Digital Signal Processing Chapter 7: Parametric Spectrum Estimation

• recursive expression for the foreward-prediction errore r+1 (k) = e r (k) − γ r+1 · b r (k − 1)• We need a similar expression for the backward prediction error!apply the Levinson-Durbin equation to b r+1 (k):∑r+1b r+1 (k) = x(k − 1 − r) + a ∗ r+1,µ · x(k − 1 − r + µ)= x(k − 1 − r) +µ=1r∑a ∗ r,µ · x(k − 1 − r + µ)µ=1⇐ b r (k − 1)} {{ }r∑− γr+1 ∗ · [x(k) + a r,r + 1 − µ · x(k −1} {{ } }−{{r + µ})] ⇐ e r (k)µ=1 ν−ν} {{ }• recursive expression forthe backward prediction errorb r+1 (k) = b r (k − 1) − γ ∗ r+1 · e r (k)Lattice Structure Page 20