Digital Signal Processing Reference
In-Depth Information
of the columns of
W
(
k
). To remedy this instability, another implementation of this
algorithm based on the numerically well behaved Householder transform has been
proposed [6]. This Householder FRANS algorithm (HFRANS) comes from (4.36)
which can be rewritten after cumbersome manipulations as
W
(
k þ
1)
¼H
(
k
)
W
(
k
) with
H
(
k
)
¼ I
n
2
u
(
k
)
u
T
(
k
)
def
p
(
k
)
kp
(
k
)
k
2
. With no additional numerical complexity, this Householder
transform allows one to stabilize the noise subspace version of the FRANS algorithm.
6
The interested reader may refer to [75] that analyzes the orthonormal error propagation
[i.e., a recursion of the distance to orthonormality
kW
T
(
k
)
W
(
k
)
I
r
k
with
u
(
k
)
¼
2
Fro
from a
nonorthogonal matrix
W
(0)] in the FRANS and HFRANS algorithms.
Another solution to orthonormalize the columns of
W
0
(
k þ
1) has been proposed
in [29, 30]. It consists of two steps. The first one orthogonalizes these columns
using a matrix
G
(
k þ
1) to give
W
00
(
k þ
1)
¼ W
0
(
k þ
1)
G
(
k þ
1), and the second
one normalizes the columns of
W
00
(
k þ
1). To find such a matrix
G
(
k þ
1) which is
of course not unique, notice that if
G
(
k þ
1) is an orthogonal matrix having as first
column, the vector
y
(
k
)
ky
(
k
)
k
2
with the remaining
r
2
1 columns completing an orthonor-
mal basis, then using (4.33), the product
W
00T
(
k þ
1)
W
00
(
k þ
1) becomes the follow-
ing diagonal matrix
W
00T
(
k þ
1)
W
00
(
k þ
1)
¼G
T
(
k þ
1)[
I
r
þd
k
y
(
k
)
y
T
(
k
)]
G
(
k þ
1)
2
e
1
e
1
¼ I
r
þd
k
ky
(
k
)
k
def
def
[0,
...
,0]
T
. It is fortunate that there exists
such an orthonogonal matrix
G
(
k þ
1) with the desired properties known as a
Householder reflector [35, Chap. 5], and can be very easily generated since it is of
the form
2
and
e
1
¼
where
d
k
¼
+
2
m
k
þm
k
kx
(
k
)
k
2
ka
(
k
)
k
2
a
(
k
)
a
T
(
k
) with
a
(
k
)
¼ y
(
k
)
ky
(
k
)
ke
1
:
G
(
k þ
1)
¼ I
r
(4
:
37)
This gives the Fast Data Projection Method (FDPM)
W
(
k þ
1)
¼
Normalize{[
W
(
k
)
+ m
k
x
(
k
)
x
T
(
k
)
W
(
k
)]
G
(
k þ
1)},
(4
:
38)
where Normalize
fW
00
(k
þ
1)
g
stands for normalization of the columns of
W
00
(
k þ
1),
and
G
(
k þ
1) is the Householder transform given by (4.37). Using the independence
assumption [36, Chap. 9.4] and the approximation
m
k
1, a simplistic theoretical
6
However, if one looks very carefully at the simulation graphs representing the orthonormality error [75,
Fig. 7], it is easy to realize that the HFRANS algorithm exhibits a slight linear instability.
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