G. Baribaud/AB-BDI

1G. Baribaud/AB-BDI

Digital Signal Processing-2003

6 March 2003DISP-2003

The z-transform

The sampling process

The definition and the properties

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The z-transform

• Classification of signals• Sampling of continuous signals• The z-transform: definition • The z-transform: properties• Inverse z-transform• Application to systems• Comments on stability

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• Continuous (or analogue) signals

Classification of signals

• Sampled signals

• Discrete (or digital or time) signals

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Continuous (or analogue) signals

Continuous signals (Single or multiple)

t

y(t)

u(t) Discontinuity

G(s)u(t)U(s) y(t) Y(s)

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x(t)

t

x[kT]

t

Amplitudemodulated

t

x[kT]Pulsewidthmodulated

Originalsignal

Sampled signal: modulation techniques

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• Noted x’(t)• Samples exist only at sampling times• The relative height represents the value (information)• The sampling T period is constant• Able to drive a physical system

t

x’(t)

T

kT(K+1)T

Sampled signal

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Physical sample

t

h Information Area=h

Dirac (or unit) pulseAs (Distribution)

NB: The energy of a sample pulse is finite(able to drive a physical system)

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• Noted x*(t)• Values defined only at sampling times• The relative height represents the numerical value • The sampling T period is constant• Usable for arithmetic operations• Unable to drive a physical system

t

x*(t)

T

kT(K+1)T

Discrete (or digital or time) signals

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•Application

G(s)u(t) y(t)

Continuous signal

Measure(ADC)

Driver(DAC)

Computer

Discrete signal Sampling

Samples or steps

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• Continuous system

G(s)x(t)

x(t)

t

y(t)

t

y(t)




• Sampled system

x’(t)

t

y(t)

t

G(s)x’(t) y(t)




• Sampled system with hold circuit

G(s)x’(t) y(t)

u(t)

t

y(t)

t

Hold

x’(t)

t

u(t)




• Discrete system

D(z)x*(t) y*(t)

y*(t)

t

x*(t)

t

D(z) defined by difference equations or by transfer function




The z-transform





Sampling time and delay

t

t=kTt=(k-1)T t=(k+1)T

t=(k+)T

x[k-1]

x[k]x[k+1]x[k,]

x(kT)=x[k,]=x[k]=xk

Several notations




t

Sampling times

(k-1)T (k+1)T

kT

)t(

Sampling function

n=integer………-2,-1,0,1,2,3,4,………..

n

n

)Tnt(T)t( Periodic function




1dte)t(TT

1dte)t(f

T

1c T

tnj2

2/t

2/t

T

tnj2

2/t

2/t

n

Decomposition in Fourier series T

tnj2n

nnec)t(

n,1cn

n=0 n=1n=-1

T

n2

0T

2

T

2




n

n

n

n

)T

n2()

T

nf()(

t

5

t

5

0

t

4

t

3

t

2

t

1

t

4

t

3

t

2

t

1

2

f

n

n

n

n

)T

nf()Tnt(




Ideal sampler

)t()t(f)t(f '

t

f(t)

T T2 T3 T4 T5T2T3T4T5 T 0

)Tnt()t(fT)t(fn

'

T T2 T3 T4 T5T2T3T4T5 T 0

t)t(




Sampling process

ContinuousFunction

f(t)

Sampler(t)

Series ofsamples

T

Reconstruction ?




)t()t(f)t(f '

Analysis in frequency domain

n

n

)Tnt(T)t(

• F()= Fourier transform of f(t)• ()= Fourier transform of (t)• F’()= Fourier transform of f’(t)

)()(F)(F' Convolution in the frequency domain




d)T

n2()(Fd)()(F)('F

n

)T

n2(Fd)

T

n2()(F)('F

nn

)T

n2(F)('F

n

Analysis of F’()




The spectra are overlapping (Folding)

0

Primary componentsFundamental components

ss

Complementarycomponents

Complementarycomponents

)('F

2s

2s

2

sFolding frequency

Aliasing




Sampler Filterf(t)

f’(t) f °(t)

)(W)('F)(F c In the frequency domain

0 s

ss2 s2

)('F

cc

Window )(Wc c2

1

Reconstruction

d)t(w)('f)t(f cIn the time domain




d)t(

)t(sin)Tn()Tn(f)t(f

c

c

n

n c

c

n c

c

)Tnt(f2

)Tnt(f2sin)Tn(f)t(f

)Tnt(

)Tnt(sin)Tn(f)t(f

Tn

tf2

)tf2sin(

t

)tsin(de

2

1)t(w

c

c

c

ctj2

cc

c

c

Reconstruction




Reconstruction

t

f(t)Interpolation functions

nT (n+1)T (n+2)T (n+3)T




The z-transform





Factors like Exp(-sT) are involvedUnlike the majority of transfer functions of continuous systemsIt will not lead to rational functions

1k

Tsk

0k

Tsk

0

st

0k

e)Tk(f2

)0(fe)Tk(f

2

)0(fdte)Tkt()t(f)]t('f[L

Apply Laplace transform of f’(t)

0k0k

)Tk(f)Tkt()t(f)t('f

t)t('f

Definition

For sampledor discrete

signals




Tsez

Definition

0k

k

0k

k z)k(fz)kT(f)z(F

)k('F)0(f2

1)s(F'

)Tsin(e)zIm(

)Tcos(e)zRe(

j)zln(T

1s

T

T

)]zln(T

1s)[s('F)z(F

Laplace z

t k




1z )k(x )1k(x

Representation of a delay




The operation of taking the z-transform of a continuous-datafunction, f(t), involves the following three steps:

1- f(t) is sampled by an ideal sampler to get f’(t)

2- Take the Laplace transform of f’(t)

0k

Tkse)Tk(f)s('F

3- Replace by z in F’(s) to get Tse

0k

kz)Tk(f)z(F

f(t) F(s) F(z)

)t(

A few z-transforms

11

)Tkt( Tske kz

s

11z

z

)t(u

t2s

12)1z(

Tz

2t 3s

23

2

)1z(

)1z(zT

as

1

Taez

z

ate

atte2)as(

1

2Ta

Ta

)ez(

Tze




Mapping Tsez S-plane

z-plane

T

2s

2s

2s

Primary strip

j

Re[z]

Im[z]

1

23

4 5

The left half of the primary strip is mapped inside the unit circle

12

5

3

4 1




MappingS-plane

Z-plane

2s

2s

Primary strip

j

1

2 3

45

1 Rez

Imz

2

5

3

4 1

Tsez

The right half of the primary strip is mapped outside the unit circle




S-plane

Z-plane

)k2

1(s

)k2

1(s

Complementary strip

j

Rez

Imz

1

The left half of thecomplementary stripis also mapped inside the unit circle

Tskj2TsTjkTsT)jks( eeeeee ss

1

23

4 5




s-plane properties of F’(s)

Primary strip

j

2/s

2/s

2/3 s

2/5 s

2/3 s

2/5 s)s('F)jms('F s

0s

s0 js

s0 2js

s0 2js

s0 js

Complementary strip

Complementary strip

Complementary strip

Complementary strip




Primary strip2/s

2/s

2/3 s

2/5 s

2/3 s

2/5 s

0s

s0 js

s0 2js

s0 2js

s0 js

Complementary strip

Complementary strip

Complementary strip

Complementary strip

X

X

X Poles of F’(s) in primary strip

X

X

X




Primary strip2/s

2/s

2/3 s

2/5 s

2/3 s

2/5 s

0s

s0 js

s0 2js

s0 2js

s0 js

Complementary strip

Complementary strip

Complementary strip

Complementary strip

X

X

X Poles of F’(s) in complementary strips

X

X

X

Folded back poles




The constant damping loci

s-plane z-plane

1

2

j

TjTeez 1

TjTeez 2




The constant frequency loci

s-plane z-plane

1j

1j

jTj 1ez

Tj 1ez

2jT1

T2




Mapping between the s-plane and the z-plane

Conclusion:

All points in the left half of the s-plane are mapped into theRegion inside the unit circle in the z-plane.

The points in the right half of the s-plane are mapped into theRegion outside the unit circle in the z-plane

G. Baribaud/AB-BDI

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