Digital Signal Processing Chapter 7: Parametric Spectrum Estimation

7.6.4 Orthogonality of the Backward Prediction Errorb r (k) = b r−1 (k − 1) − γ ∗ r · e r−1 (k) ← e r (k) = e r−1 (k) − γ r · b r−1 (k − 1)γ r · b r (k) = e r−1 (k) − e r (k) − γ r γr ∗ · e r−1 (k) = [1 − |γ r | 2 ] ·e} {{ } r−1 (k) − e r (k)σr/σ 2 r−12⇒γ r· bσr2 r (k) = 1 · e r−1 (k) − 1 · eσr2 r (k) *conventional description: b q (k) =σr−12q+1∑ν=1a ∗ q,q+1−ν · x(k + 1 − ν) with a q,0 = 1; *• target: proof that backward prediction errors b q (k) and b r (k) are orthogonal.• not restricting generality q ≤ r • inserted into CCF * and * :r Br B q(0) = E{B r (k) · B ∗ q(k)}= σ2 ∑q+1rγ rν=1a q,q+1−ν[ 1σ 2 r−1E{X ∗ (k + 1 − ν) · E r−1 (k)} − 1 } {{ } σr2g r−1 (ν − 1)]E{X ∗ (k + 1 − ν) · E r (k)}} {{ }g r (ν − 1)Lattice Structure Page 24