Digital Signal Processing Reference
In-Depth Information
This means that we would choose the solution of (
5.7
) with minimum canonical
Euclidean norm. Then, given a particular solution
w
p
, we need to search over the
ˆ
vectors
w
h
having minimum norm. Although
this seems to be difficult, the fact is that the solution to problem (
5.11
) is very simple
and is given by
w
h
ˆ
∈
N (
C
)
that lead to
w
ˆ
= ˆ
w
p
+ ˆ
C
†
d
w
ˆ
=
,
(5.12)
where
C
†
denotes the Moore-Penrose pseudoinverse of
C
. The pseudoinverse of a
general rectangular matrix
C
is a generalization of the concept of inverse. Its calcula-
tion in the most general case can be obtained from the
singular value decomposition
(SVD) of a matrix
C
[
5
]. The SVD is a very powerful technique to decompose a
general rectangular matrix.
5.1.1.1 Singular Value Decomposition and Pseudoinverse
n
×
L
be an arbitrary rectangular matrix. Matrix
C
can be written as:
∈ R
Let
C
•
≤
If
L
n
:
U
0
V
T
L
×
L
C
=
,
∈ R
.
(5.13)
•
If
L
≥
n
:
0
]
V
T
n
×
n
C
=
U
[
,
∈ R
.
(5.14)
n
×
n
and
V
L
×
L
being orthonormal matrices
6
whose columns are
with
U
∈ R
∈ R
the eigenvectors of
CC
T
and
C
T
C
respectively. Square matrix
is diagonal with
positive entries, that is:
=
diag
(σ
1
,σ
2
,...,σ
K
,
0
,...,
0
) ,
(5.15)
2
where
σ
i
,
i
=
1
,...,
K
, with
K
=
rank
(
C
)
≤
min
(
n
,
L
)
, are the non-null
eigenvalues of either
CC
T
or
C
T
C
.
The SVD is a very important tool in linear algebra, matrix analysis and signal
processing. The reader interested in more details on SVD can see [
5
,
6
] and the
references therein. The pseudoinverse
C
†
can be written is terms of its SVD as [
7
]:
•
If
L
≤
n
:
V
0
U
T
C
†
−
1
=
.
(5.16)
•
If
L
≥
n
:
V
−
1
0
U
T
C
†
=
.
(5.17)
6
U
T
U
=
UU
T
=
I
n
and
V
T
V
=
VV
T
=
I
L
.
