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KarhunenLoeve expansion
Let
X
be a zeromean
m
x 1 random vector, and let
{
}
H
R
E
=
XX
. Let
R
have the
factorization
H
R U U
=
Λ
, where the columns of
U
are the normalized eigenvectors of
R.
Let
H
U
=
Y
X
. Then
Y
is a zeromean random vector with uncorrelated
components:
{
}
{
}
{
}
H
H
H
H
H
H
H
E
E U
U
U E
U
U U U U
=
=
=
Λ
= Λ
YY
XX
XX
(1)
We can write
1
m
i
i
i
U
y
=
=
=
X
Y
u
(2)
where
i
u
is the
i
th column vector of
Y
.
This says that we can construct
X
as a linear combination of orthogonal vectors
i
u
,
where the coefficients
i
y
are uncorrelated random variables. This representation is
called KarhunenLoeve (KL) expansion of
X
. In this case, the eigenvectors of
R
are
used as the basis vectors.
KL expansion can be used in signal coding (data compression). In the
encoder, the encoding signal
X
is KL transformed by calculating
H
U
=
Y
X
.
(3)
Then,
i
y
are encoded and transmitted. In the decoder, the reconstructed
X
is
obtained by
1
ˆ
ˆ
m
i
i
i
y
=
=
X
u
.
(4).
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 Fall '08
 SinHorngChen

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