- Dans cet article, nous proposons un codage spatio-temporel en bloc non orthogonal de rendement symbole 1 pour 3 antennes à l'émission. Ce code permet d'améliorer les performances d'un système de communications sans fil. Le code proposé peut être décodé soit simplement par forçage à zéro soit en utilisant un décodeur à maximum de vraisemblance à complexité réduite. Finalement, nous avons montré que la capacité du code est très proche de la capacité du canal en boucle ouverte

BELLANGER, Maurice

LE RUYET, Didier

OZBEK, Berna

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Non-Orthogonal Space-Time Block Coding Design for 3 Transmit
Antennas
Berna ÖZBEK1, Didier LE RUYET2, Maurice BELLANGER2,
1Izmir Institute of Technology
Electrical and Electronics Eng. Dep., Urla, Izmir, Turkey
2CNAM-Electronique
292 rue Saint-Martin 75141 Paris cedex 03, France
bernaozbek@iyte.edu.tr
leruyet@cnam.fr
bellang@cnam.fr
Résumé – Dans cet article, nous proposons un codage spatio-temporel en bloc non orthogonal de rendement symbole 1 pour 3 antennes à
l’émission. Ce code permet d’améliorer les performances d’un système de communications sans fil. Le code proposé peut être décodé soit
simplement par forçage à zéro soit en utilisant un décodeur à maximum de vraisemblance à complexité réduite. Finalement, nous avons montré
que la capacité du code est très proche de la capacité du canal en boucle ouverte.
Abstract – In this paper, we present a non-orthogonal space-time block code for 3 transmit antennas with symbol rate 1. This code improves
the performance of the wireless link. The proposed code can be decoded using a simple zero forcing receiver or with a low complexity maximum
likelihood decoding. Moreover, we have shown that the achievable capacity of the scheme is almost equal to the open loop channel capacity.
1 Introduction
In the context of wireless personal communications, the ob-
jective is to improve the performance of the link by achieving
space diversity using M transmit and N receiver antennas. For
M = 2 transmit antennas, Alamouti [1] proposed the com-
plex orthogonal space-time block code (STBC) that achieves
maximum diversity. This is the only symbol rate 1 code which
allows to reach the full capacity.
For the downlink of UMTS and the future communication
systems, it will be interesting to use more than 2 transmit an-
tennas per sector at the base station. For more than 2 transmit
antennas, it has been shown from the Hurwitz-Radon theorem
that complex orthogonal STBC designs cannot achieve both
maximum diversity and symbol rate 1 [2]. For 3 and 4 transmit
antennas, generalized complex orthogonal schemes that give
maximum diversity with rates 3/4 and 1/2 were proposed in
[3, 4]. Because of the symbol rate decrease, these schemes are
not suitable for high data rate applications. However, it is pos-
sible to achieve rate 1 for complex constellations for more than
2 transmit antennas by sacrificing orthogonality. These non-
orthogonal schemes have been proposed for 3 transmit anten-
nas in [5] at different symbol rates and for 4 transmit antennas
with symbol rate 1 in terms of capacity in [6] and diversity gain
in [7] at the expense of performance loss.
In this paper, we will propose a non-orthogonal complex
space-time block code for 3 transmit antennas with symbol
rate 1. After the presentation of the coding scheme and the
associate receiver structure, performance criteria aspects and
capacity calculations will be explained. The simulation results
and comparison with existing schemes will be given.
2 The Coding Scheme
Let define two different channel transfer matrices H1 and H2
as,
H1 =
[
h1 h2
h∗2 −h
∗
1
]
H2 =
[
−h3 0
0 −h∗3
]
(1)
The two channel matrices are combined in a system channel
transfer matrix Hc by Hadamard transform,
Hc =
[
H1 H2
−H2 H1
]
=


h1 h2 −h3 0
h∗2 −h
∗
1 0 −h
∗
3
h3 0 h1 h2
0 h∗3 h
∗
2 −h
∗
1

 (2)
where hi be the complex channel coefficient from the ith trans-
mit antenna to the receiver antenna which can be represented
as hi = αie
jθi
. The channel coefficients for each antenna are
assumed to be i.i.d zero mean complex Gaussian variables with
variance 0.5 per real dimension and fixed during T time periods
(quasi static flat fading channel).
The received signals of the scheme are given as
R = Hc


s1
s2
s3
s4

+Nc (3)
where R =
[
r1 r
∗
2 r3 r
∗
4
]T is the received vector and
Nc =
[
n1 n
∗
2 n3 n
∗
4
]T is the noise vector, elements of
which are assumed to be independent samples of zero mean
complex Gaussian random variables with variance σ2 per di-
mension. Since the average energy of the symbols transmitted
from each antenna is normalized to one, at each receiver an-
tenna, the average power is equal to M.
These scheme is implemented by using the code matrix given
below,
Sc =


s1 s2 −s3
−s∗2 s
∗
1 −s
∗
4
s3 s4 s1
−s∗4 s
∗
3 s
∗
2

 (4)
In equation (4), sinceQ = 4 symbols are sent over 3 transmit
antennas during T = 4 time interval, the symbol rateR = Q/T
is equal to 1.
It should be noted that this code can be derived from the
non-orthogonal code for 4 transmit antennas proposed in [6]
by setting h4 = 0 in the channel transfer matrix H .
3 Performance Criteria
The design criteria for space-time codes in Rayleigh fading
channels are formulated in terms of the codeword difference
matrix Dce = Sc − Se. Here Sc and Se are the code matrices
corresponding respectively to the encoded and possibly erro-
neously detected sets of information bits. Minimizing the pair-
wise error probability of deciding in favor of Se when trans-
mitting Sc leads to the rank and determinant criterion, which
determine respectively the diversity order and the coding gain
[8]. Furthermore, the trace criterion is an important parameter
for designing non-orthogonal STBCs [9]. The distance matrix
which defines the Hermitian square of the difference matrix is
given by
DHceDce =
4∑
k=1
|sck − s
e
k|
2
IM +N (5)
where N =
4∑
m<k
(Dce)
H
mk(Dce)mk is a non-orthogonal mat-
rix.
The Rank Criterion: The minimum rank of DHceDce for any
possible pair of codewords determines the maximum diversity
order. The minimum rank of the proposed code is 2 with a
mean close to 3. For QPSK symbol constellation, there ex-
ist 44(44 − 1) = 65280 error events, Sc → Se, where the
pairs Sc, Se are counted with Sc 6= Se. It is found that only
2080 error events have rank 2 instead of rank 3. The degrada-
tion is due to the self-symbol interference, resulting from non-
orthogonality of the code matrix. It should be remembered that
the rank of existing codes for 3 and 4 transmit antenna [5, 6] is
also equal to 2.
The Determinant Criterion: The maximum value ofmin
c→e
det(DHceDce)
determines the coding advantage, which measures the effec-
tive product distance of the code. The criterion is satisfied
if the eigenvalues of DHceDce are close to each other and the
off-diagonal elements which result self-interference are mini-
mized.
The Trace Criterion: To maximize the minimum Euclidean
distance between all possible codeword pairs, the minimum
value of Tr(DHceDce) should be maximized. If Tr(N) 6= 0,
the Euclidean distance between a symbol and its nearest neigh-
bors differs from the distance between an equivalent rotated
symbol and its nearest neighbors. In order to avoid this, the
non-orthogonality matrix should be traceless Tr(N) = 0 and
Euclidean distance squared should be proportional to a sum
of |sck − sek|
2
value. Since the proposed code has a traceless
nonorthogonality matrix, the criterion is fulfilled.
4 The Receiver Structure
The reconstructed signals are obtained by applying the matched
filter HHc to the received vector,
Y = HHc HcS +H
H
c Nc (6)
Y =


G 0 ε 0
0 G 0 ε
−ε 0 G 0
0 −ε 0 G




s1
s2
s3
s4

+HHc Nc (7)
where ε = h∗3h1 − h3h∗1 = j2Im {h∗3h1} and G = |h1|
2
+
|h2|
2
+ |h3|
2
.
Equation (7) can be rewritten in terms of {s1, s3} and {s2, s4},
since the pairs are decoupled from each other completely.[
y1
y3
]
= ∆
[
s1
s3
]
+
[
n′1
n′3
]
(8)
[
y2
y4
]
= ∆
[
s2
s4
]
+
[
n′2
n′4
]
(9)
where ∆ =
[
G ε
−ε G
]
.
In order to reconstruct the symbols at the receiver side, a
maximum likelihood (ML), a zero forcing (ZF) and a minimum
mean squared error (MMSE) processing can be applied to each
decoupled pair of matched filter outputs.
The generalization of the proposed code to N=2 receiver an-
tennas is derived from the structure given in [1]. The com-
bined signals from the two receiver antennas are a simple ad-
dition of the combined signals from each antenna, where the
diversity gain and interference term in Equation (7) are equal
to G2 = |h11|
2
+ |h21|
2
+ |h31|
2
+ |h12|
2
+ |h22|
2
+ |h32|
2
and
ε2 = j2Im {h31h
∗
11 + h32h
∗
12} respectively, where hij is the
channel coefficient from ith transmit antenna to jth receiver
antenna.
4.1 Zero Forcing Processing
The ZF receiver is applied to the matched filter outputs in (8)
and (9) as, [
y1,ZF
y3,ZF
]
= ∆−1
[
y1
y3
]
(10)
[
y2,ZF
y4,ZF
]
= ∆−1
[
y2
y4
]
(11)
where ∆−1 = 1
G2+ε2
[
G −ε
ε G
]
.
Since only the inversion of 2x2 matrix (instead of 4x4 ma-
trix) is required ZF processing is a computationally simple re-
ceiver, however it comes at the cost of noise enhancement.
4.2 Maximum Likelihood Processing
The ML receiver is implemented to recover the symbols by tak-
ing into account the effect of self symbol interference for each
decoupled symbol pairs.
min
{s1,s3}∈A×A
∥∥∥∥
[
y1
y3
]
−∆
[
s1
s3
]∥∥∥∥
2
(12)
min
{s2,s4}∈A×A
∥∥∥∥
[
y2
y4
]
−∆
[
s2
s4
]∥∥∥∥
2
(13)
where A is the alphabet shared by all the substreams.
The use of decoupling structure leads to less computational
complexity compared to the full ML processing.
5 Capacitive Derivations
The comparison between the open-loop (M,N) channel capac-
ity and the maximum mutual information that a (M,N) STBC
code can achieve is important to evaluate the efficiency of the
code. The channel capacity of a (M,N) antenna system under
the condition that the channel is known at the receiver and no
feedback is sent back to the transmitter, is calculated in [10] as,
C(ρ,M,N) = E log2 det
(
IN +
ρ
M
HHH
)
(14)
where ρ is the signal-to-noise ratio at each receive antenna and
H is the channel coefficient matrix.
In [10], the achievable maximum mutual information by a
(T,M,N) space-time code is given as,
C(ρ, T,M,N) =
1
T
E log2 det
(
INT +
ρ
M
HcH
H
c
)
(15)
It can be shown that for the (4,3,1) proposed code using ML
receiver, the achievable capacity is equal to
CML(ρ) =
1
4
E
[
log2 det
(
I4 +
ρ
3
∆
)]
(16)
When a zero forcing detector is used, the capacity is reduced
to the following expression:
CZF (ρ) = E
[
log2
(
1 +
ρ
3
(
G2 + ε2
G
))]
(17)
The open loop channel capacity and maximum mutual infor-
mation that the proposed and existing codes can be obtained by
using equations (14) and (15).
6 Simulation Results
In this section, we provide simulation results which are ob-
tained by using QPSK over Rayleigh fading channels for the
proposed code and existing codes.
In Figure 1, we show the bit-error-rate (BER) performance of
several schemes. According to results, the proposed code with
ML detection gives better performance than the ZF processing
at the cost of some computational expense. For BER= 10−4,
the (3,1) proposed code with ML provides 2dB diversity gain
compared to the (2,1) scheme in [1]. It has almost the same
BER performance as the non-orthogonal code for 4 transmit
antennas given in [7]. The (3,2) proposed scheme provides an
additional diversity gain compared to the (2,2) code in [1].
0 2 4 6 8 10 12 14 16 18 20
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
Eb/N0
Un
co
de
d 
Bi
t E
rro
r R
at
e
(2,1) Code in [1]
(3,1) Proposed Code with ML
(3,1) Proposed Code with ZF
(4,1) Code in [7] with ML
(2,2) Code in [1]
(3,2) Proposed Code with ML
FIG. 1: Bit error rate performances in Rayleigh fading channel
for proposed and existing code schemes
Figure 2 shows the open loop channel capacity compared to
maximum mutual information obtained by the proposed code
and the codes given in [5], [7] and [11]. According to these
results, the proposed code which has low complexity encod-
ing and decoding scheme, achieves a higher mutual informa-
tion than the R=1 code in [5] and gives the same mutual in-
formation as the LD code given in [11] which was obtained
by using information-theoretic optimization criterion. Further-
more, compare to the LD code, all the elements of the diagonal
HHc Hc of the proposed code are identical. Also, the proposed
code has almost the same capacity with the (4,1) code in [7].
For the (3,2) code, the achievable mutual information is sig-
nificantly lower than the open loop channel capacity with M=3
and N=2 as shown in Figure 3. For example, at SNR = 20dB
the capacity of (3,2) proposed scheme is only 61.36% of the
open loop channel capacity while the capacity of the (3,1) pro-
posed code is 97.48%.
7 Conclusion
In this paper, we introduced a non-orthogonal space-time block
code scheme for complex symbols with symbol rate 1 employ-
ing 3 transmit antennas by constructing low complexity re-
ceiver structure. Simulation results demonstrate that the scheme
achieves diversity gains without bandwidth loss. The achieved
capacity of this scheme is almost equal to the capacity of the
open loop channel.
8 Acknowledgement
The authors would like to thank for their support the Institut
Aeronautique et Spatial (IAS).
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0 2 4 6 8 10 12 14 16 18 20
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1
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6
7
SNR (dB)
C(
bit
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(3,1) Open Loop Capacity
(4,3,1) Proposed Code with ML
(4,3,1) Proposed Code with ZF
(4,3,1) LD Code in [11]
(3,3,1) Code in [5]
(4,4,1) Code in [7] with ML
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SNR (dB)
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bit
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(3,2) Open Loop Capacity
(4,3,2) Proposed Code with ML
(2,2) Open Loop Capacity
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Non-orthogonal space-time block coding design for 3 transmit antennas

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