1. International Journal of Electronics and Communication International Journal of Electronics and Communication Engineering & Technology (IJECET),Engineering & Technology (IJECET) IJECET ISSN 0976 – 6464(Print), ISSN 0976 – 6472(Online) Volume 2, Number 1, Jan - Feb (2011), © IAEMEISSN 0976 – 6464(Print), ISSN 0976 – 6472(Online)Volume 2, Number 1, Jan - Feb (2011), pp. 01-10 ©IAEME© IAEME, http://www.iaeme.com/ijecet.html COMPARISON AND ANALYSIS OF COMBINING TECHNIQUES FOR SPATIAL MULTIPLEXING/SPACE TIME BLOCK CODED SYSTEMS IN RAYLEIGH FADING CHANNEL Prof. Vijay K. Patel Assistant Professor & HOD, Electronics & Communication Department U. V. Patel College of Engineering Ganpat vudyanagar, Mehsana, Gujarat E-Mail: vijayk_patel@yahoo.com Dr D. J. Shah Principal L. C. Institute of Technonogy, Bhandu, Mehsana, Gujarat E-Mail: research@dharmeshshah.orgABSTRACT Space divisional multiplexing (SDM) and space-time block coding (STBC) havebegun to appear in the latest wireless communication systems. In recent years differentdetection strategies for these schemes have been proposed which can be broadlycategorized as group based or direct-detection techniques. This paper presents thedetection technique that gives superior bit error rate compared to conventionaltechniques. STBC coded signals are detected using different combining techniques likeequal gain combining, selection combining. No diversity scheme (i.e. single input singleoutput (SISO), Alamouti and Maximum Ratio Combining (MRC) schemes are compared.But Maximum Ratio Combining (MRC) outperforms all the other techniques for any typeof STBC code configuration. This is true even in Rayleigh faded environment.Keywords: space division multiplexing, space-time block code, Rayleigh fading.1. INTRODUCTION The capability of the wireless systems have been greatly improved by theintroduction of spatial diversity mechanisms at the transmitter side and at the receiverside (i.e. multiple transmit and receive antennas). The two common forms of spatial 1
2. International Journal of Electronics and Communication Engineering & Technology (IJECET),ISSN 0976 – 6464(Print), ISSN 0976 – 6472(Online) Volume 2, Number 1, Jan - Feb (2011), © IAEMEtransmit processing are space division multiplexing [1] and space-time block coding.[2][3] While SDM increases the data rate by transmitting independent data streamsthrough the different transmit antennas, in STBC, the rate is kept constant or evenreduced but the performance (bit error rate (BER)) is improved by means of a space-timecode that indicates the transmission pattern of a block of symbols over the availabletransmit antennas and several time periods. The Alamouti coding scheme [2] hasachieved great popularity due to its simple optimal decoding and large performanceimprovement it offers. There is a problem in combining the two approaches, SDM andSTBC. [4]-[8], with the aim of achieving high transmission rates at low BERs. Alsovarieties of combining techniques are developed at the receiver side. [9][10] The receiver with multiple receive antennas can combine signals from differentantennas by various techniques. These combining techniques include selection combining(SC), maxilla ratio combining (MRC) and equal gain combining (EGC). In SC, thereceived signal with the highest SNR among NR receive antennas is selected fordecoding. In MRC all NR branches are combined by weighted sum of all the branches.Weight factor of each branch must be matched to the corresponding channel and thenadded. EGC is a special case of MRC in the sense that all signals from multiple branchesare combined with equal weights. The paper is organized as follows: Section II describes the basics of STBC.Section III describes the Alamouti Space Time Code (STC) with decoding. Thecombining techniques are described in IV. Simulation results and discussions arepresented in section V and conclusion is reflected in Section VI.2. SPACE TIME BLOCK CODES: OVERVIEW This section describes the mathematical description of STBC. Based on themathematical model, a pairwise error probability is derived. Finally a space-time codedesign criterion is described by using a pairwise error probability. Consider NT transmit antennas and NR receive antennas. In space-time codedMultiple Input Multiple Output (MIMO) systems, bit streams is mapped into symbol N { }stream xi i =1 . As shown in figure 1, a symbol stream of size N is space-time-encoded NTinto { xi(t ) } , t = 1, 2, K , T , where i is the antenna index and t is the symbol time index. i =1 2
3. International Journal of Electronics and Communication Engineering & Technology (IJECET),ISSN 0976 – 6464(Print), ISSN 0976 – 6472(Online) Volume 2, Number 1, Jan - Feb (2011), © IAEMENote that the number of symbols in a space-time codeword is NT •T (i.e., N= NT •T )resulting in symbol rate of N/T. Figure 1 Scenario of Different MIMO Channels. As shown in Figure 1, there are three different systems like SISO with singleinput and single output. Other system is Single Input Multiple Output (SIMO) andMultiple input Single Output (MISO). Third system is Multiple Input, Multiple Output(MIMO) system. N At the receiver side, the symbol stream {x } i i =1 is estimated by using the received NRsignals { y (jt ) } , t = 1, 2, K , T . Let h(jit ) denote the Rayleigh-distributed channel gain from j =1the ith transmit antenna to the jth receive antenna over the tth symbol period (i=1,2,…,NT,j=1,2,…,NR, and t=1,2,…,T). If we assume that the channel gains do not change during Tsymbol periods, the symbol time index can be omtted. Furthermore, as long as thetransmit antennas and receive antennas are spaced sufficiently apart, NR × NT fading gains{h( ) } can be assumed to be statistically independent. If x( ) is the transmitted signal from t ji i tthe ith transmit antenna during tth symbol period, the received signal at the jth receiveantenna during tth symbol period is 3
4. International Journal of Electronics and Communication Engineering & Technology (IJECET),ISSN 0976 – 6464(Print), ISSN 0976 – 6472(Online) Volume 2, Number 1, Jan - Feb (2011), © IAEME  x1( t )   (t )  Ex ( t )   x2   h j1 , h j 2 , K , h jN   + z (jt ) --------------------------------------- (1) y (j ) = (t ) ( t ) t N 0 NT  T   M  x(t )   NT  where z (j ) is the noise process at the jth receive antenna during tth symbol period, twhich is modeled as the zero mean circularly symmetric complex Gaussian (ZMCSCG)noise of unit variance, and Ex is the average energy of each transmitted signal.Meanwhile, the total transmitted power is considered as NT ∑E i =1 { x } = N , t = 1, 2,K ,T (t ) i 2 T ----------------------------------- (2)Note that when variance is assumed to be 0.5 for real and imaginary parts of hji, theprobability density function (PDF) of each channel gain is given as ( f H ji ( h ji ) = f H ji Re {h ji } , Im {h ji } ) ------------------------------------- (3) = 1 π exp − h ji( 2 )In a similar manner, the PDF of the additive noise can be expressed as j ( ) f Z (t ) z (j ) = t 1 π ( exp − z (j ) t 2 ). ------------------------------------- (4)Pairwise Error Probability: Assuming that Channel State Information (CSI) is exactly known at the receiverside and the noise components are independent, the conditional PDF of the receivedsignal is given as fY (Y H , X ) = f ( Z ) Z NR ( ) T 1 2 = ∏∏ exp − z (jt ) --------------------------------------- (5) j =1 t =1 π 1 = πN T R ( exp −tr ( ZZ H ) ) Using the above conditional PDF, the ML codeword XML can be found by maximizing (5), that is 4
5. International Journal of Electronics and Communication Engineering & Technology (IJECET),ISSN 0976 – 6464(Print), ISSN 0976 – 6472(Online) Volume 2, Number 1, Jan - Feb (2011), © IAEME X ML = arg max f y (Y H , X ) X --------------------------- (6)      H 1 EX EX = arg max exp  −tr  Y −  HX  Y −  HX      X πN T R   N 0 NT  N 0 NT      EX  EX   H = arg max tr  Y − HX  Y − HX    N 0 NT  N 0 NT       X Note that the detected symbol X is erroneous (i.e. XML≠X ) when the following condition is satisfied:  EX  EX   H tr  Y − HX  Y − HX   ≥  N 0 NT  N 0 NT       ------------------------------- (7)  EX  EX   H tr  Y − HX ML  Y − HX ML    N 0 NT  N 0 NT       Probability that X is transmitted but XML ≠ X is given as  EX  Pr ( X → X ML ) = Q   2N N H ( X − X ML ) F    0 T  This is upper-bounded as 1  E H ( X − X ML ) F 2  Pr ( X → X ML ) ≤ exp  − X  2  N 0 NT 4   3. ALAMOUTI SPACE TIME CODE DESIGN The very first and well-known STBC is the Alamouti code, which is a complexorthogonal space-time code specialized for the case of two transmit antennas [2]. In thissection, we first consider the Alamouti STBC and it can be generalized to the case ofthree antennas or more [11]. In the Alamouti encoder, two consecutive symbols x1 and x2 are encoded with thefollowing space-time code word matrix:  x1 − x2  * X = *  ----------------------------------- (8)  x2 x1  Alamouti encoded signal is transmitted from the two transmit antennas over twosymbol periods. During the first symbol period, two symbols x1 and x2 are simultaneouslytransmitted from the two transmit antennas. During the second symbol period, these *symbols are transmitted again, where − x2 is transmitted from the first transmit antenna *and x1 transmitted from the second antenna. The Alamouti code has been shown to have a 5
6. International Journal of Electronics and Communication Engineering & Technology (IJECET),ISSN 0976 – 6464(Print), ISSN 0976 – 6472(Online) Volume 2, Number 1, Jan - Feb (2011), © IAEMEdiversity gain of 2. We now discuss ML signal detection for Alamouti space-time codingscheme. Here, we assume that two channel gains h1(t) and h2(t) are time-invariant overtwo consecutive symbol periods, that is, h1 ( t ) = h1 ( t + Ts ) = h1 = h1 e jθ1 ------------------------------------- (9) h2 ( t ) = h2 ( t + Ts ) = h2 = h2 e jθ2 Where │h1│and θi denote the amplitude gain and phase rotation over the twosymbol periods, i=1,2. Let y1 and y2 denote the received signals at time t and t + Ts,respectively, then y1 = h1 x1 + h2 x2 + z1 * * y2 = −h1 x2 + h2 x1 + z2 where z1 and z2 are the additive noise at time t and t + Ts, respectively. MLreceiver structure for this scheme can be given by:  y  xi ,ML = Q  2 i 2  , i = 1, 2. ---------------------------------------- (10) h +h   1 2  Where Q (.) denotes a slicing function that determines a transmit symbol for thegiven constellation set. The above equation implies that x1 and x2 can be decidedseparately, which reduces the decoding complexity of original ML-decoding algorithmfrom │C│2 to 2│C│ where C represents a constellation for the modulation symbols x1and x2.4. RECEIVE DIVERSITY Consider a receive diversity system with NR receiver antennas. Assuming a singletransmit antenna as in the single input multiple output (SIMO) channel, the channel isexpressed as T h =  h1h2 L hN R    For NR independent Rayleigh fading channels. Let x denote the transmitted signalwith the unit variance in the SIMO channel. The received signal y ∈ C N R ×1 is expressed as Ex y= hx + z ------------------------------------------------ (11) N0 Where z is ZMCSCG noise with E { zz H } = I N R . The received signals in thedifferent antennas can be combined by various techniques. These combining techniquesinclude selection combining (SC), maximal ratio combining (MRC), and equal gain 6
7. International Journal of Electronics and Communication Engineering & Technology (IJECET),ISSN 0976 – 6464(Print), ISSN 0976 – 6472(Online) Volume 2, Number 1, Jan - Feb (2011), © IAEMEcombining (EGC). In SC, the received signal with the highest SNR among NR branches isselected for decoding. Let γi be the instantaneous SNR for the ith branch, which is givenas Ex 2 yi = hi , i = 1, 2, L , N R -------------------------------------------------- (12) N0Then the average SNR for SC is given as ρ SC = E max hi{ ( )} ⋅ N , i = 1, 2,L , N i E 2 x 0 R -------------------------------------------------- (13)In MRC, all NR branches are combined by the following weighted sum: NR yMRC =  w1( MRC ) w2MRC ) L wNMRC )  y = ∑ wi( MRC ) yi ------------------------------- (14)  ( ( R  i =1 where y is the received signal in (11) and wMRC is the weight vector. As Exyi = hi x + zi from (11), the combined signal can be decomposed into the signal N0and noise parts, that is,  Ex  yMRC = w T  MRC  hx + z    N0  ---------------------------------------------------- (15) Ex T = w MRC hx + w T z MRC N0 Average power of the instantaneous signal part and that of the noise part in equation(15) are respectively given as  E  2  E  E { } 2 Ps = E  x w MRC hx  = x E w T hx = x w T h and T MRC MRC  N0   N0  N0 { Pz = E w T z MRC 2 }= w T 2 MRC 2 -----------------------------------------------(16)From equation (16), the average SNR for the MRC is given as T 2 Ps Ex w MRC h ρ MRC = = Pz N 0 w T 2 MRC 2Which is upper bounded as T 2 2 Ex w MRC 2 h 2 Ex 2 ρ MRC ≤ 2 = h 2 ---------------------------------------------------- (17) N0 wT N0 MRC 2 Here the SNR in above equation is maximized at wMRC=h*, which 2yields ρ MRC = E x h 2 / N 0 . In other words, the weight factor of each branch in equation 7
8. International Journal of Electronics and Communication Engineering & Technology (IJECET),ISSN 0976 – 6464(Print), ISSN 0976 – 6472(Online) Volume 2, Number 1, Jan - Feb (2011), © IAEME(14) must be matched to the corresponding channel for MRC. Equal gain combining(EGC) is a special case of MRC in the sense that all signals from multiple branches arecombined with equal weights.5. SIMULATION RESULTS AND DISCUSSIONS Simulation is carried out by considering the Rayleigh faded channel with STBC inMIMO environment. It is assumed that the channel response remains constant during theperiod of observation. Symbols are transmitted by multiple transmit antennas andreceived by multiple receive antennas. QPSK modulation scheme is used in all theresults. Perfect channel state information is assumed to be available at the receiver.Comparison is done among SISO, MRC with one transmit and two receive antennas andMRC with one transmit and four receive antennas. This result is shown in figure 3. Performance of MRC for Rayleigh faded Channel SISO -1 MRC(Tx=1,Rx=2) 10 MRC(Tx=1,Rx=4) -2 10 B E R --> -3 10 -4 10 0 2 4 6 8 10 12 14 16 18 20 SNR dB --> Figure 2 The Performance of MRC for Rayleigh fading Channels As shown in Figure 2, MRC with multiple receive antennas give betterperformance compared to SISO channel. Also as the number of receive antennasincreases, BER improves by large amount as SNR increases. Figure 3 shows the comparison of SISO, Alamouti with two transmit and onereceive (2 × 1), Alamouti with two transmit and two receive antennas (2 × 2), MRC (1 ×2) and MRC (1 × 4). Note that the Alamouti coding achieves the same diversity orderas 1×2 MRC technique (implied by the same slope of the BER curves). 8
9. International Journal of Electronics and Communication Engineering & Technology (IJECET),ISSN 0976 – 6464(Print), ISSN 0976 – 6472(Online) Volume 2, Number 1, Jan - Feb (2011), © IAEME 0 BER Performance of Alamouti scheme and MRC with equal diversity 10 SISO Alamouti (2x1) -1 10 Alamouti (2x2) MRC(1x2) MRC(1x4) -2 10 BER --> -3 10 -4 10 -5 10 -6 10 0 2 4 6 8 10 12 14 16 18 20 SNR dB --> Figure 3 Performance Comparison of Alamouti and MRC schemes Also shown is the 2×2 Alamouti technique which achieves the same diversityorder as 1×4 MRC technique.CONCLUSION From these results it is found that Alamouti scheme achieves the same diversityorder as 1 × 2 MRC technique. Due to a total transmit power constraint (i.e., totaltransmit power split into each antenna by one half in the Alamouti coding), however,MRC technique gives better performance than Alamouti technique in providing a powercombining gain in the receiver. Also with the same diversity order of 2×2 Alamouti and1×4 MRC technique, MRC outperforms Alamouti scheme.REFERENCES1. G. Foshini, ’Layered space-time architecture for Wireless communication in a fading environment when using multi-element antennas,’ Bell Labs Technical J., vol. 1, pp.41-59, 1996.2. A. Alamouti, ‘A simplet transmit diversity technique for wireless communications,’ IEEE J. Sel. Areas Comm. vol 16, pp. 1451-1458, 1998.3. V. Tarokh, H. Jafarkhani and A. Calderbank, ‘Space-time block codes from orthogonal designs,’ IEEE trans. Inform. Theory, vol. 45, pp. 1456-1467, July 1999.4. A. F. Naguib, N. Seshadri, and A. R. Calderbank, “Applications of space-time block codes and interference supression for high capacity and high data rate wireless 9
10. International Journal of Electronics and Communication Engineering & Technology (IJECET),ISSN 0976 – 6464(Print), ISSN 0976 – 6472(Online) Volume 2, Number 1, Jan - Feb (2011), © IAEME systems," in Proc. Asilomar Conf. Signals, Systems Computers, Asilomar, USA, Nov. 1998, pp. 1803-1810.5. A. Stamoulis, N. Al-Dhahir, and A. R. Calderbank, “Further results on interference cancellation and space-time block codes," in Proc. Asilomar Conf. Signals, Systems Computers, Asilomar, USA, Nov. 2001, pp. 257-261.6. E. N. Onggosanusi, A. G. Dabak, and T. M. Schmidl, “High rate spacetime block coded scheme: performance and improvement in correlated fading channels," in Proc. IEEE Wireless Commun. Networking Conf., Orlando, USA, Mar. 2002, pp. 194-199.7. S. N. Diggavi, N. Al-Dhahir, and A. R. Calderbank, “Algebraic properties of space- time block codes in intersymbol interference multiple access channels," IEEE Trans. Inf. Theory, vol. 49, pp. 2403-2414, 2003.8. W. F. Jr., F. Cavalcanti, and R. Lopes, “Hybrid transceiver schemes for spatial multiplexing and diversity in MIMO systems," J. Commun. Inf. Syst., vol. 20, pp. 141-154, 2005.9. Thomas E., Ning K., Laurence B. M., ”Comparison of Diversity Combining Techniques for Rayleigh –fading channels”, IEEE Trans. Comm., vol. 44, No. 9, Sept. 1996.10. Todd K. Moon, “Error Correction Coding Mathematical Methods and Algorithms,” Willey Indian Edition, 2006.11. Tarokh, V., Jafarkhani, H., and Calderbank, A. R.,”Space-time block coding for wireless communications: performance results,” IEEE J. Sel. Areas Commun., vol. 17, pp. 451-460. 10
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