Space-Time Trellis Code Construction for Fast Fading Channels

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1 6 Space-Tie Trellis Code Construction for Fast Fadin Channels Zoltan Safar and K. J. Ray Liu Departent of Electrical and Coputer Enineerin University of Maryland, Collee Park, MD Abstract The need for bandwidth- and power-efficient wireless counication systes has raised considerable interest in spacetie codes. In this work, we propose a systeatic space-tie code construction procedure for fast fadin channels. The ethod can be used to desin space-tie codes for an arbitrary nuber of transit antennas and any eoryless odulation. We introduce a new desin criterion that ensures full spatial diversity and develop the code desin ethod based on this criterion. The fleibility of the proposed approach is deonstrated by desinin spacetie trellis codes for 2, 3 and 4 transit antennas with QPSK, 8PSK and 4ASK odulations. I. INTRODUCTION In wireless counications, diversity techniues have been used etensively to iprove the uality of transission at hih data rates. Spatial diversity corresponds to addin redundancy in the spatial doain: buildin a syste with ultiple transit andor receive antennas can iprove the perforance and throuhput of the wireless link by akin use of the lare nuber of propaation paths between the transitter and the receiver. Space-tie (ST) trellis codes represent a cobination of forward error correction, transit diversity and odulation. The perforance criteria for both uasi-static and fast fadin channels were derived in [1], characterizin the ST codes with two uantities: the diversity advantae, which describes the asyptotic error rate decrease as a function of the sinal to noise ratio (SNR), and the codin advantae, which deterines the vertical shift of the error perforance curve. Until recently, the focus of ST code research has been ainly on the uasi-static channel odel. The authors of [1] proposed desin rules for two transit antennas to achieve the aiu diversity advantae. Later works [2], [3], [7] described systeatic code desin ethods for an arbitrary nuber of transit antennas. The proble of code desin for fast fadin channels was first addressed in [4]. ST codes for 2 transit antennas and QPSK odulation were desined usin the idea of sinal set partitionin. In [], the desin of ST codes for fast fadin channels was also considered. The authors found ST codes for 2 transit antennas and QPSK and 8PSK odulations throuh coputer search. In this paper, we develop a code construction procedure that jointly considers diversity advantae and codin advantae for an arbitrary nuber of transit antennas and any eoryless constellation. Section II will introduce the notation and the atheatical odel of the counication syste. The relevant results of previous works will be briefly restated in Section III. Section IV will describe a new desin criterion and analyze the properties of the ST codes that satisfy this criterion. The code desin ethod will be developed in Section V. Section VI will provide the siulation results, and soe conclusions will be drawn in the last section. II. SSTEM MODEL Consider a wireless counications syste with transit and receive antennas. The input bit strea is divided into bit lon blocks, forin -ary ( ) source sybols. The ST encoder works as a finite state achine with states: it takes the current source sybol, ( ), at discrete tie # and, overned by this input and the current state, $ ($ %&'()' ), oves to the net state, $ *,. Durin this state transition, the encoder outputs -ary channel sybol indices. We denote the channel sybol inde for antenna -, -., enerated durin the state transition fro $ throuh the branch correspondin to source sybol by 10324$. We will also use the channel sybol inde vector, defined as: 98:7< '? 1 <@BA These channel sybol indices are apped onto channel sybols (or constellation points) by the odulators and transitted throuh the transit antennas. In the seuel, C2 will represent the constellation point correspondin to channel sybol inde. (For eaple, in case of -ary PSK, C2 ED'F<GH2JI "KLMN, where IO9P.) All the constellations are assued to be noralized so that the averae enery of the constellation is unity (if the channel sybols are eually likely). C2J10324$ will denote the constellation point output by antenna - when the current state is $ and the current input is. In vector notation: Q 24$ 8C2J7< 1 C2 1 C2J1? 24$ 13@JA The transission ediu is assued to be a flat (freuency nonselective), fast Rayleih fadin channel. At each discrete tie instant, the path ains between the transit and the receive antennas are odeled as independent, cople, zero ean, circularly syetric Gaussian rando variables with unit variance. These path ains are assued to be known by the receiver. At the receiver side, the received sinals at each receive antenna are deodulated, and the ST decoder produces the decoded bit strea. The receiver noise is odeled as independent, cople, zero ean, circularly syetric Gaussian rando variables $ IEEE 63

2 b P A A # III. PERFORMANCE CRITERIA Assue that the previously described transitter sends R (RTS ) -ary source sybols to the receiver. The ST encoder, while encodin the source sybol seuence, oes throuh the seuence of states $, and produces U, the by R channel sybol atri: UVW8 Q Q ''' Q A X <@ The decoder, due to decodin errors, oes throuh a different seuence of states, $Z, N producin the erroneously decoded source sybol seuence and the by R channel sybol atri U? : U 98 Q Q 24$ ''' Q 24$ A 3@ Define [, the channel sybol difference atri, as [\]UE^U_ cb. Assue that, for ` tie instants # X'' # a ( # d R ), the # d th colun of [ is nonzero, i.e. Q fe fe1 Q 24$ e e h ji for k ()'' `l " Given the earlier described channel odel, the probability that the decoder erroneously decodes U_ if U was sent can be upper bounded as [1]: on U b p arts a w Q dvu e e1 Q fe fe wo r (1) where p M' is the averae sinal to noise ratio per source wytw y{z?y sybol at the receive antennas and. The perforance criteria [1] were derived to iniize _ n U : 1. Desin for diversity advantae (distance criterion) : The iniu nuber of nonzero coluns in [ (the nuber ` ) taken over all distinct U and U atrices ust be aiized. 2. Desin for codin advantae (product criterion) : The iniu of the nor products a d}u taken over all distinct U possible. w Q 24$ e e1 Q fe fe w and U? atrices ust be as lare as IV. A NEW DESIGN CRITERION For ST encoders havin ~? states, we propose a new desin criterion that is based on the distance criterion described above. The desin criterion (uniueness criterion) is: Every channel sybol inde vector ust be uniue. That is: (a) The channel sybol inde vectors assined to different branches eanatin fro the sae state ust be different. (b) Any channel sybol inde vector assined to any state ust be different fro any channel sybol inde vector assined to any other state. Each channel sybol inde vector contains -ary sybols, so there are different channel sybol inde vectors. Since there are branches eanatin fro each state and the encoder is assued to have? states, we need eactly different channel sybol inde vectors. Therefore, it is possible to assin channel sybol inde vectors to state transitions accordin to the uniueness criterion. The followin two subsections will analyze the properties of the ST codes that satisfy the proposed criterion. A. Diversity Advantae Without loss of enerality, we can assue that the first decodin error occurs at $, so the correct and decoded paths divere at this point (i.e. $ $Z and ). The two paths are assued to ere later at tie#^`. As a conseuence of criterion (a), the channel sybol vectors correspondin to the two paths diverin at $ will be different. Moreover, as a result of criterion (b), the the channel sybol vectors correspondin to the correct and the decoded paths oin throuh different states will be different. Therefore, [ a, the channel sybol difference atri correspondin to any` -lon error event (the first ` coluns of [ ), iven by [ a 98 Q 24$ Q 24$ ' Q a a Q a will have ` nonzero coluns. Fro the perforance criteria described in Section III, one can conclude that any` -lon error event will achieve a diversity advantae of `Z. In case of ultiple error events, the total diversity advantae will be the su of the diversity advantaes of the individual error events. Conseuently, the the iniu diversity advantae of the ST code is deterined by the shortest error event. In case of ST encoders havin? states, the shortest error event is state transitions lon [7], so the ST code is uaranteed to achieve a diversity advantae of. B. Node Error Probability In this section, we will derive an upper bound on the node error probability (the probability that at any iven discrete tie instant, a decodin error occurs, akin the correct and the decoded paths divere). For the purpose of the analysis, we will assue that the trellis is infinite. Usin the union bound, the node error probability can be upper bounded as: ƒ b a'u ˆ 2oU a n U 4 (2) In (2), the first suation is over all possible sent channel sybol atrices U, the second suation is over all possible error event lenths, and the third is over all possible erroneously decoded channel sybol atrices U?a correspondin to ` -lon error events. Assuin that the decodin error occurs at state $, the coluns of [ a will be nonzero, i.e. for#š "'' `, at least one coordinate of the channel sybol vectors Q 24$ 64

3 p and Q 24$ will be different. Therefore, if we denote the iniu distance of the chosen constellation by, the suared nors of the coluns of the [ a atri can be lower bounded as w Q 24$ Q w _ for #h ()''' `t (3) Substitutin (3) into (1), we obtain an upper bound on the pairwise error probability: a n U b p ar ar Since the upper bound (4) does not depend on U_a, (2) can be rewritten as: L b where &_a a'u & a p is the nuber of ` -lon error paths for a iven correct path. Ž_a can be strictly upper bounded by a, so the upper bound siplifies to: Š a'u a p Finally, reconizin that the ters on the riht side of the suation over` do not depend on U and that the probability ass function of U sus to unity, the final epression for the upper bound on the node error probability becoes: ƒ au Fro (7), one can see that the probability that a ` state transitions lon error event occurs (the probability of the union of the ` -lon error events) decreases at least eponentially with `. Conseuently, at hih SNR, the shortest error events will doinate the loner error paths will have neliible contribution to the node error probability. V. THE DESIGN METHOD In the ST encoder, channel sybol inde vectors are assined to each state, accordin to the branches eanatin fro that state. The current source sybol selects one of the, and the - th (- ()' ) inde of the chosen vector deterines the constellation point for antenna -. Fiure 1 depicts an eaple ST code for antennas and any 4-ary constellation ). In this case, if the current state is ( state and the value of the current source sybol is, the ST encoder selects the rd channel sybol inde vector, A, and oves to state. The th, st and nd antennas will transit the channel sybols correspondin to the indices 1,3 and 1, respectively. P ar ar ar (4) () (6) (7) Fi ,111,222, ,232,303, ,313,020, ,030,101, ,223,330, ,300,011, ,021,132, ,102,213, ,331,002, ,012,123, ,133,200, ,210,321, ,003,110, ,120,231, ,201,312, ,322,033,100 state 0 state 1 state 2 state 3 state 4 state state 6 state 7 state 8 state 9 state 10 state 11 state 12 state 13 state 14 state 1 Eaple ST code for 3 antennas, 4-ary odulation This section addresses the proble of assinin channel sybol inde vectors to state transitions. The ost iportant objective is to aiize the diversity advantae, so the ST codes are reuired to satisfy the uniueness criterion. The reainin freedo can be used to increase the value of the iniu nor product ( ). Since the available channel sybol inde vectors can be arraned in 24 1 different ways, the coputational copleity of ehaustive search becoes prohibitive, as the nuber of transit antennas and the constellation size increase. Therefore, we propose an approach that does not uarantee optiality, but is siple and fleible. The basic idea behind the ethod is that it attepts to aiize the iniu nor product correspondin to the shortest error events by aiizin the nuber of nonzero entries in the channel sybol difference atri. Since the shortest error events achieve the iniu diversity advantae, it is a reasonable objective to aiize the nor product of these error events. Moreover, it was shown in Section IV-B that at hiher SNR, the shortest error events will doinate the perforance of the ST codes. '()''X Since the ST encoder has? states, any state $ ($? ) can be uniuely represented as a diit -ary nuber with diits '' '()''X? ( N 0 ): $^9? š??œ N '? ž X (8) The proposed desin rules are: 1. The th inde of the channel sybol inde vectors (the channel sybol indices for the th transit antenna) corre- 6

4 spondin to input (. ()''X Ÿ ) at state $ is deterined as: 424$ 2 %? š? ' {c < 2. The rest of the indices (the channel sybol indices for the rest of the transit antennas) are calculated as: $ 0 {c < for - " 'X (10) It can be verified that the ST codes produced by the above described construction ethod satisfy the uniueness criterion for any -ary constellation. As an eaple, consider the ST code shown in Fiure 1. Since V, state $j can be represented as $ E with and V. The channel sybol indices correspondin to the 2nd branch ( ) eanatin fro state 6 are deterined as: W2 {c 2J W2 24$ ƒc < ƒc ^ Therefore, the channel sybol inde vector assined to this state transition will be A. Assue that the correct and the erroneously decoded paths divere at $ and ere at $. Thus, $ $,, and $ Ÿ$. The followin theore characterizes [, the channel sybol difference atri correspondin to the shortest error events (the first coluns of [ ), iven by [ W8 Q Q Z ' '' Q X?? in case of the proposed desin rules, for any -ary odulation. Theore 1: For any, the 0th colun of the [ atri contains only nonzero entries. If (binary odulation), the diaonal eleents of [ in the 1st, 2nd,..., 2 st coluns will be zero, the rest of the eleents will be nonzero. If S (hiher order odulation) and 92 M' (c <, the the diaonal eleents of [ in the 1st, 2nd,..., 2 st coluns will be zero, the rest of the eleents will be nonzero. For each correct path, there is eactly one such -lon error path. If SE and 2 žm' ª <, the atri [ will contain only nonzero entries. For each correct path, there are eactly 9& such -lon error paths. VI. SIMULATION RESULTS To illustrate the perforance of the codes desined usin the above described ethod, we present soe siulation results. The siulated counication syste had one receive antenna. The source sybols were transitted in fraes of lenth 130, and the Viterbi alorith with decodin depth of 20 state transitions was used to decode the received sinals. For each discrete tie instant, the path ains between the transit antennas (9) TABLE I MINIMUM NORM PRODUCTS FOR 2 ANTENNAS, QPSK ` a a [4] TABLE II MINIMUM NORM PRODUCTS FOR 3 ANTENNAS, QPSK ` a a [4] and the receive antenna were odeled as independent, cople, zero ean, circularly syetric Gaussian rando variables with unit variance. Since the frae error probability depends on the lenth of the frae, we present probability of bit error curves as functions of the averae sinal to noise ratio (SNR) per source sybol at the receive antenna. Fiure 2 depicts the perforance of the ST codes desined for 2 transit antennas and QPSK constellation ( «). We copare our ethod with the ST codes of [4], [] and [6]. All of these codes satisfy the uniueness criterion, even the ST code iven in [6], despite the fact that this code was desined for the uasi-static fadin channel odel. It is observed that our ethod produced a ST code that has the sae perforance as the ST code of [], which was found by ehaustive search. The theoretical perforance of the above codes was also copared. Table I shows the iniu nor product values a for the ` -lon error events. The entries in the second row of the table correspond to the ST codes of [4], [] and our ethod, while the third row contains the a values for the ST code of [6]. The table entries were obtained by perforin coputer search. These a values predict the siilar perforance of the ethods described in [4], [] and the proposed approach, alon with the slihtly worse perforance of the ST code provided in [6] at hih SNR. The bit error rate curves for 3 transit antennas and QPSK X odulation ( ) are depicted in Fiure 3. We have not been able to find ST codes desined for ore than 2 transit antennas and fast fadin channels, so we copare our ethod (the code iven in Fiure 1) with the ST code described in [6]. This ST code also satisfies the uniueness criterion. The curves show that our ST code outperfors the ST code of [6] at hiher SNR. The iniu nor product values for the proposed ethod are iven in the second row of Table II, and the third row contains the values for the ST code of [6]. The theoretical nor product values confir the tendencies observed in Fiure 3. Finally, Fiures 4 and show the perforance of the ST codes desined by our approach for 3 transit antennas and 66

5 10 0 Bit Error Probability 2 antennas, QPSK K2, b2, our ethod K2, b2, ST code fro [] K2, b2, ST code fro [6] K2, b2, ST code fro [4] 10 0 Bit Error Probability 3 antennas, 8PSK K3, b3, our ethod Fi. 2. ST codes for 2 antennas, QPSK Fi. 4. ST code for 3 antennas, 8PSK 10 0 Bit Error Probability 3 antennas, QPSK K3, b2, our ethod K3, b2, ST code fro [4] 10 0 Bit Error Probability 4 antennas, 4ASK K4, b2, our ethod Fi. 3. ST codes for 3 antennas, QPSK 8PSK odulation ( ), and 4 antennas and 4ASK odulation ( ), respectively. To the best of our knowlede, no ST codes have been published that we could copare aainst. As discussed in Section IV-A, the achieved diversity advantae depends on the lenth of the decodin error events. As the SNR increases, the averae lenth of the decodin error events decreases. Therefore, the steepness of the error perforance curve is epected to decrease at hiher SNR. This phenoenon can be observed in Fiures 3 and. VII. CONCLUSION We proposed a new ST code desin criterion for fast Rayleih fadin channels and analyzed its properties. Based on this criterion, we developed a systeatic ST trellis code desin ethod for an arbitrary nuber of transit antennas and any eoryless odulation. Even thouh the proposed ethod does not uarantee optiality, both the theoretical iniu nor product values and the siulations show that our syste Fi.. ST code for 4 antennas, 4ASK atic desin procedure results in ST codes that perfor very well copared to the ones previously found by coputer search. REFERENCES [1] V. Tarokh, N. Seshadri, A. Calderbank, Space-Tie Codes for Hih Data Rate Wireless Counication: Perforance Criterion and Code Construction, IEEE Transactions on Inforation Theory, Vol. 44, No. 2, pp , March [2] J. Gri and M. Fitz and J. Kroeier, Further Results on Space-Tie Codin for Rayleih Fadin, Proceedins of the 36th Allerton Conference on Counications, Control and Coputin, pp , [3] A. R. Haons and H. El Gaal, On the Theory of Space-Tie Codes for PSK Modulation, IEEE Transactions on Inforation Theory, Vol. 46, No. 2, pp , March [4] S. Zuo and S. Al-Seari, Space-Tie Coded QPSK for Rapid Fadin Channels, Proceedins of PIMRC, Vol. 1, pp , [] W. Firanto, B. Vucetic and J. uan, Space-Tie TCM with Iproved Perforance on Fast Fadin Channels, IEEE Counications Letters, Vol., No. 4, pp , April [6] Q. an and R. Blu, Optiu Space-Tie Convolutional Codes, Proceedins of WCNC, Vol. 3, pp , [7] Z. Safar and K.J.R. Liu, Systeatic Desin of Space-Tie Trellis Codes for Diversity and Codin Advantaes, to appear, EURASIP Journal on Applied Sinal Processin, Special Issue on Space-Tie Codin and Its Applications, March