1 g. baribaud/ab-bdi digital signal processing-2003 6 march 2003 disp-2003 the z-transform the...

1G. Baribaud/AB-BDI

Digital Signal Processing-2003

6 March 2003DISP-2003

The z-transform

The sampling process

The definition and the properties

2G. Baribaud/AB-BDI



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

3G. Baribaud/AB-BDI



Convolution

)z(F)z(F])TnTk(f)Tn(f[Z)]t(f)t(f[Z 21

k

0n2111

])TnTk(f)Tn(f[Zk

0n21

]z)TnTk(f)Tn(f[])TnTk(f)Tn(f[Z k

0k

k

0n21

k

0n21

]z)TnTk(f)Tn(f[])TnTk(f)Tn(f[Z k

0k 0n21

k

0n21

nkm

m

0m2

0n

n1

k

0n21 z)Tm(fz)Tn(f])TnTk(f)Tn(f[Z

Analogous to Laplace convolution theorem

)z(F)t(f 11 )z(F)t(f 22

4G. Baribaud/AB-BDI



0k

k** z)k(f)z(F

0k

k1

0k

kk* )ze(ze)z(F

ez

z

ze1

1)k(F

1*

k* e)k(f

0k,0)k(f *

0

k

1

Apply z-transform

5G. Baribaud/AB-BDI



2

ee)kcos()k(f

kjkj*

)ez

z

ez

z(

2

1)z(F

jj*

]1)(zCos2z

)(zCos2z2[

2

z]

1)ee(zz

)ee(z2[

2

z)z(F

2jj2

jj*

1cosz2z

)cosz(z)z(F

2*

Discrete Cosine

ez

z]e[Z k

])ez)(ez(

)ez()ez([

2

z)z(F

jj

jj*

6G. Baribaud/AB-BDI



Another approach jke)ksin(j)kcos()k(y

sinjcosz

z

ez

z)z(Y

j

)sinjcosz)(sinjcosz(

)sinjcosz(z)z(Y

1cosz2z

)sinjz)cosz(z)z(Y

2

1cosz2z

)cosz(z[cos]Z

2

1cosz2z

)sin(z[sin]Z

2

7G. Baribaud/AB-BDI



Dirac function

1)]t([F

)t(

1)0(z)t()]t([F0k

Ts

8G. Baribaud/AB-BDI



Sampled step function

t

u(t)

T T2 T3 T4 T5

1

0

1z

z

z1

1)z(U

e1

1e)s(U

1

Ts0k

Tsk

1z

z

z1

1z)z(U

e)s(U

10k

k

0k

Tsk

NB: Equivalent to Exp(-k) as 0

9G. Baribaud/AB-BDI



t

k

T )Tkt(

t

T

T

0k

ke

Ts

0k

T)k(se

'e z)T)k[(xee)T)k[(x),s(X

k

e'e ]T)k(t[)t(xx

Delayed pulse train

10G. Baribaud/AB-BDI



Complete z-transform

0k

k

0k

k z),k(fz]T)k[(f),z(F

Example:exponential function

1T,0,e),k(f )k(

e

ez

zzeeze),z(F

0k

kk

0k

k)k(

eez

z),z(F




Addition and substraction

k

0k222

k

0k111

z),Tk(f)],k(f[Z),z(F

z),Tk(f)],k(f[Z),z(F

),z(F),z(F)],k(f),k(f[Z 2121




Multiplication by a constant

ttanconsa

z),Tk(f)],k(f[Z),z(F k

0k

),z(aF)],k(af[Z




-Linearity

),k(F),,k(F),k(f),,k(f 2121

tstanconsKK 2,1

),z(FK),z(FK)],k(fK),k(fK[Z 22112211

Application

1K

2K+

1f

2ff 1K

2K+

1f

2f

f




Right shifting theorem

t

),k(f ),nk(f

0k

)nk(

0k

nk z),nk(fzz),nk(f)],nk(f[Z

)],k(f[Zzz),m(fz)],nk(f[Z0m

nmn

)],k(f[Zz)],nk(f[Z n 0n




Right shifting theorem

Application

Unit step function which is delayed by one sampling period

1z

1)

1z

z(z)]Tt(u[Z 1




Left shifting theorem

0k

)nk(

0k

nk z),nk(fzz),nk(f)],nk(f[Z

0m

1n

0m

mnmn z),m(f)],k(f[Zzz),m(fz)],nk(f[Z

t

),k(f

),nk(f

1n

0m

mn z),m(f)],k(f[Zz)],nk(f[Z




Complex translation or damping

f(t) is multiplied in continuous domain by Exp(-t)And then sampled at the rate T

)ze(F]e)t(f[Z)z(G Tt

Laplace transform

dte)t(f)s(F0

st

)s(Fdte)t(fdtee)t(f)s(G0

t)s(

0

stt

T1TTsT)s( ezeee




Application

Find the z-transform of sampled at T knowing that)tsin(e at

Ta2

zez,1)Tcos(z2z

)Tsin(z[sin]Z

1)Tcos(ze2ez

)Tsin(zesin]e[Z

TaTa22

Taat

Ta2Ta2

Taat

e)Tcos(ze2z

)Tsin(zesin]e[Z




k

0n

),n(f),k(s

t

S,f

),k(Sz),k(F),k(S 1

),1k(f),k(f),n(f),k(f),k(s1k

0n

),k(F1z

z),k(S

Sum of a function




Difference equation

),1k(f),k(f),k(f

),z(Fz),z(F)],k(f[Z 1

),z(F)z1()],k(f[Z 1

),z(Fz

1z)],k(f[Z




),1k(f),k(f),k(f

kt

kt

kt

Example step function




]z1)[,z(F)],1k(f[Z)],k(f[Z)],k(f[Z 1

),z(Fz

1z)],k(f[Z

kt

11z

z

z

1z)z(V

u(t)

-u(t-T)

V(t)=u(t)-U(t-T)




0k

kz),k(f),z(F

Initial-value theorem

If f(t) has a z-transform F(z) and if lim F(z) as z exists

),z(Flim),0(f),k(flim z0k

2

0k

1k z)T2(fz)T(f)0(fz)k(f)z(F




Final-value theorem

),z(F)1zlim(),k(flim

),z(Fz

1zlim),z(F)z1lim(),k(flim 1

k 1z




Application:example

)208.0z416.0z)(1z(

z792.0)z(F

2

2

Initial value 0z

z792.0)z(F

3

2

Final value 1)208.0416.01(

792.0)k(f

Expanding F(z)

....z99.0z989.0z983.0z01.1z091.1z12.1z792.0)z(F 7654321




The z-transform





kz),k(f

),k(f ),z(F

?

-Reference to tables-Practical identification-Analytic methods-Decomposition-Numerical inversion

Inverse




)z(G1z

z

ez

z

1z

z)z(F

Discrete exponential g(k)

Practical identification

Sum of a function




0k

kz),k(f),z(F

dzz),z(Fj2

1),k(f 1k

x

xx

xx

xo Re z

Im zLaurent series Cauchy theorem

Analytic method

Enclosing all singularities of F(z,)




Partial fraction expansion

With Laplace transform

....cs

C

bs

B

as

A)s(F

....CeBeAe)t(f ctbtat

With z-transform no such an expansion, one looks for terms like:

TakTa

Aeez

Az

The function F(z)/z is developed by partial-fraction expansion




The power series method

...z)T3(fz)T2(fz)T(f)0(f)z(F 321

The coefficients of the series expansion represent the values of f(t)(usually a series of numerical values)

TaTa2

Ta

Ta

Ta

ez)e1(z

z)e1(

)ez)(1z(

z)e1()z(F

....z)e1(z)e1()z(F 2Ta21Ta

)Tkt()e1()t('f0k

Tak




The z-transform





Continuous Systems in series with an ideal sampler at each input

)s(Ga )s(Gbex '

ex ix 'ix sx

T

)z(X),z(G),z(X 'eai

)z(X)0,z(G)0,z(X)z(X 'eai

'i

)z(X),z(G),z(X 'ibs

'e

'eabs X),z(G)z(X)0,z(G),z(G),z(X

)0,z(G),z(G),z(G ab




)s(Ga )s(Gbex '

ex ix sx

T

)z(X),z(G),z(X 'es

)s(G)s(G)s(G ba

)]s(G)s(G[Z)]s(G[Z),z(G ba

In general ),z(G),z(G),z(G ba

Continuous Systems in series with an ideal sampler at first input




)z(X),z(G),z(X 'ibs

)s(Ga )s(Gbex ix

sx

T

'ix

)0,z(X)z(X 'i

'i

)s(X)s(G)s(X eai

),z(Gb ),z(Xs given by and by )]s(X)s(G[Z ea

Continuous Systems in series with an ideal sampler at second input




)z(X)z(D)z(X *ea

*i

)z(D),z(G),z(G ab

)z(X),z(G),z(X 'ibs

)z(X)z(X *i

'i

)s(Da )s(Gbex *

ix sx

T

'ix*

ex

)z(X),z(G)z(X)z(D),z(G),z(X *e

*eabs

Discrete and continuous Systems in series with an ideal sampler




Continuous and discrete Systems in series with an ideal sampler

)z(X),z(G),z(X 'eai

)0,z(G)z(D)z(G ab*

)z(X)z(D)z(X *ib

*s

)s(Db)s(Ga

ex ix *sx

T

*ix'

ex

)z(X)z(G)z(X)0,z(G)z(D)z(X 'e

*'eab

*s

)0,z(X)z(X i*i




Discrete Systems in series with an ideal sampler

)z(X)z(D)z(X *ea

*i

)z(D)z(D)z(D ab

)z(X)z(D)z(X)z(D)z(D)z(X *e

*eab

*s

)s(Db)s(Da

ex *sx

T

*ix*

ex




Continuous Systems in parallel with an ideal sampler

),z(G),z(G),z(G ba

),z(X),z(X),z(X bas

)s(Gb

)s(Ga

exsx

T

'ex

ax

bx

+'ex

)z(X),z(G)z(X),z(G),z(X 'eb

'eas

)z(X),z(G)z(X)],z(G),z(G[),z(X 'e

'ebas




Discrete Systems in parallel with an ideal sampler

)z(D)z(D)z(D ba

)z(X)z(X)z(X *b

*a

*s

)z(X)z(D)z(X)z(D)z(X *eb

*ea

*s

)z(X)z(D)z(X)]z(D)z(D[)z(X *e

*eba

*s

)s(Db

)s(Da

ex*sx

T

*ex *

ax

*bx

+*ex




The z-transform





Continuous Systems in series with zero-order holdTransfer function via impulse response

)s(Gssxex

T

T

'ex ix

)t(x 0e 'ex

tix

0ex

T t

)]Tt(u)t(u[xx 0ei




Laplace transform

s

e1

)s(X

(s)X(s)G

Ts

e

im

)]Tt(u)t(u[xx 0ei

)s

e1(x]

s

e

s

1[x)s(X

Ts

0e

Ts

0ei




s

)s(G)e1()s(G

s

e1

)s(X

(s)XG(s) sTs

s

Ts

e

s

Global transfer function

s

)s(G(s)G s

I Equal to G(s) with an integrator

Z-transform of G(s) )s(Ge)s(GG(s) ITs

I

),z(G)s(G II ),z(Gz)s(Ge I1

ITs

),z(Gz

1z),z(G)z1(),z(G II

1




Consequences on the behaviour

n

1i i

is ps

r)s(G

]ps

1

s

1[

)p(

r

)ps(s

r)s(G

i

n

1i i

in

1i i

iI

Z-transform1z

z

s

1

tp

tpi

i

ie

ez

z

ps

1

]eez

1z1[

)p(

r),z(G

z

1z),z(G tp

tp

n

1i i

iI

i

i

There are n poles of G(z,), they depend on n the poles

of the transfer function of the continuous system

1 g. baribaud/ab-bdi digital signal processing-2003 6 march 2003 disp-2003 the z-transform the...

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