Linear Prediction Coefficients

Question

I want to learn how to compute linear coefficient, but to be honest - I can't. I know how to use them if i have it and why I want them.

Suppose this toy example:

A signal $x=[-1, -2, 0, 1]$

According to matlab function lpc(X,p) if I want to compute the 2th order coefficients of this signal the result is $a = [1.0, -0.50, 0.50]$ and $ g=1.0$

I Know how to compute the g using $A(k)$ formula and that $a_0=1$ always, but can someone explain me why $a_1=-0.5, a_2=0.5$ how to compute them without using matlab or any other software.

P.S It there any method which allow me using $A(k)$ to compute directly the coefficients?

thanks

Answer 1

First you need to compute the signal's autocorrelation:

$$R(k)=\sum_nx(n)x(n+k),\quad k=0,1,\ldots,p$$

In your case with $p=2$ you get

$$\begin{align}R(0)&=(-1)(-1)+(-2)(-2)+0\cdot 0+1\cdot 1=6\\ R(1)&=(-1)(-2)+(-2)\cdot 0+0\cdot 1=2\\ R(2)&=(-1)\cdot 0+(-2)\cdot 1=-2\end{align}$$

Then you have to solve a system of linear equations:

$$\begin{bmatrix}R(0)&R(1)\\ R(1)&R(0)\\ \end{bmatrix} \begin{bmatrix}a_1\\a_2\end{bmatrix}= -\begin{bmatrix}R(1)\\R(2)\end{bmatrix}$$

which gives the values $a_1=-0.5$ and $a_2=0.5$. Of course $a_0=1$, as you've already noted.

The matrix that occurs in the system of linear equations is a Toeplitz matrix, i.e. all its diagonals have the same elements. As a result, such systems can be solved by specialized and efficient algorithms, such as Levinson's algorithm.