Digital Signal Processing Chapter 7: Parametric Spectrum Estimation

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Prediction of Bandlimited White NoiseS VV (jω)A bandlimited white noise process V (k) sampled atf A = 1/T has the ACFN 02−ω gω gωr V V (κT) = σ 2 V · sin(ω gTκ)ω g Tκ , with σ2 V = f g · N 0 .For a first order predictor we getp 1 = r V V (T)r V V (0) = sin(ω gT)ω g Tand σ 2 E = σ2 V − r V V (T) · rV V (T)r V V (0) .60a) PrädiktionsgewinnG/dB →504030n=1620n=210n=100 0.2 0.4 0.6 0.8 1Ω g /π →The prediction gain G dB then is( ) σ2G dB = 10log V10G dB = 10log 10[(1 −σ 2 E(sin(ωg T)ω g T) 2)] −1Linear Prediction Page 12
7.5 Levinson-Durbin Recursion∑A r (z) = r a r,ν · z −ν → iterat. increment of order → A r+1 (z) = r+1 ∑a r+1,ν · z −νν=0design of the gapped function r + 1: g r (κ) = 0 for 1 ≤ κ ≤ rg r (r + 1 − κ) = 0g r (κ) − γ r+1 · g ∗ r(r + 1 − κ)→ g r (r + 1) − γ r+1 · g ∗ r(0)ν=0for 1 ≤ κ ≤ r!= 0 for 1 ≤ κ ≤ r + 1!= 0 → γ r+1 = g r(r + 1)gr(0)∗r + 1: → g r+1 (κ) = g r (κ) − γ r+1 · g ∗ r(r + 1 − κ)PARCOR-coefficients (PARtial CORrelation):γ r+1 =g r (κ) = E{X ∗ (k − κ) · E r (k)} = r ∑ν=0r∑ν=0a r,ν r XX (r+1−ν)r∑ν=0a ∗ r,ν r XX(ν)whereby:a r,ν E{X(k − ν)X ∗ (k − κ)}} {{ } ; }{{} = r XX(κ − ν)Levinson-Durbin Recursion Page 13
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Page 5 and 6: Past values x(k − 1), · · · ,x
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Page 9 and 10: 7.4.2 The Principle of Orthogonalit
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Page 19 and 20: 7.6 The Lattice-Structure7.6.1 Anal
Page 21 and 22: Lattice Structure:• (r + 1)th sta
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Page 34 and 35: • rth iteration:⎡⎢⎣1â r,1
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Digital Signal Processing Chapter 7: Parametric Spectrum Estimation

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