The autocorrelation method is an efficient way of finding the [[linear prediction coefficients]] by assuming that the input signal is a [[finite-duration discrete-time signal]] resulting from a [[windowing]] process: $ s_{m}(n) = s(m+n) w(n) $ where $s_{m}(n)$ is a signal [[frame]] and $w(n)$ a [[window function]] such as a [[Hann window]]. In this case, the [[residue]] signal for each frame is also a finite-duration signal so that the sum of the square error becomes: $ \mathcal{E}_{m} = \sum_{n=0}^{N-P-1} \left( s_{m}(n) - \sum_{k=1}^{P} a_k s_{m}(n-k) \right)^2 $ The correlation coefficients become $ \begin{align} \phi(i,k) & = \sum_{n=0}^{N-1} s_{m}(n-i) s_{m}(n-k) \\ &= \sum_{n=-i}^{N-1-i} s_{m}(n) s_{m}(n-(k-i)) \\ &= \phi(k,i) \end{align} $ $ $ The correlation coefficients only depend on the difference $l=|k-i|$ and can thus be expressed in terms of the autocorrelation function: $ R_{m}(l) = \sum_{n=-\infty}^{+\infty} s_{m}(n) s_{m}(n-l) $ where: $ \phi(i,k) = \phi(k,i) = R_{m}(|k-i|) $ This makes the $\mathbf{R}$ a Toeplitz matrix, where all the elements along the diagonals of the matrix have the same value: ![[Pasted image 20230510194428.png]] When $\mathbf{R}$ is a Toeplitz matrix, the Levinson-Durbin recursion can be used to find the [[linear prediction coefficients]] $a_{k}$: $ \mathbf{a} = \mathbf{R}^{-1} \mathbf{\gamma} $