Digital Signal Processing Chapter 7: Parametric Spectrum Estimation
Digital Signal Processing Chapter 7: Parametric Spectrum Estimation Digital Signal Processing Chapter 7: Parametric Spectrum Estimation
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
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- Page 5 and 6: Past values x(k − 1), · · · ,x
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- Page 25 and 26: Br B q (0) = σ2 ∑q+1rγ rν=1a q
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- Page 32 and 33: N−1∑k=rN−1∂ ∑N−1∑[e r
- Page 34 and 35: • rth iteration:⎡⎢⎣1â r,1
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- Page 42: example 5:comparison parametric ←
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