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[IEEE The 57th IEEE Semiannual Vehicular Technology Conference, 2003. VTC 2003Spring.  Jeju, Korea..
[IEEE The 57th IEEE Semiannual Vehicular Technology Conference, 2003. VTC 2003Spring.  Jeju, Korea (April 2225, 2003)] The 57th IEEE Semiannual Vehicular Technology Conference, 2003. VTC 2003Spring.  A simple transmit antenna diversity technique for OFDM and its detection using viterbi algorithm
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Volume:
4
Année:
2003
Langue:
english
DOI:
10.1109/VETECS.2003.1208861
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A simple transmit antenna diversity technique for OFDM and its detection using Viterbi algorithm Jinho Choi School of Electrical Engineering and Telecommunications The U n i v e r s i t y of New South Wales UNSW, Sydney, NSW 2052, Australia Email; j.choi@itnsw.edu.au Abstract We investigate a transmit antenna diversity (TAD) technique for orthogonal frequency division multiplexing (OFDM). Multiple cyclicshifted signal vectors are transmitted through multiple antennas to introduce transmit diversity. From this, the intersymbol interference ( I S ) has been occurred at the receiver and it can provide the diversity gain. To fully achieve the transmit diversity gain, the maximum likelihood sequence estimation (MLSE) is used at the receiver. It is significant that cyclicshift allows the use of the Viterbi algorithm (VA) for the MLSE. Hence, the resulting complexity of the receiver is reasonable. From computer simulations, in comparison to the case of single transmit antenna (i.e., no transmit diversity), it has been observed that there is about 3dB (5dB) signaltonoise ratio (SNR) gain at a bit error rate (BER) of lo' when 2 (resp., 4) transmit antennas are employed, respectively. Furthermore, the SNR gain increases with decreasing the target BER. Keywords: Transmit antenna diversity, orthogonal frequency division multiplexing, maximum likelihood sequence estimation (MLSE), Viterbi algorithm 1 Introduction Orthogonal frequency division multiplexing (OFWM) has been considered in various wired and wireless communication systems [ I , 21. While OFDM is attractive when delay spread is large in wireless communications, there are some disadvmtages of OFDM. Since the multipath diversity is not exploited, a higher bit error rate (BER) is obtained for uncoded bit sequences [I]. To alleviate this problem, transmit diversity has been utilized in [3,4,5,6] by employing spacetime code, It is shown that the spacetime transmit diversity (STTD) scheme proposed in [7] can be readily applied to OFDM. It is noteworthy that; since two transmit antennas are used, the diversity order becomes 2 . To increase diversity order with more transmit antennas, however, some more sophisticated spacetime codes have to be used. ~78037757510311617.2w022w3 IEEE In this paper, we consider a simple transmit antenna diversity (TAD) technique for OFDM using cyclic shift of the signal vector. Multiple cyclicshifted signal vectors are transmitted through multiple antennas to introduce transmit diversity. It results in the intersymbol interference (ISI) at receiver. In other words, spatial diversity due to multiple transmit antennas is transformed into a kind of multipath diversity. The order of this transmit diversity becomes the number of transmit antennas. Hence, a better diversity gain can be achieved by simply increasing the number of transmit antennas. Since the resulting channel matrix has a structured form due to the transmission of cyclicshifted signal blocks, the Viterbi algorithm (VA) can be used for the maximum likelihood sequence estimation (MLSE) with reasonable complexity to fully achieve transmit diversity gain. Note that the TAD method can increase the complexity of receiver and it is contrary to an important advantage of OFDM which converts IS1 (memory) channel into memoryless channel so that the complexity of receiver decreases. However, since the order of channel memory is controllable (by the number of transmit antennas), we can enjoy the tradeoff between the complexity of receiver and the diversity gain. A performance analysis is considered for multipath Rayleigh fading channels. A tight upper bound for the pairwise error probability (PEP) is obtained. From this, an approximate symbol error probability (SER) can be also derived. 2 A TAD technique As in [ 5 ] or [6],the use of STTD in [7]can be readily applied to OFDM.In addition, other coding schemes can be used to introduce transmit diversity for OFDM [SI.In this section, in order to have transmit diversity without spacetime coding, we consider a simple method which allows the use of the VA for MLSE at the receiver. Let x(n) = [zo(n)q ( n ) . . z ~  i ( n be ) ]the ~ nth signal vector (or block) of size N x 1to be transmitted. Here, the superscript T stands for the transpose. If binary phase shift key).(,. € { + l ,1). ing (BPSK) has been used for signaling, 2594 Let S denote the set of symbols. We have S = {+l,1) for BPSK signaling and S = { i lfj } for quadrature phase shift keying (QPSK) signaling. Assume that there are L transmit antennas. Let Cyc(x,l) be the cyclic Ishift operation with the vector x as (1 where JI are the rotation matrices. For example. N = 3, we 1. I=1 and .Il = J1' and J o = I. Through the OFDM modulator for transmit antenna I , the signal vector x l ( n ) = Jlx(n) is modulated as ?I(.) = FHxr(n), (2) ho,o(n) hdn) ho,l(n) hI,NI(n) kz,NI(n) ... _ ' ' LO,NI(n) For example, if N = 4 and L = 2, the channel matrix H(n) where [F],,, = &ejzn(nl)("l)/N is the Fourier trans is written as form matrix and the superscript H stands for the Hermitian transpose. The transmission of cyclic shifted vector is important to introduce the TAD in OFDM. Each signal block is transmitted with cyclic prefix (CP). At the receiver, the received signal vector after removing CP, which is denoted by i ( n ) ,is transformed by discrete Fourier Clearly, the signal vector y ( n ) has the ipzrsymbol interfertransform (DFT) after removing CP and becomes ence (ISI) because the channel matrix H(n) is not diagonal. However, the diversity gain due to multiple antenna transy(n) = F i b ) mission can be obtained. If the same vector x is transmitted L = c m n , X l ( n , +Wb), ( 3 ) through multiple transmit antennas, we cannot achieve the diversity gain. Note that it is not necessary to use the cyclic I=1 shift. In order to introduce the TAD, any permutation of x ( n ) where w(n) = [wo(n) ... WN1(n)lT is the background can he used. However, as will be shown later, the VA can he noise vector which is zeromeancircular complex white Gaus used for the MLSE at the receiver due to the cyclic shift. The cyclicshift TAD is used for singlecarrier c d multisian random vector. The covariance matrix of w(n) is given = NoI. The (frequency domain) channel carrier systems in [9]. Since the channel matrix H ( n ) beas E[w(n)wH(n)] matrix from the Ith transmit antenna to the receive antenna, comes a circular matrix, the detection or decoding complexity can be significantly decreased thrOugh the (additional) Fourier Hl(n), is a diagonal matrix that is written as transform. The channel matrix H ( n ) becomes diagonal and = ~ ~ ~ g ( ~ ~ , ~ ( .n. . ,) L, l~, . ~l (, n~) )(, n (4) ) , it results in the decrease of the decoding complexity. In this case, however, the diversity gain in the ML (hard) detection where {hl,p(n)}is the Fourier transform of the channel im or decoding cannot be achieved due to the same reason as in pulse response (CIR) from the lth transmit antenna to the re OFDM [I]. ceive antenna, {hl,,,,(n)}E:;, i.e., eh) 3 Performance analysis m=0 Here, M is the length of the CIR and it is assumed that the lengths of all L C I R s are the same. Define the matrix N x L matrix Fr. as 2595 In this section, we consider the performance for multipath Rayleigh fading channels. For the simplicity of brevity, we omit the symbol index n hereafter. We assume that hl,* are zeromean complex Gaussian random variables and the vari= so that the total transance is normalized as E[lh& mission power is independent of the number of transmit antennas, L. To see the performance, we consider the PEP for & the symbol vector x. Once we have the PEP, we can find an approximate SER. This upper bound becomes tight as SNR increases. From [IO], it can he shown that 3.1 Upper bound of error probability Given 8.the conditional PEP for the error event that x = 81 ktA, = 1 . is transmitted and x = sz is incorrectly decided is given as Then, it follows that where e = sz  SI. Using the following relationship: Q(z) = 6r'Zexp (17) Hence, the upper hound of the PEP is written as (9) (L) 2 sin' 0 do, a tighter upper hound is available. From (9).the average PEP from (X) can be written as Q P,(sI+ sz) 5 f1X) \  P(sl = E[P(sl + s,/H)] + sz) 1=IZ (e"HHHe 4No sinZ EIexp 7r ]dO.(IO) 3.2 Let El = h = [hT: h diag(Jlle), . .. h:lT. Define = 2 = ' 1,2,..., L Error probability for one symbol difference and Suppose that two symbol vectors sI and s2 are the same except for one symbol. Then, the PEP for a signal block becomes the PEP for one symbol difference. Let e be the difference vector (II) which is a zero vector except for one element. That is, ' . (ELFM)l' Then, from (6).(8) and (II), it follows that e=[0..0 e"HHHe = (Ch)HCh. where 1 5 i Using this. Eq. (IO) is rewritten as P ( S I+ s x ) = ["E[exp I Note that a similar result has been obtained for spacetime code in [II]using moment generating function. ( e v the j t h element o...o]~, 5 N and e # 0. Then, it can be shown that In addition, it can he shown that Since h is a zeromean complex Gaussian random vector with the covariance matrix E[hhH]= &I, it can he shown that EHE  ,jiag(O  .. . 0 0 ... O), let' v the ( j  p + l)#th element where (z)# = [(z  1)moduloNl + 1. From (19), we can show that (13) where A, is the nonzero qth largest eigenvalue and Q is the rank of c"C. 0 CHC = Using (13). an upper bound of Eq. (IO) is given as 1 P(Sl + s2) = 1 . (20) F ~ E ~ E ' F ~ Each diagonal block matrix is rankone and the (largest) noneigenvalue is Mle/'. Hence, the eigenvalues are given 2596 Since Q = L and A, = Mlel'. the upper hound is given as 5 SimUlatiOn RSUltS L We consider BPSK signaling for uncoded symbols for simulations. That is, each element of x is one of {l,+l) with L ' equal probability. The length of signal block, N , is set to 16. It shows that the diversity order is L . At high SNR, the upper We assume that the channel coefficients of the CIR are zerobound of onesymbol difference PEP in (21) can approximate mean circular complex white Gaussian random variables (i.e., the SER. If BPSK is used, it becomes the BER with le[* = 4. multipath Rayleigh fading is considered). The variance of the channelcoefficientisnormalizedasEllhr 1'1 = l / I L M \ f o r .......,, all I and m. Since the bit energy Ea = 1. the signaltonoise 4 MMSE equalization and MLSE with ratio (SNR) . . can he defined as 'dsl t '2) 5 ( 2L1 ) (&) I~ VA SNR I M1 C L E,=, E[lhl,mIZl No As in [I,21, the (frequencydomain) minimum mean square I  _ error (MMSE) equalizer can he used to estimate the symbol (25) No. vector. From (7). the MMSE equalizalion matrix for the symbol vector with given channel matrix H and noise variance NO ~h~ length of the CIR, hi,is set 3 and the length of c p is is written as set to 2. In all simulation. it is assumed that the receiver knows the channel exactly. K,,,, = argm$nE[IlKy x1l21 In Fig. I , the simulation results are presented. It is shown = ( H H H NoI)'HH. (22) that thecyclicshift TAD technique with the MLSE can improve the bit errorrate (BER) significantly. The MMSE equalThe output of the MMSE equalizer for the symbol vector is izer can provide reasonable performance, hut it does not pergiven as form better than the MLSE. When 4 transmit antennas are f,,,, = K,,,,Y. (23) used, we can get about 5dB SNR gain at a BER of lo'. H ~ the performance ~ ~ in (18)~or (21)~may not~ be achiev. , F u r t h e r " the SNR gain is improved as the target BER beable with the MMSE equalizer, ~h~ optimal performance can comes lower. In addition, there exists error flooring effect for higher SNR. only he achieved through the MLSE. ~ + To fully achieve the transmit diversity gain generated by multiple transmit antennas, we can consider the MLSE for the symbol vector in the constrained domain as i,l = arg min (Iy Hxll', XESN IO' (24) Ib where SN is the Ndimensional Cartesian product of S.The MLSE is optimal to minimize the symbol (vector) error. The 5 computational complexity is order of O ( P N ) ,where P is the number of the elements in S. Hence, it seems that the computational complexity of the MLSE in (24) is quite high and practically prohibitive as N increases. However, fortunately, due to the structure of H, the MLSE can be cmied out by the VA. As shown in (8). there is the IS1 from L  1 consecutive symbols. Hence, the VA for the MLSE over IS1 channels can he used with complexity 0 ( P L  ' ) . I 0 5 10 lb 20 To carry out the VA, define the virtual CIR for symbol xm Yll,"a, as {&,,,,)&,'. The length of this virtual CIR is L and the CIR varies from symbol to symbol. The VA is performed in reverse.order (i.e., from the data symbol ziy1 to ZO)with time Figure I : Performance of the TAD OFDM system with the variant virtual CIR. Note that the L  1consecutive symbols MMSE equalization and the MLSE (using VA). Note that "Onebit PEP" means the approximate BER in (21). ofxiv1 become {ZO, X I , . . . , x L  z ) . It is noteworthy that if L = 1, the hard decision of the outFor performance comparison, the STTD in [7] has been put of the MMSE equalizer becomes the optimal solution of the MLSE. However, as L increases, the performance of the considered with two transmit antennas. The BER is shown in Fig. 2. It is shown that the cyclicshift TAD method with MMSE equalizer is worse than that of the MLSE. 2597 + the MLSE can provide the same performance as the STTD. = r ( n f ) As mentioned earlier, the cyclicshift TAD method is applicafin! . Of transmit antennas, ble for any the STTD Or From the following another property of the Gaussian function similar methods is only available for small number of transmit 1 Wn) antennas. r ( n )r(n) = 2 As in Figs. 1 and 2, we can see that the approximate BER in (2 I ) is in good agreement with simulation results, especially and noting that r(n) = (n  l)!,we have at high SNR. 1 2n1 A, = n + J;;m m( ). References [I] H. Sari, G. Karam, and I. Jeanclaude, “Transmission techniques for digital terrestrial TV broadcasting,’’ IEEE Comm. Mag., pp.100109,Fehruary 1995. [2] A.R.S. Bahai and B.R. Saltzberg, MultiCarrier Digiral Communications, Kluwer Academic, New York, N.Y., 1999. *“,“e, Figure 2: Performance comparison with the STTD when two transmit antennas are used. 6 Concluding remarks A TAD technique for OFDM systems has been considered without relying on spacetime coding. Due to the cyclic shift of the sienal block. the IS1 is induced at the receiver and the VA can he utilized for the MLSE with reasonable complexity. It is shown that the performance of the cyclicshift TAD technique can provide the same performance as the STTD when two antennas are used, while the cyclicshift TAD technique can he easily applied to the case of more than two transmit antennas. A tight upper bound o f the PEP has been derived. From this, an approximate SER or BER has been obtained. It is shown that the approximate SER or BER is accurate when S N R is high.  A Derivation of Eq. (17) In this appendix, we will use some properties of the Gamma functionin [IO]. Using the following property of the Gamma function r(n + 1 1.3...(2n1) ) = 2 2“ 6, we can show that A, = 1 . 3 . ’. (2n  1) 2 . 4 . ..(Zn) [3] D. Agrawal, V. Tarokh, A. Naguih, and N. Seshadri, “Spacetime coded OFDM for high datarate wireless communication over widehand channels:’ in Pmc. IEEE mc 1998, 1998, pp, 2232.2236, 141 K.F. Lee and D.B. Williams, “A spacefrequency transmitter diversity technique for OFDM systems,” in Pmc. IEEE Globecom 2000,2000, pp. 14731477. [ 5 ] M. Uysal, N. AIDhahir, and C.N. Georghiades, “A spacetime blockcoded OFDM scheme for unknown frequencyselective fading channels:’ IEEE Comm. Lerters, vo1.5, .. DD. 393395, October 2001. [6] J. Yue and 1.D. Gibon, “Performance of OFDM systems with spacetime coding:’ in Pmc, IEEE wcNc 2o02, 2002. pp. 280284. 171 S.M. Alamouti, “A simple transmitter diversity scheme for wireless communications,” IEEE Jour: Sel. Areas Comm., “01.1 6 , pp.145 I 1458, October 1998. [SI Y.Li, 1.C. Chuang, and N.R. Sollenherger, “Transmitter diversity for OFDM systems and its impact on highrate data wireless networks:’ IEEE Jour: Sel. Areas Comm.. ~01.17,pp.12331243, July 1999. [9] D. Gore, S . Sandhu, and A. Paulraj, “Delay diversity codes for frequency selective channels,” in Pmc. IEEE ICC2002,2002, pp.19491953. [IO] M.R. Spiegel, Schaum’s Mathemarical Handbook of Formulas and Tables, McGrawHill, New York, NY, 1968. [ I l l S. Siwamogsatham, M.P. Fitz, and J.H. Grimm, “A new view of performance analysis of transmit diversity schemes in correlated Rayleigh fading,” IEEE Trans. Info,: Theory, ~01.48,pp.950956, April 2002. 2598