Digital Signal Processing Chapter 7: Parametric Spectrum Estimation

Conjugating all elements yields inIntroducing the quadratic extensionE{|E(k)| 2 } = p H R XX p − p H r XX − r H XXp + σ 2 X .E{|E(k)| 2 } = (p H R XX − r H XX)R −1XX(R XX p − r XX ) − r H XXR −1XXr XX + σ 2 XThe global minimum can be found by setting the term containing p equal to zero. IfR XXis non-singular (in general fulfilled for autocorrelation-matrices) we can set thefollowing linear equation as the condition for a minimum of the prediciton-error :R XX p − r XX = 0.This results in the Wiener-Hopf-Equation providing the coefficient vector.p = R −1XXr XX .• The solution for the nonrecursive prediction-filter is given by the inversionof the autocorrelation-matrix.Linear Prediction Page 7