Digital Signal Processing Chapter 7: Parametric Spectrum Estimation

comparison of the coefficients left and right of ∗) (left side: separation of 2 terms):∑r+1ν=0r∑a r+1,ν r XX (κ − ν) = r XX (κ) + a r+1,r+1 r XX (κ − (r + 1)) + a r+1,ν r XX (κ − ν)= r XX (κ)−γ r+1 r XX (κ − (r + 1)) +ν=1r∑[a r,ν − γ r+1 · a ∗ r,r+1−ν] · r XX (κ − ν)ν=1a r+1,ν = a r,ν − γ r+1 · a ∗ r,r+1−ν(for ν = 1 ,...,r)a r+1,r+1 = −γ r+1vectorized expression:⎡ ⎤ ⎡1 1a r+1,1a r+1,2.=.⎢⎣ a r+1,r⎥ ⎢⎦ ⎣a r+1,r+1 0a r,1a r,2a r,r⎤⎥⎦⎡− γ r+1 ⎢⎣0a ∗ r,ra ∗ r,r−1.a ∗ r,11⎤⎥⎦Levinson-Durbin Recursion Page 16