Databases Reference
In-Depth Information
1, we add an amount
proportional to the magnitude of the derivative with a sign that is opposite to that of the
derivative of
d
n
at time
n
:
At any given time, in order to adapt the coefficient at time
n
+
d
n
−
α
∂
a
(
n
+
1
)
1
a
(
n
)
1
=
(41)
∂
a
1
where
α
is some positive proportionality constant.
d
n
∂
∂
a
1
=−
2
(
x
n
−
a
1
ˆ
x
n
−
1
)
ˆ
x
n
−
1
(42)
=−
2
d
n
ˆ
x
n
−
1
.
(43)
Substituting this into (
41
), we get
a
(
n
+
1
)
1
a
(
n
)
1
=
+
α
d
n
ˆ
x
n
−
1
(44)
α
where we have absorbed the 2 into
. The residual value
d
n
is available only to the encoder.
Therefore, in order for both the encoder and decoder to use the same algorithm, we replace
d
n
by
d
n
in (
44
) to obtain
a
(
n
+
1
)
1
a
(
n
)
1
+
α
d
n
ˆ
x
n
−
1
(45)
Extending this adaptation equation for a first-order predictor to an
N
th-order predictor is
relatively easy. The equation for the squared prediction error is given by
=
x
n
−
2
N
d
n
=
a
i
ˆ
x
n
−
i
(46)
i
=
1
Taking the derivative with respect to
a
j
will give us the adaptation equation for the
j
th predictor
coefficient:
a
(
n
+
1
)
j
a
(
n
)
j
+
α
d
n
ˆ
=
x
n
−
j
(47)
We can combine all
N
equations in vector form to get
A
(
n
+
1
)
=
A
(
n
)
+
α
d
n
X
n
−
1
(48)
where
⎡
⎣
⎤
⎦
x
n
ˆ
x
n
−
1
.
ˆ
X
n
=
(49)
x
n
−
N
+
1
ˆ
This particular adaptation algorithm is called the least mean squared (LMS) algorithm [
175
].
11.6 Delta Modulation
A very simple form of DPCM that has been widely used in a number of speech-coding appli-
cations is the delta modulator (DM). The DM can be viewed as a DPCM system with a 1-bit
