Digital Signal Processing Reference
In-Depth Information
Under this formulation we do not have a strict definition of the block size
T
B
because,
although we still use (2.15) for any
n
, we allow
h
q
(
l
)'s to change subblock by subblock
based on the training symbols.
2.4.2.1 Subblock Tracking Using Kalman Filtering
Define ε(
n
) := [
e
-
j
ω
1
n
e
-
j
ω
2
n
…
e
-
j
ω
Q
n
]
T
. If at time
n
the
p
th
subblock is being received, by
(2.63), (2.15)-(2.19), and (2.74) and (2.75), the received signal can be written as
H
()
=
()
⊗
()
()
+
()
T
yn
s
n
I
1
ε
n
h
pvn
,
(2.77)
L
+
where
s
(
n
) := [
s
(
n
)
s
(
n
- 1)
…
s
(
n
-
L
)]
T
. Treating (2.76) and (2.77) as the state and the
measurement equations, respectively, Kalman filtering can be applied to track the coeffi-
cient vector
h
(
p
) for each subblock.
We will employ the time-multiplexed training scheme proposed in [36] (see
section
length
l
d
symbols) and a succeeding training session (of length
l
b
= 2
L
+ 1 symbols).
Using (2.73), at time
n
p
+
l
(
p
= 0, 1,
…
and
l
= 0, 1,
…
,
L
)
( )
=
( ) ()
+
( )
.
H
yn l
γ
Enl
h
pvnl
(2.78)
p
p
l
p
We intend to use only training sessions for subblock-wise channel tracking. Defining
T
( )
()
:=
( )
y
p
y n
y n
1
y nL
,
p
p
p
T
,
()
:
( )
( )
v
p
=
vn
vn
1
vn L
p
p
p
H
En
p
( )
En
1
()
:=
p
Ψ
p
,
( )
En L
p
by (2.78), we have
()
=
()()
+
()
.
y
p
γΨ
p
h
p
v
p
(2.79)
Using this optimal training scheme, the measurement equation (2.77) is now simpli-
fied as (2.79). We have obtained a linear discrete-time system represented by (2.76) and
(2.79). Kalman filtering is applied to track the channel BEM coefficients via the follow-
ing steps [49]:
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