Digital Signal Processing Chapter 7: Parametric Spectrum Estimation

7.6 The Lattice-Structure7.6.1 Analysis-Filter using the Lattice-FormLevinson-Durbin demonstrates: Predictor fully determined through PARCOR-coefficients→ structure of an implemented predictor using only PARCOR-coefficients∑r+1e r+1 (k) = x(k) + a r+1,ν x(k − ν)} {{ }ν=1→Lev.-Durb.r∑= x(k) + [a r,ν − γ r+1 · a ∗ r,r+1−ν] · x(k − ν) − γ r+1 · x(k − (r + 1))ν=1splitting in two terms (and substitution: µ = r + 1 − ν → k − ν = k − (r + 1) + µ)r∑r∑e r+1 (k) = [x(k) + a r,ν · x(k − ν)] −γ r+1·[x(k − (r + 1)) + a ∗ r,µ · x(k − (r + 1) + µ)]ν=1} {{ }e r (k)definition backward prediction error:µ=1} {{ }?b r (k) = x(k − r) + ∑ rµ=1 a∗ r,µ · x(k − r + µ) → ? = b r (k − 1)Lattice Structure Page 19