Digital Signal Processing Reference
In-Depth Information
It is now possible to determine the estimates by minimizing the mean squared
error, i.e.
2
p
e
2
(n)
}=
s(n)
−
E
{
E
α
j
s(n
−
j)
(4.15)
j
=
1
Setting the partial derivatives of the above with respect to
α
j
to zero for
j
=
1
, ... ,p
we get,
=
p
s(n)
s(n
E
−
α
j
s(n
−
j)
−
i)
0
,
for
i
=
1
, ... ,p
(4.16)
j
=
1
That is,
e(n)
is orthogonal to
s(n
−
i)
for
i
=
1
, ... ,p
. Equation (4.16) can be
rearranged to give,
p
α
j
φ
n
(i, j)
=
φ
n
(i,
0
),
for
i
=
1
, ... ,p
(4.17)
j
=
1
where
φ
n
(i, j)
=
E
{
s(n
−
i)s(n
−
j)
}
(4.18)
In the derivation of equation (4.17), a major assumption is that the signal
of our model is stationary. For speech, this is obviously untrue over a long
duration. However, for short segments of speech the assumption that it is
stationary is reasonable. Consequently, our expectations in equation (4.18)
are replaced by finite summations over a short length of speech samples.
In this section the equation for LPC analysis was derived from the Least
Mean Square approach. An equally valid result can be obtained using the
Maximum Likelihood method and other formulations [6]. An interesting
aspect of LPC analysis is that it applies not only to speech processing, but also
to a wide range of other fields such as control and radar. However, it is in
speech processing that LPC analysis has been perhaps the most successful, as
it allows very accurate representation of speechwith a small set of parameters.
4.3.2 SolutionstoLPCAnalysis
As mentioned above, in order to model the time-varying nature of the speech
signal whilst stayingwithin the constraint of our LPC analysis, i.e. a stationary
signal, it is necessary to limit our analysis to short blocks of speech. This is
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