6996 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 66, NO. 8, AUGUST 2017
with
H
a,n
SD
(k)=μ
r,D
μ
t,S
H
n
SD
(k)+ν
r,D
ν
∗
t,S
× (H
n
SD
(N − k + 2))
∗
(15)
H
b,n
SD
(k)=μ
r,D
ν
t,S
H
n
SD
(k)+ν
r,D
μ
∗
t,S
× (H
n
SD
(N − k + 2))
∗
(16)
H
a,n
DD
(k)=μ
r,D
μ
t,D
H
n
DD
(k)+ν
r,D
ν
∗
t,D
× (H
n
DD
(N − k + 2))
∗
(17)
and
H
b,n
DD
(k)=μ
r,D
ν
t,D
H
n
DD
(k)+ν
r,D
μ
∗
t,D
× (H
n
DD
(N − k + 2))
∗
. (18)
Similar to (14), we have
y
∗
(n, N −k + 2)=(H
a,n
SD
(N − k + 2))
∗
x
∗
S
(n, N −k + 2)
+
H
b,n
SD
(N − k + 2)
∗
x
S
(n, k)
+(H
a,n
DD
(N − k + 2))
∗
x
∗
D
(n, N −k + 2)
+
H
b,n
DD
(N − k + 2)
∗
x
D
(n, k)+w
∗
(n, N −k + 2).
(19)
In the following, we call the two subchannels k and N −k+2
as the kth subchannel/subcarrier pair. Combining (14) and (19)
yields
z
n,k
= Γ
k
SD
x
n,k
S
+ Γ
k
DD
x
n,k
D
+ w
n,k
(20)
where
z
n,k
=
y(n, k)
y
∗
(n, N −k + 2)
(21)
x
n,k
S
=
x
S
(n, k)
x
∗
S
(n, N −k + 2)
(22)
x
n,k
D
=
x
D
(n, k)
x
∗
D
(n, N −k + 2)
(23)
Γ
n,k
SD
=
⎛
⎜
⎝
H
a,n
SD
(k) H
b,n
SD
(k)
H
b,n
SD
(N − k + 2)
∗
(H
a,n
SD
(N − k + 2))
∗
⎞
⎟
⎠
(24)
Γ
n,k
DD
=
⎛
⎜
⎝
H
a,n
DD
(k) H
b,n
DD
(k)
H
b,n
DD
(N − k + 2)
∗
(H
a,n
DD
(N − k + 2))
∗
⎞
⎟
⎠
(25)
and
w
n,k
=
w(n, k)
w
∗
(n, N −k + 2)
. (26)
Fig. 2. Pilot pattern for the full-duplex OFDM system.
Applying the vec(•) operator to both sides of (20) yields
z
n,k
= vec
z
n,k
= vec
Γ
n,k
SD
x
n,k
S
+ vec
Γ
n,k
DD
x
n,k
D
+ vec
w
n,k
= vec
I
2
Γ
n,k
SD
x
n,k
S
+ vec
I
2
Γ
n,k
DD
x
n,k
D
+ vec
w
n,k
(27)
with k ∈{2, 3,...,N/2}. By using the property of vec operator
in [23]
vec (AY B )=
B
T
⊗ A
vec (Y) (28)
we can rewrite (27) as
z
n,k
=
x
n,k
S
T
⊗ I
2
vec
Γ
n,k
SD
+
x
n,k
D
T
⊗ I
2
× vec
Γ
n,k
DD
+ w
n,k
=
x
n,k
S
T
×⊗I
2
x
n,k
D
T
⊗ I
2
⎡
⎣
vec
Γ
n,k
SD
vec
Γ
n,k
DD
⎤
⎦
Γ
n,k
+w
n,k
.
(29)
Apparently, the above equation is under-determined because we
use two equations to compute eight unknowns. Obviously, we
need at least eight equations to calculate eight unknowns. This
implies that at least four pilot OFDM symbols are required to
do one-time frequency-domain channel estimation.
As shown in Fig. 2, each frame consists of N
F
blocks and pi-
lot sub-block of N
P
pilot OFDM symbols. Each block is made
up of N
B
OFDM symbols. In each block, the first N
P
OFDM
symbols (pilot sub-block) are used as pilot OFDM symbols and
the remaining N
D
ones (data sub-block) are data OFDM sym-
bols, where N
B
= N
P
+ N
D
, and N
P
is larger than or equal to
four according to the above analysis. In t erms of N
B
, N
P
, and
N
D
,forthenth OFDM symbol, we may determine it belongs