Digital Signal Processing Reference
In-Depth Information
Fig. 5.1
Orthogonal projec-
tion onto
S
(
C
)
d
opt
is unique. This means that it is
the unique vector that satisfies (
5.21
), which can be put more compactly as:
It is easy to check that the optimal
∈
S
(
C
)
C
T
e
=
0
.
(5.22)
It is straightforward to see that the previous expression can be written as:
C
T
C
C
T
d
w
ˆ
=
,
(5.23)
which is the normal equation in (
5.7
). Assuming that
C
has full column rank, the
solution for the optimal
w
is given by (
5.8
) and the value of
d
opt
is given by:
ˆ
C
C
T
C
−
1
C
T
d
d
opt
=
.
(5.24)
The
n
×
n
matrix:
C
C
T
C
−
1
C
T
P
[
S
(
C
)
]
=
,
(5.25)
is called
projection matrix
, and gives a representation for the operator that computes
the unique orthogonal projection of any vector in
n
onto
S
R
(
C
)
. Projection matrices
have important properties [
3
]:
P
T
•
P
[
S
(
C
)
]
=
[
S
(
C
)
].
P
2
[
S
•
(
C
)
]
=
P
[
S
(
C
)
].
P
S
⊥
(
)
=
•
I
n
−
].
The second property says that a projection matrix is an
idempotent
operator. That is,
applying the operator twice or more times is equivalent to applying it just once. This
is equivalent to saying that if
d
C
P
[
S
(
C
)
∈
S
(
C
)
, then
P
[
S
(
C
)
]
d
=
d
, which gives
e
=
0
.
The third property says that the projector onto the orthogonal complement of
S
(
C
)
,
(denoted as
S
⊥
(
C
)
, can be calculated in a very simple manner using the projector
