Low Delay General Complex Orthogonal Space-Time Block Code for Seven and Eight Transmit Antenna

Low Delay General Complex Orthogonal Space-Time Block Code for Seven and Eight Transmit Antenna

N. S. Murthy Associate Professor,	Dr. S. Srigowri Professor& HOD	Dr. B. Prbhakara Rao Professor,
ECE Dept,	ECE Dept,	ECE Dept
VR Siddhartha Engg. College	SRK Institute of Technology	JNTU kakinada
Vijayawada-7,INDIA	Vijayawada, INDIA	Kakinada,INDIA

Abstract

Z , Z ,….., Z or their conjugates Z*, Z*,…….., Z*

or the

1 2 k 1 2 k

1 2

1 2

Space time block codes using orthogonal designs have full code rate, maximum diversity at the receiver simple decoding algorithm. Complex orthogonal designs of maximum possible rate of full, 3/4, and 3/4 have been presented for two, three, and four transmit antennas respectively. For five, six, seven and eight transmit antennas, four generalized complex orthogonal space-time block codes of rates 2/3,

negative of these complex variables and their conjugates, satisfying the following complex orthonormality condition.

2/3,5/8, and 5/8 were proposed recently. Complex orthogonal designs STBCs for other numbers of transmits antennas exhibit rates of 1, 1for four, eight antennas respectively. In this paper we achieved low delay

O HO Z 2 Z 2 …….. Z

2

I

I

k nn

generalized complex orthogonal space time block code for 7& 8 transmit antenna.

Where O H

represents the Hermitian transpose of O

and In×n the n n identity matrix. The matrix O is said to be a

Index Terms Diversity, (generalized) complex

[ p, n, k]

complex orthogonal STBC. For example,

orthogonal designs, space time block codes.

Altamontes code [1] for 2 transmit antennas is a [ p, n, k] = [2, 2, 2] complex orthogonal STBC given by

1. INTRODUCTION

   For two transmit antennas full-rate OSTBC is

   z1

   z*

   z2

   z*

   Alamoutis transmit diversity scheme [1] for given a complex- valued modulation scheme. For half-rate OSTBC the complex- valued modulation scheme was constituted for any number of transmit antennas which is shown in[6]. The generalized Space- time block codes exist with symbol transmission rate 3/4 for 3 and 4 transmit antennas with linear processing [6] or from GCODs without linear processing

   2 1

   The rate of complex orthogonal STBC O is defined as R p . For example, Alamoutis code in (2) for 2 transmit antennas has the rate R p 2 1. Clearly, a complex orthogonal STBC O with high rate can improve the bandwidth efficiency.. In the recent work [5], we have demonstrated that,

   for any number of transmit antennas n 2m 1 and 2m with

   Let k , n k, n, and p be positive integers. A complex

   orthogonal space-time block code (STBC) for any number of transmit antennas n may be described by a p n matrix O ,

   any given positive integer m, the maximum achievable rate

   R p for a [p, n, k] complex orthogonal STBC is the same

   m 1

   the nonzero entries of which are the k complex variables

   value

   2m

   . For example, two complex orthogonal STBCs of

   [p, n, k] = [4, 3, 3] and [p, n, k] = [4, 4, 3] with the same

   maximal rate 3/4 for 3 and 4 transmit antennas, respectively, were constructed in [6]. A specific complex orthogonal STBC of [p, n, k] = [15, 5, 10] with maximal rate 2/3 for 5 transmit antennas was successfully handcrafted in [4]

   The first spacetime block code from complex orthogonal design was proposed in Alamouti [1] for two transmit antennas. It is the following 2 × 2 COD in variables x1 and x2

   x x2

   For any given number of transmit antennas, we have presented in [5] a simple construction procedure with initial diagonal arrangement for complex orthogonal STBCs with various rates and decoding delays In particular, the construction procedure can generate complex orthogonal STBCs with the maximal

   m 1

   G2= 1

   x

   x

   2 1

   2 1

   x* *

   Clearly, the rate of G2 achieves the maximum rate 1. For space time block codes from (generalized) complex orthogonal designs, rate 1 is achievable only for two transmit antennas.

   rate

   2m

   for any number of transmit antennas n = 2m 1 and

   For n = 3 and n = 4 transmit antennas, there are complex orthogonal designs of rate R = 3/4 for example,

   2m. For example, for 6, 7, and 8 transmit antennas, we have constructed the complex orthogonal STBCs of [p, n, k] = [30, 6,

   20], [p, n, k] = [56, 7, 35], and [p, n, k] = [112, 8, 70] with the

   x1 x2

   x3

   maximal rates 2/3, 5/8, and 5/8, respectively. Note that the decoding delay of the above complex orthogonal STBC for 8

   x*

   =

   =

   G3 x*

   2 x*

   1

   1

   0

   0

   0

   x*

   transmit antennas is twice of that of the complex orthogonal

   STBC for 7 transmit antennas, i.e., 112 = 56 × 2. From practical

   3

   0 x*

   x*

   1

   point of view, it is significant for a [p, n, k] complex orthogonal STBC O with the maximal rate to have the memory length or decoding delay p as small as possible

   3 2

   1 2 3 0

   1 2 3 0

   x x x

   x

   x

   0

   0

2. COMPLEX ORTHOGONAL DESIGNS

   x*

   2

   2

   G4 = x*

   * 1

   0 x*

   x3

   x

   x

   2

   3 1

   0 x* x* x

   Definition 1: A generalized complex orthogonal design (GCOD) in variables x1, x2, ., xk is a p × n matrix G such that:

   1 2 k

   1 2 k

   • the entries of G are complex linear combinations of x1, x2, ., xk and their complex conjugates x *,x *,., x *

     • GHG = D, where GH is the complex conjugate

       3 2 1

       The theory of space-time block codes was further developed by Weifen Su and Xian-Gen Xia [7]. They defined space time block codes in terms of orthogonal code matrices. The properties of these matrices ensure rate 7/11 and 3/5 for 5 and 6 transmit antenna.

       and transpose of G, and D is an n×n diagonal matrix with the (i, i) th diagonal element of the form

       x1 x2 x3

       x * x * 0

       0 x4

       x x

       2 2 2 2

       2 1 3 5

       li,1 x1

       li,2 x2

       li,3 x3

       ……. li,k xk

       where all the

       x *

       0 x *

       x x

       3 1 2 6

       coefficients li,1,li,2 ,li,3 ,…..,li,k are strictly positive numbers.

       0 x * x * x x

       3 2 1 7

       x * 0 0 x * x *

       *

       *

       *

       *

       *

       *

       The rate of G is defined as R = k/p. If

       4 7 1

       GHG x 2 x 2 ….. x 2 I

       Then G is called a

       G5= 0 x4 0 x6 x2

       1 2 k nxn

       0 0

       x * x *

       x *

       complex orthogonal design (COD).

       4 5 3

       0 x * x * 0 x

       5 6 1

       Tarokh, Jafarkhani, and Calderbank [6] first mentioned that the rate of space-time block codes from generalized complex orthogonal designs cannot be greater than 1, i.e., R = k/p 1.

       * *

       x 0 x 0 x

       x 0 x 0 x

       5 7 2

       x * x * 0 0 x

       7 6 5 4

       7 6 5 4

       6 7 3

       Later, it was proved in [9] that this rate must be less tha 1 for more than two transmit antennas. For a fixed number of transmit antennas n and rate R, it is desired to have the block length p as small as possible.

       x x x x 0

3. EXISTED COMPLEX ORTHOGONAL STBC FOR 7 TRANSMIT ANTENNAS

x1 x2 x3

0 x7

0 x21

x* x*

0 x*

0 x* 0

2 1 4 11

A complex orthogonal STBC of [p, n, k] = [56, 7, 35]

x* 0

x* x*

0 x* 0

3 1 5 12

0 x* x* x* 0 x* 0

with rate 1/2 and decoding delay 56 for 7 transmit antennas is

3 2 6 13

0 x

x x x

0 x

4 5 1 8 22

given as shown in Tabe.1.

x4

x6 x2 x9

x23

x x

0 x x

0 x

5 6 3 10 24

4 . A NEW COMPLEX ORTHOGONAL STBC FOR 7

x*

x* x* 0 0

x* 0

6 5 4 14

6 5 4 14

x* 0 0 x* x* x* 0

TRANSMIT ANTENNAS

7 8 1 15

0 x* 0 x* x* x* 0

7 9 2 16

0 0 x* x* x* x* 0

A complex orthogonal STBC of [p, n, k] = [42, 7, 21]

7 10 3 17

x* x* 0 0 x* x* 0

with rate 1/2 and decoding delay 42 for 7 transmit antennas is

9 8 4 18

10 8 5 19

10 8 5 19

x* 0 x* 0 x* x* 0

given as shown in Tabe.2.

0 x* x*

0 x* x* 0

10 9 6 20

x8

x9 x10

x7

0 0 x25

A complex orthogonal STBC of [p, n, k] = [15, 8, 9]

0 x11

x12

0 x15

x1 x26

with rate 3/5 and decoding delay 15 for 8 transmit antennas is

x11

0 x13

0 x16

x2 x27

x x 0 0 x x x

given as shown in Tabe.3.

12 13 17 3 28

0 0

x14 x11

x18 x4 x29

0 x

0 x

x x x

14 12 19 5 30

x 0 0 x x x x

14 13 20 6 31

x15

x16

x17 0 0

x7 x32

1. Conclusion

   0 x

   x x

   0 x x

   18 19 15 8 33

   x18

   0 x20

   x16 0

   x9 x34

   x x 0 x 0 x x

   19 20 17 10 35

   Here in this Paper the Complex orthogonal space-time block

   x* x*

   x*

   x*

   0 0 0

   codes (COSTBC) satisfy full diversity as well as fast ML

   13 12 11 14

   x* x* 0 x* x* 0 0

   16 15 18 11

   decoding conditions. In the previous work the designs of rate

   x* 0 x* x* x*

   0 0

   17 15 19 12

   greater than ½ and less than 1 were give only for three or four

   0 x* x* x* x*

   0 0

   transmit antennas with rate of ¾ and only the code rate 1 was for

   x*

   x* x*

   0 x*

   0 0

   17 16 20 13

   17 16 20 13

   20 19 18 14

   two transmit antenna and 4 transmit antennas. In this work we

   x*

   0 0 x*

   0 x*

   x*

   21 22 26 1

   21 22 26 1

   0 x*

   0 x*

   0 x*

   x*

   Propose the new complex orthogonal design with low delay

   21 23 27 2

   0 0

   x*

   x*

   0 x*

   x*

   code rate ½ using 7 transmit antennas. By increasing number of

   21 24 28 3

   23 22

   23 22

   29 4

   29 4

   x* x*

   0 0 0

   x*

   x*

   transmit antennas the bit error rate decreases and hence the

   x*

   0 x*

   0 0 x*

   x*

   Performance of the Wireless Communication system increases

   24 22 30 5

   24 23 31 6

   0

   0

   x*

   x*

   x*

   x*

   x*

   x*

   0 0 0

   x* x*

   x*

   x*

   0 0

   0 0

   x*

   x*

   25 21 32 7

   x*

   0 0 0

   x*

   x*

   x*

2. REFERENCES

25 22 33 8

25 23 34 9

25 23 34 9

0 x*

0 0 x*

x*

x*

1.S. Alamouti, A simple transmit diversity technique for wireless

0 0

x*

0 x*

x*

x*

communications, IEEE J. Select. Areas Commun., vol. 16, pp.

25 24 35 10

27 26 29 11

27 26 29 11

x* x*

0 x*

0 0 x*

1451 1458, Oct. 1998.

x*

0 x* x*

0 0 x*

28 26 30 12

2.H. Kan and H. Shen, A counterexample for the conjecture

0 x* x* x*

0 0 x*

28 27 31 13

on the minimal delay of orthogonal designs with maximal rates,

x*

x* x*

0 0 0

x*

31 30 29 14

submitted to IEEE Trans. Inform. Theory, preprint, Mar. 2004.

x*

0 0 x* x*

0 x*

32 33 26 15

3.E. G. Larsson and P. Stoica, Space-Time Block Coding for

Wireless Communications. Cambridge, UK: Cambridge University

0 x*

32 34 27 16

32 34 27 16

0 0

0 x* x*

x* x* x*

0 x*

0 x*

32 35 28 17

x* x*

0 0 x*

0 x*

Press, May 2003.

34 33 29 18

x*

0 x*

0 x*

0 x*

4. X. B. Liang, A high-rate orthogonal space-time block code,

35 33 30 19

0 x* x*

0 x*

0 x*

IEEE Commun. Lett., vol. 7, pp. 222-223, May 2003

35 34 31 20

.5. X. B. Liang, Orthogonal designs with maximal rates, IEEE Trans. Inform. Theory, vol. 49, pp. 2468-2503, Oct. 2003.

x22

x26

x23 x27

x24 x28

x21 x25

0 x32

0 0

x21 0

0 x x x x x 0

6.V. Tarokh, H. Jafarkhani, and A. R. Calderbank, Space-time

29 30 26 33 22

block codes from orthogonal designs, IEEE Trans. Inform.

x29

0 x31

x27

x34

x23 0

x x 0 x x x 0

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

30 31 28 35 24

x x

x x

0 x 0

7. Weifeng Su; Xiang-Gen Xia Two generalized complexorthogonal space-time block codes of rates 7/11 and 3/5 for

5 and 6transmit antennas information theory,IEEE transactions Volume49page313-316

33 34 35 32 25

Table.1

z1 0 0 0 0

z2 z3

Table.3

0 z 0 0 0 z z

1 4 5

0 0 z1 0 0 z6 z7

x* x* 0 0 x x* x 0

0 0 0 z 0 z z

2 5 4 3 9

1 8 9

x x 0 x x* 0 x* 0

0 0 0 0

z1 z10

z11

5 2 7 9 4

z*

z*

z*

z*

z*

z* 0

x* x*

0 x* 0 x

0 0

2 4 6 8 10 1

1 6 3 7

z*

z*

z*

z*

z*

0 z*

* * *

3 5 7 9 11 1

x8 0

x2 x4

0 0 x6

x9

z4 z2 0 0 0 0

z12

0 x x* x x* x x* x*

z 0 z

0 0 0 z

3 9 6 8 5 7 1

6 2 13

z

0 0 z

0 0 z

x x x*

0 x x

0 0

8 2 14

9 8 1 3 4

z10 0 0 0 z2 0 z15

x* x 0 x* 0 x* x x*

0 z* z* z* z* z* z*

7 9 5 6 8 2

8

8

12 13 14 15 3 2

G 0 x* x* x x x* x x

*

*

z5 z3 0 0 0

z12 0

4 7 9 5 8 3 6

z 0 z 0 0 z* 0

0 0 x 0 x x* 0 x*

7 3 13

3 7 9 5

z9 0 0 z3 0 z* 0

x x x 0 0 x x* 0

14 6 1 8 7 2

z11 0 0 0

z3 z* 0

0 x* x x* 0 x* 0 x*

0 z6 z4

15

0 0 0 z16

7 4 2 1 8

0 z 0 z 0 0 z

0 0 x* 0 x 0 x x*

8 4 17

6 2 5 7

0 z10 0 0 z4 0 z18

0 0 x* 0 x 0 0 0

* 0

* * * * * 5 1

z12

z16

z17

z18

z5

z4

x*

0 0 x* x

0 x* x

0 z z

0 0 z 0

4 8 6 1 3

o

7 5 16 *

9 5 17

9 5 17

z 0 z 0 z 0 z 0

x3

0 0 x1 0

x2 0

x4

0 z 0 0 z z 0

11 5 18

0 0 z8 z6 0 0 z19

0 0 z 0 z 0 z

10 6 20

z* z*

0 z*

z*

z*

z*

13 16 19 20 7 6

0 0

0 0

z9 z7

z11 0

0 z19 0

z7 z20 0

0 0 0

z10 z8

0 z21

z* z* z*

0 z*

z*

z*

14 17 19 21 9 8

0 0 0 z11 z9 z21 0

z* z* z* z* 0 z* z*

15 18 20 21 11 10

z16

z13

z12

0 0 0 0

z z 0 z 0 0 0

17 14 12

z18

z15

0 0 z12

0 0

z19 0

z14

z13

0 0 0

z 0 z 0 z 0 0

20 15 13

z21 0 0 z15

z14 0 0

0 z z z 0 0 0

19 17 16

0 z20 z18 0 z16 0 0

0 z 0 z z 0 0

21 18 17

0 0

z21

z20

z19

0 0

Table.2