We construct signal constellations for differential transmission with multiple basestation antennas. The signals are derived using the theory of fixed-point-free groups and are especially suitable for mobile

cellular applications because they do not require the handset to have more than one antenna or to know the time-varying propagation environment. Yet we achieve full transmitter diversity and excellent performance gains over a single-antenna system

Hassibi, Babak

Hochwald, Bertrand

Shokrollahi, Amin

Sweldens, Wim

Caltech Authors - Main

Codes for Differential Signaling with Many Antennas 
Babak Hassibi, Bertrand Hochwald, 
Amin Shokrollahi and Wim Sweldens 
Mathematical Sciences Center 
Lucent Technologies 
600 Mountain Avenue 
Murray Hill, NJ 07974 
e-mail: 
{hassibi,hochwald,mshokrollahi,wim}@lucent.com 
Abstracr - We construct simal constellations for differential 
transmission with multiple basestation antennas. The signals are 
derived using the theory of fixed-point-free groups and are espe- 
cially suitable for mobile cellular applications because they do 
not require the handset to have more than one antenna or to 
know the time-varying propagation environment. Yet we achieve 
full transmitter diversity and excellent performance gains over a 
single-antenna system. 
I. INTRODUCTION 
Differential phase-shift keying (DPSK) is a well-known technique 
for transmitting digital information across an unknown time-varying 
channel. Let the data consist of a sequence of integers t i ,  1 2 , .  . . E 
(0,. . . , L  - 1) where L is the size of our alphabet (often a power 
of two). DPSK with a single transmitter transmits complex baseband 
signals that obey the recursion 
S t  = ZtrtSt-l t = 1,2,.. . , 
where SO = 1 and the vzt E (1, e Z m i I L , .  . . ,e2ri(L-i)’L } are L 
points around the complex unit circle. 
We can extend this differential scheme to M > 1 transmit anten- 
nas by transmitting M x M matrices that obey the recursion 
s,=v*,s,-i 7 = 1 , 2  ,..., (1) 
where SO is the M x M identity matrix and V, are complex unitary 
data matrices [l]. (See also [3] for a similar differential scheme and 
[2] for a differential scheme based on orthogonal designs.) Each row 
of the transmission matrix S, specifies what is transmitted on the 
M transmit antennas; hence kach S, specifies what the M antennas 
do for M time samples. The index T marks each block of M time 
samples. 
To use this method effectively, we need to design L = Z R M  uni- 
tary data matrices V = {VO,. . ., VL-~}, where R is the data rate. 
We also need a simple decoding algorithm at the receiver. It can be 
shown that across an unknown flat-fading channel, the received sig- 
nals on N antennas for M units of time obey the recursion 
where W, is a M x N matrix of additive independent wmplex- 
Gaussian noise. Maximum likelihood decoding simply chooses the 
V-matrix that minimizes the Frobenius norm of the difference be- 
tween X, and V .  XT-i. 
Unlike with DPSK, the choice of effective unitary data matrices 
V,, is nontrivial. At high SNR, the performance of a constellation 
can be shown to depend on 
Gmuptype L it4 Comments 
1-  Gm,r mn n 
2- Dm,r,t 2mn 2n Contains quaternions 
3- Em,r 8mn 2n 
4. Fm,r,t 16mn 4n ifn > l o r  
e $1 mod m/3 
Fm,1,f 16mn 2 if!= 1modm/3  
5. Jm,r 120mn 2n 
6. Km,r,t 240mn 4n 
Table 1: There are six types of groups with < > 0: For each 
group G, L is the order of G (the size of the constellation) 
and M is the dimension of the representation of G (number of 
transmit antennas). n is the smallest integer such that rn 
1 (mod m). 
We wish to choose the constellation V to have 0 6 cv 6 1 as large 
as possible. 
11. GROUP CONSTELLATIONS 
Equation (1) is the fundamental differential transmission and we 
see that if the set V forms a group, the transmitted signal S, is always 
an element of V. In fact, the multiplication in (1) is replaced by a 
group table-lookup. We therefore ask if there are groups of unitary 
matrices that have large 5. Perhaps surprisingly, this question was 
examined in the early 20th century in the mathematical literature (in 
a different context!) and answered in large part in [5] using the theory 
of fixed-point-free groups. 
Briefly, a group that has a “representation” in unitary matrices 
with nonzero < is called fixed-point-free. Zassenhaus in [SI classifies 
most of the fixed-point-free groups. We complete the classification 
in [6]. We defer a detailed discussion of these groups to [6] and refer 
instead to Table 1 where the classes of the groups are outlined. Note 
that there are only six classes and any group with 6 > 0 must fall 
within one of them. We also list their orders (size of constellation L)  
and the dimension of their representations (number of antennas M). 
We see from the table that only the class G,,, gives signal con- 
stellations for odd M. The class G,,, also contains the diagonal 
signals corresponding to the cyclic groups studied in [l]. The class 
Dm,r,l contains the generalized quaternion (also called dicyclic) 
0-7803-6596-8/00/$10.00 0 2000 IEEE 23 
Figure 1: Performance of M = 1, 2, 3, 4, and 5 transmit 
antennas and N = 1 receiver antenna as a function of SNR 
p. The channel has unknown Rayleigh fading that is chang- 
ing continuously according to Jakes’ [4] model with parameter 
fDTB = 0.0025. The data rate is R = 1. 
groups studied in [3). Within the classes are many groups that per- 
form very well when simulated on a wireless channel, as can be seen 
in the next section. 
111. PERFORMANCE CURVES 
Figure 1 shows the performance of using cyclic ‘groups (corre- 
sponding in Table 1 to G,+) for M = 1,. . . , 5  antennas transmit- 
ting across a continuously fading channel to one receive antenna with 
transmission rate R = 1. (With M = 1 antenna, standard differential 
BPSK is used.) The advantages of using more than one antenna are 
readily seen. 
For higher rates, non-Abelian groups seem to provide better per- 
formance. As an example of a high-rate fixed-point-free group code, 
we plot in Figure 2 the performance of the group ~ 1 5 , 1 , 1 1 ,  whose 
details defy a simple description and appear in [6]. This group 
has a representation as 240 complex 2 x 2 unitary matrices (rate 
R = log(240)/2 x 3.95) suitable for differential transmission over 
a two-antenna fading channel. We also plot the performance of the 
best cyclic group with the same rate [l], a 2 x 2 orthogonal design [2] 
(which is not a group) and a generalized quaternion group code [3] 
with similar rates. All of these codes can also be used with a known 
channel with a performance gain of approximately 3 dB. 
Another example, appearing in Figure 3, is the group K i , i , - i  
(again defying a simple description) with a representation in 240 
complex 4 x 4 matrices (rate R = log(240)/4 x 1.98). We also 
plot the performance of the best cyclic group with the same rate and 
see that the performance gains are appreciable. 
REFERENCES 
[l] B. Hochwald and W. Sweldens, “Differential unitary spacetime mod- 
ulation,’’ to appear in IEEE Trans. CO”. Download available at 
http://mars.bell-labs.com 
[2] V. Tarokh and H. Jafarkhani, “A differential detection scheme for transmit 
diversity,” to appear in J. Sel. Area Comm 
[3] B. L. Hughes, “Differential space-time modulation,” submitted to IEEE 
Trans. Info. Theory, 1999. 
Figure 2: Block-error rate performance of the group F15,1,11 
for M = 2 transmitter antennas and N = 1 receiver antenna. 
The solid line is F15,1,11, which has L = 240 unitary matrices 
(R x 3.95). The dashed line is an orthogonal design with 16th 
roots of unity (R = 4). The dash-dotted line is the best diago- 
nal (Abelian group) construction (R m 3.95). The dotted line 
is the quatemion group (&~,1,-1) with L = 256 matrices 
(R = 4). 
[4] W. C. Jakes, Micnnvave Mobile Commwticarions, Piscataway, NJ: IEEE 
[5] H. Zassenhaus, “Uber endliche Fastkorper,’’ Abh. Math. Sem Hamburg, 
[6] B. Hassibi, B. Hochwald, A. Shokrollahi and W. Sweldens, 
“Representation theory for high-rate multiple-antenna code design,” 
submitted to IEEE Tmns. Info. Theory. Download available at 
http://mars.bell-labs.com. 
Res, 1993. 
vol. 11. pp. 187-220, 1936. 
10-5 lo  t 
I 1 0“ 
O 5 10 15 20 25 30 35 
SNR (a) 
Figure 3: Block-error rate performance of the group K t , l , - 1  
compared with the best diagonal code for M = 4 transmit- 
ter antennas and N = 1 receiver antenna. The solid line is 
K l , l , - l  having L = 240 unitary matrices (R x 1.98). The 
dashed line is a diagonal construction with the same rate. 
24 


Codes for differential signaling with many antennas

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Codes for differential signaling with many antennas

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