 Open Access
 Total Downloads : 314
 Authors : N. S. Murthy, Dr. S. Srigowri, Dr. B. Prbhakara Rao
 Paper ID : IJERTV2IS60068
 Volume & Issue : Volume 02, Issue 06 (June 2013)
 Published (First Online): 01062013
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Low Delay General Complex Orthogonal SpaceTime 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 
Vijayawada7,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 spacetime 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 InÃ—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

INTRODUCTION
For two transmit antennas fullrate OSTBC is
z1
z*
z2
z*
Alamoutis transmit diversity scheme [1] for given a complex valued modulation scheme. For halfrate 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 spacetime 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 samemaximal 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 Ã— 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 Ã— 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

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 Ã— 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 spacetime block codes was further developed by Weifen Su and XianGen 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 nÃ—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 spacetime 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



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

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 spacetime 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 Â½ and less than 1 were give only for three or four
0 x* x* x* x*
0 0
transmit antennas with rate of Â¾ 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 Â½ 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*

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, SpaceTime 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 highrate orthogonal spacetime block code,
35 33 30 19
0 x* x*
0 x*
0 x*
IEEE Commun. Lett., vol. 7, pp. 222223, May 2003
35 34 31 20
.5. X. B. Liang, Orthogonal designs with maximal rates, IEEE Trans. Inform. Theory, vol. 49, pp. 24682503, 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, Spacetime
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; XiangGen Xia Two generalized complexorthogonal spacetime block codes of rates 7/11 and 3/5 for
5 and 6transmit antennas information theory,IEEE transactions Volume49page313316
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