Digital Signal Processing Chapter 7: Parametric Spectrum Estimation

ecursion for the predictive coefficients (Levinson-Durbin recursion)into the recursion for the “Gapped Function”: g r+1 (κ) = g r (κ) − γ r+1 · gr(r ∗ + 1 − κ)∑insert the definition g r (κ) = r a r,ν r XX (κ − ν) :∑r+1ν=0a r+1,ν r XX (κ − ν) =ν=0r∑a r,ν r XX (κ − ν) − γ r+1 ·ν=0r∑a ∗ r,ν rXX(r ∗ + 1 − κ − ν) ∗)convert the right side: (second term substitute r + 1 − ν = µ)r∑γ r+1 a ∗ r+1 ∑r,ν r XX (κ − (r + 1 − ν)) = γ r+1 a ∗ r,r+1−µ r XX (κ − µ)ν=0µ=1r∑entirely ∗) : a r,0 r XX (κ) + a r,ν r XX (κ − ν)r∑ ν=1− γ r+1 a ∗ r,r+1−µ r XX (κ − µ) − γ r+1 a ∗ r,0 r XX (κ − (r + 1))µ=1= r XX (κ) −γ} {{ } r+1 r XX (κ − (r + 1)) +a r,0 = 1ν=0r∑[(a r,ν − γ r+1 · a ∗ r,r+1−ν) · r XX (κ − ν)]ν=1Levinson-Durbin Recursion Page 15