Digital Signal Processing Chapter 7: Parametric Spectrum Estimation

ecursive calculation of the prediction errors powerr∑r∑g r (0) = a}{{} r,0 ·r XX (0)+ a r,ν ·r XX (−ν) = σX+2 a r,ν rXX(ν) ∗ = σX 2 +r H XX · }{{} a=1ν=1ν=1−R −1XXr XXWiener-Hopf solution: min { E{|E(k)| 2 } } = σ 2 E = σ2 X −rH XX R−1 XX r XX;recursion: g r+1 (0) = σr+1 2 = g r (0) − γ r+1 gr(r ∗ + 1)} {{ } = σ2 r − γ r+1 gr(r ∗ + 1)using the definition γ r+1 = g r(r+1)g}{{} r(0)∗ → σr 2 γ r+1 = g r (r + 1)σr2σ 2 r+1 = σ 2 r − γ r+1 · γ ∗ r+1 σ 2 r → σ 2 r+1 = σ 2 r[1 − |γ r+1 | 2 ] thus:• |γ r+1 | < 1, because the power must be σ 2 r+1 > 0• σ 2 r+1 ≤ σ 2 r• limr → ∞r > nγ r = 0; σ 2 r+1 → σ 2 r → σ 2 Q if X(k) nth order AR-modelg r (0) = σ 2 E =: σ2 rLevinson-Durbin Recursion Page 14