Digital Signal Processing Chapter 7: Parametric Spectrum Estimation

Wiener-Approach for solving the problem of prediction:e(k) → E(k) x(k) → X(k)E{|E(k)| 2 } = E{E(k) · E ∗ (k)} = ! Min.E{|E(k)| 2 } = E{(X(k) − ¯p H X(k − ))(X ∗ (k) − X H (k − )¯p)}= E{X(k) · X ∗ (k)} − ¯p H E{X(k − ) · X ∗ (k)}− E{X(k) · X H (k − )}¯p+¯p H E{X(k − ) · X H (k − )}¯p (1)It is E{X(k)·X ∗ (k)} = σX 2 and furthermore E{X(k− )·X H (k − )} = △ ¯R XX the autocorrelationmatrixwith conjugate complex elements.With the definition of the autocorrelation-vector we obtain:¯r XX = E{X ∗ (k)X(k − )} = [rXX(1), ∗ ...,rXX(n)] ∗ T⇒ E{|E(k)| 2 } = ¯p H ¯R XX¯p − ¯p H¯r XX −¯r H XX¯p + σX 2 .Linear Prediction Page 6