Digital Signal Processing Chapter 7: Parametric Spectrum Estimation

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For p we insert the solution of the Wiener-Hopf-Equation and since the crosscorrelationvectorvanishesThe principle of orthogonality is:r EX = r XX − R XX [R −1XXr XX ] = 0• The prediction-error-process E(k) and the past values of the processX(k − 1),...,X(k − n) are orthogonal to each other.The ’gapped function’ for a predictor of order n is defined asg n (κ) = E{E(k) · X ∗ (k − κ)}.Since r EX = r XX − R XX [R −1XXr XX ] = 0 the gapped function has the propertyg n (κ) = 0 , 1 ≤ κ ≤ n ;Linear Prediction Page 10
7.4 Linear Prediction (Overview)approach: E{|E(k)| 2 } = min; → σE 2 = σ2 X − rH XX R−1 XX r XXsolution: p = R −1XX · r XX using R XX = E{XX ∗ },and r H XX = [r∗ XX (1), · · ·,r∗ XX (n)]orthogon.: E{E ∗ (k)[X(k − 1), · · · ,X(k − n)] T } = 0gappedfunction:relationship between the linear prediction and the Yule-Walker-equation:a = −R −1XX · r XX→ σ 2 Q · |Pe(ej Ω )| 2|A(e j Ω )| 2= σ 2 QLinear Prediction Page 11
Page 1 and 2: transparencies - lecture: Digital S
Page 3 and 4: 7.2 Markov Process as an Example fo
Page 5 and 6: Past values x(k − 1), · · · ,x
Page 7 and 8: Conjugating all elements yields inI
Page 9: 7.4.2 The Principle of Orthogonalit
Page 13 and 14: 7.5 Levinson-Durbin Recursion∑A r
Page 15 and 16: ecursion for the predictive coeffic
Page 17 and 18: Levinson-Durbin Recursion• initia
Page 19 and 20: 7.6 The Lattice-Structure7.6.1 Anal
Page 21 and 22: Lattice Structure:• (r + 1)th sta
Page 23 and 24: 7.6.3 Minimal Phase - Stability•
Page 25 and 26: Br B q (0) = σ2 ∑q+1rγ rν=1a q
Page 30 and 31: 2. Covariance Method• disadvantag
Page 32 and 33: N−1∑k=rN−1∂ ∑N−1∑[e r
Page 34 and 35: • rth iteration:⎡⎢⎣1â r,1
Page 36 and 37: 7.8 Examples for Parametric Spectru
Page 38 and 39: • application for speech codingso
Page 40 and 41: example 3:vowel ”a”, german mal
Page 42: example 5:comparison parametric ←

Digital Signal Processing Chapter 7: Parametric Spectrum Estimation

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