4.7.2 Autoregressive Models

1. 4.7.2 Autoregressive Models • A wide-sense stationary autoregressive process of order p is special case of an ARMA(p, q) process in which q = 0  all-pole filter • Input  unit variance white noise • all-pole filter • autocorrelation sequence of an AR process

2. matrix form for k = I , 2, . . . P, using the conjugate symmetry of rx(k) • given the autocorrelations rx(k)  find the AR coefficients ap(k) • to determine the coefficient b(0)  use the Yule-Walker equation for k = 0

3. But, In most applications, the statistical autocorrelation rx(k) is unknown and must be estimated from a sample realization of the process • given x(n) for 0<n<N  using the sample autocorrelation • two approaches 1. spectral factorization 2. Durbin's method

4. these equations are nonlinear in the model coefficients, bq(k)  spectral factorization of the power spectrum Px(z) • Specifically, since the autocorrelation of an MA(q) process is equal to zero for 4.7.3 Moving Average Models • MA(q) process • unit variance white noise v(n) with an FIR filter of order q • autocorrelation sequence

5. power spectrum • spectral factorization • the output of the minimum phase FIR filter

6. Example 4.7.2 Moving Average Model Using Spectral Factorization Consider the MA(1) process • autocorrelation sequence • The power spectrum : second-order poIynomial • spectral factorization  system function or

7. Durbin's method  by finding a high-order all-pole model Ap(z) for the moving average process • let x(n) be a moving average process of order q with , w(n) is white noise  • p is large enough 

8. For example and (expanded in a power series )  , • If p is sufficiently large 

9. ,b(0)=1 Example 4.7.3 The Durbin Algorithm • signal x(n) : z-transform • find a high-order all-pole model for x(n) • all-pole model, p >> 1 • first-order all-pole model for this sequence (Example 4.6.1) , and

10. assum , • covariance method (Example 4.6.2 ) and first-order moving average model

11. 4.7.4 Application: Power Spectrum Estimation • power spectrum of a wide-sense stationary random process x(n) autocorrelation • power spectrum of an AR(p) process

12. Example 4.7.4 AR Spectrum Estimation • N = 64 samples of an AR(4) process < Fig. 4.23 > • unit variance white noise with the fourth-order all-pole filter

13. Estimating the autocomelation sequence for and • pair of poles at

14. Yule-Walker equations  find,  using estimated autocorrelations 과 using the Yule-Walker method의 power spectrum 이 거의 동일하다.