ZHANG et al.: TIME VARYING CHANNEL ESTIMATION FOR DSTC-BASED RELAY NETWORKS 5025
where
(l)
h
k
and
(l)
g
k
are the correlation matrices for h
k,m
and
g
k,m
, respectively, whose entries can be derived from (1-a) and
(1-b) as
(l)
h
k
i,j
= σ
2
h
J
0
(
2πf
ds
(i −j +2Nl)T
s
)
×J
0
2πf
dr,k
(i −j + 2Nl)T
s
, (8-a)
(l)
g
k
i,j
= σ
2
g
J
0
(
2πf
dd
(i − j +2Nl)T
s
)
×J
0
2πf
dr,k
(i −j + 2Nl)T
s
. (8-b)
Furthermore, the in-channel modeling mean square error
(MSE) for the m-th time segment can be expressed as
MSE
bem
=
1
2KN
K−1
k=0
E
e
g
k,m
H
e
g
k,m
+
e
h
k,m
H
e
h
k,m
=
1
2KN
K−1
k=0
tr
(I −
†
)
2
(0)
h
k
+
(0)
g
k
. (9)
Plugging (5) into (4), we can reexpress y
m
through P-BEM as
y
m
=
K−1
k=0
α(μ
k,m
)U
k,m
(λ
k,m
s
m
)+μ
k,m
t
k,m
+
K−1
k=0
αμ
k,m
U
k,m
v
m
+ n
m
. (10)
Before proceeding, let us define N × N diagonal matrices S
m
=
diag(s
m
) and T
k,m
= diag(t
k,m
) for further use.
Remark: There is one specific case for the precoding matri-
ces U
k,m
, i.e., U
k,m
is diagonal. For diagonal U
k,m
, it can be
obtained
(μ
k,m
) U
k,m
(λ
k,m
s
m
)
= U
k,m
diag
⎛
⎜
⎝
(λ
k,m
⊗ μ
k,m
)
c
k,m
⎞
⎟
⎠
s
m
, (11)
where c
k,m
is one (2Q −1) × 1 vector, and
is the N ×(2Q −
1) P-BEM matrix with the similar structure with . With (11),
y
m
can be rewritten as
y
m
=
K−1
k=0
U
k,m
S
m
c
k,m
+ T
k,m
μ
k,m
+
K−1
k=0
αdiag(μ
k,m
)U
k,m
v
m
+ n
m
. (12)
With U
k,m
being diagonal, the original signal model in (4) can
be reexpressed as
y
m
= α
K−1
k=0
U
k,m
S
m
(h
k,m
g
k,m
) + α
K−1
k=0
T
k,m
g
k,m
+
˜
n
m
.
(13)
Comparing (12) and (13), we have the following observations
when the precoding matrices U
k,m
are diagonal: the co-channel
vectors h
k,m
g
k,m
for the link from S to R
k
and then to D can
be approximated by P-BEM with 2Q −1 compositive BEM-
CV c
k,m
. If we consider valid data transmission process with no
superimposed training from R
k
, only the composite BEM-CV
c
k,m
are needed for ML data detection, which is similar to the
case under time-invariant scenario [14]. Notice that the above
results can be applied for CE-BEM but not for DKL-BEM.
Nonetheless, in order to gain better diversity performance,
the precoding matrices U
k,m
should be optimized according to
different criteria, i.e., data detection MSE, bit-error-rate (BER),
throughput and others, and the resultant U
k,m
’s are usually
not diagonal. Unfortunately, for non-diagonal matrices U
k,m
,
we cannot obtain results similar with that for diagonal pre-
coding matrices in (12), (13). From (10), it can be found
that the equivalent transmission channel from S to R
k
and
then to D can only be characterized by one N × N matrix
diag(μ
k,m
)U
k,m
diag(λ
k,m
) but not the equivalent composite
BEM-CV for non-diagonal U
k,m
. Thus if D implements
ML data detection with non-diagonal U
k,m
at R
k
,the
estimation of in-BEM-CVs λ
k,m
, μ
k,m
should be ob-
tained to construct the estimated transmission matrices, i.e.,
diag( ˆμ
k,m
)U
k,m
diag(
ˆ
λ
k,m
),where
ˆ
λ
k,m
( ˆμ
k,m
) represents
the estimation of λ
k,m
(μ
k,m
). The above phenomenon is quite
different from the case under time-invariant fading scenario
[14] and that with diagonal precoding matrices for time-varying
fading scenario.
For generality, we will adopt (10) as the observation signal
model, and focus on the estimation of in-BEM-CVs λ
k,m
, μ
k,m
,
k = 0,...,K − 1, m = 0,...,M − 1. Moreover, the general
model (10) can be written as the compacted matrix form as
y
m
= α[λ
m
](I ⊗)μ
m
+ ϒ
m
(I ⊗ )μ
m
+ α[μ
m
]
˜
v
m
+ n
m
= α[μ
m
]
m
(I ⊗)λ
m
+ ϒ
m
(I ⊗)μ
m
+ α[μ
m
]
˜
v
m
+ n
m
˜w
m
, (14)
where
[λ
m
]=
diag(U
0,m
S
m
λ
0,m
),...,diag(U
K−1,m
S
m
λ
K−1,m
)
,
[μ
m
]=
diag(μ
0,m
),...,diag(μ
K−1,m
)
,
m
= blkdiag(U
0,m
S
m
,...,U
K−1,m
S
m
),
ϒ
m
=[T
0,m
,...,T
K−1,m
],
˜
v
m
=
(U
0,m
v
0,m
)
T
,...,(U
K−1,m
v
K−1,m
)
T
T
, (15)
and
˜
w
m
is the equivalent noise at D corresponding to
˜
n
m
in (4).
Note that y
m
is a nonlinear function with respect to μ
m
, λ
m
due
to the first term on right hand side (RHS) of (14).
III. I
N-BEM-CV ESTIMATION
A. MAP In-BEM-CV Estimator
Define the KQ × 1 in-BEM-CVs λ
m
=[λ
T
0,m
,...,λ
T
K−1,m
]
T
,
μ
m
=[μ
T
0,m
,...,μ
T
K−1,m
]
T
,2KQ × 1 vector τ
m
=[λ
T
m
, μ
T
m
]
T
,