Block Layering Approach in TAST Codes

doi:10.4236/ijcns.2010.310105

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Int. J. Communications, Network and System Sciences, 2010, 3, 788-792

doi:10.4236/ijcns.2010.310105 Published Online October 2010 (http://www.SciRP.org/journal/ijcns)

Block Layering Approach in TAST Codes

Zahoor Ahmed, Jean Pierre Cances, Vahid Meghdadi

Université de Limoges - Ecole Nationale Supéri eure d'Ingénieurs de Limoges (ENSIL)

XLIM-Dept. C2S2, UMR CNRS 6172 16, Rue Atlantis Parc ESTER-BP 6804-87068 Limoges cedex, France

E-mail: zahoor.ahmed@ensil.unilim.fr, ca nces@ensil.unilim.fr, meghdadi@ensil.unilim.fr

Received July 26, 2010; revised August 20, 2010; accepted September 21, 2010

Abstract

Threaded Algebraic Space Time (TAST) codes developed by Gamal et al. is a powerful class of space time

codes in which different layers are combined and separated by appropriate Diophantine number



. In this

paper we introduce a technique of block layering in TAST codes, in which a series of layers (we call it Block

layers) has more than one transmit antenna at the same time instant. As a result we use fewer layers (Dio-

phantine numbers) for the four transmit antennas scheme, which enhances the coding gain of our proposed

scheme. In each block layer we incorporate Alamouti’s transmit diversity scheme which decreases the de-

coding complexity. The proposed code achieves a normalized rate of 2 symbol/s. Simulation result shows

that this type of codes outperforms TAST codes in certain scenarios.

Keywords: TAST Code, Block Layer, Space Time Coding

1. Introduction

It is well know that wireless communications systems over

Rayleigh fading channels can benefit from the simultane-

ous use of multiple antennas at both the transmitter and

receiver to convey information either more reliably or at

higher rates than would be possible for single antenna sys-

tem. The remarkable paper of Alamouti [1] which is con-

sidered a benchmark in space time coding, is based on or-

thogonal design for two transmit antennas offer full diver-

sity and simple linear maximum likelihood (ML) detectors

that decouple the transmitted symbols.

Unfortunately, the Hurwitz-radon theorem showed that

square complex linear processing orthogonal designs can-

not achieve full diversity and full rate simultaneously for

more than two antennas. Later on such type of proof has

also been shown in [2]. In [3] Jaffarkhani et al. has gener-

alized the scheme of orthogonal STBC codes construction

for more than two transmit antennas by compromising ei-

ther the diversity or coding gain. Some researchers have

also introduced codes with higher rates and better per-

formances by sacrificing the simplicity of ML decoding

and thus orthogonality. In [4] a layering concept in STBC,

called vertical Bell Lab layered space-time (V-BLAST),

was introduced, but the main draw back of this code was its

inflexibility with number of antennas.

Extending the work of layering concept of [4], H. Gamal

et al. [5] introduced a new architecture in STBC codes,

known as Threaded Space Time (TST) codes. In this ar-

chitecture, independent codes streams are distributed

throughout the transmission resource array in different

threads. Of course the efficient separation of individual

layers from one another was the primary objectives in the

design of such codes. The main draw back of this type of

code is the complexity of ML decoder which rises expo-

nentially with number of transmit antennas.

Threaded Algebraic codes [5] based on Diophantine ap-

proximation theory and number field were further general-

ized in [6,7] for arbitrary number of transmit and receive

antennas, retaining full rate and maximum diversity. Such

types of high rate STBC codes have also been constructed

using division algebras [8,9].

In this paper we propose a technique of construction

TAST codes within the framework of [6]. The proposed

codes are flexible both in term of usage of antennas (at

both ends) and Diophantine numbers. We use term AF

TAST code for being flexible in term of antennas and DNF

TAST code for being flexible in term of Diophantine

numbers. As a result the DNF TAST code for four transmit

antennas scheme provides higher coding gain and higher

code rate retaining maximum diversity as that of original

layered codes. This framework is based on TAST code

with a slight modification in the definition of layer that in

this scheme we may use more than one transmit antenna

for transmitting same block or a series of layers

The rest of paper is organized as follow:

Z. AHMED ET AL.

789

A brief review of previous work on TAST code is out-

lined in Section 2. In Section 3 we present the new ap-

proach of flexible TAST code construction in term of an-

tennas’ flexibility. In Section 4 we discuss flexible TAST

codes in term of Diophantine numbers. The decoding is

presented in Section 5 and finally Section 6 presents our

conclusion.

2. Preliminaries

As our proposed framework is based on the threaded

space time architecture [6], so for sake of completeness

we review some notation from [6].

A layer in an T

NT (where T

N denote the num-

ber of transmit antennas) transmission resource array is

identified by an indexing setlT

I whereT





1, 2,...,T

N and the t-th symbol interval on antenna a

belongs to the layer if and only



,at l. This indexing

set must satisfy the requirement that if



,at l



, then

either tt



 or aa



 (i.e., that a is a function of t).

The definition will be clearer from Table 1 given below,

which depicts a view for four transmit antennas having

four layers.

Where for layerl, 1,...,ln of the codeword, the set

of matrix entries in positions are given by

(,(1)mod()1), 1,...,tl tnforkn

With an arbitrary number of threads, the TAST codes

are constructed by transmitting a scaled DAST code [10]

in each thread, i.e.,

llllll



xMs (1)

is transmitted over thread l

l. Where l

x are encoded

symbols, l

M is an TT

NN real or complex rotation

matrix, lll

xMs are rotated complex information

symbol vectors and , 1,...,

llL



 are the Diophantine

numbers chosen to ensure full diversity and maximize

the coding gain of the component codes. In [6] l



given by

(1)/T





 (2)

where (0)





 is an algebraic number

3. AF TAST Codes

In some communication systems (for example UMTS),

the number of antennas varies among base stations and

Table 1. Thread distribution.

1 4 3 2

2 1 4 3

3 2 1 4

4 3 2 1

mobile devices, so it is vital to design a flexible MIMO

transmission scheme supporting various multi-element

antennas. As a minimum requirement, the mobile station

might only be informed about the number of transmit

antennas at the base station. Based on its own number of

receive antennas, it can then decide which decoding al-

gorithm to apply. Conventional STBC codes offer great

complexity in varying the number of receive/transmit

antennas. The TAST codes [6] are flexible with respect

to number of transmit/receive antennas. In this section

we introduce a different and simple technique of flexible

ST codes construction which are also flexible with re-

spect to number of transmit/receive antennas and reduc-

ing decoding complexity.

We start with basic simple Alamouti code.























(3)























(4)























(5)























(6)

For 1lL



 (L being the numbers of layers)

where l



is Diophantine number and it is not difficult

to verify that taking any one matrix from (3) to (6) re-

sults a simple Alamouti codes as we know from (2) that





As our proposed scheme is flexible with respect to

number of transmit and receive antennas, so by simple

reshuffle of (3) to (6) we get different structure of TAST

codes for different set up of transmit/ receive antennas.

Below is a body of a simple program that might be used

for this purpose.

Let NT, NR, L, A, denote number of transmit antennas,

number of receive antennas, number of layers, and num-

ber of Alamouti matrices (given in (3) to (6)), respec-

tively.

Initialization, NT, NR

Condition (No. of transmit & receive antenna)

Select (value for L and A)

Process (build TAST codeword matrix with given no. of

L ,and NT)

end

Note that for all the following structure of codes we

Z. AHMED ET AL.

790

consider Diophantine number l



same as in (2).

For case of NT = 2, (Alamouti code) we simply take

any one matrix from ((3) to (6)). For NT = 2 and NR ≥ 2,

we shall add any two matrices from ((3) to (6)) with a

minor manipulation. To save space we avoid going in

detail. Likewise for NT = 3 and NR ≥ 2, we add any three

matrices from ((3) to (6)) with a slight modification. In

same way we can develop a code for NT = 4. In next sec-

tion we discuss one of such type of code for NT = 4 and L

= 2.

4. DNF TAST Code

In case of Diophantine numbers flexibility, the case is

interesting for NT = NR = 4 and L = 2. Therefore in what

follows, we discuss a case for NT = 4 and L = 2, and at

the end of this section we give the numeral representa-

tions for others set up as well.

The necessary condition of layering concept in [5] that

the more than one antenna cannot transmit symbols from

a given layer at a given time instant has been relaxed. A

group of transmit antennas may now belong to a series of

layers (for simplicity we call a series of layers as block)

for a given symbol period.

A block layer is indexed by a set b, T

bb b



(,)1, 2,....,T

bwt N . Like TAST [6] and DAST [10]

schemes, the idea is to map each block layer to a differ-

ent subspace so that they are as far away from each other

as possible. With the concept of block layers, the total

number of layers becomes less and consequently a less

number of Diophantine numbers are required which in-

creases the coding gain. Also, real or complex rotated

symbols are used to further increase the coding gain. In

each block we use Alamouti’s transmit diversity scheme

that ensures simple decoding at the receiver.

Combining (3) to (6)

112 2

24 13

()( )

() ()



(7)

Or more precisely

1112232 4

12112423

2728 1516

28 27 1615

ss s

sss

ss s

sss















(8)

It is straightforward to verify that the modified repre-

sentation in (8) has the same property as the original

Alamouti code. However, this modified representation

clearly falls within the scope of the threaded coding

framework.

In (8) 1



and 2



are two Diophantine numbers

and 1, 28

[,..., ]

ss is the rotated information vector to

be transmitted.

In matrix form the DNF-TAST code for NT = NR = 4

and L = 2 is given in (9) which uses 2q PSK or QAM

signal constellation, and has a rate of R = 2q. For TAST

code we use the notation T

N,L,R

T while for flexible

TAST code we use T

N,L,R

T, where the subscripts in both

cases show the numbers of transmit antennas, number of

layers, and symbols per channel use, respectively.

4,2,4

1122

2211



























T (9)

The transmitted symbol l

x corresponding to source

information symbol l

s over th

l block layer is

() , 1,...,

lllll ll





sxMs

where L represents the total number of block layers and

lll



xMs are the rotated information symbol vectors.

Here l

is an TT



real or complex rotation matrix

built on an algebraic number field ()





with



algebraic number of degree n, and the numbers,



1,...,lL



are the Diophantine numbers. Both for real

and complex rotation matrices we use the matrices same

as given in [6].

In general, one can use different rotation matrices in

different blocks. A general and simple MATLAB pro-

gram which generate rotation matrix Md of any dimen-

sion 2q



on a number field



cos 2/ 8Qd



is given

in [10].

sqrt(2 /)*cos(/ (4*)

*(4*[1: ]'1)*(2*[1: ]1));

dpid





M (10)

To construct a rotation matrix Md of higher dimen-

sions in d the following recursive approach can be

used [11].

/2 /2



MM

MMM

(11)

where 1

Mis the optimal real rotation in dimension d/2

and 2

Mis an orthogonal transformation in dimension

d/2. The Diophantine approximation intends to achieve

full diversity and maximize the coding gain [6]. For a

DNF-TAST code with L layers the Diophantine numbers

are chosen same as (2) with L denoting number of block

layers.

For a neat comparison for NT = 4 and L = 2, we repro-

duce the code as given in [8] in (12). It is crystal clear

that the performance of the code in (8) is much better

Z. AHMED ET AL.

791

than in (12), as the former contain no zeros in transmis-

sion matrix.

4,2,2

1002

2100

0210

0021











T (12)

For NT = 3 and L = 2, we can get flexible TAST code

by deleting last row and adding last and second last

columns in (12).

3,2,2

102

210

021











T (13)

For NT = 3 and L = 3, we can get flexible TAST code

by adding third thread on empty layer in (13).

3,3,3

132

213

321











T (14)

For NT = 2 and L = 2,we get a code by deleting last

row and adding last and second last columns in (13).

2,2,2







T (15)

5. Decoding

For the set up with one and three Diophantine numbers,

we can use simple decoding schemes given in [1] and [6]

respectively. Here we elaborate decoding scheme for our

proposed code in (9).

The received signal can be written as

NLR

YH NT (16)

where

is the RT

NN complex Gaussian random

channel matrix with element , , 1,2,...,

ij R

hi N and

1, 2,...,T

jN, and N is a complex Gaussian random

noise vector.

Let

()

vec Y (17)

arranges the matrix T

Y in one column vector by stack-

ing its columns one after other, and let

, ,....,RT

yy y





y (18)

Simplifying equation (1) and (16), we get







y (19)

where

MA

MAM

(20)

where A is a 44



matrix, 



and



are respectively

rotation matrix, Diophantine matrix and the channel ma-

trix given in (20), (21) and (22), and n is obtained by

converting T

vec(N)into column vector by stacking its

columns one after other, and u is a vector carrying source

information symbols.















 (21)

where

0000000

0 000000

00 00000

000 0000









and

0000000

0 000000

00 00000

000 0000













= hh





(22)

where

1,2,3,4

0 00000

0000 00

0000 0

ij ij

ijij ij

hhh











































and

1,2,3,4

0 00000

0 0 0000

0000 0 0

000000

ij ij

ijijij

hhh











































Z. AHMED ET AL.

792

Note that 1

h and 2

h are stacked into column for

different values of i.

The simulation results given in Figure 1 confirm our

mathematical analysis for obtaining better performances

of our proposed code in (9) over his brethren codes with

L = 2 and 4. When comparing with the code when L = 2

as given in (12), our proposed codes absolutely, however

in case when L = 4, our code outperform at low SNR.

Due to the hardware constraint we could not carried out

simulation for large no of antennas but we intelligently

guess that our code gains better performance over the

both codes for large no of antennas.

6. Conclusions

TAST codes with different number of transmit/ receive

antennas and Diophantine numbers have been proposed.

A code having four transmit antennas and two layers is

discussed which attains a better performance as com-

pared to same class of code having four layers in certain

scenarios. For four receive antennas our proposed code

outperform TAST code at low SNR, but for higher SNR

TAST code works better. Due to limitation of hardware

we could not simulate for higher number of receive an-

tennas but we guess intelligently that increasing the

number of receive antennas may enhance the perfor-

mance of our proposed code. In addition ML decoding is

Figure 1. Comparison of different class of TAST codes.

another positive point of our scheme.

7. References

[1] S. M. Alamouti, “A Simple Transmit Diversity Tech-

nique for Wireless Communication,” IEEE Journal on

Selected Areas in Communications, Vol. 16, No. 8, 1998,

pp. 1451-1458.

[2] X. -B. Liang and X. -G. Xia, “On the Nonexistence of

Rate-One Generalized Complex Orthogonal Designs,”

IEEE Transactions on Information Theory, Vol. 49. No.

11, 2003, pp. 2984-2988.

[3] V. Tarokh, H. Jafarkhani and A. R. Calderbank, “Space

Time Block Codes from Orthogonal Designs,” IEEE

Transactions on Information Theory, Vol. 45 No. 5, 1999,

pp. 1456-1467.

[4] G. J. Foschini, “Layered Space Time Architecture or

Wireless Communication in a Fading Environment when

Using Multiple Antennas,” Bell Labs Technical Journal,

Vol. 1, No. 2, 1996, pp. 41-59.

[5] H. Gamal and A. R. Hammon, “A New Approach to

Layered Space Time Coding and Signal Processing,”

IEEE Transactions on Information Theory, Vol. 47, No. 6,

2001, pp 2321-2334.

[6] H. E. Gamal and M. O. Deman, “Universal Space Time

Coding,” IEEE Transactions on Information Theory, Vol.

49, No. 5, 2003, pp. 1097-1119.

[7] M. O. Damen, A. Tewfik and J.-C. Belfiore, “A Con-

struction of a Space-Time Code Based on Number The-

ory,” IEEE Transactions on Information Theory, Vol. 48,

No. 3, 2002, pp. 753-760.

[8] F. Oggier, J. Belfiore and E. Viterbo, “Cyclic Division

Algebras: A Tool for Space Time Coding,” Boston, Delft,

2007.

[9] B. A. Sethuraman, B. S. Rajan and V. Shashidhar, “Full-

Diversity, High-Rate Space Time Block Codes from

Division Algebras,”IEEE Transactions on Information

Theory, Vol. 49, No. 10, 2003, pp. 2596-2616.

[10] M. O. Damen, K. A. Meraim and J.-C. Belfiore, “Diago-

nal Algebraic Space-Time Block Codes,” IEEE Transac-

tions on Information Theory, Vol. 48, No. 3, 2002, pp.

628-636.

[11] J. Boutros and E. Viterbo, “Signal Space Diversity: A

Power and Bandwidth Efficient Diversity Technique for

the Rayleigh Fading Channel,” IEEE Transactions on In-

formation Theory, Vol. 44, No. 4, 1998, pp. 1453-1467.