Soner Bekleric Title of Thesis: Nonlinear Prediction via Volterra Ser
Soner Bekleric Title of Thesis: Nonlinear Prediction via Volterra Ser
Soner Bekleric Title of Thesis: Nonlinear Prediction via Volterra Ser
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2.3. LINEAR PREDICTION 17<br />
x1 = a1x2 + a2x3 + a3x4 + ε1<br />
x2 = a1x3 + a2x4 + a3x5 + ε2<br />
x3 = a1x4 + a2x5 + a3x6 + ε3<br />
x4 = a1x5 + a2x6 + a3x7 + ε4<br />
x4 = a1x3 + a2x2 + a3x1 + ε5<br />
x5 = a1x4 + a2x3 + a3x2 + ε5<br />
x6 = a1x5 + a2x4 + a3x3 + ε6<br />
x7 = a1x6 + a2x5 + a3x4 + ε7.<br />
(2.27)<br />
Notice that only data that has been recorded is used to compute the prediction<br />
coefficients ap. In other words, no assumption about samples <strong>of</strong> non-recorded data<br />
is made. In addition, I are conveniently using forward and backward prediction to<br />
avoid any type <strong>of</strong> truncation or aperture artifact. Data that cannot be predicted<br />
with equation (2.12) is predicted <strong>via</strong> equation (2.13) and and vice versa (Marple,<br />
1987).<br />
The equations in (2.27) can be written in matrix form as follows:<br />
⎡<br />
⎤<br />
⎡<br />
⎢ xl ⎥ ⎢ xl+1<br />
⎢ ⎥ ⎢<br />
⎢ ⎥ ⎢<br />
⎢ .<br />
⎥ ⎢<br />
⎥ ⎢ .<br />
⎢ ⎥ ⎢<br />
⎢ ⎥ ⎢<br />
⎢ ⎥<br />
⎢ xm ⎥ = ⎢ xm<br />
⎢ ⎥ ⎢<br />
⎢ ⎥ ⎢<br />
⎢ . ⎥ ⎢<br />
⎥ ⎢ .<br />
⎢ ⎥ ⎢<br />
⎣ ⎦ ⎣<br />
xn xn−1<br />
<br />
⎛ ⎞<br />
xl+2<br />
. ..<br />
xm<br />
.<br />
xn−2<br />
⎛<br />
. . .<br />
.<br />
. ..<br />
.<br />
.<br />
<br />
. . .<br />
.<br />
. . .<br />
. ..<br />
.<br />
⎞<br />
xl+p<br />
.<br />
xm<br />
.<br />
xn−p<br />
⎥ ⎢ a1 ⎥ ⎢ εl ⎥<br />
⎥ ⎢ ⎥ ⎢ ⎥<br />
⎥ ⎢ ⎥ ⎢ ⎥<br />
⎥ ⎢ ⎥ ⎢ ⎥<br />
⎥ ⎢ a2 ⎥ ⎢ εl+1 ⎥<br />
⎥ ⎢ ⎥ ⎢ ⎥<br />
⎥ ⎢ ⎥ ⎢ ⎥<br />
⎥ × ⎢ .<br />
⎥ + ⎢ .<br />
⎥<br />
⎥ ⎢ ⎥ ⎢ ⎥<br />
⎥ ⎢ ⎥ ⎢ ⎥<br />
⎥ ⎢<br />
⎥ ⎢ . ⎥ ⎢<br />
⎥ ⎢ . ⎥<br />
⎥ ⎢ ⎥ ⎢ ⎥<br />
⎦ ⎣ ⎦ ⎣ ⎦<br />
ap<br />
εn<br />
<br />
⎛ ⎞ ⎛ ⎞<br />
⎜<br />
⎝<br />
d ⎟<br />
⎠<br />
N × 1<br />
⎜<br />
⎝<br />
A<br />
N × P<br />
⎟<br />
⎠<br />
⎜ m ⎟<br />
⎜ ⎟<br />
⎝ ⎠<br />
P × 1<br />
⎜<br />
⎝<br />
ε ⎟<br />
⎠<br />
N × 1<br />
⎤<br />
⎡<br />
⎤<br />
⎡<br />
⎤<br />
(2.28)