Digital Signal Processing Reference
In-Depth Information
T
m
O
OO
m
A
P
m
(
∨
)
P
m
=
.
Let
U
be the
m
×
m
unitary matrix given by
⎡
⎤
1
⎣
⎦
.
1
U
=
1
Then
U
∗
T
m
U
is an
m
×
m
matrix. For
α, β
∈
-entry of
U
∗
T
m
U
Q
n
−
m
,
n
the
(α, β)
.Let
U
be the
n
+
m
−
1
m
×
n
+
m
−
1
m
matrix given by
is per
A
(α
|
β)
UO
OI
U
=
.
m
A
, the matrix
U
∗
(
∨
U
. Then
m
A
t
∨
We denote by
)
U
∗
T
m
UO
OO
m
A
P
m
(
∨
)
P
m
=
.
(5.3.13)
In particular for
m
=
n
−
1 this becomes
(
t
O
OO
padj
A
)
n
−
1
A
P
n
−
1
(
∨
)
P
n
−
1
=
.
1
matrix
XO
, equation (
5.3.2
)
n
matrix
X
with
2
n
−
1
n
1
×
2
n
−
1
Identifying an
n
×
−
n
−
OO
can be written as
n
−
1
A
(
)(
)
=
(
P
n
−
1
(
∨
)
P
n
−
1
)
.
D per
A
X
tr
X
(5.3.14)
Its generalisation for higher order derivatives can be given as follows.
Theorem 3.4
Fo r
1
≤
m
≤
n
tr
P
n
−
m
(
∨
P
n
−
m
n
−
m
A
D
m
per
X
1
X
m
(
A
)(
,...,
)
=
m
!
)
P
m
(
P
m
X
1
X
m
∨···∨
)
.
(5.3.15)
In particular,
tr
P
n
−
m
(
∨
P
n
−
m
P
m
(
∨
P
m
n
−
m
A
D
m
per
m
X
(
A
)(
X
,...,
X
)
=
m
!
)
)
.
