zyxwv zyxwvu zyxw zyxwv A Pragmatic Approach to Space-Time Coding Marco Chiani *, Andrea Conti *, Velio Tralli O * DEIS, CSITE-CNR, University of Bologna, V.le Risorgimento 2,401 36 Bologna, Italy; email: {mchiani, aconti} @deis.unibo.it O DIF, CSITE-CNR, University of Ferrara, Via Saragat 1 , 44 100 Ferrara, Italy. email: vtralli@ing.unife.it Absrruct-A pragmatic approach to space-time codes (STC) over block fading channels (BFC) is proposed. The new approach consists in using common convolutional codes to obtain STC, simplifying the encoder and the decoder. It is shown that pragmatic STC (P-STC) achieve a good performance, similar to that the best known STC, and that they are suitable for systems with different spectral efficiencies and fading velocity (taken into account by the BFC model). To design P-STC we propose a search algorithm based on a new formulation of the pairwise error probability and the error enumerating function for geometrically uniform STC over BFC. I. INTRODUCTION In radio communication the received power level fluctuates due to multipath propagation. It is known from many years that the use of multiple receiving antennas, sufficiently spaced apart each other, to obtain independent copies of the transmitted signal is an efficient way to mitigate the effects of this impairment [I]. However, only in the last decade it has been realized that even the use of multiple transmitting antennas can give similar improvements [2][3]. In this regard, an information theoretic study of transmit diversity techniques is given in [4]. More recently, with the introduction of Space-Time Codes (STC) [5][6][7], it has been shown how, with the use of proper trellis codes, multiple transmitting antennas can be exploited to improve system performance, without sacrificing spectral efficiency. The design of STC over quasi-static flat fading (i.e. fading level constant over a frame and independent frame by frame) has been first addressed in [6], which proposed some handcrafted trellis codes for two transmitting antennas and slow fading channel. In successive works [8] new STC with improved coding advantage have been found by means of computer searches, for two transmitting antennas. The work [9] points out that the diversity diversity achievable by STC for BPSK and QPSK modulation can also be investigated by a binary design criteria, instead of looking at the complex baseband differences between modulated codewords. The problem of determining the best STC (i.e. maximum diversity advantage and good coding advantage) still remain a difficult task, especially for a large number of transmitting antennas and large number of states in the underlying trellis code. Moreover, the design of STC over fast fading channels is still an open problem. In this paper we use the block fading channel (BFC) model [IO] [ 1 13 to investigate the design and the performance of STC. The BFC model represents a simple and powerful mean to include a variety of fading velocities, from ”fast” fading (i.e. perfectly interleaved) to quasi-static. We first derive the pairwise error probability of STC over block fading channels. Then, we propose a method based on a modified trellis transfer function to evaluate the performance of geometrically uniform STC over a BFC. In order to find good codes, the method given in [6] is cumbersome, especially with an increasing number of fading levels per codeword. Moreover, the STC in [6] require ad-hoc encoders and decoders. For these reasons, as major contribution of this paper, we present another possible approach to ST coding, denominated Pragmatic STC (P-STC). Here, the “pragmatic” approach (following [ 131) is the use of common convolutional codes over multiple transmitting antenna as space-time codes. We show that P-STC, obtained with common convolutional codes, can be jointly used with QPSK or BPSK to achieve maximum diversity and very good performance. They do not require any specific encoder or decoder different from those used for convolutional codes; the decoder requires only a modification in the metric computation. zyxwvutsrqpo Work performed under contract with CNR (Rome) and MURST A new algorithm to search for optimum pragmatic STC over BFC is employed to obtain the polynomial generators for various constraint lengths and fading velocities. The numerical results, which compares our codes with the best known STC found in [6][8], confirm the validity of the approach. The paper is organized as follows: in Section I1 the channel model and the general architecture of a system with STC are described; in Section I11 the Pairwise Error Probability (PEP) for STC over BFC is derived, whereas in Section IV geometrically uniform STC are investigated, with an algorithm which evaluates the modified weight enumerating function. The class of Pragmatic STC is proposed in Section V and, finally, some numerical results are given in Section VI, zyxwvutsr 0-7803-7097-1/01/$10.00 02001 IEEE 2794 zyxwvutsrqpo zyxwvuts zyxwvuts zyxwvuts zyxwvuts zyxwv zyxwvutsrqpo zyxwvuts 11. SYSTEM ARCHITECTURE AND CHANNEL MODEL 111. THEPAIRWISE ERROR PROBABILITY FOR SPACE-TIME CODES OVER A general scheme for STC is depicted in Fig. 1, where n and BFC m denote the number of transmitting and receiving antennas, respectively. Given two codewords c, g , the conditional pairwise error probability (PEP) can be wri%en as: a super-symbol, We denote' by C(t)= [cf)c;) . cf'] i.e. a vector of symbols simultaneously transmitted by the n transmitting antennas at the time t. A frame is composed by a sequence of N super-symbols (every T, seconds, n symbols are sent in parallel on the n transmitting antennas). The generic sequence c transmitted during a frame is: d2 (G,21 a:::}),is: q-, where, as in [6], the conditional squared distance, { At the time t, the complex envelope of the signal received by antenna s can be written as: The squared distance can be rewritten as i=l where the noise components Q?), ...,q$ are independent complex Gaussian r.v., with zero mean and variance N0/2 per dimension. The constellations are contracted by a factor fi(E,signal energy), in order to have average energy of the constellation equal to one. Here, a!:! represents the complex Gaussian fading coefficient between antenna s and i, at time t , assumed to be independent for each transmitting-receiving antenna pair. We assume that aifb have zero mean, and variance 0.5 per dimension; the envelope of each coefficient results to be a normalized Rayleigh r.v., but the extension to Ricean fading is straightforward.. To extend the analysis of STC to correlated fading channels, we will use the Block Fading Channel (BFC) model [I I][lO], where fading coefficients ai::, for each couple (i, s), are constants in blocks of B consecutive time instants, and independent from block to block . The number of fading blocks per codeword is denoted by L, so that N = L . B. When the block length, B, is equal to one, we have the common 'ideally interleaved' fading channel, whereas for L = 1 we have the 'quasi-static' fading channel. By varying L we can describe channels with different correlation degrees [ 111. Over a Rayleigh fading channel, a system with diversity div has an asymptotic error probability: s=l t=1 Qr) where = [at:,a;:, ..., is the vector of the fading coefficients related to the receiving antenna s at the time t, and C(t),G(t) are the super-symbols at the time t of the sequences c, g , respectively. The resulting (n x n ) matrix with elements - E Qp), {z!'),~?), ..,zLL)} , s = 1, ...,m. BY these vectors, we can rewrite eq.(5) as P, z k ~ (a> diu % where is the signal-to-noise ratio and k is a constant. In other words, a system with diversity div is described by a curve of (bit or frame) error probability with a slope approaching, for large signal-to-noise ratios, lO/diw [dB/decade]. means transpose conjugate, OT stands for transpose, ()* for conju- gate. 0-7803-7097-1 /01/$10.0002001 IEEE zy is Hermitian and non-negative definite. Due to the BFC model, for each frame and each receiving antenna, the fading channel is described b only L different vectors that we indicate as &l),,7$), ..,ZiL),i.e. 2795 where grouping zyxwvutsrqpon zyxwvutsrq zyxwvu zyxwvutsrqpo zyxwvutsrq zyxwvu zyxw zyxwvutsrqpon and T ( v )= { t : C):l = g?)} is the set of indexes t where Er); the channel has the same fading level this set depends on the interleaving strategy used. Note that in our scheme the interleaving is done "horizontally" for each transmitting antenna and that the set T ( v )is independent on s. The new matrix E("),being the sum of Hermitian nonnegative definite ma&es, is also Hermitian non-negative definite. It has, therefore, real non-ne ative eigenvalues. Moreover, it can be written as @") =_ _ _ where @") is a unitary matrix and A('-') - is a real diagonal matrix, whose di- uP")n(")u(v)H, agonal elements are the eigenvalues Xiv), i = 1,2, ..,n of E(") counting multiplicity. Note that E(")and its eigenvalues Xi"' are a function of s - g. As a re& we can express the squared distance d2 (c, 91 {ai::}) by utilizing the eigenvalues of E(") as follows: Equation (10) can be seen as the generalization for BFC of the results in [6]. Even in this case, for each pair of sequences, the product and the number of non-zero eigenvalues of the suitably defined matrix E(")(accounting for the block fading channel and the interleaGng rule for each possible pair c - g) must be computed to obtain the Pairwise Error Probability. Iv. PERFORMANCE EVALUATION FOR GEOMETRICALLY UNIFORM STC OVER BFC For geometrically uniform codes, the performance can be evaluated by assuming the transmission of a particular sequence z, i.e. the all-zero sequence generated when all the information bits are 0. Therefore, the conditional frame error probability can be written, by using the standard approach, as: By introducing the error sequence e = g - z , the unconditional bound on the frame error rate becomes s=l v = l i=l NOW, since - , zyxwvutsrqpon where in (1 2 ) the dominant terms, for large signal-to-noise ratio, are those with minimum 77. Hence, the asymptotic bound is , represents a unitary transformation, Zp)@") - has the same statistical description of in the case of Rayleigh distribution, has independent, complex Gaussian elements, with zero mean and variance 0.5 per dimension. Moreover, due to BFC model, the vectors are independent for v # W . E:-'), By using the position r = ES/4N0,the unconditional pairwise error probability becomes: where E[.]means statistical average. By evaluating the asymptotic behavior of (9) we obtain: where qv is the number of non-zero eigenvalues of E(") - and v = v(c - g) = cv=l rank [E(')] = cv=, vv is the sum L over v of the ranks of E("). 0-7803-7097-1/01/$10.00 02001 IEEE L zyxw = zp).Hence, where Qmin = min, ~ ( gand ) Emin = {e # O : ~ ( e=)vmin} i:; the set of error sequences with minimum diversity. The asymptotic bound shows that the achievable diversity (diversity gain), vmin . m, increases linearly with the number of receiving antenna. Note that the asymptotic performance of a code depends on both the achievable diversity, vmin-m,and the gain factor given by nf=.=, n:Ll However, vmin and the weights Xi")do not depend on the number of receiving antenna. Therefore, when a code is found to reach the maximum diversity (vmin) in a system with one receiving antenna, the same code reaches the maximum diversity in a system with multiple receiving antenna (vmin . m). However, due to the presence of the exponent m in each term of the sum in (14), the best code (i.e. the code having the smaller gain factor (14)) for a given number of antennas is not necessarily the best for a different number of 2796 zyxwvutsrqp zyxwvutsrq zyxwvut zyxwvutsrqp zy zyxwvutsr zyxwvutsr receiving antennas. Thus, a search for optimum codes must be pursued for each m. In conclusion, the derivation of the asymptotic behavior of a given code with a given length, requires the evaluation, for each non zero error sequence, of the matrices @")(e)and, for each matrix, of both the rank and the productGf the non-zero eigenvalues. Fortunately, this procedure can be carried out in an effective way, as for convolutional codes over block fading channels [ 141, through the concept of the generalized transfer function for ST-codes over BFC. With respect to convolutional codes, some modifications are required as explained below. The generalized transfer function can be obtained with the following steps: - construct the error state diagram, having on the edges the difference between the output super-symbol associated to that transition, and the input super-symbol for the all-zero sequence. Denote by E the resulting error super-symbol on a generic edge; - construct the modified error state diagram having on the E edges the variables Df,where E = E . EH is the matrix related with the error super-symba on that transition. - evaluate the code weight profile as in [ 141 with the rules: At this point, by using a standard procedure, starting from and ending into the state zero, we can derive the modified transfer function for a frame of a given length zyx where It is important to remark that this bound can, sometimes, be large if a lot of terms have minimum diversity, as all union bounds. However, this happens when the channels have a large amount of available diversity ( L > 1);here the bound is expected to give good results. For slow fading channels, where the union bound is expected to be too loose, the same method as in [6] can be used, looking at the term with minimum eigenvalues product, A!.'. The enumeration of the minimum diversity terms can be also pursued for long codewords following the approach in [ 141, by using a proper error trellis diagram. In this case, at each iteration in the error trellis diagram some "purging" of the survivors is possible, leaving only the paths giving rise to large PEP. This is due to the following property [ 121: the (ordered) eigenvalues of the sum of two non-negative definite Hermitian matrices are greater than, or equal to, the eigenvalues of each matrix. n:==,nI:l v. A PRAGMATIC APPROACH TO SPACE-TIME CODES Another, possible approach to consider STC over block fading channel is here presented. The "pragmatic" approach (following [ 131) consists in using common convolutional codes as space-time codes, with the architecture presented in Fig.2. Here, k information bits are convolutionally encoded by a rate klnh code. The n x h output bits are divided in n streams. one for each transmitting antenna, of QPSK ( h = 2) or BPSK ( h = 1) symbols, obtained by natural mapping of h bits. This means that for BPSK an information bit b is mapped to symbol 1 - 2b; for QPSK a pair of information bits a, b are mapped to symbol (1 - 2a) j (1- 2b). Finally, each stream of symbols is eventually interleaved. In this section we restrict our attention to symbol interleaving, leaving the case of bit interleaving for another contribution. Therefore, for each block of k information bits, a supersymbol is transmitted. The spectral efficiency is E s = k [bit/s/Hz]. Now, the advantages of our proposal with respect to the original in [6] are: the encoder is a common convolutional encoder; the (Viterbi) decoder is the same as for a convolutional code, except for a change in the metric evaluation; this kind of space-time codes are geometrically uniform. While the first two properties give practical, implementation benefits, from a theoretical point of view the last property is even more important. In fact, since all codes are geometrically uniform, the search for good codes is feasible even with long frames and fast channels. This is possible by using the methodology previously presented to exhaustively investigate the whole class of possible codes, choosing those polynomial generators giving, for a given frame length, rate, channel memory L , and interleaving, the highest diversity gain and the best zyxw + (17) ,-., For each - - ...,# (2, ...,2) it provides the _ _ ...,number w(F('), E ( L ) )of error sequences producing F(1),@2), - Among the terms in eq.(17), the most important are those related to matrices _ ...,- hav- ing minimum diversity, i.e. minimum L zyxwvut rank [E")]. For nu=, n,=, these, it is important to evaluate the weight L Xiv), i.e. the product of all the non-zero eigenvalues of F(1),F(2), By using the transfer function, the union boundbecomes 0-7803-7097-1/01/$10.00 02001 IEEE 2797 zyxwvutsrqp zyxwvuts zyxwvuts zyxwvutsrq zyxwvutsr zyxw zyxwvutsrqpo gain factor. For slow fading channels ( L = 1) since the union bound may be large, the approach of [6] can be used, looking, in the modified transfer function (17), at the terms with minimum product of eigenvalues instead of the gain factor in (1 4). VI. NUMERICAL RESULTS In this section the performance, in terms of Frame Error Rate (FER) vs. the Signal-to-Noise Ratio per receiving antenna element (SNR), of some P-STC and STC are investigated by computer simulation with perfect channel state information. In order to evaluate the suitability of the “pragmatic” approach, let us first consider some P-STC obtained by using the best, over AWGN, convolutional codes. For example, for n transmitting antenna and BPSK, a rate l / n binary code can be used. In this regard, for n = 2 and frame length N = 200, in Fig.3 we show the performance of the de-facto standard 1/2, 64 states, convolutional encoder, with generators (133, 171)s. The spectral efficiency is 1 bit/s/Hz. It can be noted that the maximum achievable diversity (ne m ) in a quasi-static channel ( L = 1) is achieved. We cannot compare these results with other STC, because no results are available in the literature for BPSK. Similarly, it is possible to design P-STC for QPSK, where we need a rate 2/2n convolutional code. The simpler way to design it is to use two rate l / n encoders, in parallel. Then, in order to exploit diversity, the outputs of each encoder are split to different antenna. The number of states of the resulting encoder is the product of that of the constituent encoders. In the following we will consider a frame-length of N = 130, to compare P-STC with known STC in [6][S]. In Fig.4 the FER vs. SNR for a slow fading channel ( L = 1) are presented, for n = m = 2 , and 4 states codes. In the figure, the results of [8][6] are compared with our P-STC with generators (06,13,11,16)8, obtained by a search algorithm. It can be noted that the proposed P-STC performs better than the best known STC [SI. In Fig.5 the performance of P-STC is compared to that of the STC of [6] over a faster BFC with L = 5 and symbol interleaving. The searching for good generator polynomials suggests the generators (05,06,11, 13)s for n = 2 , m = 1 and (05,06,13,17)8 for n = m = 2. The Figure clearly shows that the performance is significantly better using P-STC, that appear able to take a greater advantage from temporal diversity, even if the STC codes from [?I are not optimized for this kind of channel. We have also checked that the generators that we have found have good performance both for m = 1and m = 2. At the moment, due to difficulty in designing STC, only results for two transmitting antenna elements are available in literature. Here we present P-STC for n = 3. In Fig.6 the performance of 16 states P-STC with generators (57,47,65,75,71, 77)8 over slow fading channel ( L = 1) is shown for QPSK, 2 bit/s/Hz, n = 3 transmitting antenna elements and m = 1 , 2 , 4 receiving antenna elements. 0-7803-7097-1/01/$10.00 02001 IEEE VII. CONCLUSIONS A new method has been presented to derive the pairwise error probability for STC and the asymptotic performance for geometrically uniform STC, over block fading channels. Then, the class of pragmatic STC, using common binary convolutional codes, have been introduced. Finally, search algorithm for good pragmatic STC, made possible by the asymptotic analysis, has been shown to be possible, leading to good polynomial generators. The comparison between P-STC and STC for slow fading channel shows that P-STC achieve better results of the best known STC, due to the possibility of a search taking into account the gain factor, validating the proposal. Moreover, some P-STC for three transmitting antenna, and over BFC are investigated. REFERENCES [31 I41 [51 I61 181 r91 J.G.Proakis, “Digital Communications”, Third ed., McGraw-Hill Int Ed. 1995. G.J.Foschini. M.J.Gans, “On limits of wireless communications in a fading environment when using multiple antennas”. Wireless personal Communications, vol. 6, pp 31 1-335, 1998. I.E.Telatar, “Capacity of multi-antenna Gaussian channels”, AT&T Bell Labs repon. A.Nmla, M.D.Trott. G.W.Womell. ”Performance Limits of Coded Diversity Methods for Transmitter Antenna Arrays,’’ IEEE Trans. on Information Theory, vo1.45, n.7, pp.2418-2433, Nov. 1999. J.-C.Guey,M.P.Fitz, M.R.Bel1, and W.-Y. Kuo, ”Signal Design for transmitter diversity wireless communication systems over Rayleigh fading channels,” Proc. IEEE Vehic. Techn. Conf., Atlanta, GA. pp. 136-140, 1996. V.Tarokh,N.Seshadri,A.R.Calderbank,“Space-Time Codes for High Data Rate Wireless Communication: Performance Criterion and Code Construction“. IEEE Trans. on Information Theory, vo1.44. n.2, pp.744-765, March1998. V.Tarokh.A.Naguib.N.Seshadri,A.R.Calderbank. ”Space-Time Codes for High Data Rate Wireless Communication: Performance Criteria in the Presence of Channe1 Estimation Errors. Mobility, and Multiple Paths“, IEEE Trans. on Comm.. vo1.47, n.2, pp. 199-207, Feb. 1999. S.Baro,G.Bauch, and A.Hansmann, ”Improved codes for space-time trellis-coded modulation,” IEEE Commun. Lett., vol. 4, pp. 20-22, Jan. 2000. A.R.Ha”ons, and H. El Gamal. ”On the Theory of Space-Time Codes for PSK Modulation,” IEEE Trans. on Inform. Theory, vol. 46, n. 2, pp. 524-542, March 2000. E. Malkamaki, H.Leib, “Coded Diversity on Block-Fading Channels”. IEEE Trans. on Information Theory, vo1.45, march 1999. M.Chiani, “Error Probability for Block Codes over Channels with Block Interference”, IEEE Trans. on Information Theory, vo1.44. nov. 1998. R.A.Horn, C.R.Johnson. Mauix Analysis. Cambridge University Press, UK, 1999. A.J.Viterbi, J.K.Wolf. E.Zehavi, R.Padovani “A Pragmatic Approach to TrellisCoded Modulation”, IEEE Comm. Mas., vo1.27, pp.11-19, July 1989. MChiani, A.Conti and V.Tralli, ”On the Design of Convolutional Codes over Block Fading Channels,” proc. of IEEE International Symposium on Information Theory (ISIT2000), Sorrento, Italy, June 2000. 2798 .................................. zyxwvutsrqp DIIMOII zyxwvutsrqponmlkjihgfedcbaZYX .............................. SPACli-Tlhll: ENrOnER Fig. 1. Block diagram for space-time encoding and decoding. MAPPER - I SYMBOL MOD INI.ERI.EAV.- Il-i I' ENCOI) zyxwvutsr zyxwvut I oo I ~ 10.' MAPPER MOI). 4 Fig. 2. Block diagram for the proposed pragmatic space-time codes 1oo zyxwvutsrqp zyxw zyxwvutsrqpo 6 4 8 10 12 14 16 SNR dB per rx antenna element 18 20 lo-' Fig U U w S Codes with 4 states, QPSK, 2 bit/s/Hz, n=2. L=S; a)P-STC m=l, b)STC from [6] m=l, c)P-STC m=2, d)STC from [6] in=2. lo-* 0 2 4 6 8 SNR dB per rx antenna element 10 12 1oo Fig. 3. Pragmatic 64 states STC, BPSK, I bit/s/Hz, generators (133,171)s. L= 1 ;a) n=2 m= 1. b) n=m=2. 1oo lo-' I I I . . . . . . . '. ... ........ ... .': .:.. . \ ...\. ...... \, . "..: . . . . . . . . . II a: . 2 .............. ........... ,I . . . . . . . . . . . . . . . . ; . - I I lo-* ~ I 0 0 2 4 6 8 10 12 14 16 SNR dB per rx antenna element 18 20 Fig. 6. Pragmatic S T C with 16 states, QPSK 2 bit/s/Hz, n=3, L=l; a ) m = l , b)m=2, c)m=4. zyxwvutsrqponm 2 4 6 8 SNR dB per rx antenna element 10 12 Fig. 4. Codes with 4 states, QPSK, 2 bit/s/Hz, n=m=2, L=l; a) P-STC, b) S T C from [6],c) STC from [8]. 0-7803-7097-1/01/$10.00 02001 IEEE 2799