Soner Bekleric Title of Thesis: Nonlinear Prediction via Volterra Ser

2.3. LINEAR PREDICTION 10
E(z) 1
X(z)
A(z)
Figure 2.1: AR IIR filter representation. After Makhoul (1975) and Ulrych and
Sacchi (2005).
2.3 Linear Prediction
The relationship between input/output for a linear time -invariant system is given
by the classical convolution integral (Rugh, 1981; Oppenheim and Schafer, 1989;
Schetzen, 2006),
y(n) =
∞
−∞
h(σ)x(n − σ)dσ . (2.8)
In the last expression h(σ) is the impulse response of the system or kernel function
that defines the input x(n) /output y(n) of the system.
Equation 2.8 is often used in the discrete form. The convolution integral is
replaced by a convolution sum and signals are replaced by a finite length discrete
time series:
yn =
N
aixn−i . (2.9)
i=1
The latter is the discrete convolution sum that arises very frequently in geophysics
and other sciences. It basically relates the output of a system that is excited with
an input signal xn via the convolution of the signal with the impulse response of
the system, in this case called an.
In linear prediction theory, one attempts to predict a signal by its past values
(or future values) by replacing the output signal yn by a future (one step ahead)