1. Lecture 12: Parametric Signal Modeling XILIANG LUO 2014/11 1

2. Discrete Time Signals

3. Representation of Sequences by FT  Many sequences can be represented by a Fourier integral as follows:  x[n] can be represented as a superposition of infinitesimally small complex exponentials  Fourier transform is to determine how much of each frequency component is used to synthesize the sequence

4. Z-Transform a function of the complex variable: z

5. Periodic Sequence  Discrete Fourier Series For a sequence with period N, we only need N DFS coefs

6. Discrete Fourier Transform DFT is just sampling the unit-circle of the DTFT of x[n]

7. Parametric Signal Modeling 7 A signal is represented by a mathematical model which has a Predefined structure involving a limited number of parameters. A given signal is represented by choosing the specific set of parameters that results in the model output being as close as possible in some prescribed sense to the given signal.

8. Parametric Signal Modeling 8 LTI H(z) v[n] s’[n]

9. All-Pole Modeling 9 All-pole model assumes the signal can be approximated as a linear combination of its previous values!  this modeling is also called: linear prediction

10. All-Pole Modeling 10  Least Squares Approximation

11. All-Pole Modeling 11  Least Squares Inverse Model LTI A(z) s[n] g[n]

12. All-Pole Modeling 12  Least Squares Inverse Model LTI A(z) s[n] g[n]

13. All-Pole Modeling 13  Least Squares Inverse Model Yule-Walker equations

14. Linear Predictor 14 1.if input v[n] is impulse, the prediction error is zero 2.if input v[n] is white, the prediction error is white Linear Predictor + s[n] s’[n] e[n] -

15. Deterministic Signal 15 Minimize total error energy will render the following definitions:

16. Random Signal 16 Minimize expected error energy will render the following definitions:

17. All-Pole Spectrum 17 All-pole method gives a method of obtaining high-resolution estimates of a signal’s spectrum from truncated or windowed data!

18. All-Pole Spectrum 18 For deterministic signal, we have the following DTFT:

19. All-Pole Analysis of Speech 19

20. Solution to Yule-Walker Eq. 20

21. Solution to Yule-Walker Eq. 21

22. All-Zero Model 22 Moving-Average Model:

23. ARMA Model 23

24. Wold Decomposition 24 Wold (1938) proved a fundamental theorem: any stationary discrete time stochastic process may be decomposed into the sum of a general linear process and a predictable process, with these two processes being uncorrelated with each other.