Soner Bekleric Title of Thesis: Nonlinear Prediction via Volterra Ser

3.1. NONLINEAR PROCESSES VIA THE VOLTERRA SERIES 28
ries is equivalent to the convolution representation of a system. The last expression
can be represented in a more general form as follows:
y(t) =
∞
k=1
1 ∞ ∞
dσ1 · · ·
k! −∞
−∞
k
dσkhk(σ1, σ2 . . . , σk) x(t − σp) (3.2)
p=1
where, again, x(t) is the input, y(t) is the output of the system. This functional
form was first studied by the Italian mathematician Vito Volterra, so is known as
the Volterra series, and the functions hk(σ1, . . . , σk) are known as Volterra kernels
of the system. Norbert Wiener (1942) first applied these series to the study of
nonlinear systems. As seen in equation (3.2) the Volterra series can be regarded as
a nonlinear extension of the classical linear convolution.
H
x(t) y(t)
H
H
1
2
3
Figure 3.1: Schematic representation of a system characterized by a third-order
Volterra series. Modified from Schetzen (1980). H1, H2 and H3 represent the
impulse responses of the first, second and third-order Volterra kernels, respectively.
Figure 3.1 illustrates a schematic representation of a system is characterized by
a third-order Volterra series.
3.1.2 Frequency domain representation of Volterra kernels
The representation of a Volterra kernel in the Fourier domain is given by
+