Soner Bekleric Title of Thesis: Nonlinear Prediction via Volterra Ser

2.3. LINEAR PREDICTION 14
which can be written in matrix form as:
⎡
⎤ ⎡ ⎤ ⎡
⎢ r0
⎢ r1
⎢ .
⎢
⎣
rp
r−1
r0
. ..
rp−1
· · ·
· · ·
. ..
· · ·
r−p ⎥ ⎢ 1 ⎥ ⎢ σ
⎥ ⎢ ⎥ ⎢
⎥ ⎢ ⎥ ⎢
⎥ ⎢ ⎥ ⎢
r−p+1 ⎥ ⎢ a1 ⎥ ⎢
⎥ ⎢ ⎥
⎥ ⎢ ⎥ = ⎢
.
⎥ ⎢
⎥ ⎢ . ⎥ ⎢
⎥ ⎢
⎥ ⎢ ⎥ ⎢
⎦ ⎣ ⎦ ⎣
r0 ap
R
a
2 ⎤
ε ⎥
0
⎥
. ⎥
⎦
0
e
(2.20)
where the matrix is the data autocorrelation matrix and σ 2 ε is the innovation vari-
ance .
The system above is written as
Ra = σ 2 ε e1, (2.21)
where e T 1 = [1, 0, · · · , 0] is the zero-delay spike vector. The AR Yule-Walker equa-
tions are formed with an autocorrelation matrix (Marple, 1987). Since R−i = R ∗ i
the autocorrelation matrix, which is almost always invertible in equation (2.21) is
both Toeplitz and Hermitian. The Levinson recursion solves the resulting system
in (p + 1) 2 operations.
2.3.3 Estimating AR coefficients via Burg’s algorithm
Burg (1975) developed an AR algorithm to estimate AR prediction coefficients when
the autocorrelation sequence of the system is unknown. The primary advantage of
Burg’s method is to estimate prediction coefficients directly form the data, in con-
trast to the least squares solution and the Yule-Walker method. In addition, the
method provides a stable AR model in a time series with low noise levels, and is