Np y(k) = SUM p(i) x(k-i) , i=1where x(k) is the input signal. The prediction error is
e(k) = x(k) - y(k) .
The mean-square prediction error is E[e(k)^2] or in vector-matrix notation
E = Ex - 2 p'r + p'R p ,
The mean-square value Ex, matrix R and vector p are defined as follows
Ex = E[x(k)^2] R(i,j) = E[x(k-i) x(k-j)], for 1 <= i,j <= Np, r(i) = E[x(k) x(k-i)], for 1 <= i <= Np.
For this routine, the matrix R must be symmetric and Toeplitz. Then
R(i,j) = rxx(|i-j|) r(i) = rxx(i)
Predictor coefficients are usually expressed algebraically as vectors with 1-offset indexing. The correspondence to the 0-offset C-arrays is as follows.
p(1) <==> pc predictor coefficient corresponding to lag 1 p(i) <==> pc[i-1] 1 <= i < Np