Signal Compression
114
Copyright © 2005 – 2008
Hayder Radha
The Karhunen-Loeve Transform
The KLT is the leading transform in terms of
theoretical importance and in being optimal under
certain criteria.
In particular, the KLT is an orthogonal, normal, linear
transform that generates an uncorrelated random vector.
Signal Compression
115
Copyright © 2005 – 2008
Hayder Radha
As before, we have the linear system:
[
]
[
] [
]
1
1
n
n n
n
Y
T
X
×
×
×
=
,
where
[
]
1
2
t
n
X
X
X
X
=
"
is a random source
vector of jointly random variables (in general,
dependent and correlated), and which is being
transformed into another random vector
[
]
1
2
t
n
Y
Y
Y
Y
=
"
. For a particular instance
X
x
=
Signal Compression
116
Copyright © 2005 – 2008
Hayder Radha
of the source, we have a particular instance of the
output
Y
y
=
. Hence, we also have:
[
]
[
] [ ]
1
1
n
n n
n
y
T
x
×
×
×
=
. In
short, we will use both expressions
Y
T X
=
and
y
T x
=
,
interchangeably.
The KLT only assumes knowledge of the covariance
matrix
X
K
of the source. More importantly, the
Signal Compression
117
Copyright © 2005 – 2008
Hayder Radha
development of the KLT is tightly coupled with the
basic linear algebraic tools for the decomposition of
symmetric matrices. In other words, the KLT exploits
the symmetric nature of the covariance matrix
X
K
to
generate the desired diagonal matrix
Y
K
.

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