Digital Signal Processing Reference
In-Depth Information
With this solution at hand, it is suitable to discuss two interesting
properties mentioned in the beginning of this chapter.
3.8.1 The Linear Prediction-Error Filter as a Whitening Filter
As previously discussed, the prediction error
e
(
n
)
depends on the present
and past values of the input signal
x
. Assuming that we are using an
infinite number of past samples to estimate the current sample, we have
(
n
)
∞
a
k
x
e
f
(
n
)
=
x
(
n
)
−
(
n
−
k
)
(3.106)
k
=
1
where the prediction coefficients are obtained by minimizing the MSE cost
function defined in (3.6). Therefore, the following relationship holds
E
e
f
(
2
∂
n
)
=
0
(3.107)
∂
a
k
which can be expressed as
2E
e
f
(
)
∂
e
f
(
n
)
n
=
0
(3.108)
∂
a
k
Differentiating (3.106) and substituting in (3.108) yields
E
e
f
(
)
=
n
)
x
(
n
−
k
0,
k
≥
1
(3.109)
This last expression shows that the PEF produces an output sample that
is orthogonal to all past samples of
x
(
n
)
. On the other hand, according to
(3.106), any past output sample,
e
f
(
n
−
k
)
, also represents a linear combination
of
x
(
n
−
k
)
and all its past samples. Thus,
e
f
(
n
−
k
)
is also orthogonal to
e
f
(
n
)
, i.e.,
E
e
f
(
)
=
n
)
e
f
(
n
−
k
0,
k
≥
1
(3.110)
Equation 3.110 reveals that a PEF with a sufficiently large number of coeffi-
cients will produce uncorrelated output samples, hence acting as a
whitening
filter
[32].
