Digital Signal Processing Reference
In-Depth Information
def
W
i
CW
i
directly from this predecessor
T
i
2
1
. Prove that the following
iterations
T
i
¼
T
0
¼Q
0
CQ
0
where
Q
0
is an arbitrary orthonormal matrix
for
i ¼
1, 2,
... T
i
1
¼Q
i
R
i
QR factorization
T
i
¼ R
i
Q
i
produce a sequence (
T
i
,
Q
0
Q
i
:::Q
i
)
that converges to (Diag(
l
1
,
...
,
l
n
),
[
+u
1
,
...
,
+u
n
]).
4.5
Specify what happens to the convergence and the convergence speed, if the
step
W
i
¼
orthonorm
fCW
i
2
1
g
of the orthogonal iteration algorithm (4.11)
is replaced by the following
fW
i
¼
orthonorm
f
(
I
n
þmC
)
W
i
2
1
g
. Same
questions, for the step
fW
i
¼
orthonormalization of
C
2
1
W
i
2
1
g
, then
fW
i
¼
orthonormalization of (
I
n
2
mC
)
W
i
2
1
g
. Specify the conditions that must
satisfy the eigenvalues of
C
and
m
for these latter two steps. Examine the specific
case
r ¼
1.
4.6
Using the EVD of
C
, prove that the solutions
W
of the maximizations and mini-
mizations (4.7) are given by
W ¼
[
u
1
,
...
,
u
r
]
Q
and
W ¼
[
u
n
2
rþ
1
,
...
,
u
n
]
Q
respectively, where
Q
is an arbitrary
r r
orthogonal matrix.
def
E(
kxWW
T
xk
2
)of
W ¼
[
w
1
,
...
,
4.7
Consider the scalar function (4.14)
J
(
W
)
¼
def
E(
xx
T
). Let
7
W
¼
[
7
1
,
...
,
7
r
] where (
7
k
)
k¼
1,
...
,
r
is the gradient
operator with respect to
w
k
. Prove that
w
r
] with
C¼
7
W
J ¼
2(
2
CþCWW
T
þWW
T
C
)
W:
(4
:
90)
Then, prove that the stationary points of
J
(
W
) are given by
W ¼ U
r
Q
where
the
r
columns of
U
r
denote arbitrary
r
distinct unit-2 norm eigenvectors
among
u
1
,
...
,
u
n
of
C
and where
Q
is an arbitrary
r r
orthogonal matrix.
Finally, prove that at each stationary point,
J
(
W
) equals the sum of eigenvalues
whose eigenvectors are not involved in
U
r
.
Consider now the complex valued case where
J
(
W
)
¼
def
E(
kxWW
T
xk
2
)
def
E(
xx
H
) and use the complex gradient operator (see e.g., [36])
defined by
7
W
¼
with
C¼
2
[
7
R
þ i7
I
] where
7
R
and
7
I
denote the gradient operators
with respect to the real and imaginary parts. Show that
7
W
J
has the
same form as the real gradient (4.90) except for a factor 1
/
2 and changing the
transpose operator by the conjugate transpose one. By noticing that
7
w
J ¼ O
is equivalent to
7
R
J ¼ 7
I
J ¼ O
, extend the previous results to the complex
valued case.
1
4.8
With the notations of Exercise 4.7, suppose now that
l
r
.
l
rþ
1
and consider first
the real valued case. Show that the (
i
,
j
)th block
7
i
7
T
j
J
of the block Hessian
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