Digital Signal Processing Chapter 7: Parametric Spectrum Estimation
Digital Signal Processing Chapter 7: Parametric Spectrum Estimation Digital Signal Processing Chapter 7: Parametric Spectrum Estimation
7.4.1 Linear Prediction: Derivation of the Wiener-Hopf-EquationFrom a sample fuction x(k) of a stationary, mean-free process X(k) an estimate ˆx(k)shall be computed based on past values of x(k) .^( x k)x( k) z –1 P( z)e( k)–+Pe ( z )A prediction-filter P(z) is excited by x(k − 1) ; e(k) is the prediction error.The prediction-error-filter is defined bySample function of the prediction errorP(z) = ∑ nν=1 p ν · z −ν+1 , p ν ǫ CP e (z) = 1 − z −1 · P(z) = 1 − ∑ nν=1 p ν · z −ν .e(k) = x(k) − ∑ nν=1 p νx(k − ν)Linear Prediction Page 4
Past values x(k − 1), · · · ,x(k − n) are summarized in a vector⎡ ⎤x(k − 1)x(k − x(k − 2)) =⎢⎣. ⎥⎦x(k − n)For the coefficients of the predictor definitions of vectors are⎡ ⎤ ⎡ ⎤p 1 p ∗ 1p 2p ∗ 2p =; ¯p =⇒ ¯p H = [p⎢⎣. ⎥ ⎢⎦ ⎣. ⎥1 , p 2 , · · · , p n ]⎦p np ∗ nWith the above definitions we can rewrite the convolution as a scalarproducte(k) = x(k) − ¯p H x(k − ) ;e ∗ (k) = x ∗ (k) − x H (k − )¯pLinear Prediction Page 5
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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
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Past values x(k − 1), · · · ,x(k − n) are summarized in a vector⎡ ⎤x(k − 1)x(k − x(k − 2)) =⎢⎣. ⎥⎦x(k − n)For the coefficients of the predictor definitions of vectors are⎡ ⎤ ⎡ ⎤p 1 p ∗ 1p 2p ∗ 2p =; ¯p =⇒ ¯p H = [p⎢⎣. ⎥ ⎢⎦ ⎣. ⎥1 , p 2 , · · · , p n ]⎦p np ∗ nWith the above definitions we can rewrite the convolution as a scalarproducte(k) = x(k) − ¯p H x(k − ) ;e ∗ (k) = x ∗ (k) − x H (k − )¯pLinear Prediction Page 5
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