Digital Signal Processing Chapter 7: Parametric Spectrum Estimation

7.4.1 Linear Prediction: Derivation of the Wiener-Hopf-EquationFrom a sample fuction x(k) of a stationary, mean-free process X(k) an estimate ˆx(k)shall be computed based on past values of x(k) .^( x k)x( k) z –1 P( z)e( k)–+Pe ( z )A prediction-filter P(z) is excited by x(k − 1) ; e(k) is the prediction error.The prediction-error-filter is defined bySample function of the prediction errorP(z) = ∑ nν=1 p ν · z −ν+1 , p ν ǫ CP e (z) = 1 − z −1 · P(z) = 1 − ∑ nν=1 p ν · z −ν .e(k) = x(k) − ∑ nν=1 p νx(k − ν)Linear Prediction Page 4