowpass filter with gain

T and cutoff frequency

)()(

)(

jXjX cr

NscN

c


6

Aliasing distortion

Due to the overlap among the copies of , due to

not recoverable by lowpass filtering

)(c jX

Ns 2

)( jX not recoverable by lowpass filtering)(c jX


Aliasing – an example

ttxc 0cos)(

a 1a.1

b.1

k

sc kjXT

jX ))((1

)(s


ttxr 0cos)(

ttx sr )cos()( 0b.2

a.2

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Nyquist sampling theorem

Given bandlimited signal with

Nc jX ||for ,0)(

)(txc

Then is uniquely determined by its samples

If

is called Nyquist frequencyN

nnTxnx c ),(][

Ns T 2

2

)(txc


is called Nyquist frequency

is called Nyquist rateN

N2

Fourier transform of x[n]

From to ][ )( nxtxs

cs nTtnTxtx )()()(

From to

By taking continuous-time Fourier transform of xs(t)

)( )( js eXjX

Tnjcs enTxjX )()(

n

nnTxnx c ),(][

))()(( dtetxjX tj


By taking discrete-time Fourier transform of x[n]

n

cs enxj )()(

n

njj enxeX ][)( )(|)()( TjT

js eXeXjX

) )()( ( dtetxjX

8

Fourier transform of x[n]

)(|)()( TjT

js eXeXjX

k

sc kjXT

jX ))((1

)( s

From Slide 14,

From Slide 8,

i.e. is simply a frequency-scaled version of

kc

kc

Tksc

T

sj

T

kjX

TT

k

TjX

T

kjXT

jXeX

)2

(1

))2

((1

|))((1

|)()(

k

)( jX s)( jeX

So,


p y q ywith

retains a spacing between samples equal to the sampling period T while always has unity space.

T

][nx)(txs

Sampling and reconstruction of Sin Signal

)cos())3/2cos(()4000cos()(][

aliasing no 12000/26000/1

4000)4000cos()(

0

0

nnnTnTxnx

TT

ttx

c

s

c

)4000()4000()()( jXtx cc

)())(()()(][ 0c

TTjXjXeX Tj frequency normalized with )/(|)()( s/s

k

sc kjXT

jX ))((1

)( s


)16000cos()(

about How

ttxc

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Part I-C: Reconstruction



Reconstruction



Requirement for reconstruction

On the basis of the sampling theorem, samples represent the signal exactly when:

B dli it d i l Bandlimited signal

Enough sampling frequency

+ knowledge of the sampling period recover the signal


10

Reconstruction steps

Given x[n] and T, the impulse train is

cs nTtnxnTtnTxtx )(][)()()(

(1)

i.e. the nth sample is associated with the impulse at t=nT.

The impulse train is filtered by an ideal lowpass CT filter with impulse response

nn

cs )(][)()()(

(2)

)(thr )( jHr


n

rr nTthnxtx )()()(

)()()( Tjrr eXjHjX

Ideal lowpass filter

Tsc /2/

asfrequncy cutoff chooseCommonly


Tt

Ttthr /

)/sin()(

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Ideal lowpass filter interpolation

CT signal

Modulated impulse train


n

r TnTt

TnTtnxtx

/)(

)/)(sin(][)(

Ideal discrete-to-continuous-time converter


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discrete-to-continuous-time converter

“Practical DACs do not output a sequence of dirac impulses (that, if ideally low-pass filtered, result in the original signal before sampling) but instead output a sequence of piecewise constant values or rectangular pulses”rectangular pulses


From http://en.wikipedia.org/wiki/Digital-to-analog_converter.

Ideally sampled signal. Piecewise constant signal typical of a practical DAC output.

Applications


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Part II System analysis

Part I: sampling and reconstruction

Part II: system analysis Frequency response

System functions

All-pass systems

Minimum-phase systems

Linear systems with generalized linear phase


System analysis

Three domains

Time domain: impulse response, convolution sum

Frequency domain: frequency response

z-transform: system function)()()( jjj eHeXeY

k

knhkxnhnxny ][][][*][][


LTI system is completed characterized by …

)()()( zHzXzY

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Part II-A: Frequency response



System functions

All-pass systems




Frequency response

Relationship btw Fourier transforms of input and output

)()()( jjj eHeXeY

In polar form

Magnitude magnitude response, gain, distortion

|)(||)(||)(| jjj eHeXeY

)()()( eHeXeY


Phase phase response, phase shift, distortion

)()()( jjj eHeXeY

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Ideal lowpass filter – an example

Frequency response

||0

,|| ,1)( cjeH

Frequency selective filter Impulse response

Noncausal, cannot be implemented!

|| ,0 c

nn

nnh c

lp ,sin

][

0 ,0][?

nnh


p How to make a noncausal system causal?

In general, any noncausal FIR system can be made causal by cascading it with a sufficiently long delay!

But ideal lowpass filter is an IIR system!

Phase distortion and delay

Ideal delay system][][ did nnnh Delay distortion

Ideal lowpass filter with linear phase

1|)(| jid eH

dnjjid eeH )(

||,)( dj

id neH Linear phase distortion


| ,0

,|| ,)(

c

cnj

jlp

deeH

nnn

nnnh

d

dclp ,

)(

)(sin][

Ideal lowpass filter isalways noncausal!

16

Group delay

A measure of the linearity of the phase

Concerning the phase distortion on a narrowbandsignalsignal

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

)cos(][][ 0nnsnx

)( j neH

0 w0


and the output is approximately

Group delay

0)( dneH

))(cos(][|)(|][ 000 dd

j nnnnseHny

)]}({arg[)]([

jj eH

d

deHgrd

An example of group delay

Figure 5.1, 5.2, 5.3


17

An example of group delay


Part II-B: System functions



System functions

All-pass systems




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System function of LCCDE systems

Linear constant-coefficient difference equation

M

m

N

kk mnxbknya

00

][][

z-transform format

N

M

m

mmzb

X

zYzH 0

)(

)()(

M

m

mm

N

k

kk zXzbzYza

00

)()(

mk 00


N

kk

M

mm

N

k

kk

zd

zc

a

b

zazX

1

1

1

1

0

0

0

)1(

)1(

)(

)(

k

k

m

m

dzz

zd

zcz

zc

at pole a 0at zero a

r denominato in the )1(

0at pole a at zero a

numerator in the )1(

1

1

Stability and causality

Stable h[n] absolutely summable

H( ) h ROC i l di h i i l 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

M

zc 1)1(

][][*][][

)(

1)(

1)()()(

nnhnhng

zHzH

zHzHzG

i

i

i

M

N

kk

i

N

kk

mm

zc

zd

b

azH

zd

zc

a

bzH

1

1

1

0

0

1

1

1

0

0

)1(

)1(

)()(

)1(

)1(

)()(


Poles become zeros and vice versa.

ROC: must have overlap btw the two for the sake of G(z).

m

mzc1

)1(

Example

11

1

1

901901

9.0||,9.01

5.01)(

zz

zzH

So,

1

1

11

1

5.01

9.0

5.01

1

5.01

9.01)(

z

z

zz

zzHi

]1[)5.0(9.0][)5.0(][

5.0||1

nununh

znn

i


20

Part II-C: All-pass systems



System functions

All-pass systems




All-pass systems

Consider the following stable system function

1

*1

1)(

az

azzHap

ll t f hi h th1|)(| jH

j

jj

j

jj

ap

ae

eae

ae

aeeH

1

1

1)(

*

*


all-pass system: for which the frequency response magnitude is a constant.

1|)(| jap eH

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Example: First-order all-pass system

P275 Example 5.13


Part II-D: Minimum-phase systems



System functions

All-pass systems




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Magnitude does not uniquely characterize the system

St bl d l l i id it i l Stable and causal poles inside unit circle, no restriction on zeros

Zeros are also inside unit circle inverse system is also stable and causal (in many situations, we need inverse systems!)

such systems are called minimum-phase


such systems are called minimum-phase systems (explanation to follow): are stable and causal and have stable and causal inverses

Minimum-phase and all-pass decomposition

Any rational system function can be expressed as:

S ppose H( ) has one ero o tside the nit circle at)()()( min zHzHzH ap

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

1

*11

1

*11

1)1)((

))(()(

cz

czczzH

czzHzH

1||,/1 * ccz


minimum-phase all-pass

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Frequency response compensation

When the distortion system is not minimum-phase system:

)()()( HHH 1)()()( min zHzHzH apdd )(

1)(

min zHzH

dc

)()()()( zHzHzHzG apcd


Frequency response magnitude is compensated

Phase response is the phase of the all-pass

Properties of minimum-phase systems

From minimum-phase and all-pass decomposition

)()()( min zHzHzH ap

From the previous figure, the continuous-phasecurve of an all-pass system is negative for

So change from minimum-phase to non-minimum-phase (+all pass phase) always decreases the

0

)](arg[)](arg[)](arg[ min j

apjj eHeHeH


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

24

Part II-E: GLP systems



System functions

All-pass systems




Design a system with non-zero phase

System design sometimes desires Constant frequency response magnitude

Z h h ibl 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


25

Ideal delay

1|)(| jeH

||,)( jjid eeH

)(

)(sin][

n

nnhid

1|)(| jid eH

)]([

||,)(j

id

jid

eHgrd

eH

h

phaselinear with lowpass Ideal


][][ did nnnh

dnwhen

)(

)(sin][

d

dclp nn

nnnh

Generalized linear phase

Linear phase filters

|)(|)( jjj eeHeH

Generalized linear phase filters

constantsrealareand

, offunction real a is )(

)()(

j

jjjj

eA

eeAeH


constants real are and

26

Summary


F d i i Frequency domain representation

Reconstruction


System functions


All-pass systems



Course at a glance

Discrete-time signals and systems

MM1 System

Fourier transform and Z-transform

Filt d i

MM2

Sampling andreconstruction

MM3

Systemanalysis


DFT/FFT

Filter design

MM5

MM4

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