Databases Reference
In-Depth Information
2
d
n
a
1
F I GU R E 11 . 11
A plot of the residual squared versus the predictor coefficient.
DPCM with Backward Adaptive Prediction (DPCM-APB)
Forward adaptive prediction requires that we buffer the input. This introduces delay in the
transmission of the speech. As the amount of buffering is small, the use of forward adaptive
prediction when there is only one encoder and decoder is not a big problem. However, in
the case of speech, the connection between two parties may be several links, each of which
may consist of a DPCM encoder and decoder. In such tandem links, the amount of delay can
become large enough to be a nuisance. Furthermore, the need to transmit side information
makes the system more complex. In order to avoid these problems, we can adapt the predictor
based on the output of the encoder, which is also available to the decoder. The adaptation is
done in a sequential manner [
172
,
174
].
In our derivation of the optimum predictor coefficients, we took the derivative of the
statistical average of the squared prediction error or residual sequence. In order to do this, we
had to assume that the input process was stationary. Let us now remove that assumption and
try to figure out how to adapt the predictor to the input algebraically. To keep matters simple,
we will start with a first-order predictor and then generalize the result to higher orders.
For a first-order predictor, the value of the residual squared at time
n
would be given by
d
n
2
=
(
x
n
−
a
1
ˆ
x
n
−
1
)
(40)
If we could plot the value of
d
n
against
a
1
, we would get a graph similar to the one shown in
Figure
11.11
. Let's take a look at the derivative of
d
n
as a function of whether the current value
of
a
1
is to the left or right of the optimal value of
a
1
—that is, the value of
a
1
for which
d
n
is
minimum. When
a
1
is to the left of the optimal value, the derivative is negative. Furthermore,
the derivative will have a larger magnitude when
a
1
is further away from the optimal value. If
we were asked to adapt
a
1
, we would add to the current value of
a
1
. The amount to add would
be large if
a
1
was far from the optimal value, and small if
a
1
was close to the optimal value.
If the current value was to the right of the optimal value, the derivative would be positive, and
we would subtract some amount from
a
1
to adapt it. The amount to subtract would be larger if
we were further from the optimal, and as before, the derivative would have a larger magnitude
if
a
1
were further from the optimal value.
