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Question: Explain the covariance method
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Subject: Speech Processing

Topic: LPC and Parametric Speech Coding

Difficulty: Medium

sp(38) • 194 views
 modified 3 months ago by written 4 months ago by awari.swati831 • 160
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(i) Another method for solving the normal equation is the co-variance method. In this method, the original signal s(n) is used instead of using the windowed signal.

To minimize the error: $$E_m = \sum_{n=m}^{m+N-1} [s(n) - \sum_{p=1}^{k} a_p s(n-p)]^2$$

Solving the equation $\frac{\partial E_m}{\partial a_p} = 0$, we get, $$\phi_m(i,0) = \sum_{p=1}^{k} a_p \phi_m(i,p)$$

where, $\phi_m(i,p) = \sum_{n=m}^{m+N-1} s(n-i) s(n-p)$

The equation can also written in the form $\gamma = \bar{\bar{\phi}} \,\,\bar{a}$

$$\bar{\gamma} = \begin{bmatrix} \phi (1,0) \\ \phi (2,0) \\ . \\ . \\ . \\ \phi (k,0) \end{bmatrix} \\ \bar{\bar{\phi}} = \begin{bmatrix} \phi (1,1) & \phi (1,2) & . & . & . & \phi (1,k) \\ \phi (2,1) & \phi (2,2) & . & . & . & \phi (2,k) \\ . & . & . & . & . & . \\ . & . & . & . & . & . \\ . & . & . & . & . & . \\ \phi (k,1) & \phi (k,2) & . & . & . & \phi (k,k) \end{bmatrix}$$

$\bar{\bar{\phi}} \to$ symmetric, but is not Toeplitz matrix.