On single-symbol and double-symbol decodable stbcs

(1)

lSlT 2003, Yokohama, Japan, June 29

-

JUIY 4, 2003

On Single-Symbol and Double-Symbol Decodable STBCs

Md.

Zafar Ali

Khan B.Sundar Rajan’ Moon Ho Lee

Indian Institute of Science Chonbuk National University

Bangalore, India 560 012 Bangalore, India 560 012 Chonju, Korea

zafar(0protocol.ece.iisc.ernet.in e-mail: bsrajanQece.iisc.ernet.in e-mail: moonho(0chonbuk. ac

.

^kr

Indian Institute of Science

Abstract

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A characterization of all single-symbol decodable designs (SSDD) (with or without full- diversity) is presented and a rate-one, double-symbol decodable, full-diversity coordinate-interleaved or- thogonal design based on CIOD is given for eight transmit antennas.

I. SINGLE-SYMBOL

DECODABLE LINEAR STBCs

A linear Design S is a L x N matrix whose entries are complex linear combinations of K complex indeterminates xk = xkl

+

j x k g , k = 0,.

. . ,

K

-

1 and their complex con- jugates. Any Space-Time Block Code (STBC) obtained by letting each indeterminate to take all possible values from a complex constellation A is called a linear STBC over A. The rate of the code is given b y . K / L symbols/channel use. The ML decoding metric for multiple transmit and receive antenna system in general results in exponential decoding complexity with the rate of transmission in bits/sec/Hz. If it can be writ- ten as sum of several terms with each term involving only one (two) variable(s) then S is said to be a SSDD (Double-Symbol Decodable Design (DSDD)). STBCs from Generalized Linear Processing Complex Orthogonal Designs (GLPCOD) [l] are well known due to their ML single-symbol decoding and full- diversity. These designs can be used with any complex constellation. With a minor restriction on the allowed constellations, a variant of these designs called co-ordinate interleaved orthogonal designs (CIOD) have been shown to admit single- symbol decoding along with full-diversity [2].

Theorem 1 characterizes all linear designs that lead to STBCs admitting single-symbol decoding for arbitrary complex constellations.

Theorem 1: For a linear STBC using the design S in K complex variables,

s

⁼ X k r A z k +xkQAzk+l, with arbitrary complex constellation, the ML metric M ( S ) decomposes as M ( S ) =

E&‘

Mk(Xk)+MC where MC = -(K-l)tr

( v ~ v )

that is independent of all the variables (V is the received signal matrix) and h f k ( x k ) is a function of only the variable X k ,

iff

A f A i = O , O 5 k # 15 2 K - 1. (1) Examples of SSDDs are OD, in-particular the Alamouti code, and the CIODs of [2]. That the CIODs constitute a subclass of SSDD can be verified by substituting the weight matrices of CIOD, S, in (1).

Definition 1:[2] The Coordinate Product Distance (CPD) between any two signal points U = ur

+

j u g and v = vr

+

^{j u g ,}

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

pairs is defined as the CPD of A.

Proposition 2: A square linear STBC S in K complex variables whose weight matrices satisfy (1) exits if€ there exists a

‘This work was partly funded by the DRDO-11% Program on Mathematical Engineering through a grant to B.S.Rajan.

Square linear STBC ,!? =

c&,l

X k I a 2 k -k X k Q a z k + i such that

afa, +

⁼^0,^k

#

1, and = D k , t / k , where v k is a diagonal matrix. Further ,!? achieves full diversity iff either (i) is of full rank for all k or (ii) v 2 k + 2 ) 2 k + l

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 the square GLPCODs. The second condition (ii) gives the important consequence of the Proposition 2: There can exist designs that are not covered by GLPCODs offering full- diversity and single symbol decoding provided the associated signal set has non-zero CPD. The SSDDs that are not GLP- CODS are such that VZk and/or 2&?k+l is not full rank for at least one k . The designs of [2] belong to this category. We call such SSDD codes Generalized Complex Restricted Design (GCRD), since any full-rank design within this class can be there only with restrictions on the complex constellation from which the indeterminates take values. When a GCRD satisfy- ing the conditions for full-rank is used along with a signal set with non-zero CPD we simply refer the design as Full-Rank GCRD (FRGCRD).

Theorem 2: There exists square FRGCRDs with the max- imal rate for N = 2O antennas whereas only rates up to

%

is possible with square GLPCODs with the same number of antennas. Moreover, rate-one square FRGCRD of size N exist, iff N = 2 , 4 .

A rate-one DSDD for Eight Transmit Antennas: Let

20, $1, X Z , 2 3 , ~ 4 ~ 2 5 ~ x 6 and 27 denote eight complex indeterminates, where xi =

+

jxig, i = 0 , 1 , . . . , 7

.

Define, Eo = xor

+

^jzzg; ²¹⁼^zir

+

^jx3g; ^{E 2}⁼^xzr

+

j x o q ; E 3 =

2 3 1

+

j x I g ; E 4 = 2 4 1

+

j x ~ g ; 25 = E51

+

j x r g ; 56 =

2 6 1 +jx4Q; 57 = 271 +jxSQ to be the new complex indetermi-

nates. The following design for eight transmit antennas is of rate-one and double-symbol decodable. It gives full-diversity when the variables X O , X ~ , 22,23,24,25, E6 and 27 takes values from a signal set with nonzero CPD.

’ E o Z l 0 0 5 4 E 5 0 0

-5; 5; 0 0 -E; 5: 0 0

0 0 -E; 2.2’ 0 0 -E; 2;

E 4 E5 0 0 Eo 41 0 0

-z;

5: 0 0 -E? ^E: ⁰ 0

. 0 0 -2; 2; 0 0 -5; 2.2’

0 0 E 2 2 3 0 0 56 2 7

0 0 5 6 2 7 0 0 2 2 23

REFERENCES

[l] H. J. V. Tarokh and A. R. Calderbank, “Space-time block codes from orthogonal designs,” ZEEE ”hnsaction on Information

[2] Md. Zafar Ali Khan and B. Sundar Rajan, “Space-time block codes from co-ordinate interleaved orthogonal designs,” ZSZT 2002, pp 316,Lausanne, Switzerland, June 30

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July 5 also sub- mitted to IEEE Zhnsaction o n Information Theory.

Theory, vol. 45, pp. 1456-1467, July 1999.

0-7803-7728-1/03/$17.00 02003 IEEE. 127