MAP Equalization of SpaceTime Coded Signals over Frequency Selective Channels Invited Paper Gerhard auch, Ayman.F. Naguib, and Nambi Seshadri Institute for Communication Engineering LNT Information Sciences Research Shannon Laboratory Munich University of Technology AT&T Labs Research Archisstraße 8 Park Avenue, uilding, Room C88 89 Munich, Germany Florham Park, NJ 79, USA ASTRACT This paper addresses the problem of equalization spacetime codes with transmit diversity. We derive a symbolbysymbol MAP equalizer/decoder for spacetime coded signals over frequency selective channels. We describe a turbo equalization/decoding scheme where the results of the decoding are fed back to the equalizer. Simulation results show an improved equalizer performance due to the feedback. Information Source s k SpaceTime Encoder c k c N k r k r k Receiver ~ s k Figure : Transmitter Diversity with SpaceTime Coding. INTRODUCTION Future wireless communications systems promise to offer a variety of multimedia services. In order for future wireless systems to offer such services, high data rates needs to be reliably transmitted over wireless channels. The main impairments of wireless communication channels include interference and time varying fading due to multipath propagation and time dispersion. Therefore, new transceiver techniques are being developed so that bandwidth efficient transmission will be possible. The interference problem can be solved by using careful frequency reuse and/or array signal processing with multiple antennas, which exploits the correlation between the impairments interference + noise on different antennas to suppress the interference. Antenna diversity is widely used to reduce the effect of multipath fading by combining signals from spatially separated antennas. The time dispersion problem can be solved by any of the equalization techniques, such as linear, decision feedback DFE, and maximum likelihood sequence estimation MLSE [, ]. It has been a standard practice to use multiple antennas at the receiver with some sort of combining of the received signals, e.g. selection or maximal ratio combining. Recently, there have been a number of proposals that use multiple antennas at the receiver with the appropriate signal processing to jointly combat the above wireless channel impairments [, 4]. However, it is hard to efficiently use receive antenna diversity at the remote unites since they are supposed to be relatively simple, small, and inexpensive. Therefore, receive antenna diversity and array signal processing with multiple antennas have been almost exclusively used or proposed for use at the base station resulting an asymmetric improvement of the reception quality only in the uplink. Recently, transmit antenna diversity techniques have been introduced to benefit from the antenna diversity also on the downlink while placing most of the diversity burden on the base station. Substantial benefits can be achieved by using channel codes that are specifically designed taken into account multiple transmit antennas. The first bandwidth efficient transmit diversity scheme was proposed. by Wittneben [5] and it included the transmit diversity scheme of [6] as a special case. In [7] spacetime trellis codes were introduced, where a general theory for design of combined trellis coding and modulation for transmit diversity is proposed. An input symbol to the spacetime encoder is mapped into modulation symbols, each is transmitted simultaneously from transmit antennas. These codes were shown were shown to achieve the maximal possible diversity benefit for a given number of transmit antennas, modulation constellation size, and transmission rate. Another transmit diversity schemes for two transmit antennas and a simple decoding scheme was proposed in [8] and later generalized to an arbitrary number of antennas as a spacetime block coding in [9]. Here the input data are encoded using spacetime block code with an orthogonal structure. The orthogonal structure of the spacetime block code allows a simple decoding algorithm by decoupling of the modulation symbols transmitted from different antennas. The performance analysis of the spacetime codes in [7] and [] was done assuming a flat fading channel. The analysis in [] shows that the design criteria of spacetime trellis codes in [7] is still optimum when used over a frequency selective channel, assuming that the receiver performs the optimum matched filtering for that channel. In addition, although the spacetime coding modem in [] was designed assuming a flat fading channel, it performed remarkably well when used over channels with delay spreads that are relatively small as compared to the symbol period T s. However, when the delay spread is large enough T s /4, there was a severe performance degradation, which pointed out the need for equalization. This paper address the equalization problem for communication systems employing spacetime coding with multiple transmit antennas. As we mentioned earlier, for each input symbol, the spacetime encoder produces modulation symbols that are transmitted simultaneously using transmit antennas. The received signal will be, therefore, the superposition of the transmitted signal after going through the channel perturbed by the noise. Therefore, an equalizer in this case, will have to equalize all the channels from the each of the transmit antennas to the receiver relying only on this superposition of signals. Hence, it should be obvious that a new frame work for equalization needs to be developed. This paper is an attempt towards this development. The organization
Figure : A 4state spacetime code for 4PSK with transmit antennas 4 5 6 7 4 5 6 7 4 5 6 7 5 5 5 5 54 55 56 57 4 5 6 7 7 7 7 7 74 75 76 77 4 4 4 4 44 45 46 47 4 5 6 7 6 6 6 6 64 65 66 67 4 5 6 7 Figure : An 8state spacetime code for 8PSK with transmit antennas of this paper is as follows. In the next Section, we will describe the spacetime coding and signal model. In Section we describe a joint turbo MAP equalization/decoder scheme for spacetime codes. Finally, Section 5 includes our concluding remarks.. SPACETIME CODING AND SIGNAL MODEL The baseband equivalent of the transmission scheme under consideration is shown in Figure. We consider a scenario of transmitter with transmit antennas and a receiver with N r receive antennas in a frequency selective Rayleigh fading environment. The impulse response of the overall transmission channel between the ith transmitting to the jth receiving antennas is modeled with a time varying FIR impulse response: D α ij k = a ij k, dδk d d= which includes the effects of the transmitter and receiver pulse shaping filters and the physical multipath channel between the ith transmitting antenna to the jth receiving antenna. The channel model order is D +. The tap gains a ij k, d are modeled as iid complex Gaussian random variables with zero mean and variance σa d. The channel is assumed to be passive, that is D σa d = d= It is assumed that all the N r channels have the same model order D +. This is a reasonable assumption since the number of individual multipath components is dictated by large structures and reflecting objects in the propagation environment. Without loss of generality, an important assumption that we will make through this paper is that the multipath channel parameters {a ij k, d, i =, j = N r } are invariant within a data burst, although they may be varying from burst to burst. This assumption relaxes the necessity of timevarying channel models and simplifies the development of equalization techniques for spacetime codes that would require channel tracking. In cellular systems such as GSM, the length of a data burst is about of.58 ms. Compared to the coherence time of the channel at 6 MPH mobile velocity, which is.5 ms, the burst length is small enough such that the block timeinvariant channel model is valid. This assumption is satisfied in most of the GSM environment. On the contrary, the burst length of an IS6 is about 6.67 ms, which is about the same order of the coherence time. Hence, in this case, we must use timevarying channel models. At a given time k, Let sk be the input to the spacetime encoder and let the corresponding output of the spacetime encoder be {c k, c k,, c Nt k}, where the code symbol c i k is transmitted from antenna i at time k. Then we can write the received signal at receive antenna j as D r j k = a ij dc i k d + n j k, j N r. d= The term n j k is a sequences of i.i.d. complex Gaussian noise samples with zero mean and variance σn. The above equation can be put in a matrix form as r j k = a ij c i k + n j k 4 where a ij = [a ij, a ij,, a ij d] and c i k = [c i k, c i k,, c i k D] T. Consider the output of the N r receive antennas at time k, where rk = [r k, r k,, r Nr k] T = H i c i k + nk 5 nk = [n k, n k,, n Nr k] T a i H i = a i.. = [h i, h i, h i D] a inr h i d = [a i d, a i d,, a inr d] T The noise vector nk has a zero mean and covariance R n = σ n I N r N r. We can extend 5 into a spacetime data model N r receive antennas and L + time taps by staking L + taps of rk into an N r L + vector x k = [rk T rk T rk L T ] T. The new spacetime data model is then given by xk = H i c i k + nk 6
where c i k = [c i k, c i k,, c i k D L] T, nk = [nk T nk T nk L T ] T, and H i H i =... 7 H i is an N r L+ D+L+ block toeplitz matrix. The noise vector nk has a zero mean and covariance R n = σ n I N r L+ N r L+. Equations, 5, and 6 will serve as a received signal model in developing different equalization techniques later in this paper. Figures and show two examples of spacetimes codes designed for two transmit antennas = for 4PSK and 8PSK constellations, respectively. transmit antennas. For example, for the 8PSK 8state spacetime code in Figure, consider the 8PSK constellation as labeled in the same figure. Figure also shows the trellis description for this code. The input bit stream to the ST encoder is grouped into groups of bits and each group is mapped into one of 8 constellation points. Each row in the matrix shown in Figure represents the edge labels for transitions from the corresponding state. The edge label c c indicates that symbol c is transmitted over the first antenna and that symbol c is transmitted over the second antenna. This codes has a spectral efficiency of bits/sec per channel use.. MAP EQUALIZATION OF ST CODES For data rates on the order of the coherence bandwidth of the channel, or larger, an equalizer must be used to compensate for the intersymbol interference induced by the resolvable multipath receptions. Two powerfully equalization techniques are known []:The probabilistic symbolbysymbol MAP algorithm provides the MAPprobabilities for each individual symbol [5], whereas the Viterbi algorithm VA [, 6, ] is a maximum likelihood sequence estimator that outputs the MLchannel path. oth techniques have the advantage that they gather energy from all channel tap gains therefore taking full advantage of the diversity gain offered by the multipath propagation without suffering from noise enhancement or error propagation. This is rather an important feature since in wireless propagation environments, the reflections may be stronger than the direct path. There are, however, two main problems for these approaches. First, complexity in terms of the number of equalizer states, M D, for both algorithms increases exponentially with the channel memory D, where M is the cardinality of the spacetime code signal space. Secondly, the Viterbi equalizer outputs the ML path in a form of hard decisions, where the symbolbysymbol MAP algorithm provides soft decisions, but is more complex. In the classical equalization problem, the first disadvantage can be solved by using reduced complexity approach [7]. However, reduced complexity techniques will suffer from performance degradation if the channel response is not minimum phase or nearly so. Since wireless channels are time varying and hence the minimum phase condition is not guaranteed all the time, a whitened matched filter or a precurser equalizer must be used to render the channel minimum phase all the time. While designing a whitened matched filter is well known [] for the classical equalization problem, it is not yet known when spacetime coding with transmit diversity is used. This is because, as mentioned earlier, the received signal at the receiver is the superposition of all transmitted signals, that propagated through totally independent channels, and hence the job of the whitened matched filter in his case is to render all these channels minimum phase. The second disadvantage r ln p c r Π MAP MAP L s Equalizer Decoder Π r Nr ln p c Nt r Channel State Information CSI Π Π ln p c r ln p c Nt r L a s Apriori Figure 4: MAP Equalization of SpaceTime Codes can be avoided by avoided by providing reliability information along with the hard decisions. Let us assume that information are transmitted in finite blocks or bursts of length L + and let ck represent the code vector transmitted at time k, that is ck = [c k c k c Nt k] T. Note that ck is related to {c i k, i } defined earlier such that ck is the first raw ofck = [c k c k c Nt k], ck is the second raw of Ck, and so on. Let us also assume that the spacetime encoder uses a modulation constellation with size M. Then we can easily see that there are M possible code vectors that can be transmitted at any given time. Given the FIR impulse response of the channel between the transmitter and the receiver defined in, The following definitions are used to describe the channel trellis. State: A state µk at time k is defined as µk = ck D, ck D +,, ck, ck. ranch or Transition: a branch or transition ζk at time k is defined as ζk = µk, µk + = ck D, ck D +,, ck, ck = µk, ck = ck l, µk +. D The channel has S = M states and there are M branches from and to each state.. ranch Output and ranch Squared Euclidean Distance: a branch ζk determines a noiseless output yζk yζk = H i c i k 8 The squared Euclidean distance of a branch ζ k, denoted by,dk, is defined as Dk = rk yζk 9 4. Path: A trellis path πk at time k is defined as πk = ζ, ζ,, ζk = µ, µ,, µk, µk + = µ, c, c,, ck
k β k µ is the backward probability of state µ at time k and is given by β k µ = γ k µ, µ β k µ, µ States with forward probabilities ' a kcm h c k c r i L = m States with backward probabilities b kbmg and γ k µ, µ is the branch transition probability for the existing transitions and can be expressed as the product of an a priori probability and the channel transition probability: γ k µ, µ = prk µ, µ Pµ µ. 4 The a priori information is obtained by the feedback of the extrinsic information P e c i r L of the decoder: m ' c k c r i L = m Figure 5: Equalizer Trellis The task of the MAP equalizer/decoder shown in Figure 4 is to compute the probabilities P c i k r L for k L and i, given the received signal vectors r L := r, r,, rl. For this, we use a "Turbo" scheme for joint equalization and decoding as shown in Figure 4. As we will see later from the simulations, this will significantly improve the results. We use a softin/softout algorithm, e.g. the symbol by symbol MAP, for both equalization and decoding. esides the received signal vectors r,..., rl the equalizer accepts an additional a priori information P c i k r L, i =,, about the code symbols c i k, i =,,. This a priori information is obtained by a turbo feedback from the decoder. In the first equalization step iteration all symbols c i k are assumed to be equally likely. Let M be the number of constellation points and {c, c,, c M } be the constellation points. For each possible symbol c i k = c m, m =,..., M the MAP equalizer computes the a poste riori information P cording to P c k = c m r L c i k = c m r L =,, P µ,µ c i k=cm p m c Nt k = c m r L µ, µ, r L ac where µ = µk is the state of the trellis at time k and µ = µk is the state of the trellis at time k for the respective transition. The sum is taken only over those transitions ζk = µ, µ which are labeled with the code symbol c i k = c m. The joint probability p µ, µ, r L b is the product of three factors, two of which can be calculated by recursive formulae: p µ, µ, r L = p µ, r k }{{} α k µ P µ µ p rk µ, µ }{{} γ k µ,µ p r L k+ µ } {{ } β k µ where the α k µ is the forward probability of state µ at time k and is given by α k µ = µ γ k µ, µ α k µ, N T Pµ µ = P e c i r L, 5 where the transition ζk = µ, µ is labeled with the code symbols c i, i =,..., N T. In iteration zero, all transitions are assumed to be equally likely. The channel transition probability is pr k µ, µ = πσn exp Dk σn. 6 The a posteriori information of the equalizer P c r L,, P c NT r L is the channel input to the decoder. If an independent a priori information L a u = log Pu=+ Pu= about the information bits u t is available e.g. from knowledge about the source statistics, it can be used by the decoder. The decoder computes the a posteriori loglikelihood ratios L uˆ t = log Pu t = + r L Pu t = r L pµ, µ, r L = log µ,µ ut =+ µ,µ ut = pµ, µ, r L 7 about the information bits u. Additionally the decoder delivers a posteriori probabilities P c i k = c m r L b = p µ, µ, r L µ,µ = µ,µ α k µ γ k µ, µ β k µ about the transmitted code symbols. From this information we can obtain the a priori information for the equalizer in the first iteration. We know from turbo decoding and turbo equalization [8, 9] that it is important to feed back only the extrinsic part of the a posteriori information because the correlation between the a priori information used by the equalizer and previous decisions of the equalizer should be minimized. Ideally the a priori information would be an independent estimation. The extrinsic information about a certain symbol is the incremental information about the current symbol obtained through the decoding process from all information which is available for the other symbols in the block. It can be calculated by subtracting the logarithm of the input probabilities
ER of the decoder from the logarithm of the output probabilities. Therefore, the extrinsic information to be fed back is [ P e c i k = c m r L P c i k = c m r L ] decoder = [ P c i k = c m r L ]. 8 equalizer Performance of MAP Equalizer for 8PSK 8State STC with Transmit Antennas and a Tap Channel and No Interleaving Iteration Iteration For the same reason we have to subtract the logarithm of the a priori probabilities of the equalizer from the logarithm of its a posteriori probabilities. This makes sure that we supply the decoder with channel information and extrinsic information of the equalizer only. The a posteriori information of the equalizer in the first iteration second equalization step is: Pc i k = c m r L = = µ,µ α k µ N T [ j= µ,µ pµ, µ, r L a priori feedback from decoder {}}{ P e c j k r L ] pr k µ, µ β k µ }{{} γ k µ,µ = P e c i k = c m r L α k µ P e c j k r L pr k µ, µβ k µ µ,µ j= j i } {{ } extr. info of equalizer + channel info The information which is passed to the decoder is P c i k = c m r L P ext+channel = c i k = c m r L P e c i k = c m r L 9 It is important to note that the interleaving rule before transmission has to be the same for all transmit antennas in order not to destroy the rank property of the spacetime code [7]. In a flat fading environment we don t gain using the interleaver. However, in a frequency selective environment the interleavers help to break up burst errors which occur after the equalizer. The interleavers are also very important in the turbo scheme in order to provide the equalizer with a nearly independent a priori information. 4. SIMULATION RESULTS In this Section, we present simulation results for the turbo MAP equalization and decoding for the 8PSK 8state spacetime code shown in Figure with two transmit and receive antennas. The channel from each of the two transmit antennas to the receive antennas was assumed to have taps with equal energy. oth channels were assumed to be fixed over a frame. Each frame was 4 8PSK symbols long. In these simulation experiments, we assumed perfect knowledge of both channels. The simulation results are shown in Figures 6 and 7. Figure 6 show the performance of the above MAP turbo equalizer/decoder and no interleaving. We plot the results for the first two iterations. We can easily see the improvement due to the feedback of the extrinsic information of the decoder. At ER, the required SNR energy per symbol to noise ratio is about d. After feedback the required SNR is about 7 d, i.e. a 4 d gain. Figure 4 6 8 4 6 8 E s /N o d Figure 6: Performance of MAP Equalizer for 8PSK SpaceTime Code with Two Transmit Antennas and a Tap Channel with no Interleaving 7 shows the same results but with random interleaving. We can that in the case of no feedback iteration, the improvement due to interleaving at ER is at most d as compared to the case when interleaving is used. However, with feedback, the improvement due to interleaving is.5 d. 5. SUMMARY A joint MAP equalization/decoding iterative scheme is proposed for spacetime coded signals. The extrinsic information from the decoder is fed back to the equalizer. Preliminary simulation results for the 8PSK 8state spacetime code with transmit and receive antenna and a two tap channel show a 4 d improvement in SNR required for ER without interleaving and a 6.5 d improvement with interleaving. The problem, however, of the above MAP scheme is that it will have a large number of states even for moderate channel length because of the multiple transmissions of spacetime codes. Therefore, we a reduced complexity approach is needed. 6. REFERENCES [] P. alaban and J. Salz, Optimum Diversity Combining and Equalization in Digital Data Transmission with Application to Cellular Mobile Radio, IEEE Trans. Veh. Tech., vol. VT4, pp. 4 54, May 99. [] P. alaban and J. Salz, Optimum Diversity Combining and Equalization in Data Transmission with Application to Cellular Mobile Radio Part I: Theoretical Considerations, IEEE Trans. Commun., vol. COM45, pp. 885 894, May 99. [] J. H. Winters, Optimum Combining in Digital Mobile Radio with Cochannel Interference, IEEE J. Select. Areas Commun., vol. JSAC4, pp. 58 59, July 984. [4] J. H. Winters, On the Capacity of Radio Communication Systems with Diversity in a Rayleigh Fading Environment, IEEE J. Select. Areas Commun., vol. JSAC55, pp. 87 878, June 987.
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