Digital Signal Processing Chapter 7: Parametric Spectrum Estimation

7.5 Levinson-Durbin Recursion∑A r (z) = r a r,ν · z −ν → iterat. increment of order → A r+1 (z) = r+1 ∑a r+1,ν · z −νν=0design of the gapped function r + 1: g r (κ) = 0 for 1 ≤ κ ≤ rg r (r + 1 − κ) = 0g r (κ) − γ r+1 · g ∗ r(r + 1 − κ)→ g r (r + 1) − γ r+1 · g ∗ r(0)ν=0for 1 ≤ κ ≤ r!= 0 for 1 ≤ κ ≤ r + 1!= 0 → γ r+1 = g r(r + 1)gr(0)∗r + 1: → g r+1 (κ) = g r (κ) − γ r+1 · g ∗ r(r + 1 − κ)PARCOR-coefficients (PARtial CORrelation):γ r+1 =g r (κ) = E{X ∗ (k − κ) · E r (k)} = r ∑ν=0r∑ν=0a r,ν r XX (r+1−ν)r∑ν=0a ∗ r,ν r XX(ν)whereby:a r,ν E{X(k − ν)X ∗ (k − κ)}} {{ } ; }{{} = r XX(κ − ν)Levinson-Durbin Recursion Page 13