(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.

**Advantages:**

If s(n) is in steady state, the co-variance method can find the true spectrum.

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