lSlT 2003,Yokohama, Japan, June 29 - JUIY4, 2003
On Single-Symbol and Double-Symbol Decodable STBCs
Md. Zafar Ali Khan
Indian Institute of Science
Bangalore, India 560 012
zafar(0protocol.ece.iisc.ernet.in
B.Sundar Rajan’
Moon Ho Lee
Indian Institute of Science
Bangalore, India 560 012
e-mail: bsrajanQece.iisc.ernet.in
Chonbuk National University
Chonju, Korea
e-mail: moonho(0chonbuk.ac .kr
c&,l
Abstract - A characterization of all single-symbol Square linear STBC ,!? =
X k I a 2 k -k X k Q a z k + i such that
decodable designs (SSDD) (with or without full= 0, k # 1, and
= Dk,t/k,
diversity) is presented and a rate-one, double-symbol
decodable, full-diversity coordinate-interleaved or- where v k is a diagonal matrix. Further ,!? achieves full diverthogonal design based on CIOD is given for eight sity iffeither (i)
is of full rank for all k or (ii) v 2 k + 2 ) 2 k + l
transmit antennas.
is full rank for all k = 0 , 1 , . . . ,K - 1 and the C P D of A # 0.
SSDDs that correspond to the condition (i) are precisely
I. SINGLE-SYMBOL
DECODABLE
LINEARSTBCs
the square GLPCODs. The second condition (ii) gives the
A linear Design S is a L x N matrix whose entries are important consequence of the Proposition 2: There can excomplex linear combinations of K complex indeterminates ist designs that are not covered by GLPCODs offering fullxk = xkl j x k g , k = 0,. . . ,K - 1 and their complex con- diversity and single symbol decoding provided the associated
jugates. Any Space-Time Block Code (STBC) obtained by signal set has non-zero CPD. The SSDDs that are not GLPletting each indeterminate to take all possible values from a CODS are such that VZk and/or 2&?k+l is not full rank for at
complex constellation A is called a linear STBC over A. The least one k . The designs of [2] belong to this category. We
rate of the code is given by.K/L symbols/channel use. The call such SSDD codes Generalized Complex Restricted Design
ML decoding metric for multiple transmit and receive antenna (GCRD), since any full-rank design within this class can be
system in general results in exponential decoding complexity there only with restrictions on the complex constellation from
with the rate of transmission in bits/sec/Hz. If it can be writ- which the indeterminates take values. When a GCRD satisfyten as sum of several terms with each term involving only one ing the conditions for full-rank is used along with a signal set
(two) variable(s) then S is said to be a SSDD (Double-Symbol with non-zero CPD we simply refer the design as Full-Rank
Decodable Design (DSDD)). STBCs from Generalized Linear GCRD (FRGCRD).
Processing Complex Orthogonal Designs (GLPCOD) [l]are Theorem 2: There exists square FRGCRDs with the maxfor N = 2O antennas whereas only rates up to
well known due to their ML single-symbol decoding and full- imal rate
diversity. These designs can be used with any complex con- % is possible with square GLPCODs with the same number
stellation. With a minor restriction on the allowed constella- of antennas. Moreover, rate-one square FRGCRD of size N
tions, a variant of these designs called co-ordinate interleaved exist, iff N = 2,4.
orthogonal designs (CIOD) have been shown to admit single- A rate-one DSDD for Eight Transmit Antennas: Let
20,$1, XZ,2 3 , ~ 4 ~ 2 5 ~and
x 6 27 denote eight complex indetersymbol decoding along with full-diversity [2].
+ jxig, i = 0 , 1 , . . . , 7. Define,
Theorem 1 characterizes all linear designs that lead to minates, where xi =
STBCs admitting single-symbol decoding for arbitrary com- Eo = xor jzzg; 21 = zir jx3g; E 2 = xzr j x o q ; E 3 =
231
jxIg;E4 = 2 4 1 j x ~ g ; 25 = E51 j x r g ; 56 =
plex constellations.
Theorem 1: For a linear STBC using the design S in K com- 2 6 1 +jx4Q; 57 = 271 +jxSQ to be the new complex indeterminates. The following design for eight transmit antennas is of
plex variables, =
X k r A z k +xkQAzk+l, with arbitrary
complex constellation, the ML metric M ( S ) decomposes as rate-one and double-symbol decodable. It gives full-diversity
,
E6 and 27 takes values
M ( S ) = E&‘ Mk(Xk)+MC where MC = -(K-l)tr ( v ~ v when
) the variables X O , X ~22,23,24,25,
that is independent of all the variables (V is the received sig- from a signal set with nonzero CPD.
nal matrix) and hfk(xk) is a function of only the variable Xk,
E5
0
0
Zl
0
0
’
Eo
54
iff
0
0
5;
0 0 -E; 5:
-5;
0
0
E2
23
0
0
56 2 7
AfAi
=O,O 5 k # 15 2 K - 1.
(1)
0 0 -E;
2.2’ 0 0 -E;
2;
Examples of SSDDs are OD, in-particular the Alamouti
0
0
0 0
Eo 41
E4 E5
code, and the CIODs of [2]. That the CIODs constitute a
0
0
0 0 -E? E:
-z; 5:
subclass of SSDD can be verified by substituting the weight
0
0
5 6 27
0
0
22
23
matrices of CIOD, S, in (1).
.
0 0 -2; 2;
0
0 -5; 2.2’
Definition 1:[2] The Coordinate Product Distance (CPD)
between any two signal points U = ur j u g and v = vr j u g ,
REFERENCES
U # U, in the signal set A is defined as C P D ( u , v ) = (ur vr I ~ U Q - vg I and the minimum of this value among all possible [l] H. J. V.Tarokh and A. R. Calderbank, “Space-time block codes
from orthogonal designs,” ZEEE ”hnsaction on Information
pairs is defined as the CPD of A.
Theory, vol. 45,pp. 1456-1467, July 1999.
Proposition 2: A square linear STBC S in K complex vari[2] Md. Zafar Ali Khan and B. Sundar Rajan, “Space-time block
ables whose weight matrices satisfy (1)exits if€ there exists a
codes from co-ordinate interleaved orthogonal designs,”ZSZT
2002, pp 316,Lausanne, Switzerland, June 30 - July 5 also sub‘This work was partly funded by the DRDO-11% Program on
mitted to IEEE Zhnsaction o n Information Theory.
Mathematical Engineering through a grant to B.S.Rajan.
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0-7803-7728-1/03/$17.00
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