Block-wise Alamouti Schemes for OQAM-OFDM Systems J. Li
et al.
Ta b l e I . The
c
0
p
,
q
value of the PHYDYAS prototype filter.
c
0
p
,
q
q
D3
q
D2
q
D1
q
D 0
q
D 1
q
D 2
q
D 3
p
D1 0.0429
j
0.1250
j
0.2058
j
0.2393
j
0.2058
j
0.1250
j
0.0429
j
p
D 0 0.0668
j
0 0.5644
j
1 0.5644
j
0 0.0668
j
p
D 1 0.0429
j
0.1250
j
0.2058
j
0.2393
j
0.2058
j
0.1250
j
0.0429
j
3. ORTHOGONALITY CONDITION
OF ALAMOUTI-BASED
OQAM-OFDM SYSTEMS
In general, a code C is a 2 2 matrix with columns cor-
responding to antennas and rows corresponding to either
time indexes for STBC or frequency tones for SFBC. The
elements of C are linear combinations of S
1
, S
2
and their
conjugates. The code C provides a full diversity gain of two
if it satisfies the orthogonality condition of the Alamouti
code [21,22] as
C
H
C D
j
S
1
j
2
C
j
S
2
j
2
I (8)
where I is the 2 2 identity matrix.
In this section, the orthogonality condition is derived
for Alamouti-based OQAM-OFDM systems. Because the
SFBCschemeandtheSTBCschemehavethesamestruc-
ture, we take the SFBC scheme as an example for analysis.
3.1. Conventional space-frequency block
coding-orthogonal frequency division
multiplexing systems
For example, in a conventional SFBC-OFDM system,
there is
C D
"
S
1
S
2
S
2
S
1
#
(9)
Obviously, (9) satisfies (8).
Supposing the channel frequency response does not vary
over the subcarriers of the code C, the received signals
could be expressed in the frequency domain as
R
1
R
2
D
H
1
H
2
H
2
H
1
„
ƒ‚ …
N
H
S
1
S
2
C
1
2
(10)
where R
1
and R
2
are the received signals on the two sub-
carriers, H
1
and H
2
are the channel frequency responses
of the two antennas, the frequency index m and time index
n are omitted for simplicity. It is observed that the equiv-
alent channel matrix
N
H is orthogonal with the help of the
orthogonal code C. Then, the received signals could be pro-
cessed by a space-frequency combiner, which computes
the decision variables as
Q
R
1
Q
R
2
D
N
H
H
R
1
R
2
D
H
1
H
2
H
2
H
1
H
1
H
2
H
2
H
1
S
1
S
2
C
N
H
H
1
2
D
j
H
1
j
2
C
j
H
2
j
2
S
1
S
2
C
N
H
H
1
2
(11)
Then, the maximum likelihood detection is the optimum
way to make the symbol decisions, and a full diversity gain
of two is achieved.
3.2. Orthogonality condition of
Alamouti-based offset quadrature
amplitude modulation-based orthogonal
frequency division multiplexing systems
Considering the real-valued data symbols transmitted on
the subcarriers m
1
and m
2
at the time index n,thereal
SFBC code could be written as
C
r
D
"
d
m
1
,n,1
d
m
1
,n,2
d
m
2
,n,1
d
m
2
,n,2
#
(12)
where d
m,n,l
represents the real-valued data transmitted on
the subcarrier m at the time index n from the antenna l
(l D 1, 2). Because the noise component is not unique,
hence, the noise component is not given in the following
for simplicity.
According to (5), the signals after the analysis filter bank
at the receive antenna are
R
1
D H
1
d
m
1
,n,1
C jd
.i/
m
1
,n,1
C H
2
d
m
1
,n,2
C jd
.i/
m
1
,n,2
R
2
D H
1
d
m
2
,n,1
C jd
.i/
m
2
,n,1
C H
2
d
m
2
,n,2
C jd
.i/
m
2
,n,2
(13)
Then, the equivalent complex code is written as
C D
2
4
d
m
1
,n,1
C jd
.i/
m
1
,n,1
d
m
1
,n,2
C jd
.i/
m
1
,n,2
d
m
2
,n,1
C jd
.i/
m
2
,n,1
d
m
2
,n,2
C jd
.i/
m
2
,n,2
3
5
(14)
and
C
r
D<
f
C
g
(15)
That C
r
satisfies the orthogonality condition of the Alam-
outi code seems sufficient for OQAM-OFDM systems
because the transmitted data symbols are real valued. How-
ever, it is not true as explained in the following example.
2978
Wirel. Commun. Mob. Comput.
2016; 16:2975–2990 © 2016 John Wiley & Sons, Ltd.
DOI: 10.1002/wcm