Fully-diverse constellations, i.e., a set of unitary matrices whose pairwise differences are nonsingular, are useful in multi-antenna communications, especially in multi-antenna differential modulation, since they have good pairwise error properties. Recently, group theoretic ideas, especially fixed-point-free (FPF) groups, have been used to design fully-diverse constellations of unitary matrices. Here we construct four-transmit-antenna constellations appropriate for differential modulation based on the symplectic group Sp(2) These can be regarded as extensions of S.M. Alamouti's celebrated two-transmit-antenna orthogonal design which can be constructed from the group Sp(1) (see IEEE J. Sel. Area Commun., p.1451-8, 1998). We further show that the structure of the code lends itself to efficient maximum likelihood (ML) decoding via the sphere decoding algorithm. Finally, the performance of the code is compared with existing methods including Alamouti's scheme, Cayley differential unitary space-time codes and group based codes

Hassibi, Babak

Jing, Yindi

Caltech Authors - Main

DESIGN OF FULLY-DIVERSE MULTI-ANTENNA CODES BASED ON SP(2)  
Yindi Jing and Babak Hassibi 
California Institute of Technology 
Department of Electrical Engineering 
Pasadena, CA 91 125 
ABSTRACT 
Fullydiverse constellations, i.e., a set of unitary matrices 
whose pairwise differences are nonsingular, are useful in 
multi-antenna communications, especially in multi-antenna 
differential modulation, since they have good pairwise er- 
ror properties. Recently, group theoretic ideas, especially 
fxed-point-free ( f p f ,  groups, have been used to design fully- 
diverse constellations of unitary matrices. Here we con- 
struct four-transmit-antenna constellations appropriate for 
differential modulation based on the symplectic group Sp(2 ) .  
These can he regarded as extensions of Alamouti's cele- 
brated two-transmit-antenna orthogonal design which can 
he constructed from the group Sp(1). We further show 
that the structure of the code leads itself to effcient max- 
imun likelihood (ML) decoding via the sphere decoding al- 
gorithm. Finally, the performance of the code is compared 
with existing methods including Alamouti's scheme, Cay- 
ley differential unitary space-time codes and group based 
codes. 
1. INTRODUCTION 
It is well known in theory that multiple antennas can greatly 
increase the data rate and the reliability of a wireless com- 
munication link in a fading environment. In practice, how- 
ever, one needs to devise effective space-time transmission 
schemes. This is particularly challenging when the propaga- 
tion environment is unknown to the sender and the receiver, 
which is often the case for mobile applications when the 
channel changes rapidly. 
A differential transmission scheme called differenrial uni. 
rary space-time modulation was proposed in [ 1,2,3],  which 
is well-tailored for unknown continuously varying Rayleigh 
mt-fading channels. The signals transmitted are unitary 
matrices. In this scheme the probability of error of mis- 
taking one signal Si for another s,,, at high SNR, is proved 
to he inversely proportional to I det(Si - Si,)l. Therefore 
Thanks to AFOSR for MURI "Mathematical Infrastructure for Ro- 
bust Virtual Engineering" and URI "Protecting Infrastructures from Them- 
sdvcs" and Caltech's Lee Cenler for Advanced Networking for Funding. 
the quality of the code is measured by its diversity product 
0-7803-7663-3/03/$17.00 02003 IEEE IV - 33 
where M is the number of transmit antennas and C is the 
set of all possible signals. We therefore say that a code is 
fully-diverse or has f i l l  diversify if the determinants of the 
pairwise differences are all nonzero. The design problem is 
thus the following: "Given the number of transmitter an- 
tennas, M ,  and the transmission rate, R, fnd a set C of 
L = Z M R  A4 x M unitary matrices, such that the mini- 
mum of the absolute value of the determinant of their pair- 
wise differences is as large as possible." 
The design problem, as just stated, appears to be in- 
tractable since frst  the signal set and the cost function are 
non-convex and second, the size of the problem can be huge, 
especially at high data rates. Therefore, in [4,51, it was pro- 
posed to enforce a group structure on the constellation. This 
has several advantages that are discussed in [4, 51. More- 
over, it is shown that a constellation is fully-diverse iff the 
corresponding group is €xed-point-free (fpf), i.e. all non- 
identity matrices have no eigenvalue at one. In [41, all E- 
nite fully-diverse constellations that form a group are clas- 
sifed. And also, in [5], it is proved that the only fpf infnite 
Lie groups are U(l) ,  the group of unit-modulus scalars, and 
SU(2) .  the group of unit-determinant 2 x 2 unitary matrices. 
However, no good constellations are obtained for very 
high rates from the fnite fpf groups classifed in [41, and 
constellations based on U(1) and SU(2)  are constrained 
to one and two-transmit-antenna systems. In this paper, 
to get high rate constellations which work for 4-transmit- 
antenna systems, we relax the fpf condition by considering 
Lie groups with non-identity elements having no more than 
k > 0 unit eigenvalues instead of no unit eigenvalues. It can 
he shown that if a Lie group has rank n, then it has at least 
one element with n - 1 eigenvalues at 1. (The rank of a Lie 
group equals the maximum number of commuting basis el- 
ements of its Lie algebra and it can he shown that fpf groups 
have rank 1. See [5] . )  The lower the rank, the more possi- 
ble it is to get a subset with no unit eigenvalue elements, that 
is, the more possible for us to fnd a fully-diverse subset of 
ICASSP 2003 
it. There are only three simply-connected, simple, compact 
Lie groups of rank 2 ,  the Lie group of unit-determinant 3 x 3 
unitary matrices SU(3) ,  the Lie group of unit-determinant 
4 x 4 unitary, symplectic matrices Sp(2)  and one of the ex- 
ceptional groups &. In this paper, we focus on Sp(2) .  The 
codes designed based on it are fully-diverse, can be used 
in four transmit antenna and any number of receive antenna 
systems, exist for almost any rate and lead themselves to 
polynomial-time ML decoding via the sphere decoder. 
1.1. Differential Unitary Space-time Modulation 
Consider a wireless communication system with M trans- 
mit antennas and N receive antennas. The channel is used 
in blocks of M transmissions (for more on this model, see 
[6, 71). the system equations of block T can be written as: 
X, = JIIS,H, + V'
Here, S denotes the M x M transmitted signal with stm 
the signal sent by the mth transmit antenna at time t. H is 
the A4 x N complex-valued propagation matrix ,which is 
unknown to both the transmitter and the receiver, and h,, 
is the propagation coeffcient between the mth transmit an- 
tenna and the nth receive antenna and has an iid CN(0,l) 
distribution. V is the M x N noise matrix with utn, the noise 
at the nth receive antenna at time t ,  iid CN(0,l) distribu- 
tion. X is the &I x N received signal matrix. The transmit- 
tedpowerconstraintisC,=,E 1stml = 1, t = 1, ..., M 
so p represents the expected SNR at each receive antenna. 
In differential modulation, the transmitted matrix S, at 
block T equals to the product of the previously transmitted 
matrix and a unitary data matrix Vz7 taken from our signal 
set C. In other words, S, = Vz7S,-, where So = I M .  The 
transmission rate is R = & log, L, where L indicates the 
cardinality of our code. Further assume that the propagation 
environment keeps approximately constant for 2 M  consec- 
utive channel uses, that is, H ,  H,-,, we may get the 
fundamental differential receiver equations [8] 
M 2 
x, = Vz7X7-,  + w: (2) 
where W: = W, - VZ7W7-,. We can see that the channel 
matrix H does not appear in (2). This implies that differ- 
ential transmission permits decoding without knowing the 
channel information. The ML decoder of z7 is given by 
It is shown in [ l ,  31 that, at high SNR, the pairwise proba- 
bility of error (of transmitting V, and erroneously decoding 
I+) has an upper bound that is inversely proportional to the 
diversity product of the code. 
2. MATH FUNDAMENTALS 
Defnition 1 (Fixed-point-free Group)  [ 5 J A  group G is called 
fxed-point-free ( fpo iff it has a representation a s  unitary 
matrices with the property that the representation of each 
non-unit element of the group has no eigenvalue a t  unity 
It can be proved easily that constellations that form a group 
are fully-diverse iff the group is fpf. In [4], all fnite fpf 
groups, are classifed. These fnite fpf groups are few and far 
between although there exists an infnite number of them. 
Although these yield very good constellations at low to mod- 
erate rates, no good constellations are obtained for very high 
rates from them. This motivates the search for infnite fpf 
groups, in particular, their most interesting case, Lie groups. 
Defnitinn 2 (Lie Group) [9] A Lie group is a differential 
manifold which is also a group such that the group multipli- 
cation and inversion map are differential maps. 
Here are some examples of Lie groups. GL(n,C) is the 
group of nonsingular n x TI complex matrices. SL(n,C) 
is the group of unit-determinant nonsingular n x n com- 
plex matrices. U ( n )  is the group of n x n complex unitary 
matrices and SU(n)  is the group of unit-determinant n x n 
unitary matrices. The following result shows that the groups 
of interest to us are compact semi-simple Lie groups. 
Theorem 1 (Lie groups with Unitary Representations) [ S I  
A Lie group has a representation as unitary matrices iff i f  is 
a compact semi-simple group or  the direct sum of U (  1) and 
a compact semi-simple gmup.  
It is  proved in [ 5 ] ,  that the only fpf infnite Lie groups 
are U(1) and S U ( 2 ) .  Due to their dimensions, constella- 
tions based on the two Lie groups are constrained to one and 
two-transmit-antenna systems. To obtain a four-transmit- 
antenna constellation, we relax the fpf condition and con- 
sider compact semi-simple Lie groups whose non-identity 
elements have no more than k > 0 unit eigenvalues (k = 0 
corresponds to fpf groups.) In designing a constellation of 
fnite size, we need to sample the Lie group's underlying 
manifold. When k is small, there is a good chance that, 
sampling appropriately, the resulting code is fully-diverse. 
In general, it does not seem that there is a straightforward 
way to analyze the number of unit eigenvalues of a matrix 
element of any given Lie group. However, it is possible 
to relate the number of unit eigenvalues to the rank of the 
group. We have proved that if a matrix Lie group G has 
rank r ,  then it  has at least one non-identity element with 
T - 1 unit eigenvalues. Therefore, instead of exploring Lie 
groups whose non-identity elements have no more than IC 
unit eigenvalues, we study compact semi-simple Lie groups 
with rank no more than k+ 1 and design codes that are fully- 
diverse subsets of it. Since semi-simple Lie groups can he 
IV - 34 
written as a direct product of simple Lie groups, we frst 
consider simple, simply connected, compact Lie groups in- 
stead of semi-simple ones with rank 2. As mentioned in the 
introduction, there are three of them: SU(3) .  Sp(2) and '&. 
Since Sp(1) = SU(2).  and SU(2)  constitutes the orthog- 
onal design of Alamouti [IO], the symplectic group Sp(2 )  
can be regarded as a generalization of orthogonal designs. 
DeBnition 3 (Symplectic Group) [ I 1 1  Sp(n), the ntli or- 
der symptectic group, is the set of complex 271 x 2n matrices 
S obeying the Unitary condition: S'S = SS' = I2,, and 
the symplecric condition: S 'JS  = J .  
where J = [ -? 1. S' denotes the transpose of S 
and S' denotes its conjugate transpose. 
Sp(n)  has dimension n(2n + 1) and rank n. We are 
most interested in the case of n = 2.  Actually, it is readily 
shown that the maximum number of unit eigenvalues of any 
non-identity element in Sp(2)  is 2. 
3. SP(2)  FULLY-DIVERSE CODE DESIGN 
From Defnition 3, it is easy to see that any 2n x 2n ma- 
trix in Sp(n) has the form - B  A for some complex 
n x n matrices A and B. The group can be identifed as the 
subgroup of unitary matrices with a structure that is simi- 
lar to Alamouti's 2-dimensional orthogonal design [lo], but 
here each entry is an n x n matrix instead of a scalar. Using 
the unitary condition of S and singular value decomposition 
of A and B,  the following theorem can be proved. 
Theorem 2 (Parametrization of Sp(n)) Any matrix S be- 
longs to Sp (n )  iff it can be written as 
[ A  " 1  
where U and V are any n x n unitary matrices, and 
CA = diag(cosgl, ... cosO,),CB = diag(sin0, .... sing,,) 
for some real angles 81, 82 ..., 8,. 0 and V denote the con- 
jugates of U and V .  
Since any n x n unitary matrix has dimension nz, there 
are all together 2n2 degrees of freedom in the unitary ma- 
trices U and V .  Together with the n real angles, t?,, the di- 
mension of S is, therefore, n(2n + l), which is exactly the 
same as that of Sp(n) .  Based on Theorem 2, the matrices 
in Sp(n)  can be parameterized by U, V and 0,s. 
Now, let us look at the easiest case of n = 2. For sim- 
plicity, we frst let C A  = C s  = h12. by which 2 degrees 
of freedom are lost. We further choose U and V as orthog- 
onal designs with Ad-PSK and shifted I\'-PSK entries. The 
following code is obtained 
L -  
M ,  0 5 m, n < N and A I  and N are integers. 8 is an an- 
gle to be chosen later. The rate of the code is $(log, iZf + 
log2 N). The angle 9, an extra degree of freedom added to 
the code to gain diversity product, is crucial in the proof of 
the full diversity of the code although simulation results in- 
dicates that the code always get its highest diversity product 
at 9 = 0. 
Since the U and V in our code have an orthogonal de- 
sign structure, it is not diffcult to calculate the determinant 
of the difference of any two signals in the code directly. Us- 
ing this calculation, we can prove the following theorem. 
Theorem 3 (Condition for full diversity) There exists a 0 
such that the code C A C , ~ .  in (4) is fully-diverse iff M and N 
are relatively prime. 
To get codes at higher rates, we can add the two degrees 
of freedom in diagonal matrices C A  and C B  in by letting 
C A  = cosyi12,CB = siny,12 for T~ E r. The full diver- 
sity of the modifed codes can be proved similarly when 8 
and the set r are properly chosen. 
4. DECODING OF THE SP(2) CODE 
One of the most prominent properties of our Sp(2)  code 
is that it can be seen as a generalization of orthogonal de- 
signs. This property can be used to get linear decoding, 
which means that the receiver can be made to form a sys- 
tem of linear equations in the unknowns. 
From (3), the ML decoder is equivalent to, 
Note that the formula is quadratic in the entries of U and 
V .  Using the property that U and V constitutes orthogonal 
designs, it can be shown that the ML decoder reduces to, 
where A, B, C,  D are 4 x 4 real matrices which only de- 
pandonX,andX,-i a n d a =  [ c o s ~ , s i n ~ ~ c o s ~ ,  
s i n ~ , c o s ~ , s i n ~ , c o s ~ , s i n ~ ]  bl is the vector 
of unknowns. 
t .  
IV - 35 
We can see form formula (5) that the decoding crite- 
rion is quadratic in the sine and cosine of the unknowns. 
Thus, it can be solved using the sphere decoder algorithm 
[12]. By choosing M odd, the map f : 0 + s in0  for 
0 E { O >  g,. . . ,v} is a one-to-one and onto map. 
Therefore, we can equivalently regard sin % and sin 
to he our unknowns instead of k and 1. And the same for 
m and n. Also notice that there are actually 4 independent 
unknowns instead of 8 in (5). We combine the 2i-th compo- 
nents (of the form cosz) and the (% + 1)-th component (of 
the form s inx )  together in the sphere decoding. 
5. SIMULATION RESULTS 
In this section, the performance of the Sp(2 )  code is com- 
pared with other codes. The block error rate (bler), which 
corresponds to errors in decoding the 4 x 4 transmitted ma- 
trices, is demonstrated as the error event of interest. 
In Fig 1, we compare our S p ( 2 )  code of M = 5, N = 
3 and rate R = 1.95 with rate 2 orthogonal design and a 
differential Cayley code at rate 1.75. The number of receive 
antenna is 1. At abler  of the Sp(2) code is 2dB better 
than the differential Cayley code, even though it has a lower 
rate, and 4dB better than the orthogonal design. 
In Fig 2, we compare our Sp(2)  code with a group- 
based diagonal code and the fpf code Kl,l,-l at rate 1.98 
[4]. The number of receive antenna is 1. At ab ler  of 
2dB improvement is obtained by using the Sp(2)  code in- 
stead of a diagonal code, but the Sp(2)  code is 1.5dB worse 
than the K1,1,-1 group code. However, decoding K ~ , I , - I  
requires an exhaustive search over the enure constellation. 
10 12 I* 18 18 2c 22 24 28 28 24 
SNR 
Fig. 1. Performance of S p ( 2 )  code with differential Cayley 
code and orthogonal design. 
Sp(2) code YS. group based codes 
t, 
1 
I 
10 l i  20 21 30 
SNR 
Fig. 2. Performance of S p ( 2 )  code with group-based diag- 
onal code and K ~ J - I  code. 
6. REFERENCES 
B. Hochwald and W. Sweldens. “Differential unitary space 
time modulation,” IEEE Trans. Comm., vol. 48, pp. 2041- 
2052, Dec. 2000, 
V. Tarakh and H. Jafarkhani, “Differential detection scheme 
for transmit diversity,” JSAC, pp. 1169-74, July 2000. 
B. Hughes, “Differential space-time modulation:’ IEEE 
Trans. IT, pp. 2567-2578, Nov. 2000. 
A. Shokrollahi, B. Hassibi, B. Hochwald, and W. Sweldens, 
“Representation theory for high-rate multiple-antenna code 
design,” IEEE Trans. IT, pp. 2335-2361, Sep. 2001. 
B. Hassibi and M. Khorrami, “Fully-diverse multi-antenna 
constellations and fxed-point-free Lie groups:’ submiffed 10 
IEEE Trans. IT, 2000, 
T. L. Marzetta and B. M. Hochwald, “Capacity of a mobile 
multiple-antenna communication link in Rayleigh Dat fad- 
ing,’’ IEEE Trans. IT, pp. 139-157, 1999. 
B. M. Hochwald and T. L. Marzetta, “Unitary space-time 
modulation for multiple-antenna communication in Rayleigh 
Qt-fading,” IEEE Trans. IT, pp. 543-564, Mar. 2000. 
B. Hassibi and B. Hochwald, “Cayley differential unitary 
spce-time codes:’ IEEE Trans. IT, pp. 1458-1503,2002. 
Brocker and Dieck, Represenrafions of compact Lie Groups, 
Springer-Verlag. 1995. 
S. M. Alamouti, “A simple transmitter diversity scheme for 
wireless communications,” IEEE J. Sel. Area Comm., pp. 
1451-1458, Oct. 1998. 
Barry Simon, Represenrations of inire and compact Lie 
Groups, Providence, R.I. : Amer. Math. Soc., 1994. 
M. 0. Damen, K. Abed-Meraim, and M. S.  Lemdani, “Fur- 
ther results on the sphere decoder algorithm,” personal com- 
municarions. 
IV - 36 


Design of fully-diverse multi-antenna codes based on Sp(2)

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Design of fully-diverse multi-antenna codes based on Sp(2)

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