Digital Signal Processing Reference
In-Depth Information
Let us consider two vectors, each of
M -
1 observations:
()
(
)
⎡
xk
⎤
⎡
xk
+ ⎤
1
⎢
⎥
⎢
⎥
(
)
(
)
xk
+
1
xk
+
2
⎢
⎥
⎢
⎥
()
(
)
x
k
=
and
x
k
+ =
1
[8.58]
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
(
)
(
)
xk M
+−
2
xk M
+−
1
⎢
⎥
⎢
⎥
⎣
⎦
⎣
⎦
According to [8.7], it is easy to see that we have:
() () ()
(
x
k
=
s
k
+
b
k
[8.59]
)
(
) (
)
x
k
+=
1
As
k
++ +
1
b
k
1
where:
Aa a
=
⎣
⎡
,
…
,
⎤
1
p
⎦
T
()
⎡
j
2
π
f T
j
2
π
f
T
⎤
s
k
=
⎣
α
e
,
…
,
α
e
[8.60]
1
e
P
e
1
P
⎦
T
() () (
)
b
k
=
⎡
bk
,
…
,
bk
+
M
−
2
⎤
⎣
⎦
and:
T
(
22
⎡
jM
−
π
fT
⎤
()
j
2
π
f T
2 2
j
π
f T
aa
=
f
=
⎢
1,
e
,
e
,
…
,
e
[8.61]
ie
ie
ie
i
i
⎥
⎣
⎦
We note that:
(
)
( )
s
k
+=
1
Φ
k
[8.62]
where:
{
}
j
2
π
f T
j
2
π
f T
Φ
=
diag
,
…
,
p
e
[8.63]
e
1
e
e
is a unitary matrix of dimension
P
×
P.
Thus, we see that matrix Φ, which is often
associated with a rotation operator, contains all the information about the
frequencies to estimate.
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