Soner Bekleric Title of Thesis: Nonlinear Prediction via Volterra Ser

3.1. NONLINEAR PROCESSES VIA THE VOLTERRA SERIES 29
Hk(ω1, ω2, · · · , ωk) =
∞
−∞
∞
· · · hk(σ1, . . . , σk)e
−∞
−i(ω1σ1+ω2σ2+···+ωkσk)
dσ1, . . . , dσk.
(3.3)
The inverse Fourier transform of k th -order Volterra kernels is as follows (Rugh,
1981)
hk(σ1, σ2, . . . , σk) = 1
(2π) k
∞ ∞
· · · Hk(ω1, · · · , ωk)e
−∞ −∞
−i(ω1σ1+ω2σ2+···+ωkσk)
dω1, . . . , dωk.
(3.4)
X(f)
Nonlinear System
H (i)
1
H (j,k)
2
j + k = i
H (l,m,s)
3
l + m + s = i
Figure 3.2: Frequency domain Volterra model of a cubic nonlinear system. After
Nam and Powers (1994) and Schetzen (2006).
Y (f)
Y (f)
Y (f)
Diagram shown in Figure 3.2 represents a discrete frequency domain third- order
Volterra model which is expressed as
L
Q
C
+
Y(f )
i
+
−
^
Y(f )
i
+
ε (f )
i