Digital Signal Processing Reference
In-Depth Information
7.2.1 Introduction to Polynomial and Volterra Filters
Discrete-time causal polynomial filters
24
are described by the input-output
relationship:
P
y
(
n
)
=
f
i
[
x
(
n
)
,x
(
n
−
1
)
,
...
,x
(
n
−
N
+
1
)
,y
(
n
−
1
)
,
...
,y
(
n
−
M
+
1
)
]
,
i
=
0
(7.1)
where the function
f
i
[
]isapolynomial of order
i
in the variables within the
parentheses. The linear filter is a particular case of polynomial filters because
the relationship in Equation 7.1 becomes linear if
f
i
[
···
1.
With reference to Equation 7.1, polynomial filters can be classified into recur-
sive and nonrecursive filters. Recursive filters are characterized by (possibly
nonlinear) feedback terms in their input-output relationships and, as infinite
impulse response linear filters, possess an infinite memory. A simple example
of a recursive filter is the bilinear filter represented by the following equation:
···
]
=
0 for all
i
=
N
1
N
2
N
3
N
4
y
(
n
)
=
a
i
x
(
n
−
i
)
+
b
j
y
(
n
−
j
)
+
c
ij
x
(
n
−
i
)
y
(
n
−
j
).
(7.2)
i
=
0
j
=
1
i
=
0
j
=
1
In consideration of its infinite memory, a recursive polynomial filter admits,
within some stability constraints, a convergent
Volterra series expansion
of the
form
24
∞
∞
∞
y
(
n
)
=
h
0
+
h
1
(
m
1
)
x
(
n
−
m
1
)
+
h
2
(
m
1
,m
2
)
x
(
n
−
m
1
)
x
(
n
−
m
2
)
m
1
=
0
m
1
=
0
m
2
=
0
∞
∞
∞
+···+
0
···
h
p
(
m
1
,m
2
,
...
,m
p
)
x
(
n
−
m
1
)
m
1
=
m
2
=
m
p
=
0
0
×
x
(
n
−
m
2
)
···
x
(
n
−
m
p
)
+···
,
(7.3)
(
...
)
where
h
p
m
1
,m
2
,
,m
p
denotes the
p
th order
Volterra kernel
of the nonlinear
filter.
Nonrecursive polynomial filters, commonly called Volterra filters, are char-
acterized by input-output relationships that result from a double truncation
of the Volterra series, i.e., a memory truncation, by limiting the memory of
the filters to a finite number of terms in the summations of Equation 7.3, and
an order truncation by limiting the number of Volterra kernels
N
1
−
1
N
2
−
1
N
2
−
1
y
(
n
)
=
h
0
+
h
1
(
m
1
)
x
(
n
−
m
1
)
+
h
2
(
m
1
,m
2
)
x
(
n
−
m
1
)
x
(
n
−
m
2
)
m
1
=
0
m
1
=
0
m
2
=
0
N
p
−
1
N
p
−
1
N
p
−
1
···+
0
···
h
p
(
m
1
,m
2
,
...
,m
p
)
x
(
n
−
m
1
)
x
(
n
−
m
2
)
m
1
=
0
m
2
=
m
p
=
0
···
x
(
n
−
m
p
).
(7.4)
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