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4.7 STOCHASTIC MODELS

4.7 STOCHASTIC MODELS. In the preceding sections, we looked at approaches for modeling deterministic signal. In this section we consider the problem of modeling a wide-sense stationary random process. random process

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4.7 STOCHASTIC MODELS

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  1. 4.7 STOCHASTIC MODELS • In the preceding sections, we looked at approaches for modeling deterministic signal. • In this section we consider the problem of modeling a wide-sense stationary random process. • random process • may only be characterized statistically and the values of x(n) are only known in a probabilistic sense the errors that are minimized for deterministic signals are no longer appropriate • input signal • deterministic - unit sample • random process - random process(unit variance white noise)

  2. the Yule-Walker equations will play an important role in the development of stochastic modeling algorithms • 4.7.1 Autoregressive Moving Average Models • 1. Modified Yule-Walker Equation(MYWE) • 2. least squares MYWE method • 4.7.2 Autoregressive (all-pole) Models • 4.7.3 Moving Average Models • 1. spectral factorization • 2. Durbin's method

  3. Modeling a random process v(n) – unit variance white noise To find the filter coefficients that produce the best approximation to x(n)

  4. Yule-Walker equations sequence statistical autocorrelation h(n) is assumed to be causal, then cq(k)=0 for k>q (matrix form)

  5. Modified Yule-Walker Equation (MYWE) method to find the coefficients ap(k) in the filter H(z) from the autocorrelation sequence rx(k) for k = q, q+1, . . . , q+p Trench algorithm to find the values of the sequence cq(k) for k = 0, 1, . . . , q

  6. z-transform of cq(k) causal anticausal Recall power spectrumy(n) causal part anticausal part Find Bq

  7. Example 4.7.1 The MYWE Method for Modeling an ARMA(1,I) Process find an ARMA(1,1) model autocorrelation values rx(0)=26 rx(1)=7 rx(2)=7/2 Yule-Walker equations modified Yule-Walker equations 

  8. to find Cq causal part

  9. Using the symmetry spectral factorization

  10. If rx(k) is unknown estimated from the data for k = 0, 1, . . . , L L - q equations least squares solution

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