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Digital Signal Processing-2003. The z-transform. The sampling process The definition and the properties. 6 March 2003. DISP-2003. G. Baribaud/AB-BDI. Digital Signal Processing-2003. The z-transform. • Classification of signals • Sampling of continuous signals

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Presentation Transcript
slide1

Digital Signal Processing-2003

The z-transform

  • The sampling process
  • The definition and the properties

6 March 2003

DISP-2003

G. Baribaud/AB-BDI

slide2

Digital Signal Processing-2003

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

13 March 2003

DISP-2003

G. Baribaud/AB-BDI

slide3

Digital Signal Processing-2003

Convolution

Analogous to Laplace convolution theorem

13 March 2003

DISP-2003

G. Baribaud/AB-BDI

slide4

1

k

Digital Signal Processing-2003

Apply z-transform

13 March 2003

DISP-2003

G. Baribaud/AB-BDI

slide5

Digital Signal Processing-2003

Discrete Cosine

13 March 2003

DISP-2003

G. Baribaud/AB-BDI

slide6

Digital Signal Processing-2003

Another approach

13 March 2003

DISP-2003

G. Baribaud/AB-BDI

slide7

Dirac function

Digital Signal Processing-2003

13 March 2003

DISP-2003

G. Baribaud/AB-BDI

slide8

u(t)

1

t

Digital Signal Processing-2003

Sampled step function

NB: Equivalent to Exp(-k) as  0

13 March 2003

DISP-2003

G. Baribaud/AB-BDI

slide9

Digital Signal Processing-2003

T

Delayed pulse train

t

t

13 March 2003

DISP-2003

G. Baribaud/AB-BDI

slide10

Digital Signal Processing-2003

Complete z-transform

Example:exponential function

13 March 2003

DISP-2003

G. Baribaud/AB-BDI

slide11

Digital Signal Processing-2003

Addition and substraction

13 March 2003

DISP-2003

G. Baribaud/AB-BDI

slide12

Digital Signal Processing-2003

Multiplication by a constant

13 March 2003

DISP-2003

G. Baribaud/AB-BDI

slide13

+

+

Digital Signal Processing-2003

-Linearity

Application

13 March 2003

DISP-2003

G. Baribaud/AB-BDI

slide14

t

Digital Signal Processing-2003

Right shifting theorem

13 March 2003

DISP-2003

G. Baribaud/AB-BDI

slide15

Right shifting theorem

Application

Unit step function which is delayed by one sampling period

Digital Signal Processing-2003

13 March 2003

DISP-2003

G. Baribaud/AB-BDI

slide16

t

Digital Signal Processing-2003

Left shifting theorem

13 March 2003

DISP-2003

G. Baribaud/AB-BDI

slide17

Digital Signal Processing-2003

Complex translation or damping

f(t) is multiplied in continuous domain by Exp(-t)

And then sampled at the rate T

Laplace transform

13 March 2003

DISP-2003

G. Baribaud/AB-BDI

slide18

Find the z-transform of sampled at T knowing that

Digital Signal Processing-2003

Application

13 March 2003

DISP-2003

G. Baribaud/AB-BDI

slide19

S,f

t

Digital Signal Processing-2003

Sum of a function

13 March 2003

DISP-2003

G. Baribaud/AB-BDI

slide20

Digital Signal Processing-2003

Difference equation

13 March 2003

DISP-2003

G. Baribaud/AB-BDI

slide21

Example step function

kt

kt

kt

Digital Signal Processing-2003

13 March 2003

DISP-2003

G. Baribaud/AB-BDI

slide22

u(t)

kt

-u(t-T)

V(t)=u(t)-U(t-T)

Digital Signal Processing-2003

13 March 2003

DISP-2003

G. Baribaud/AB-BDI

slide23

Initial-value theorem

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

Digital Signal Processing-2003

13 March 2003

DISP-2003

G. Baribaud/AB-BDI

slide24

Digital Signal Processing-2003

Final-value theorem

13 March 2003

DISP-2003

G. Baribaud/AB-BDI

slide25

Initial value

Final value

Digital Signal Processing-2003

Application:example

Expanding F(z)

13 March 2003

DISP-2003

G. Baribaud/AB-BDI

slide26

Digital Signal Processing-2003

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

13 March 2003

DISP-2003

G. Baribaud/AB-BDI

slide27

Digital Signal Processing-2003

Inverse

?

-Reference to tables

-Practical identification

-Analytic methods

-Decomposition

-Numerical inversion

13 March 2003

DISP-2003

G. Baribaud/AB-BDI

slide28

Digital Signal Processing-2003

Practical identification

Discrete exponential g(k)

Sum of a function

13 March 2003

DISP-2003

G. Baribaud/AB-BDI

slide29

Digital Signal Processing-2003

Analytic method

Laurent series

Cauchy theorem

Im z

x

x

Re z

o

x

x

x

x

Enclosing all singularities of F(z,)

13 March 2003

DISP-2003

G. Baribaud/AB-BDI

slide30

Digital Signal Processing-2003

Partial fraction expansion

With Laplace transform

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

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

13 March 2003

DISP-2003

G. Baribaud/AB-BDI

slide31

Digital Signal Processing-2003

The power series method

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

(usually a series of numerical values)

13 March 2003

DISP-2003

G. Baribaud/AB-BDI

slide32

Digital Signal Processing-2003

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

13 March 2003

DISP-2003

G. Baribaud/AB-BDI

slide33

Digital Signal Processing-2003

Continuous Systems in series with an ideal sampler at each input

6 March 2003

DISP-2003

G. Baribaud/AB-BDI

slide34

Digital Signal Processing-2003

Continuous Systems in series with an ideal sampler at first input

In general

6 March 2003

DISP-2003

G. Baribaud/AB-BDI

slide35

and by

given by

Digital Signal Processing-2003

Continuous Systems in series with an ideal sampler at second input

6 March 2003

DISP-2003

G. Baribaud/AB-BDI

slide36

Digital Signal Processing-2003

Discrete and continuous Systems in series with an ideal sampler

6 March 2003

DISP-2003

G. Baribaud/AB-BDI

slide37

Digital Signal Processing-2003

Continuous and discrete Systems in series with an ideal sampler

6 March 2003

DISP-2003

G. Baribaud/AB-BDI

slide38

Digital Signal Processing-2003

Discrete Systems in series with an ideal sampler

6 March 2003

DISP-2003

G. Baribaud/AB-BDI

slide39

+

Digital Signal Processing-2003

Continuous Systems in parallel with an ideal sampler

6 March 2003

DISP-2003

G. Baribaud/AB-BDI

slide40

+

Digital Signal Processing-2003

Discrete Systems in parallel with an ideal sampler

6 March 2003

DISP-2003

G. Baribaud/AB-BDI

slide41

Digital Signal Processing-2003

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

13 March 2003

DISP-2003

G. Baribaud/AB-BDI

slide42

t

t

Digital Signal Processing-2003

Continuous Systems in series with zero-order hold

Transfer function via impulse response

6 March 2003

DISP-2003

G. Baribaud/AB-BDI

slide43

Digital Signal Processing-2003

Laplace transform

6 March 2003

DISP-2003

G. Baribaud/AB-BDI

slide44

Equal to G(s) with an integrator

Digital Signal Processing-2003

Global transfer function

Z-transform of G(s)

6 March 2003

DISP-2003

G. Baribaud/AB-BDI

slide45

Z-transform

Digital Signal Processing-2003

Consequences on the behaviour

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

of the transfer function of the continuous system

6 March 2003

DISP-2003

G. Baribaud/AB-BDI