Databases Reference
In-Depth Information
x
x
e,k
e,k
x
k
x
k
Merge
Split
x
o,k
x
o,k
F I GU R E 15 . 16
Decomposing a sequence into its odd- and even-indexed
components.
x
x
e,k
e,k
x
k
x
k
P
Merge
Split
P
x
o,k
x
o,k
F I GU R E 15 . 17
Generating the high-frequency difference sequence.
one sequence using the other. In this case, let's use the even sequence to predict the odd
sequence as shown in Figure
15.17
:
d
k
=
x
o
,
k
−
P
(
{
x
e
,
k
}
)
As long as the values of the sequences are preserved, the original sequence can be reconstructed
by reversing the operation at the encoder. If we have a good model for the signal, the prediction
residual
.Let's
suppose the input sequence is highly correlated. In this case, we can predict an odd-indexed
value by using the even-indexed neighbor,
{
d
n
}
will have a much smaller dynamic range than the original sequence
{
x
o
,
n
}
d
k
=
x
o
,
k
−
x
e
,
k
=
x
2
k
+
1
−
x
2
k
{
d
k
}
The magnitude of the
values will depend on the linear correlation between the samples
of the original signal. We can see that the sequence
{
d
k
}
corresponds, in some sense, to the
high-frequency component of the original sequence. The sequence
, on the other hand, is
simply the sampled version of the original signal. Depending on the frequency content of the
original signal, this subsampling could lead to significant aliasing. We can try and reduce this
aliasing effect by attempting to preserve some of the statistical properties of the original signal.
The simplest thing is to preserve the average value of the sequence. We can do this by using
the odd values, or rather the residual of the odd-indexed values, to update the even-indexed
values as shown in Figure
15.18
, where
{
x
e
,
k
}
s
k
=
x
e
,
k
+
U
(
{
x
o
,
k
}
)
