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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
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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
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3 G. Baribaud/AB-BDI Digital Signal Processing-2003 13 March 2003 DISP-2003 Convolution Analogous to Laplace convolution theorem
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4 G. Baribaud/AB-BDI Digital Signal Processing-2003 13 March 2003 DISP-2003 k 1 Apply z-transform
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5 G. Baribaud/AB-BDI Digital Signal Processing-2003 13 March 2003 DISP-2003 Discrete Cosine
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6 G. Baribaud/AB-BDI Digital Signal Processing-2003 13 March 2003 DISP-2003 Another approach
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7 G. Baribaud/AB-BDI Digital Signal Processing-2003 13 March 2003 DISP-2003 Dirac function
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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
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9 G. Baribaud/AB-BDI Digital Signal Processing-2003 13 March 2003 DISP-2003 t t TT Delayed pulse train
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10 G. Baribaud/AB-BDI Digital Signal Processing-2003 13 March 2003 DISP-2003 Complete z-transform Example:exponential function
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11 G. Baribaud/AB-BDI Digital Signal Processing-2003 13 March 2003 DISP-2003 Addition and substraction
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12 G. Baribaud/AB-BDI Digital Signal Processing-2003 13 March 2003 DISP-2003 Multiplication by a constant
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13 G. Baribaud/AB-BDI Digital Signal Processing-2003 13 March 2003 DISP-2003 -Linearity Application ++
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14 G. Baribaud/AB-BDI Digital Signal Processing-2003 13 March 2003 DISP-2003 Right shifting theorem t
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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
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16 G. Baribaud/AB-BDI Digital Signal Processing-2003 13 March 2003 DISP-2003 Left shifting theorem t
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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
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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
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19 G. Baribaud/AB-BDI Digital Signal Processing-2003 13 March 2003 DISP-2003 t S,fS,f Sum of a function
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20 G. Baribaud/AB-BDI Digital Signal Processing-2003 13 March 2003 DISP-2003 Difference equation
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21 G. Baribaud/AB-BDI Digital Signal Processing-2003 13 March 2003 DISP-2003 kt Example step function
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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)
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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
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24 G. Baribaud/AB-BDI Digital Signal Processing-2003 13 March 2003 DISP-2003 Final-value theorem
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25 G. Baribaud/AB-BDI Digital Signal Processing-2003 13 March 2003 DISP-2003 Application:example Initial value Final value Expanding F(z)
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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
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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
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28 G. Baribaud/AB-BDI Digital Signal Processing-2003 13 March 2003 DISP-2003 Discrete exponential g(k) Practical identification Sum of a function
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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, )
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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
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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)
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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
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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
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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
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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
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36 G. Baribaud/AB-BDI Digital Signal Processing-2003 6 March 2003 DISP-2003 Discrete and continuous Systems in series with an ideal sampler
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37 G. Baribaud/AB-BDI Digital Signal Processing-2003 6 March 2003 DISP-2003 Continuous and discrete Systems in series with an ideal sampler
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38 G. Baribaud/AB-BDI Digital Signal Processing-2003 6 March 2003 DISP-2003 Discrete Systems in series with an ideal sampler
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39 G. Baribaud/AB-BDI Digital Signal Processing-2003 6 March 2003 DISP-2003 Continuous Systems in parallel with an ideal sampler +
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40 G. Baribaud/AB-BDI Digital Signal Processing-2003 6 March 2003 DISP-2003 Discrete Systems in parallel with an ideal sampler +
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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
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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
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43 G. Baribaud/AB-BDI Digital Signal Processing-2003 6 March 2003 DISP-2003 Laplace transform
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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)
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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
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