Digital Signal Processing Chapter 7: Parametric Spectrum Estimation
Digital Signal Processing Chapter 7: Parametric Spectrum Estimation Digital Signal Processing Chapter 7: Parametric Spectrum Estimation
The power of the remaining prediction-error isMin{E{|E(k)| 2 }} = σ 2 X − rH XXR −1XXr XX .Rewriting the equation using the coefficient-vector p results inMin{E{|E(k)| 2 }} = σ 2 X − rH XXp.Linear Prediction Page 8
7.4.2 The Principle of OrthogonalityExamining the conjugate crosscorrelation-vector⎧⎡⎪⎨¯r EX = E{E ∗ (k) · X(k − )} = E⎢⎣⎪⎩and recalling thatE ∗ (k) · X(k − 1)E ∗ (k) · X(k − 2).E ∗ (k) · X(k − n)⎤⎫⎪⎬⎥⎦⎪⎭e(k) = x(k) − ¯p H x(k − ) ; e ∗ (k) = x ∗ (k) − x H (k − )¯p⇒ ¯r EX = E{[ X ∗ (k) − X H (k − ) ¯p ] · X(k − )}⇒ ¯r EX = E{X(k − ) · X ∗ (k)} − E{X(k − ) · X H (k − )} · ¯pExpressing the expectation values by the help of the autocorrelation-vector and -matrixwith further conjugation givesr EX = r XX − R XX p .Linear Prediction Page 9
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7.4.2 The Principle of OrthogonalityExamining the conjugate crosscorrelation-vector⎧⎡⎪⎨¯r EX = E{E ∗ (k) · X(k − )} = E⎢⎣⎪⎩and recalling thatE ∗ (k) · X(k − 1)E ∗ (k) · X(k − 2).E ∗ (k) · X(k − n)⎤⎫⎪⎬⎥⎦⎪⎭e(k) = x(k) − ¯p H x(k − ) ; e ∗ (k) = x ∗ (k) − x H (k − )¯p⇒ ¯r EX = E{[ X ∗ (k) − X H (k − ) ¯p ] · X(k − )}⇒ ¯r EX = E{X(k − ) · X ∗ (k)} − E{X(k − ) · X H (k − )} · ¯pExpressing the expectation values by the help of the autocorrelation-vector and -matrixwith further conjugation givesr EX = r XX − R XX p .Linear Prediction Page 9
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