Volume 02, Issue 08 (August 2013)

Performance Analysis Of Space Time Block Codes Over Rayleigh Fading Channel

DOI : 10.17577/IJERTV2IS80091

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Mandeep, Deepak Kedia, 2013, Performance Analysis Of Space Time Block Codes Over Rayleigh Fading Channel, INTERNATIONAL JOURNAL OF ENGINEERING RESEARCH & TECHNOLOGY (IJERT) Volume 02, Issue 08 (August 2013),

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Performance Analysis Of Space Time Block Codes Over Rayleigh Fading Channel

Mandeep

Department of Electronics and Communication, GJUS&T, Hisar (Haryana)

Deepak Kedia

Department of Electronics and Communication, GJUS&T, Hisar (Haryana)

Abstract

To overcome the effect of multi-path fading of the channel and to achieve full diversity, the multiple antennas seems to be an efficient solution. STBC provides a new concept of transmission over Rayleigh

channel using multiple transmit and receive antenna.

j=1,2,3,m) is constant as the quasi-static Reyleigh fading channel is assumed.

At time t the received signal , at antenna j is given by

This paper presents a detailed study of STBC scheme

= . +

(1)

which includes the Alamoutis STBC for two transmitting antennas as well as orthogonal space time

=1

codes (OSTBC) for three and four transmitting antennas. Bit Error Rate (BER) performance is simulated and analyzed for different constellation

Where are independent noise samples of zero-mean

complex Gaussian random variables. For all code words

= 12 1 2 . . 1 2 the receiver

1 1 1 2 2 2

schemes as BPSK, QPSK ,8-PSK and 16-QAM using

MATLAB.

computes the decision metric

2

Keywords Space Time Block Code (STBC), Alamouti

,

(2)

codes, Orthogonal Space Time Block Code (OSTBC)

=1 =1

=1

1. Introduction

The wireless channel suffers from time-varying impairments like multi-path fading, interference and noise. The channel statistic is usually Rayleigh which makes it difficult for the receiver to decide the exact transmitted signal unless some less attenuated replica of transmitted signal is provided to the receiver. Thus applying multiple transmitters on base stations and multiple receivers at receiving stations is a reliable solution to achieve antenna diversity.

Space Time Block Codes (STBC) were introduced to

obtain coded diversity for communication systems with

in favor of the codeword that minimizes this sum and completes the ML (Maximum Likelihood) decoding.

3. Alamoutis STBC

Alamoutis [1], [8] scheme was the first STBC. The Alamouti STBC scheme uses two transmit antennas and receive antennas and can achieve a maximum diversity order of 2 . In addition, Alamouti scheme has full rate since it transmits 2 symbols every 2 time intervals. The Alamouti scheme encoding operation is given by (3).

multiple antennas [3], [4]. The very first well-known STBC is the Alamouti code [1], which was a complex

orthogonal space-time code particularly for the case of

2

= 1 (3)

2

2

1

1

2

two transmit antennas. After that, Tarokh applied the theory of orthogonal designs, analogous to Alamouti scheme, to present Orthogonal Space Time Block Codes

At a time t, symbol 1 and 2 are transmitted from antenna 1 and 2 respectively. Assuming that each symbol has duration T, then at time t + T, the

symbols and , where (*) is the complex

(OSTBC) [2]. This feature of orthogonality makes the 2 1

detection of the received signal linear. So these codes not only provide the maximum diversity order but also a less complex decoding.

In section II, the mathematical model for the wireless communication system under consideration is represented. Section III and IV includes Alamouti code and OSTBC code along with their encoding and

conjugate, are transmitted from antenna 1 and 2 respectively.

In case of 1 Receive Antenna:

The reception and decoding of signal depends on the number of receive antennas. For the case of one receive antenna, the received signals are (4)

(1) = 1 () = 1,1 1 + 1,22 + (1) (4)

decoding procedures. The mathematical analysis 1 1

(2) = 1 + = 1,1 + 1,2 + (2) (5)

presented by Alamouti [1] and Tarokh [5] has been 1

2 1 1

reproduced in these sections of paper for better understanding. This section covers detailed expressions for both Alamoutis STBC as well as OSTBC. Simulation results and conclusion are discussed in section V and section VI respectively.

2. System Model

where 1 is the received signal at antenna 1, , is the channel transfer function from the transmit antenna and the receive antenna 1 is complex random variable representing noise at antenna 1.

These signals are sent to the decoder and are combined as follows [1]

1 = 1,1 (1)+1,2 (2) (6)

Consider a wireless communication system with n

1

= (1)+

1

(2) (7)

transmitting and m receiving antennas. At each time slot

2 1,2 1

1,1 1

t, signals , i=1,2,3,n are transmitted from n

And substituting (4) in (6) and (5) in (7) yields

transmit a

nnas. Assume the channel to be flat fading

= 2 + 2

+ (1)+

(2)

nte

1 1,1,

1,2 1

1,1 1

1,2 1

(8)

,1, 2 2 ,2 1,1

,1, 2 2 ,2 1,1

,

,

1, 1 1 1

1, 1 1 1

and path gains from transmitter antenna i to receiver antenna j is defined to be , . The path gains are modeled as samples of independent complex Gaussian random variable with variance 0.5 per real dimension [2]. The channel coefficient , (i=1,2,3..,n;

2 = 12 + 2 (1)+ (2) (9) where 2 is the squared magnitude of the channel transfer function , . The calculated 1 and 2 are then

sent to a Maximum Likelihood (ML) decoder to estimate the transmitted symbols 1 and 2 respectively. In case of 2 Receive Antenna:

For the case of two receive antennas, received symbols

3 =

1 2

2 1

3 4

4 3

3

4

1

2

(23)

are

1 =

+

+ (1) (10)

1

2

2

1

1

3

4

4

1

2

1,1 1

1,2 2

1

(2)

3

4

1

2

= 1,12 + 1,22 + 1

(11)

4 3 2

1 = 2,11 + 2,22 + (1) (12)

The decision metric minimized by the decoder for

2 2

2 2 2 2

2 2 2 2

2 = 2,1 + 2,2 + (2) (13)

detecting 1 ,2,3 , 4 is given by (24), (25), (26), (27) respectively where

And combined signals are

1 = 1,1 1 + 1,2 2 + ,1(1)

1 1 2 2 2

2

2

+ 2,2(2) (14)

= 1 + 2 ,

= 1 +

2 + (1)

=1 =1

2 1,1 1

1,2 1

2,1 2

2

2

+ 2,2(2) (15)

( 1 + 2

+ 3 + 5

which, after substituting equations (10)-(13), equation

(14) and (15) becomes:

=1

,1

,2

,3

,1

= 2 + 2 + 2 + 2

2

+ 1

1 1,1

1,2

1,2

2,1 1

1,1 1

+ 6

+ 7

)

+1,2(2) + 2 ,1 (1) + 2,2(2) (16)

,2

,3 1

1 2 2

+ 1 2 (24)

,1 2 2 1 2 ,1

,1 2 2 1 2 ,1

2 = 12 + 2 + 2 + 2 1

1, 1, 2, 1 1

1 2 (

1 2 (

+1,2 (2) 2,1 + 2,2(2) (17)

=1

1

,2

2

,1 +

4

+

+

,3

5

The ML decoder decision statistic decodes in favor of

1 and 2 over all possible values of 1 and 2 such that

(18) and (19) are minimized where is given by (20)

2

6 ,1 + 8 ,3) 2

for = 2 [2], [5].

((1) + (2)

2

+ 2 2 (25)

,1

,2 1

( 1 3

4 + 5

=1

+ 1 2 (18)

2

=1

,3

,1

,2

,3

2

( 1 (2)

7 ,3 7 ,1) 3

=1

,2

,1 2

+ 2 (26)

+ 2 2 (19)

3

= 1 +

2 (20)

( 2 + 3

4 6

=1 =1

,

=1

,3

,2

,1

,3

2

+ 7 ,2 8 ,1) 4

  1. Orthogonal Space Time Block Code

    The Alamouti code is the first known Orthogonal Space- Time Block Codes (OSTBCs). Tarokh [2] apply the mathematical framework of orthogonal designs to construct both real and complex orthogonal codes that achieved full diversity. For the case of real orthogonal

    + 4 2 (27)

    And the 3/4 rate code 3 is given by (28)

    3

    codes, the 2×2 design is

    1 2

    2

    3

    1 2 2 1 2

    (21)

    3 =

    +

    (28)

    2 1 3 3

    2 2

    1 1 2 2

    2 1

    2 1

    2

    3

    3

    3

    3

    The 4×4 design

    1 2

    3 4

    2

    2

    2 ++1

    2

    2 1

    3 4

    4 3

    4 3

    (22)

    (22)

    1 2

    1 1

    The decision statistics to minimize 1, 2 , 3 given by (29), (30), (31) respectively

    4 3

    Similarly design for 8×8 can be constructed. Further

    1 2 + 2 +

    ,3

    complex orthogonal codes can also be constructed.

    OSTBC, [5]-[7] for the case of 3 transmitting antennas, block codes can be constructed for both ½ and ¾ coding

    i=1

    ,1

    ,2 2

    3 + 4 2

    rate .The 3 represents the ½ and 3 represents the ¾ rate codes. The ½ rate code 3is given by (23)

    2

    ,3

    s1

    + s1 2 (29)

    (4) + (3)

    (1) (2) +

    ,3

    r(1)h r(2)h + r(3)h r(4)h + r(5)h

    i=1

    (3)

    ,2

    (4)

    ,1 2

    2

    i

    i=1

    i,4

    i i,3

    i i,2

    i i,1 i

    i,4

    + ,3

    r(6)hi,3 + r(7)hi,2 r(8)hi,1

    + 2 s2

    i i i

    2

    + s2 2 (30)

    s4

    + s4 2 (36)

    (1) + (2)

    (3) +

    i=1 2

    ,3

    +

    ,1

    2

    ,2

    The decoding decision metric (36) is taken from the

    2

    2

    (4)

    paper by Luis Miguel et al [9] as Tarokh [5] had

    +

    ,1

    2

    ,2

    s3

    probably mistaken in that metric.

    + s3 2 (31)

    And 4 transmit antenna block code with rate ¾ is given by (37)

    2

    2

    Similarly for the 4 transmit antennas block codes can be constructed for both ½ and ¾ coding rate .The 4

    1

    2

    3

    3

    3

    3

    2

    represents the ½ and 4 represents the ¾ rate codes. The

    4 can be represented by (32)

    =

    2 1 3 3

    2

    2

    2

    2

    4 3

    3

    1 +2

    ++

    1 2

    3 4

    2 2

    1 2 2 2 1 1

    2 1

    2 1

    1 2

    1 2

    2 2

    2 1

    3 4

    4 3

    4 3

    1 2

    2 1

    3

    2

    3

    2

    2 +1 2

    1 ++2 2

    (37)

    4 =

    (32)

    1

    2

    2

    3

    4

    4

    4

    1

    3

    The decision statistics to minimize 1 ,2, 3 is given by

    3 4 1 2

    (38), (39), (40) respectively

    4 3 2 1

    The decision metric to minimize by the ML decoder for

    (4) (3)

    (1) + (2) +

    ,3

    ,4

    detecting 1 ,2,3,4 given by (33), (34), (35), (36)

    =1

    ,1

    ,2 2

    (3) + 4 + 2

    r(1)hi,1 + r(2)p + r(3)p + r(4)h4 + r(5)hi,1

    ,3

    ,4

    i

    i=1

    i i,

    i i,

    i i, i

    2 1

    i

    i

    + r(6)h

    i,2

    + r(7)h

    i,3

    + r(8)h

    i,4

    + 1 2 (38)

    i

    i

    i

    i

    2

    (4) + (3)

    s

    + s 2 (33)

    (1) (2) +

    ,3

    ,4

    1 1

    =1

    ,2

    ,1 2

    (3) + 4 + 2

    r 1 h r 2 h r 3 h + r 4 h + r 5 h

    ,3

    ,4

    i

    i=1

    i,2

    i i,1

    i i,4

    i i,3 i

    i,2

    + 2 2

    i

    i

    r(6)h

    i,1

    r(7)h

    i,4

    + r(8)h

    i,3

    + 2 2 (39)

    i

    i

    i

    i

    2

    1 + 2

    1 2

    ,3

    ,4

    s2

    + s2 2 (34)

    =1

    +

    2 2

    3 +

    2

    +

    r(1)h + r(2)h r(3)h r(4)h + r(5)h

    ,1

    +

    ,2

    ,1

    +

    ,2

    i

    i

    i

    i

    i

    i=1

    i,3

    i i,4

    i i,1

    i i,2 i

    i,3

    2 2 3

    i

    i

    + r(6)

    2

    hi,4 r(7)

    hi,1 r(8)

    hi,2

    + 3 2 (40)

    s3

    + s3 2 (35)

  2. Simulation Results

    The mathematical expressions covered in previous section were simulated through MATLAB programming for analyzing the performance of STBC for different modulation schemes, different number of transmitting antennas and code rate. BER results obtained by simulation are shown in figures 1-4 respectively.

    Figure 1: STBC performance in BPSK

    Figure 2: STBC performance in QPSK

    Figure 3: STBC performance in 8-PSK

    Figure 4: STBC performance in 16-QAM

    The modulation schemes considered for simulation include BPSK, QPSK, 8-PSK and 16-QAM. Further, the performance was analyzed by varying the number of transmitting antennas and code rate. From figure 1, SNR requirement for 4-Tx antenna (4 ) is 13 dB lower than

    using 2-Tx antenna (2) where as it is lower 9dB lower in 3-Tx antenna ( 3 ) at the BER value of 105 as compared to 2. At the same BER value, 3-Tx antenna system with higher code rate (3 ) i.e. ¾ have the SNR value 20 dB which is 7dB lower than 2 and 2dB more than its corresponding lower rate system3 . Although

    3 and 4 have higher rate than 3 and 4 , the performance of 3 and 4 is better.

    By comparing the figure 1, 2, 3 and 4 for the same SNR, BER performance of STBC degrades with increase in the order of the constellation i.e. BPSK has the best performance followed by QPSK, 8-PSK and 16-QAM.

  3. Conclusion

    This paper provides a detailed mathematical analysis of Space Time Block Codes. An introduction to Space- Time Coding has been provided by Alamoutis scheme Then block codes schemes with different code rates for three and four transmitting antennas have been discussed. The encoding and decoding procedure have also been presented. BER performance Alamouti STBC and OSTBC have been analyzed by simulation carried out in MATLAB. Finally, it can be concluded that BPSK modulation technique provides the best results among all modulation techniques. Also, four transmitting antenna system i.e. 4 provides best BER

    performance among different number of transmitting

    antenna system with half rate codes.

  4. References

  1. M.Alamouti, A simple transmit diversity technique for wireless communications, IEEE Journal of Select Areas in Communications, vol.16 no. 8, Page(s):1451-1458, October 1998.

  2. V. Tarokh, H. Jafarkhani et al, Space-time block codes from orthogonal designs, IEEE Journal on IT., Page(s): 1456-1467, July 1999.

  3. V. Tarokh, A. R. Calderbank et al, Space-time codes for high data rate wireless communication: performance criterion and code construction, IEEE Journal on I.T., Page(s): 744-765, Year: 1998.

  4. V. Tarokh, A. R. Calderbank et al, Space-time codes for high data rate wireless communications: code construction, IEEE conference on Vehicular Technology, Page(s): 637-641, Year: 1997.

  5. V. Tarokh, A. R. Calderbank et al, Space-time block coding for wireless communications: performance results, IEEE Journal on Selected Areas in Communications, Page(s): 451-460, Year: 1999.

  6. Tuan A. Tran and Abu B.Sesay, Orthogonal Space Time Block Codes: Performance Analysis and comparison, ICICS-PCM, December 2003.

  7. Zhang Zhen-chuan, Li Ying, Li Xing-zhong Research on OSTBC over Rayleigh Fading Channels and Its Performance Simulation, 6th International Conference on Wireless Communications Networking and Mobile Computing (WiCOM), Page(s): 1

    – 4, Year: 2010.

  8. Yong Soo Cho, Jaekwon Kim, Won Young Yang and Chung G. Kang MIMO-OFDM wireless communications with MATLAB, IEEE Press, John Wiley & Sons (Asia) Pvt. Ltd. 2010.

  9. Luis Miguel, Cort´es-Pena, MIMO Space- Time Block Coding (STBC): Simulations and Results, Personal and Mobile Communication, Gorgia Tech (ECE6604), April 2009

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