Digital Signal Processing Reference
In-Depth Information
T
−
1
∑
(( )
n
=
()()
j
unune
αα
−
Ψ
mm
′
(2 . 61)
q
′
q
mQ qmQq
+,′+′
n
=
0
The estimate of D is given by
D
=
( )
.
Ψ
1
G
(2.62)
I
N
By (2.59) and (2.78) we have the channel coefficient estimate
=
( ) ( )
.
−
1
ˆ
ˆ
†
HCDCCC
H
H
−
1
Ψ
I
G
(2.63)
N
The channel estimate is acquired by regenerating the DPS-BEM:
Q
∑
1
ˆ
(
=
() ()
.
=
ˆ
h
nl
h
lun
(2.64)
q
q
q
remark 1:
Using the fact that the (infinite) DPS sequences are band limited to the nor-
malized frequency range [-
f
d
T
s
,
f
d
T
s
], the time-limited DPS sequences, obtained by rec-
tangular windowing over 0 ≤
n
≤
T
- 1, approximately satisfy
T
−
1
∑
()()
≈ ′
( )
′−
′
( )
δ
))
,
(
j
αα
δ
−
n
unune
mm
m
mqq
(2.65)
q
′
q
n
=
0
if
f
d
T
s
1/
P
and
T
are multiples of
P
or if
T
is large, so that Ψ ≈
I
PQ
. An estimate
ˆ
mq
of
d
mq
, following (2.54) and (2.65), is given as
T
−
1
∑
0
d
=
()()
.
=
−
j
α
n
y
nu ne
(2.66)
m
mq
q
n
The estimation of the channel coefficients (2.63) is then given by
=
( )
.
−1
ˆ
ˆ
H
H
HCCCD
(2.67)
remark 2:
As noted in [57], if the mean of the noise
v
(
n
) is unknown, say
({
=,
En
v
m
(2.68)
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