Digital Signal Processing Reference
In-Depth Information
and by equating the weighted Euclidean norms of both sides of this equation
we arrive again at Equation 1.10, which is repeated here for ease of reference,
+
e
a
(
)
2
+
e
p
(
)
2
2
2
2
2
·
=
·
.
u
i
w
i
i
u
i
w
i
−
1
i
(1.57)
1.12.1
Variance Relation for Error Nonlinearities
Now recall that in transient analysis we are interested in characterizing the
time evolution of the quantity E
2
w
i
, for some
of interest (usually,
=
I
R
u
). To characterize this evolution, we replace
e
p
(
or
=
i
)
in Equation 1.57
by its expression (Equation 1.55) in terms of
e
a
(
)
(
)
i
and
e
i
to obtain
2
2
2
·
2
=
2
·
2
+
µ
2
f
2
[
e
u
i
w
i
u
i
w
i
−
1
u
i
(
i
)
]
2
e
a
(
−
2
µ
u
i
i
)
f
[
e
(
i
)
]
.
2
Assuming the event
u
i
=
0 occurs with zero probability, we can eliminate
2
u
i
from both sides and take expectations to arrive at
E
e
a
(
]
+
µ
2
E
]
,
2
2
2
f
2
[
e
E
w
i
=
E
w
i
−
1
−
2
µ
i
)
f
[
e
(
i
)
u
i
(
i
)
(1.58)
which is the equivalent of Equation 1.13 for filters with error nonlinearities.
Observe, however, that the weighting matrices for E
w
i
2
w
i
−
1
2
and E
are
still identical because we did not substitute
by their expressions
in terms of
w
i
−
1
. The reason we did not do so here is because of the non-
linear error function
f
. Instead, to proceed, we show how to evaluate the
expectations
{
e
a
(
i
)
,
e
(
i
)
}
E
e
a
(
]
E
]
.
2
f
2
[
e
)
(
)
(
)
i
f
[
e
i
and
u
i
i
(1.59)
These expectations are generally hard to compute because of
f
. To facilitate
their evaluation, we assume that the filter is long enough to justify, by central
limit theorem arguments, that
and
e
a
(
e
a
(
i
)
i
)
are jointly Gaussian random variables.
(1.60)
1.12.1.1 Evaluation of
E
(
e
a
f
[
e
]
)
Using Statement 1.60 we can evaluate the first expectation, E
e
a
(
]
,
by appealing to Price's theorem.
15
The theorem states that if
x
and
y
are
jointly Gaussian random variables that are independent from a third random
variable
z
, then
i
)
f
[
e
(
i
)
E
xy
E
y
2
E
y
k
E
x
k
(
y
+
z
)
=
(
y
+
z
)
,
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