Digital Signal Processing Reference
In-Depth Information
In the following, we will use
z
-transformation for the mathematical derivation. The
(two-sided)
z
-transformation is given by:
+∞
z
−
k
S
(
z
)
=
s
(
k
)
.
(6.35)
k
=−∞
With the
z
-transformations
E
(
z
)
and
S
(
z
)
of the signals
e
(
k
)
and
s
(
k
)
, respectively,
z
−
i
and obeying the rule of the
z
-transformation that
s
(
k
−
i
)
corresponds to
S
(
z
)
in
the
z
-domain, holds:
p
a
i
z
−
i
E
(
z
)
=
S
(
z
)(
1
+
),
(6.36)
i
=
1
and for the transfer function
H
(
z
)
:
p
E
(
z
)
a
i
z
−
i
H
(
z
)
=
=
1
+
.
(6.37)
S
(
z
)
i
=
1
In the inverse case the system is excited by the error signal and produces the speech
signal—the filter then is a mere recursive filter and the transfer function the reciprocal.
This is a simple model for speech production, where the vocal tract is seen as linear
filter which is excited by regular pulses by the vocal chords. The excitation pulses
are not linearly predictable at a low number of predictor coefficients within a short
analysis interval and thus produce the prediction error. In the case of unvoiced sounds,
excitation is given by white noise. The transfer function in this case has only poles
and no zeros, i.e., the system is an all-pole model [
6
]. These poles can be determined
directly from the predictor coefficients
a
i
. One now has to determine these for a given
order
p
such that the deviation between the estimated signal and the real signal is
minimal.
The squared error
α
within the interval of analysis (for the moment running from
k
=−∞
to
+∞
; later within the open window region) is:
2
α
=
e
(
k
)
(6.38)
k
p
2
α
=
a
i
s
(
k
−
i
)
.
(6.39)
k
i
=
0
Note that, for simplification a coefficient
a
0
was introduced that equals one. In order
to determine the minimum of this error, one differentiates the error partially per
predictor coefficient and sets the derived error equal to zero:
