Determine coefficients of Nthorder forward linear predictors
DSP System Toolbox / Estimation / Linear Prediction
The Autocorrelation LPC block determines the coefficients of an Nstep forward linear predictor for the timeseries in each lengthM input channel, u, by minimizing the prediction error in the least squares sense. A linear predictor is an FIR filter that predicts the next value in a sequence from the present and past inputs. This technique has applications in filter design, speech coding, spectral analysis, and system identification.
The Autocorrelation LPC block can output the prediction error for each channel as polynomial coefficients, reflection coefficients, or both. The block can also output the prediction error power for each channel.
Data Types 

Direct Feedthrough 

Multidimensional Signals 

VariableSize Signals 

ZeroCrossing Detection 

The Autocorrelation LPC block computes the least squares solution to
$$\underset{i\in {\Re}^{n}}{\mathrm{min}}\Vert U\tilde{a}b\Vert $$
where $$\Vert \cdot \Vert $$ indicates the 2norm and
$$U=\left[\begin{array}{cccc}{u}_{1}& 0& \cdots & 0\\ {u}_{2}& {u}_{1}& \ddots & \vdots \\ \vdots & {u}_{2}& \ddots & 0\\ \vdots & \vdots & \ddots & {u}_{1}\\ \vdots & \vdots & \vdots & {u}_{2}\\ \vdots & \vdots & \vdots & \vdots \\ {u}_{M}& \vdots & \vdots & \vdots \\ 0& \ddots & \vdots & \vdots \\ \vdots & \ddots & \ddots & \vdots \\ 0& \cdots & 0& {u}_{M}\end{array}\right],\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{}\text{}\tilde{a}=\left[\begin{array}{c}{a}_{2}\\ \vdots \\ {a}_{n+1}\end{array}\right],\text{\hspace{0.17em}}\text{\hspace{0.17em}}b=\left[\begin{array}{c}{u}_{2}\\ {u}_{3}\\ \vdots \\ {u}_{M}\\ 0\\ \vdots \\ 0\end{array}\right]$$
Solving the least squares problem via the normal equations
$${U}^{\ast}U\tilde{a}={U}^{\ast}b$$
leads to the system of equations
$$\left[\begin{array}{cccc}{r}_{1}& {r}_{2}^{\ast}& \cdots & {r}_{n}^{\ast}\\ {r}_{2}& {r}_{1}& \ddots & \vdots \\ \vdots & \ddots & \ddots & {r}_{2}^{\ast}\\ {r}_{n}& \cdots & {r}_{2}& {r}_{1}\end{array}\right]\text{\hspace{0.17em}}\left[\begin{array}{c}{a}_{2}\\ {a}_{3}\\ \vdots \\ {a}_{n+1}\end{array}\right]\text{\hspace{0.17em}}=\left[\begin{array}{c}{r}_{2}\\ {r}_{3}\\ \vdots \\ {r}_{n+1}\end{array}\right]$$
where r = [r_{1}r_{2}r_{3} ... r_{n+1}]^{T} is an autocorrelation estimate for u computed using the Autocorrelation block, and * indicates the complex conjugate transpose. The normal equations are solved in O(n^{2}) operations by the LevinsonDurbin block.
Note that the solution to the LPC problem is very closely related to the YuleWalker AR method of spectral estimation. In that context, the normal equations above are referred to as the YuleWalker AR equations.
[1] Haykin, S. Adaptive Filter Theory. 3rd ed. Englewood Cliffs, NJ: Prentice Hall, 1996.
[2] Ljung, L. System Identification: Theory for the User. Englewood Cliffs, NJ: Prentice Hall, 1987. Pgs. 278280.
[3] Proakis, J. and D. Manolakis. Digital Signal Processing. 3rd ed. Englewood Cliffs, NJ: PrenticeHall, 1996.