MAP Equalization of Space-Time Coded Signals ... - Semantic Scholar

MAP Equalization of Space-Time Coded Signals over Frequency Selective Channels Invited Paper Gerhard Bauch1 , Ayman.F. Naguib2 , and Nambi Seshadri2 1

Institute for Communication Engineering (LNT) Munich University of Technology Archisstraße 21 80290 Munich, Germany

2

Information Sciences Research - Shannon Laboratory AT&T Labs - Research 180 Park Avenue, Building 103, Room C-288 Florham Park, NJ 07932, USA

ABSTRACT This paper addresses the problem of equalization space-time codes with transmit diversity. We derive a symbol-by-symbol MAP equalizer/decoder for space-time 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.

c1 ( k )

r1 ( k )

cN ( k )

rNt ( k )

Space-Time Encoder Information Source

s( k )

~ s (k )

Receiver

Figure 1: Transmitter Diversity with Space-Time Coding 1. 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) [1, 2]. 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 [3, 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] space-time 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 space-time encoder is mapped into Nt modulation symbols, each is transmitted simultaneously from Nt transmit antennas. These codes were shown were shown to achieve the maximal possible diversity benefit for a given number of transmit antennas Nt , 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 space-time block coding in [9]. Here the input data are encoded using space-time block code with an orthogonal structure. The orthogonal structure of the space-time block code allows a simple decoding algorithm by decoupling of the modulation symbols transmitted from different antennas. The performance analysis of the space-time codes in [7] and [10] was done assuming a flat fading channel. The analysis in [10] shows that the design criteria of space-time 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 space-time coding modem in [11] 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 Ts . However, when the delay spread is large enough ≥ Ts /4, there was a severe performance degradation, which pointed out the need for equalization. This paper address the equalization problem for communication systems employing space-time coding with multiple transmit antennas. As we mentioned earlier, for each input symbol, the space-time encoder produces Nt modulation symbols that are transmitted simultaneously using Nt transmit antennas. The received signal will be, therefore, the superposition of the Nt 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

0

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1

10 11 12 13

1

0

2

2

20 21 22 23

3

30 31 32 33

3

Figure 2: A 4-state space-time code for 4-PSK with 2 transmit antennas

2 3

1 0

4 5

7 6

0

00 01 02 03 04 05 06 07

1

50 51 52 53 54 55 56 57

2 3

20 21 22 23 24 25 26 27 70 71 72 73 74 75 76 77 40 41 42 43 44 45 46 47

4 5

10 11 12 13 14 15 16 17

6

60 61 62 63 64 65 66 67

7

30 31 32 33 34 35 36 37

Figure 3: An 8-state space-time code for 8-PSK with 2 transmit antennas

variables with zero mean and variance σa2 (d). The channel is assumed to be passive, that is D X σa2 (d) = 1 (2) d=0

It is assumed that all the Nt × Nr channels have the same model order D + 1. 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 {aij (k, d), i = 1 · · · Nt , j = 1 · · · Nr } are invariant within a data burst, although they may be varying from burst to burst. This assumption relaxes the necessity of time-varying channel models and simplifies the development of equalization techniques for space-time codes that would require channel tracking. In cellular systems such as GSM, the length of a data burst is about of 0.58 ms. Compared to the coherence time of the channel at 60 MPH mobile velocity, which is 12.5 ms, the burst length is small enough such that the block time-invariant channel model is valid. This assumption is satisfied in most of the GSM environment. On the contrary, the burst length of an IS-136 is about 6.67 ms, which is about the same order of the coherence time. Hence, in this case, we must use time-varying channel models. At a given time k, Let s(k) be the input to the space-time encoder and let the corresponding output of the space-time encoder be {c1 (k), c2 (k), · · · , cNt (k)}, where the code symbol ci (k) is transmitted from antenna i at time k. Then we can write the received signal at receive antenna j as rj (k) =

Nt X D X

2. SPACE-TIME CODING AND SIGNAL MODEL The baseband equivalent of the transmission scheme under consideration is shown in Figure 1. We consider a scenario of transmitter with Nt transmit antennas and a receiver with Nr receive antennas in a frequency selective Rayleigh fading environment. The impulse response of the overall transmission channel between the i-th transmitting to the j -th receiving antennas is modeled with a time varying FIR impulse response: αij (k) =

D X

aij (k, d)δ(k − d)

(1)

d=0

which includes the effects of the transmitter and receiver pulse shaping filters and the physical multipath channel between the i-th transmitting antenna to the j -th receiving antenna. The channel model order is D + 1. The tap gains aij (k, d) are modeled as iid complex Gaussian random

(3)

The term nj (k) is a sequences of i.i.d. complex Gaussian noise samples with zero mean and variance σn2 . The above equation can be put in a matrix form as Nt X aij · ci (k) + nj (k) (4) rj (k) = i=1

where aij = [aij (1), aij (2), · · · , aij (d)] and ci (k) = [ci (k), ci (k − 1), · · · , ci (k − D)]T . Consider the output of the Nr receive antennas at time k, r(k)

of this paper is as follows. In the next Section, we will describe the space-time coding and signal model. In Section 3 we describe a joint turbo MAP equalization/decoder scheme for space-time codes. Finally, Section 5 includes our concluding remarks.

aij (d)ci (k − d) + nj (k), 1 ≤ j ≤ Nr .

i=1 d=0

=

[r1 (k), r2 (k), · · · , rNr (k)]T

=

Nt X

Hi · ci (k) + n(k)

(5)

i=1

where n(k)

=

Hi

=

hi (d)

=

[n1 (k), n2 (k), · · · , nNr (k)]T   ai1  ai2     ..  = [hi (0), hi (1), · · · hi (D)]  .  aiNr

[ai1 (d), ai2 (d), · · · , aiNr (d)]T

The noise vector n(k) has a zero mean and covariance Rn = σn2 · INr ×Nr . We can extend (5) into a space-time data model (Nr receive antennas and L + 1 time taps) by staking L + 1 taps of r(k) into an Nr (L + 1) × 1 vector xk = [r(k)T r(k − 1)T · · · r(k − L)T ]T . The new space-time data model is then given by x(k) =

Nt X i=1

¯ i · c¯ i (k) + n(k) ¯ H

(6)

¯ where c¯ i (k) = [ci (k), ci (k−1), · · · , ci (k−D−L)]T , n(k) = [n(k)T n(k− T T T 1) · · · n(k − L) ] , and 

 ¯i =  H 

··· 0

Hi .. 0 ···

. Hi

   

Π

ln p(c1| r)

-

Π (7)

¯ is an Nr (L+1)×(D+L+1) block toeplitz matrix. The noise vector n(k) has a zero mean and covariance Rn¯ = σn2 · INr (L+1)×Nr (L+1) . Equations (3), (5), and (6) will serve as a received signal model in developing different equalization techniques later in this paper. Figures 2 and 3 show two examples of space-times codes designed for two transmit antennas (Nt = 2) for 4-PSK and 8-PSK constellations, respectively. transmit antennas. For example, for the 8-PSK 8-state space-time code in Figure 3, consider the 8-PSK constellation as labeled in the same figure. Figure 3 also shows the trellis description for this code. The input bit stream to the ST encoder is grouped into groups of 3 bits and each group is mapped into one of 8 constellation points. Each row in the matrix shown in Figure 3 represents the edge labels for transitions from the corresponding state. The edge label c1 c2 indicates that symbol c1 is transmitted over the first antenna and that symbol c2 is transmitted over the second antenna. This codes has a spectral efficiency of 3 bits/sec per channel use. 3. 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 [12]:The probabilistic symbolby-symbol MAP algorithm provides the MAP-probabilities for each individual symbol [12-15], whereas the Viterbi algorithm (VA) [12, 16, 13] is a maximum likelihood sequence estimator that outputs the ML-channel path. Both 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 space-time code signal space. Secondly, the Viterbi equalizer outputs the ML path in a form of hard decisions, where the symbol-by-symbol 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 [17]. 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 pre-curser equalizer must be used to render the channel minimum phase all the time. While designing a whitened matched filter is well known [12] for the classical equalization problem, it is not yet known when space-time 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

r1

ln p(c1| r)

MAP Equalizer

-

Π -

rNr

ln p(cNt | r)

Π

ln p(cNt | r)

-

-1

MAP Decoder

-1

L( s)

La ( s )

Channel State Information (CSI)

Apriori

Figure 4: MAP Equalization of Space-Time 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 LB + 1 and let c(k) represent the code vector transmitted at time k, that is c(k) = [c1 (k) c2 (k) · · · cNt (k)]T . Note that c(k) is related to {ci (k), 1 ≤ i ≤ Nt } defined earlier such that c(k) is the first raw of C (k) = [c1 (k) c2 (k) · · · cNt (k)], c(k − 1) is the second raw of C (k), and so on. Let us also assume that the space-time encoder uses a modulation constellation with size M. Then we can easily see that there are M Nt 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 (1), The following definitions are used to describe the channel trellis 1. State: A state µ(k) at time k is defined as △

µ(k) = (c(k − D), c(k − D + 1), · · · , c(k − 2), c(k − 1)) 2. Branch or Transition: a branch or transition ζ (k) at time k is defined as ζ (k)

△

= = = =

(µ(k), µ(k + 1)) (c(k − D), c(k − D + 1), · · · , c(k − 1), c(k)) (µ(k), c(k)) (c(k − l), µ(k + 1)).

 D The channel has |S| = M Nt states and there are M branches from and to each state. 3. Branch Output and Branch Squared Euclidean Distance: a branch ζ (k) determines a noiseless output y(ζ (k)) y(ζ (k)) =

Nt X

Hi ci (k)

(8)

i=1

The squared Euclidean distance of a branch ζk , denoted by, D(k), is defined as △ (9) D(k) = |r(k) − y(ζ (k))|2 4. Path: A trellis path π(k) at time k is defined as π(k)

△

= = =

(ζ (1), ζ (2), · · · , ζ (k)) (µ(1), µ(2), · · · , µ(k), µ(k + 1)) (µ(1), c(1), c(2), · · · , c(k))

k

βk−1 (µ) is the backward probability of state µ′ at time k − 1 and is given by X γk (µ′ , µ) · βk (µ), (13) βk−1 (µ′ ) =

ci ( k ) = cm1 | r0LB

and γk (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:

µ

States with forward probabilities

States with backward probabilities

bg

c h

a k -1 m

'

bk m

γk (µ′ , µ) = p(r(k)|µ′ , µ) · P (µ|µ′ ).

The a priori information is obtained by the feedback of the extrinsic L information Pe (ci |r0 B ) of the decoder: P (µ|µ′ ) =

ci ( k ) = cm2 | r

m

Figure 5: Equalizer Trellis The task of the MAPequalizer/decoder shown in Figure 4 is to com L pute the probabilities P ci (k)|r0 B for 0 ≤ k ≤ LB and 1 ≤ i ≤ Nt , L

given the received signal vectors r0 B := r(0), r(1), · · · , r(LB ) . 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 soft-in/soft-out algorithm, e.g. the symbol by symbol MAP, for both equalization and decoding. Besides the received signal vectors r(0),  ..., r(LB) the equalizer accepts an addiL tional a priori information P ci (k)|r0 B , i = 1, · · · , Nt about the code symbols ci (k), i = 1, · · · , Nt . This a priori information is obtained by a turbo feedback from the decoder. In the first equalization step (iteration 0) all symbols ci (k) are assumed to be equally likely. Let M be the number of constellation points and {c1 , c2 , · · · , cM } be the constellation points. For each possible symbol ci (k) = cm , m =  1, ..., M the MAP  equalizer  computes the a poste L L riori information P c1 (k) = cm |r0 B , · · · , P cNt (k) = cm |r0 B according to     X L L P ci (k) = cm |r0 B = p µ′ , µ, r0 B (10) (µ′ ,µ) ci (k)=cm

where µ′ = µ(k − 1) is the state of the trellis at time k − 1 and µ = µ(k) is the state of the trellis at time k for the respective transition. The sum is taken only over those transitions ζ (k) = (µ′ , µ) which are  labeled with  L the code symbol ci (k) = cm . The joint probability p µ′ , µ, r0 b is the product of three factors, two of which can be calculated by recursive formulae:    

 LB  L p µ′ , µ, r0 B = p µ′ , r0k−1 · P µ|µ′ · p r(k)|µ′ , µ · p rk+1 |µ {z } | | {z } | {z } αk−1 (µ′ )

γk−1 (µ′ ,µ)

βk (µ)

(11) where the αk (µ) is the forward probability of state µ at time k and is given by X αk (µ) = γk (µ′ , µ) · αk−1 (µ′ ), (12) µ′

NT Y

L

Pe (ci |r0 B ),

(15)

i=1

LB 0

m'

(14)

where the transition ζ (k) = (µ′ , µ) is labeled with the code symbols ci , i = 1, ..., NT . In iteration zero, all transitions are assumed to be equally likely. The channel transition probability is   1 D(k) exp − . (16) p(rk |µ′ , µ) = 2πσn2 2σn2     L L The a posteriori information of the equalizer P c1 |r0 B , · · · , P cNT |r0 B is the “channel input” to the decoder. If an independent a priori infor(u=+1) mation La (u) = log P P (u=−1) about the information bits ut is available (e.g. from knowledge about the source statistics), it can be used by the decoder. The decoder computes the a posteriori log-likelihood ratios L

L(uˆt )

=

=

log

P (ut = +1|r0 B ) L

P (ut = −1|r0 B ) P L p(µ′ , µ, r0 B ) (µ′ ,µ) ut =+1

log P

(µ′ ,µ) ut =−1

L

p(µ′ , µ, r0 B )

(17)

about the information bits u. Additionally the decoder delivers a posteriori probabilities    X  L L P˜ ci (k) = cm |r0 b = p˜ µ′ , µ, r0 B (µ′ ,µ) ci =cm

=

X

α˜ k−1 (µ′ )γ˜k (µ′ , µ)β˜k (µ)

(µ′ ,µ) ci =cm

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 [18, 19] 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

of the decoder from the logarithm of the output probabilities. Therefore, the extrinsic information to be fed back is i h  L   P ci (k) = cm |r0 B L i decoder . Pe ci (k) = cm |r0 B = h  (18) L P ci (k) = cm |r0 B

Performance of MAP Equalizer for 8-PSK 8-State STC with 2 Transmit Antennas and a 2-Tap Channel and No Interleaving 100 Iteration 0 Iteration 1

equalizer

BER

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.

10-1

10-2

10-3

The a posteriori information of the equalizer in the first iteration (second equalization step) is: X L L P (ci (k) = cm |r0 B ) = p(µ′ , µ, r0 B ) (µ′ ,µ) ci =cm

10-4 6

8

10

12

14

16

18

20

Es /No (dB)

a priori feedback from decoder

=

X

(µ′ ,µ) ci =cm

αk−1 (µ′ ) [

NT Y

j =1

z }| { L Pe (cj (k)|r0 B )]

|

p(rk |µ′ , µ) βk (µ)

{z

γk (µ′ ,µ)

  L = Pe ci (k) = cm |r0 B ·   Nt X Y  L  αk−1 (µ′ )  Pe (cj (k)|r0 B ) p(rk |µ′ , µ)βk (µ) (µ′ ,µ) ci =cm

|

}

7 shows the same results but with random interleaving. We can that in the case of no feedback (iteration 0), the improvement due to interleaving at 10−3 BER is at most 1 dB as compared to the case when interleaving is used. However, with feedback, the improvement due to interleaving is 2.5 dB.

j =1 j 6=i

{z

extr. info of equalizer + channel info

}

The information which is passed to the decoder is   L   P ci (k) = cm |r0 B L   P ci (k) = cm |r0 B = L ext+channel Pe ci (k) = cm |r0 B

Figure 6: Performance of MAP Equalizer for 8-PSK Space-Time Code with Two Transmit Antennas and a 2-Tap Channel with no Interleaving

5. SUMMARY

(19)

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 space-time 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 8-PSK 8-state space-time code shown in Figure 3 with two transmit and 1 receive antennas. The channel from each of the two transmit antennas to the receive antennas was assumed to have 2 taps with equal energy. Both channels were assumed to be fixed over a frame. Each frame was 400 8-PSK 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 10 −3 BER, the required SNR (energy per symbol to noise ratio) is about 21 dB. After feedback the required SNR is about 17 dB, i.e. a 4 dB gain. Figure

A joint MAP equalization/decoding iterative scheme is proposed for space-time coded signals. The extrinsic information from the decoder is fed back to the equalizer. Preliminary simulation results for the 8-PSK 8-state space-time code with 2 transmit and 1 receive antenna and a two tap channel show a 4 dB improvement in SNR required for 10 −3 BER without interleaving and a 6.5 dB 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 space-time codes. Therefore, we a reduced complexity approach is needed. 6. REFERENCES [1] P. Balaban and J. Salz, “Optimum Diversity Combining and Equalization in Digital Data Transmission with Application to Cellular Mobile Radio,” IEEE Trans. Veh. Tech., vol. VT-40(2), pp. 342– 354, May 1991. [2] P. Balaban 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. COM-40(5), pp. 885–894, May 1992. [3] J. H. Winters, “Optimum Combining in Digital Mobile Radio with Cochannel Interference,” IEEE J. Select. Areas Commun., vol. JSAC-2(4), pp. 528–539, July 1984. [4] J. H. Winters, “On the Capacity of Radio Communication Systems with Diversity in a Rayleigh Fading Environment,” IEEE J. Select. Areas Commun., vol. JSAC-5(5), pp. 871–878, June 1987.

Performance of MAP Equalizer for 8-PSK 8-State STC with 2 Transmit Antennas and a 2-Tap Channel and Random Interleaving 100 Iteration 0 Iteration 1

10

-1

10-2

[15] L. N. Lee, “Concatenated Coding Systems Employing a UnitMemory Conventional Code and a Byte-Oriented Decoding Algorithm,” IEEE Trans. Commun., vol. COM-25, pp. 1064–1074, October 1977. [16] G. D. Forney, “Maximum Likelihood Sequence Estimation of Digital Sequences in the Presence of Intersymbol Interference,” IEEE Trans. Inform. Theory, vol. IT-18, pp. 363–378, March 1972.

BER

[17] M. V. Eyuboglu and S. U. Qureshi, “Reduced-State Sequence Estimation with Set Partioning and Decision feedback,” IEEE Trans. Commun., vol. COM-36, pp. 12–20, January 1988.

10-3

[18] C. Douillard, M. J´ez´equel, C. Berrou, , and A. G. A. Picart, P. Didier, “Iterative Correction of Intersymbol Interference: Turbo– Equalization,” in European Transactions on Telecommunications, vol. 6, pp. 507–511, September 1995.

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Es /No (dB)

Figure 7: Performance of MAP Equalizer for 8-PSK Space-Time Code with Two Transmit Antennas and a 2-Tap Channel with Random Interleaving

[5] A. Wittneben, “A New Bandwidth Efficient Transmit Antenna Modulation Diversity Scheme for Linear Digital Modulation,” in Proc. IEEE ICC’93, vol. 3, (Geneva, Switzerland), pp. 1630–1634, 1993. [6] N. Seshadri and J. H. Winters, “Two Schemes for Improving the Performance of Frequency-Division Duplex (FDD) Transmission Systems Using Transmitter Antenna Diversity,” International Journal of Wireless Information Networks, vol. 1, pp. 49–60, Jan 1994. [7] V. Tarokh, N. Seshadri, and A. R. Calderbank, “Space-Time Codes for High Data Rate Wireless Communications: Performance Criterion and Code Construction,” IEEE Trans. Inform. Theory, pp. 744– 765, March 1997. [8] S. Alamouti, “Space Block Coding: A Simple Transmitter Diversity Scheme for Wireless Communications,” IEEE JSAC, October 1998. [9] V. Tarokh, H. Jafarkhani, and R. A. Calderbank, “Space Block Coding for Wireless Communications: Basic Theory of Generalized Orthogonal Designs.” submitted to IEEE Trans. Inform. Theory, 1998. [10] V. Tarokh, A. F. Naguib, N. Seshadri, and A. Calderbank, “SpaceTime Codes for High Data Rate Wireless Communications: Performance Criteria in the Presence of Channel Estimation Errors, Mobility, and Multiple Paths,” IEEE Trans. Commun., February 1998. [11] A. F. Naguib, V. Tarokh, N. Seshadri, and A. R. Calderbank, “A Space-Time Coding Modem for High Data Rate Wireless Communications,” IEEE JSAC, October 1998. [12] J. G. Proakis, Digital Communications. New Yor, NY: McGrawHill, second ed., 1989. [13] G. D. Forney, “The Viterbi Algorithm,” Proc. of IEEE, vol. 61, pp. 268–278, March 1973. [14] L. R. Bahl, J. Cocke, F. Jelinek, and J. Raviv, “Optimal Decoding of Linear Codes for Minimizing Symbol Error Rate,” IEEE Trans. Inform. Theory, vol. IT-20, pp. 284–287, March 1974.

[19] G. Bauch, H. Khorram, and J. Hagenauer, “Iterative Equalization and Decoding in Mobile Communications ’ Systems,” in Second European Personal Mobile Communications Conference (EPMCC), (Bonn, Germany), pp. 307–312, September 1997.