Digital Signal Processing Reference
In-Depth Information
def
bW
0T
(
k
)
x
(
k
), which implies (see Exercise 4.9) the following recursions
with
z
(
k
)
¼
1
b
[
I
n
t
1
e
1
e
1
t
2
e
2
e
2
]
G
(
k
)
G
(
k þ
1)
¼
(4
:
41)
W
(
k þ
1)
¼W
(
k
)[
I
n
t
1
e
1
e
1
t
2
e
2
e
2
]
1
b
x
(
k
)
y
T
(
k
)
G
T
(
k
)[
I
n
t
1
e
1
e
1
t
2
e
2
e
2
]
þ
(4
:
42)
where
t
1
,
t
2
and
e
1
,
e
2
are defined in Exercise 4.9.
Note that the square root inverse matrix
G
(
k þ
1) of
W
0T
(
k þ
1)
W
0
(
k þ
1) is asym-
metric even if
G
(0) is symmetric. Expressions (4.41) and (4.42) provide an algorithm
which does not involve any matrix-matrix multiplications and in fact requires only
O
(
nr
) operations.
Based on the approximation that
W
(
k
) and
W
(
k þ
1) span the same
r
-dimensional
subspace, another power-based algorithm referred to as the approximated power iter-
ation (API) algorithm and its fast implementation (FAPI) have been proposed in [8].
Compared to the NP3 algorithm, this scheme has the advantage that it can handle the
exponential (4.19) or the sliding windowed (4.22) estimates of
C
x
(
k
) in the same fra-
mework (and with the same complexity of
O
(
nr
) operations) by writing (4.19) and
(4.22) in the form
C
(
k
)
¼ bC
(
k
1)
þx
0
(
k
)
Jx
0T
(
k
)
and
10
0
b
l
with
J ¼
1 and
x
0
(
k
)
¼ x
(
k
) for the exponential window and
J ¼
x
0
(
k
)
¼
[
x
(
k
),
x
(
k
2
l
)] for the sliding window [see (4.22)]. Among the power-based
minor subspace tracking methods issued from the exponential of sliding window,
this FAPI algorithm has been considered by many practitioners (e.g. [11]) as outper-
forming the other algorithms having the same computational complexity.
4.5.2 Projection Approximation-based Methods
Since (4.14) describes an unconstrained cost function to be minimized, it is straight-
forward to apply the gradient-descent technique for dominant subspace tracking.
Using expression (4.90) of the gradient given in Exercise 4.7 with the estimate
x
(
k
)
x
T
(
k
)of
C
x
(
k
) gives
W
(
k þ
1)
¼W
(
k
)
m
k
[
2
x
(
k
)
x
T
(
k
)
þx
(
k
)
x
T
(
k
)
W
(
k
)
W
T
(
k
)
þW
(
k
)
W
T
(
k
)
x
(
k
)
x
T
(
k
)]
W
(
k
)
:
(4
:
43)
We note that this algorithm can be linked to Oja's algorithm (4.30). First, the term
between brackets
the term
x
(
k
)
x
T
(
k
)
þW
(
k
)
W
T
is the symmetrization of
Search WWH ::
Custom Search