Digital Signal Processing Chapter 7: Parametric Spectrum Estimation
Digital Signal Processing Chapter 7: Parametric Spectrum Estimation 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
Levinson-Durbin Recursion• initialization (r = 0): a 0,0 = 1σ0 2 = r XX (0)• 1st iteration (r = 1) :γ 1 = r XX (1)/r XX (0)[ ] [ ] [ ]1 1 0= − γ 1a 11 0 1→ a 11 = −γ 1σ 2 1 = [1 − |γ 1 | 2 ] · σ 2 0 .• rth iteration (r = 2, ...,n) :γ r =r−1 ∑ν=0a r−1,ν r XX (r − ν)σ 2 r−1Levinson-Durbin Recursion Page 17
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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
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