New Space-Time Code Design for Prime Number of
Transmitter Antennas
Terasan Niyomsataya
Ali Miri
Monica Nevins
School of Information Technology
and Engineering
University of Ottawa
Ottawa, Canada K1N 6N5
School of Information Technology
and Engineering
University of Ottawa
Ottawa, Canada K1N 6N5
Department of Mathematics
and Statistics
University of Ottawa
Ottawa, Canada K1N 6N5
tniyomsa@site.uottawa.ca
samiri@site.uottawa.ca
mnevins@uottawa.ca
Abstract— A full diversity constellation, that is, a set of unitary
matrices whose differences have nonzero determinant, is a design
criterion for codes with good performance using differential
unitary space-time modulation.
In this paper, we present a new Bruhat decomposition design
for constructing full diversity unitary constellations for a prime
number of transmitter antennas. We give examples of our proposed constellations for 3 and 5 transmitter antennas. Simulations
show that these proposed constellations perform well in unknown
channel, and also outperform the best abelian group designs.
Index Terms— Differential unitary space-time modulation, full
diversity, Bruhat decomposition, permutation representation
I. I NTRODUCTION
channel matrix. The fading coefficients of a channel matrix
and the additive noise on receiver antenna are independent
complex Gaussian variables with mean zero and variance one,
CN (0, 1). Here ρ is the SNR at each receiver antenna. Let
Vzτ be an M × M unitary message matrix at block τ which
is chosen from an alphabet zτ ∈ {0, 1, . . . , L − 1}. The
date rate is R = log2 L/M . The transmitted signal matrix
is Sτ = Vzτ Sτ −1 with S0 = IM where IM is an M × M
identity matrix. We assume that the channel matrix is constant
over two consecutive time periods, that is Hτ ≈ Hτ −1 . Then
the received signal matrix of (2) can be written as
√
Xτ = Vzτ Xτ −1 + 2Wτ0
(3)
The design problem of differential unitary space-time modulation [1], [2] is to maximize the minimum value of the
distance
(1)
d(l, l0 ) = | det(Vl − Vl0 )| ≥ 0,
where Wτ0 is also independent CN (0, 1). The ML decoder is
used for decode a message ẑτ to be
for all distinct Vl , Vl0 ∈ V, a set of M × M unitary matrices
where M is the number of transmitter antennas. A constellation V for which d(l, l 0 ) > 0 for all l 6= l0 is said to have full
diversity. The application of the Bruhat decomposition to the
design of a full diversity unitary subgroup constellations was
first introduced in [3] where it was developed only for 2 and
4 transmitter antennas, and used a method not applicable to
an odd number of transmitter antennas.
In this paper, we present a new Bruhat decomposition design
for constructing full diversity unitary space-time constellations
for a prime number of transmitter antennas. Simulations show
that our proposed constellations perform well in unknown
fading channel, and also outperform the best abelian group
design, a cyclic group [1], [2].
III. M ATHEMATICAL P RELIMINARIES
II. D IFFERENTIAL U NITARY S PACE -T IME M ODULATION
Consider a multiple-antenna system in a Rayleigh flat fading
channel with M transmitter and N receiver antennas. The M ×
N received signal matrix Xτ is
√
(2)
Xτ = ρSτ Hτ + Wτ , τ = 0, 1, . . .
where Sτ is the M × M transmitted signal matrix, Wτ is
the M × N additive noise matrix and Hτ is the M × N
ẑτ = arg
min
l=0,1,...,L−1
kXτ − Vl Xτ −1 k.
(4)
Let Sn denote the symmetric group of order n, that is, the
group consisting of all permutations of n elements. The symmetric group Sn has a natural representation on C n , called
the permutation representation, in which each element ω ∈ Sn
is realized as the n × n matrix of the linear transformation
which permutes the standard basis vectors e1 , e2 , . . . , en of C n
according to ω. For example, in S3 , the permutation ω = (123)
sends e1 to e2 , e2 to e3 and e3 to e1 . Hence under the natural
representation, we identify ω with the unitary matrix


0 0 1
ω = 1 0 0 .
0 1 0
The symmetric group Sn has a special relationship with the
general linear group GL(n) (that is, the group of all invertible
n × n matrices) defined by the Bruhat decomposition. The
Bruhat decomposition of GL(n) is the disjoint union of double
cosets
[
GL(n) =
BωB
(5)
ω∈Sn
where B denotes the subgroup of all invertible upper triangular
matrices, and ω is written in its natural representation.
Whereas the group B is not unitary, it contains the subgroup
T of diagonal unitary matrices. Restricting to T in equation
(5) defines a unitary subgroup of GL(n). A quick calculation
reveals that the double coset T ωT is equal to the left coset
ωT for any ω ∈ Sn . This defines an infinite unitary subgroup
of GL(n), Sn n T , a semi-direct product as:
[
[
Sn n T =
T ωT =
ωT.
(6)
ω∈Sn
OF F ULL D IVERSITY
C ONSTELLATION
U NITARY
In this section, we present the design of a full diversity
unitary constellation using Bruhat decomposition as explained
in Section III, equation (6). Let M be the number of transmitter antennas and ω be an M × M matrix representing the
full permutation (1 2 3 · · · M ) in the natural representation.
This cycle has the property that neither ω, nor any power of
ω (up to ω M = IM ), has any fixed points. Consequently, the
positions of the nonzero entries in ω i , or equivalently, in ω i T ,
are entirely disjoint from those for ω j for any i 6= j.1 Choose
a finite full diversity subgroup D of diagonal unitary matrices
2
having determinant equal to 1. Set λ = e2πj/M . Our proposed
unitary constellation V is a subset of Sn n T , given by the
disjoint union of sets of unitary matrices defined by
V=
M
−1
[
i=0
ddc (l, l0 ) = | det(D − λi ω i D0 )|.
i
i
λ ω D.
(7)
Write D ∈ D as D = diag(d1 , d2 , . . . , dM ), with |dj | = 1
and d1 d2 · · · dM = 1. Hence det(λi ω i D) = ±e2πji/M for
all i = 0, 1, . . . , M − 1. In order to construct a full diversity
constellation, we need to have d(l, l 0 ) > 0 (see (1)). This
problem can be broken to two cases:
i) Same Coset (sc) Case: Suppose Vl , Vl0 ∈ V lie in the
same coset λi ω i D. Then we can write Vl = λi ω i D and Vl0 =
λi ω i D0 for some D, D0 ∈ D. We compute
dsc (l, l0 ) = | det(Vl − Vl0 )|
= | det(λi ω i )|| det(D − D0 )|
= | det(D − D0 )|.
(8)
Thus the minimum value of dsc (l, l0 ) equals the minimum
distance between two elements of D. Hence, in order to have
dsc (l, l0 ) > 0, it suffices to require the full diversity of the
subgroup D.
ii) Distinct Coset (dc) Case: Suppose now that Vl , Vl0 ∈ V
lie in distinct cosets; says Vl = λi ω i D and Vl0 = λj ω j D0
for some D, D0 ∈ D. Then as above, we may restrict
ourselves to the case where Vl ∈ D and Vl0 ∈ λi ω i D,
1 This property makes the computation of the distance between distant cosets
more accessible to theoretical analysis.
(9)
When the number of transmitter antennas is prime, any power
ω i of the full permutation ω is a single cycle. It follows that
the distance of this case can be simply computed as
ddc (l, l0 ) = |λiM − 1|.
ω∈Sn
However this subgroup does not have full diversity, since T
itself does not. The idea of construct a full diversity subgroup
is to restrict to a full diversity subgroup of T and constraint
a constellation from its cosets.
IV. D ESIGN
since | det(λk ω k D − λj ω j D0 )| = | det(λk ω k )|| det(D −
λj−k ω j−k D0 )| = | det(D − λj−k ω j−k D0 )|. This gives
(10)
This value is equal to the Euclidean distance between two
points on a unit circle which is always greater than 0 (when
2
λiM 6= 1). It follows that the value of λ = e2πj/M maximizes
the minimum value of ddc (l, l0 ) by ensuring the M points
λiM are equally spaced along the unit circle.
Design Method Summary: We now summarize the design
method of a full diversity unitary constellation as follows:
2
1) Construct a constellation by (7) which has λ = e2πj/M
for a given M .
2) Choose D to be a unitary cyclic diagonal subgroup
D −1
{Dl }L
l=0 , such that Dl = diag(d1 , d2 , . . . , dM ) =
j2πu1 l/LD j2πu2 l/LD
, . . . , ej2πuM l/LD ) where
,e
diag(e
M
the values of {ui }i=1 ∈ {1, 2, . . . , LD − 1} must be
chosen so that Dl satisfies det Dl = 1, and also so that
the distance dsc (l, l0 ) = | det(Dl − Dl0 )| is maximized.
3) The order and the rate of our proposed constellation are
LD M and log2 (LD M )/2 respectively.
V. E XAMPLES & P ERFORMANCE
We give the examples of constructing our proposed full
diversity unitary constellations for M = 3 and 5 transmitter
antennas using the Bruhat decomposition as discussed in
Section IV, and also show their performance in unknown
Rayleigh flat fading channels. We use a differential modulation
to transmit signals (as explained in Section II). The ML
decoder of (4) is used for decoding.
A. For M = 3
Our proposed constellation for three transmitter antennas is
V3 = D ∪ λωD ∪ λ2 ω 2 D
(11)
with λ = ej2π/9 . The distance ddc (l, l0 ) may be computed as,
for i = 1, ω = (123), we have ddc = |λ3 − 1| = 1.7321 > 0,
and for i = 2, ω = (132), we have ddc = |λ6 −1| = 1.7321 >
0. The choice of Dl = diag(d1 , d2 , d3 ) can be made by setting
−1
d3 = d−1
1 d2 . Consequently the constellation, of order 3L D ,
is V3 =




0 0 1
0 1 0
D ∪ ej2π/9  0 0 1  D ∪ ej4π/9  1 0 0  D (12)
0 1 0
1 0 0
−2
where each Dl ∈ D has the form Dl = diag(d1 , d2 , d−1
1 d2 )
with d1 = ej2πu1 l/LD and d2 = ej2πu2 l/LD . The values
of u1 , u2 = {1, 2, . . . , L − 1} are also chosen to maximize
dsc (l, l0 ) for a given LD . Table I shows some of our proposed
constellations with their best diversity products (as defined
0
10
−1
10
bler
in [4]) compared with different 3 × 3 unitary constellation
designs in the literature. We can see that our proposed constellations have diversity products higher than cyclic groups
and some constellations obtained from fixed-point free groups.
Figure 1 displays the block error rate (bler) performance of
our proposed 3 × 3 constellation u1 = 1, u2 = 4 at R = 2.04
compared with the cyclic group u = (1, 11, 27) at R = 2.00 as
given in Table I with N = 1. We can see that our constellation
outperforms the cyclic group with ≈ 2 dB gain improvement.
B. For M = 5
Our proposed constellation for five transmitter antennas is
V5 = D ∪ λωD ∪ λ2 ω 2 D ∪ λ3 ω 3 D ∪ λ4 ω 4 D
−2
10
(13)
our proposed code
cyclic
j2π/25
where λ = e
. The choice of a subgroup of unitary
diagonal matrices D = {Dl }L−1
l=0 is
10
0
2
4
6
8
(14)
where d1 = ej2πu1 l/LD , d2 = ej2πu2 l/LD , d3 = ej2πu3 l/LD
and d4 = ej2πu4 l/LD . The values of u1 , u2 , u3 , u4 =
{1, 2, . . . , LD − 1} are chosen to maximize dsc (l, l0 ) for a
given LD . The order of this constellation is 5LD . Table I
also compares our proposed 5 × 5 constellation with different
5 × 5 unitary constellation designs at the data rate R ≈ 1.00.
Figure 2 shows the block error rate (bler) performance of our
proposed 5 × 5 constellation as given in Table I with N = 1.
We can see that our proposed constellation outperforms the
cyclic group and compares well with the nongroup constellation.
10
SNR in db
12
14
16
18
20
Fig. 1. Block error rate performance for M = 3, N = 1 at R ≈ 2.00 of
our proposed constellation and cyclic group
0
10
−1
10
−2
10
bler
Dl =
−1 −1 −1
diag(d1 , d2 , d3 , d4 , d−1
1 d2 d3 d4 )
−3
−3
M
3
3
3
3
3
3
3
3
3
3
3
3
3
5
5
5
LD M
9
9
57
57
63
63
63
64
69
513
513
4095
4095
32
33
35
R
1.06
1.06
1.94
1.94
1.99
1.99
1.99
2.00
2.04
3.00
3.00
4.00
4.00
1.00
1.01
1.03
ζV
0.6004
0.6004
0.4845
0.4845
0.3301
0.3851
0.3851
0.2765
0.3548
0.1353
0.1436
0.0361
0.0471
0.4095
0.5580
0.5164
Designs
fixed-point free group G9,1 [4]
proposed design u1 = 1, u2 = 1
nongroup S19,3 , u = (1, 7, 11) [4]
proposed design u1 = 1, u2 = 7
cyclic group u = (1, 17, 26) [2]
fixed-point free group G21,4 [4]
proposed design u1 = 1, u2 = 4
cyclic group u = (1, 11, 27) [2]
proposed design u1 = 1, u2 = 4
fixed-point free group G171,64 [4]
proposed design u1 = 1, u2 = 22
fixed-point free group G1365,16 [4]
proposed design u1 = 1, u2 = 121
cyclic group u = (1, 5, 7, 9, 11) [2]
nongroup S11,3 u = (1, 3, 4, 5, 9) [4]
proposed design u = (1, 1, 4, 5)
TABLE I
C OMPARISON OF DIFFERENT UNITARY CONSTELLATION DESIGNS , (O UR
PROPOSED CONSTELLATION IS HIGHLIGHTED IN GREY )
10
−4
10
our proposed code
cyclic
nongroup
−5
10
0
2
4
6
8
SNR in db
10
12
14
16
Fig. 2. Block error rate performance for M = 5, N = 1 at R ≈ 1.00 of
our proposed constellation, cyclic group and nongroup
ACKNOWLEDGMENT
This work was partially supported by grants from Natural Sciences and Engineering Research Council of Canada
(NSERC), and Communications and Information Technology
Ontario (CITO).
R EFERENCES
VI. C ONCLUSION
We have proposed the design of a full diversity unitary
space-time constellation for a prime number of transmitter antennas using Bruhat decomposition. Simulations show that our
proposed constellations perform well without the knowledge
of the information of the channel. The evaluation of the full
diversity condition of ddc (l, l0 ) in (9) can be extended to a
general number of transmitter antennas.
[1] B.M. Hochwald and W. Sweldens, “Differential unitary space-time modulation,”IEEE Trans. Info. Theory, vol. 48, pp. 2041-2052, Dec 2000.
[2] B.L. Hughes, “Differential space-time modulation,” IEEE Trans. Info.
Theory, vol. 46, pp. 2567-2578, Nov 2000.
[3] T. Konishi, “Unitary subgroup space-time codes using Bruhat decomposition and Weyl groups,” IEEE Trans. Info. Theory, vol. 49, pp. 2713-2717,
Oct 2003.
[4] A. Shokrollahi, B. Hassibi, B.M. Hochwald and W. Sweldens, “Representation theory for high rate multiple-antenna code designs,” IEEE Trans.
Info. Theory, vol. 47, pp. 2335-2367, Sept 2001.