Databases Reference
In-Depth Information
a random process
X
is a matrix whose
(
i
,
j
)
th element
[
R
]
i
,
j
is given by
[
R
]
i
,
j
=
E
[
X
n
X
n
+
|
i
−
j
|
]
(39)
We can show [
134
] that a transform constructed in this manner will minimize the geometric
mean of the variance of the transform coefficients. Hence, the Karhunen-Loéve transform
provides the largest transform coding gain of any transform coding method.
If the source output being compressed is nonstationary, the autocorrelation function will
change with time. Thus, the autocorrelation matrix will change with time, and the KLT will
have to be recomputed. For a transform of any reasonable size, this is a significant amount of
computation. Furthermore, as the autocorrelation is computed based on the source output, it
is not available to the receiver. Therefore, either the autocorrelation or the transform itself has
to be sent to the receiver. The overhead can be significant and remove any advantages to using
the optimum transform. However, in applications where the statistics change slowly and the
transform size can be kept small, the KLT can be of practical use [
190
].
Example13.4.1:
Let us see how to obtain the KLT transform of size two for an arbitrary input sequence. The
autocorrelation matrix of size two for a stationary process is
R
xx
(
0
)
R
xx
(
1
)
R
=
(40)
R
xx
(
1
)
R
xx
(
0
)
Solving the equation
|
λ
I
−
R
| =
0, we get the two eigenvalues
λ
1
=
R
xx
(
0
)
+
R
xx
(
1
)
and
λ
2
=
R
xx
(
0
)
−
R
xx
(
1
)
. The corresponding eigenvectors are
α
α
β
−
β
V
1
=
V
2
=
(41)
where
are arbitrary constants. If we now impose the orthonormality condition, which
requires the vectors to have a magnitude of 1, we get
α
and
β
1
√
2
α
=
β
=
and the transform matrix
K
is
11
1
1
√
2
K
=
(42)
−
1
Notice that this matrix is not dependent on the values of
R
xx
(
0
)
and
R
xx
(
1
)
. This is only true
of the 2
×
2 KLT. The transform matrices of higher order are functions of the autocorrelation
values.
Although the Karhunen-Loéve transform maximizes the transform coding gain as defined
by (
27
), it is not practical in most circumstances. Therefore, we need transforms that do not
depend on the data being transformed. We describe some of the more popular transforms in
the following sections.
