Presentation is loading. Please wait.

Presentation is loading. Please wait.

1 G. Baribaud/AB-BDI Digital Signal Processing-2003 6 March 2003 DISP-2003 The z-transform  The sampling process  The definition and the properties.

Published byEmil Singleton Modified over 9 years ago

Similar presentations

Presentation on theme: "1 G. Baribaud/AB-BDI Digital Signal Processing-2003 6 March 2003 DISP-2003 The z-transform  The sampling process  The definition and the properties."— Presentation transcript:

1 1 G. Baribaud/AB-BDI Digital Signal Processing-2003 6 March 2003 DISP-2003 The z-transform  The sampling process  The definition and the properties

2 2 G. Baribaud/AB-BDI Digital Signal Processing-2003 13 March 2003 DISP-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

3 3 G. Baribaud/AB-BDI Digital Signal Processing-2003 13 March 2003 DISP-2003 Convolution Analogous to Laplace convolution theorem

4 4 G. Baribaud/AB-BDI Digital Signal Processing-2003 13 March 2003 DISP-2003 k 1 Apply z-transform

5 5 G. Baribaud/AB-BDI Digital Signal Processing-2003 13 March 2003 DISP-2003 Discrete Cosine

6 6 G. Baribaud/AB-BDI Digital Signal Processing-2003 13 March 2003 DISP-2003 Another approach

7 7 G. Baribaud/AB-BDI Digital Signal Processing-2003 13 March 2003 DISP-2003 Dirac function

8 8 G. Baribaud/AB-BDI Digital Signal Processing-2003 13 March 2003 DISP-2003 Sampled step function t u(t) 1 NB: Equivalent to Exp(-  k) as   0

9 9 G. Baribaud/AB-BDI Digital Signal Processing-2003 13 March 2003 DISP-2003 t t TT Delayed pulse train

10 10 G. Baribaud/AB-BDI Digital Signal Processing-2003 13 March 2003 DISP-2003 Complete z-transform Example:exponential function

11 11 G. Baribaud/AB-BDI Digital Signal Processing-2003 13 March 2003 DISP-2003 Addition and substraction

12 12 G. Baribaud/AB-BDI Digital Signal Processing-2003 13 March 2003 DISP-2003 Multiplication by a constant

13 13 G. Baribaud/AB-BDI Digital Signal Processing-2003 13 March 2003 DISP-2003 -Linearity Application ++

14 14 G. Baribaud/AB-BDI Digital Signal Processing-2003 13 March 2003 DISP-2003 Right shifting theorem t

15 15 G. Baribaud/AB-BDI Digital Signal Processing-2003 13 March 2003 DISP-2003 Right shifting theorem Application Unit step function which is delayed by one sampling period

16 16 G. Baribaud/AB-BDI Digital Signal Processing-2003 13 March 2003 DISP-2003 Left shifting theorem t

17 17 G. Baribaud/AB-BDI Digital Signal Processing-2003 13 March 2003 DISP-2003 Complex translation or damping f(t) is multiplied in continuous domain by Exp(-  t) And then sampled at the rate  T Laplace transform

18 18 G. Baribaud/AB-BDI Digital Signal Processing-2003 13 March 2003 DISP-2003 Application Find the z-transform of sampled at  T knowing that

19 19 G. Baribaud/AB-BDI Digital Signal Processing-2003 13 March 2003 DISP-2003 t S,fS,f Sum of a function

20 20 G. Baribaud/AB-BDI Digital Signal Processing-2003 13 March 2003 DISP-2003 Difference equation

21 21 G. Baribaud/AB-BDI Digital Signal Processing-2003 13 March 2003 DISP-2003 kt Example step function

22 22 G. Baribaud/AB-BDI Digital Signal Processing-2003 13 March 2003 DISP-2003 kt u(t) -u(t-  T) V(t)=u(t)-U(t-  T)

23 23 G. Baribaud/AB-BDI Digital Signal Processing-2003 13 March 2003 DISP-2003 Initial-value theorem If f(t) has a z-transform F(z) and if lim F(z) as z  exists

24 24 G. Baribaud/AB-BDI Digital Signal Processing-2003 13 March 2003 DISP-2003 Final-value theorem

25 25 G. Baribaud/AB-BDI Digital Signal Processing-2003 13 March 2003 DISP-2003 Application:example Initial value Final value Expanding F(z)

26 26 G. Baribaud/AB-BDI Digital Signal Processing-2003 13 March 2003 DISP-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

27 27 G. Baribaud/AB-BDI Digital Signal Processing-2003 13 March 2003 DISP-2003 ? -Reference to tables -Practical identification -Analytic methods -Decomposition -Numerical inversion Inverse

28 28 G. Baribaud/AB-BDI Digital Signal Processing-2003 13 March 2003 DISP-2003 Discrete exponential g(k) Practical identification Sum of a function

29 29 G. Baribaud/AB-BDI Digital Signal Processing-2003 13 March 2003 DISP-2003 x x x x x x o Re z Im z Laurent seriesCauchy theorem Analytic method Enclosing all singularities of F(z,  )

30 30 G. Baribaud/AB-BDI Digital Signal Processing-2003 13 March 2003 DISP-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

31 31 G. Baribaud/AB-BDI Digital Signal Processing-2003 13 March 2003 DISP-2003 The power series method The coefficients of the series expansion represent the values of f(t) (usually a series of numerical values)

32 32 G. Baribaud/AB-BDI Digital Signal Processing-2003 13 March 2003 DISP-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

33 33 G. Baribaud/AB-BDI Digital Signal Processing-2003 6 March 2003 DISP-2003 Continuous Systems in series with an ideal sampler at each input

34 34 G. Baribaud/AB-BDI Digital Signal Processing-2003 6 March 2003 DISP-2003 In general Continuous Systems in series with an ideal sampler at first input

35 35 G. Baribaud/AB-BDI Digital Signal Processing-2003 6 March 2003 DISP-2003 given by and by Continuous Systems in series with an ideal sampler at second input

36 36 G. Baribaud/AB-BDI Digital Signal Processing-2003 6 March 2003 DISP-2003 Discrete and continuous Systems in series with an ideal sampler

37 37 G. Baribaud/AB-BDI Digital Signal Processing-2003 6 March 2003 DISP-2003 Continuous and discrete Systems in series with an ideal sampler

38 38 G. Baribaud/AB-BDI Digital Signal Processing-2003 6 March 2003 DISP-2003 Discrete Systems in series with an ideal sampler

39 39 G. Baribaud/AB-BDI Digital Signal Processing-2003 6 March 2003 DISP-2003 Continuous Systems in parallel with an ideal sampler +

40 40 G. Baribaud/AB-BDI Digital Signal Processing-2003 6 March 2003 DISP-2003 Discrete Systems in parallel with an ideal sampler +

41 41 G. Baribaud/AB-BDI Digital Signal Processing-2003 13 March 2003 DISP-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

42 42 G. Baribaud/AB-BDI Digital Signal Processing-2003 6 March 2003 DISP-2003 Continuous Systems in series with zero-order hold Transfer function via impulse response t t

43 43 G. Baribaud/AB-BDI Digital Signal Processing-2003 6 March 2003 DISP-2003 Laplace transform

44 44 G. Baribaud/AB-BDI Digital Signal Processing-2003 6 March 2003 DISP-2003 Global transfer function Equal to G(s) with an integrator Z-transform of G(s)

45 45 G. Baribaud/AB-BDI Digital Signal Processing-2003 6 March 2003 DISP-2003 Consequences on the behaviour Z-transform There are n poles of G(z,  ), they depend on n the poles of the transfer function of the continuous system

Download ppt "1 G. Baribaud/AB-BDI Digital Signal Processing-2003 6 March 2003 DISP-2003 The z-transform  The sampling process  The definition and the properties."

Similar presentations

LAPLACE TRANSFORMS.

LAPLACE TRANSFORMS.
Signal Processing in the Discrete Time Domain Microprocessor Applications (MEE4033) Sogang University Department of Mechanical Engineering.

Signal Processing in the Discrete Time Domain Microprocessor Applications (MEE4033) Sogang University Department of Mechanical Engineering.
H(s) x(t)y(t) 8.b Laplace Transform: Y(s)=X(s) H(s) The Laplace transform can be used in the solution of ordinary linear differential equations. Let’s.

H(s) x(t)y(t) 8.b Laplace Transform: Y(s)=X(s) H(s) The Laplace transform can be used in the solution of ordinary linear differential equations. Let’s.
Sep 15, 2005CS477: Analog and Digital Communications1 Modulation and Sampling Analog and Digital Communications Autumn 2005-2006.

Sep 15, 2005CS477: Analog and Digital Communications1 Modulation and Sampling Analog and Digital Communications Autumn
Laplace Transforms Important analytical method for solving linear ordinary differential equations. - Application to nonlinear ODEs? Must linearize first.

Laplace Transforms Important analytical method for solving linear ordinary differential equations. - Application to nonlinear ODEs? Must linearize first.
Lecture 19: Discrete-Time Transfer Functions

Lecture 19: Discrete-Time Transfer Functions
Autumn Analog and Digital Communications Autumn

Autumn Analog and Digital Communications Autumn
Bogazici University Dept. Of ME. Laplace Transforms Very useful in the analysis and design of LTI systems. Operations of differentiation and integration.

Bogazici University Dept. Of ME. Laplace Transforms Very useful in the analysis and design of LTI systems. Operations of differentiation and integration.
Lecture 25 Laplace Transform

Lecture 25 Laplace Transform
Digital Control Systems The z-Transform. The z Transform Definition of z-Transform The z transform method is an operational method that is very powerful.

Digital Control Systems The z-Transform. The z Transform Definition of z-Transform The z transform method is an operational method that is very powerful.
Analogue and digital techniques in closed loop regulation applications Digital systems Sampling of analogue signals Sample-and-hold Parseval’s theorem.

Analogue and digital techniques in closed loop regulation applications Digital systems Sampling of analogue signals Sample-and-hold Parseval’s theorem.
Computational Geophysics and Data Analysis

Computational Geophysics and Data Analysis
EE513 Audio Signals and Systems Digital Signal Processing (Systems) Kevin D. Donohue Electrical and Computer Engineering University of Kentucky.

EE513 Audio Signals and Systems Digital Signal Processing (Systems) Kevin D. Donohue Electrical and Computer Engineering University of Kentucky.
The sampling of continuous-time signals is an important topic It is required by many important technologies such as: Digital Communication Systems ( Wireless.

The sampling of continuous-time signals is an important topic It is required by many important technologies such as: Digital Communication Systems ( Wireless.
2. Z-transform and theorem

2. Z-transform and theorem
Chapter 2 Discrete System Analysis – Discrete Signals

Chapter 2 Discrete System Analysis – Discrete Signals
Signals and Systems Jamshid Shanbehzadeh.

Signals and Systems Jamshid Shanbehzadeh.
THE LAPLACE TRANSFORM LEARNING GOALS Definition The transform maps a function of time into a function of a complex variable Two important singularity functions.

THE LAPLACE TRANSFORM LEARNING GOALS Definition The transform maps a function of time into a function of a complex variable Two important singularity functions.
Properties of the z-Transform

Properties of the z-Transform
The z-Transform Prof. Siripong Potisuk. LTI System description Previous basis function: unit sample or DT impulse  The input sequence is represented.

The z-Transform Prof. Siripong Potisuk. LTI System description Previous basis function: unit sample or DT impulse  The input sequence is represented.

Similar presentations

Ads by Google