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Abstract—We study the use of turbo-coded modulation for wire- less communication systems with multiple transmit and receive an- tennas over block Rayleigh ...
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IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 19, NO. 5, MAY 2001

Turbo-Coded Modulation for Systems with Transmit and Receive Antenna Diversity over Block Fading Channels: System Model, Decoding Approaches, and Practical Considerations Andrej Stefanov, Student Member, IEEE, and Tolga M. Duman, Member, IEEE

Abstract—We study the use of turbo-coded modulation for wireless communication systems with multiple transmit and receive antennas over block Rayleigh fading channels. We describe an effective way of applying turbo-coded modulation as an alternative to the current space–time codes with appropriate interleaving. We study the performance with the standard iterative turbo decoding algorithm, as well as the iterative demodulation–decoding algorithm. In addition to the introduction of the turbo-coded modulation scheme, we consider a variety of practical issues including the case of large number of antennas, the effects of estimated channel state information, and correlation among subchannels between different transmit–receive antenna pairs. We present examples to illustrate the performance of the turbo-coded modulation scheme and observe significant performance gains over the appropriately interleaved space–time trellis codes. Index Terms—Antenna diversity, space–time coding, turbocoded modulation, turbo codes, wireless communications.

I. INTRODUCTION

I

N RECENT years, the goal of providing high-speed wireless data services has generated a great amount of interest among the research community. The main challenge in achieving reliable communications lies in the severe conditions that are encountered when transmitting information over the wireless channel. Recent information theoretic results [1], [2] have demonstrated that the capacity of the system in the presence of block Rayleigh fading improves significantly with the use of multiple transmit and receive antennas. Similar results for multiple antenna systems over quasi-static Rayleigh fading channels have previously been reported by Foschini and Gans [3] and Telatar [4]. The block fading channel model [5] is motivated by the fact that in many wireless systems the coherence time of the channel is much longer then one symbol interval, resulting in adjacent symbols being affected by the same fading value. The block fading channel model [5] assumes that a codeword of length spans blocks of length , where the group of blocks is referred to as frame. The value of the fading in each Manuscript received April 28, 2000. This work was supported in part by the National Science Foundation Grant CAREER CCR-9984237 and by Project ECS-9979403. The authors are with the Department of Electrical Engineering, Arizona State University, Tempe, AZ 85287–7206 USA (e-mail: [email protected]; [email protected]). Publisher Item Identifier S 0733-8716(01)03908-7.

block is constant, and each block is sent through an independent channel. In addition, an interleaver may be used to spread the code symbols over the blocks. The fading blocks will experience independent fades, provided we have sufficient separation in time, in frequency, or both in time and in frequency. An example of the latter is slow frequency hopping as is done in global system for mobile communications (GSM), where the spacing between the carriers is larger then the coherence bandwidth, resulting in basically uncorrelated blocks. In the case of GSM, there are four (half rate) or eight (full rate) differently faded blocks per frame. Another example is IS-54, where we have two time division multiple access (TDMA) blocks per frame, separated in time, which become less correlated as the speed of the mobile increases. Recently, there has been an explosion of interest in space– time coding. Liu and Fitz [6], considered turbo trellis coded modulation. There, the constituent codes are trellis codes as opposed to binary convolutional codes. In subsequent work, Liu et al. [7] considered design guidelines for four-phase shift keying (PSK) with multiple antenna systems with an application to turbo codes. The performance of their scheme is comparable with the performance of the system presented in this paper. Similarly, the turbo trellis coded modulation concept where the constituent codes are space–time trellis codes was investigated by Narayanan [8], and later also by Cui and Haimovich [9]. Design guidelines for turbo codes with BPSK have been investigated by Su and Geraniotis [10]. In [11], Bauch considered concatenation of a turbo code with an inner space–time block code. Finally, bit interleaved serial concatenation scheme with an emphasis on systems with a large number of antennas was recently introduced by Reial and Wilson [12]. A large number of antenna systems with a different approach to transmitter and receiver design has also recently been considered by El Gamal and Hammons [13]. On the other hand, new space–time trellis codes with modest performance gains over the codes by Tarokh et al., in terms of the probability of frame error, have been introduced by Baro et al. [14] and by Blum [15]. Bit interleaved coded modulation with convolutional codes and multiple antennas has also been studied in [16]. General design guidelines for space–time codes with PSK modulation have been presented by Hammons and El Gamal [17] and recently investigated by Blum [15]. In this paper, we present a comprehensive study of turbocoded modulation for systems with transmit and receive

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STEFANOV AND DUMAN: TURBO-CODED MODULATION

antenna diversity introduced in [18], [19]. We provide performance comparisons with the channel capacity and we also provide comparisons with the space–time codes introduced by Tarokh et al. [20], [21]. The space–time trellis codes [20] have been designed for the quasi-static fading scenario; hence, they are not necessarily optimal for the block fading channel case. Nevertheless, they provide a useful reference for multiple antenna systems. Furthermore, it has been shown in [22] that the space–time trellis codes continue to perform well in the presence of mobility depicted by the block fading channel model, as well as in the presence of channel estimation errors. This excellent performance and robustness of the space–time codes is the reason we consider them in the performance comparisons. We show that a simple turbo code performs significantly better than the interleaved space–time trellis codes for block Rayleigh fading channels. The space–time trellis codes are interleaved in order to obtain additional diversity advantage over the block fading channel. The performance improvements over the space–time codes are much greater when the block length of the turbo code is larger. However, in some cases, we obtain excellent performance even for short block lengths, appropriate for speech applications. We also present an improved version of the decoding algorithm from [18], [19], based on iterations between the demodulator and the turbo decoder. We show that the new version improves the system performance at the expense of some increase in computational complexity. Furthermore, we observe that the computational complexity at the receiver increases exponentially with the number of transmit antennas. Hence, we present a new decoding method based on array processing [23] at the receiver, and show that turbo-coded modulation outperforms the multilayered space–time trellis coded modulation over block Rayleigh fading channels [23]. Finally, we focus on the effects of estimated channel state information at the receiver and correlation among the subchannels between different transmit–receive antenna pairs. The paper is organized as follows. In the next section, we present the system model and establish notation. In Section III, we describe the application of turbo-coded modulation to systems with antenna diversity; in particular, we present the encoding process and two versions of the (sub-optimal) iterative decoding algorithm. In Section IV, we present a decoding algorithm based on array processing and turbo-coded modulation, suitable for systems with a large number of transmit and receive antennas. Section V considers the estimation of the channel state information at the receiver. Section VI focuses on the correlation among the subchannels, i.e., the path gains between different transmit–receive antenna pairs. In section VII, we present several numerical examples for the block Rayleigh fading channel and compare our results with the space–time codes, which are designed for the quasi-static fading case but nevertheless provide a useful reference for multiple antenna systems. Finally, we conclude in Section VIII.

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Fig. 1.

Block diagram of the transmitter.

information bits are encoded by a channel encoder, the coded bits are passed through a serial to parallel converter, and are mapped to a particular signal constellation. At each time slot , the output of the modulator is a signal that is transmitted . All signals are transusing transmit antenna , for mitted simultaneously, each from a different transmit antenna, and all signals have the same transmission period . The block diagram of the transmitter is given in Fig. 1. The signal at a receive antenna is a noisy superposition of the transmitted signals corrupted by Rayleigh fading. The coeffiis the path gain from transmit antenna , , to cient . Since we assume a Rayleigh the receive antenna , fading channel, the path gains are modeled as samples of independent zero mean complex Gaussian random variables with variance 0.5 per dimension. The wireless channel is modeled as a block fading channel, i.e., the path gains are constant over symbols which corresponds to information bits, where is the spectral efficiency of the system, and are independent from one block of size to the next. is At time the received signal by antenna , denoted by given by (1) where the noise samples are modeled as independent samples of a zero mean complex Gaussian random variable with variance per dimension. We define the signal-to-noise ratio (SNR) , where is the total transmitted energy at each transas , with denoting the mission interval. We have average energy of the signal constellation at the th transmit antenna. We can equivalently write (2) where

and

II. SYSTEM MODEL We consider a mobile communication system that employs antennas at the transmitter and antennas at the receiver. The

.. .

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(3)

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IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 19, NO. 5, MAY 2001

Fig. 3. Block diagram of the receiver.

B. Iterative Decoding Fig. 2.

Block diagram of the transmitter with the turbo encoder.

A. Channel Capacity We will use the capacity of the channel with multiple transmit and multiple receive antennas for comparison purposes. The Shannon capacity of the system assuming no delay constraints, i.e., the number of fading blocks may increase without bound, and perfect channel state information at the receiver, is given by [1]–[4] (4) denotes the identity matrix. where is the SNR and Note, that the capacity does not depend on the block length [1]. To compute the channel capacity, we note that since the channel is ergodic, we can use the Monte Carlo integration method, where we generate a large number of channel realizations and average over them.

In this section, we present a suboptimal decoding algorithm for the above system. The decoding algorithm is similar to the decoding algorithm given in [26] and [27]. We compute the loglikelihoods of the transmitted bits, and use them as if they are the likelihoods of the observations from a BPSK modulation over an additive white Gaussian noise (AWGN) channel. This decoding algorithm is clearly suboptimal. The block diagram of the receiver is given in Fig. 3. We now describe how the log-likelihoods of the individual bits are computed from the received signal. Assume that the number of different channel symbols at each transmit antenna, , and a two-dimensional i.e., the size of the constellation is (2-D) modulation is used. Let us denote the set of constella. Note that each constellation point cortion points by bits. Following the notation of the previous secresponds to , tion, the received signal by antenna at time , denoted by is given by

III. TURBO CODES FOR SYSTEMS WITH ANTENNA DIVERSITY In this section, we describe the use of turbo-coded modulation for wireless communication systems with multiple transmit and receive antennas.

At this point, for clarity, we drop the subscript . We have

A. Encoding The block diagram of the transmitter where a turbo code is used as a channel encoder is given in Fig. 2. The data is divided bits, and encoded by a binary turbo code. The in blocks of turbo code consists of two systematic recursive convolutional codes concatenated in parallel via a pseudorandom interleaver [24]. The turbo-coded bits are then interleaved, passed through a serial to parallel converter, and mapped to a particular signal constellation. We can obtain different spectral efficiencies by varying the code rate and the constellation. The additional interleaver is used to remove the correlation between the consecutive bits being transmitted which helps us in the decoding process. Its size is chosen such that there is no additional increase in the delay requirements of the system. Since we assume block fading, the turbo code interleaver size . Effectively, we are channel is chosen to be a multiple of coding across consecutive “differently faded” blocks. The additional interleaver is necessary in order to decorrelate the loglikelihoods of the adjacent bits. Furthermore, it distributes the burst errors due to a deeply faded block over the entire frame, which provides additional diversity. The above coded modulation scheme is obtained by concatenating a binary encoder to memoryless modulators, through a bit interleaver. Therefore, it represents a realization of bit-interleaved coded modulation [25].

Notice that the received signals correspond to coded bits; hence, we need to compute the log-likelihoods of bits using this set of signals. Let us denote the these bits that construct by

The group of bits is used to select the constellation point for the th transmit antenna, denoted by . The log-likelihood for the th element of , , is given by

(5) which can also be written as

(6)

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Note that by knowing , we also have a knowledge of ; hence

(7) Fig. 4. Block diagram of the receiver with iterative demodulation–decoding.

, and is the mapping from to where . Assuming that all constellation points are equally likely, we can write

Without the assumption that all symbols are equally likely, it follows from (7) that

(8)

are conditionally independent given

Since

(9)

(12) which, under the assumption that the transmitted symbols are independent, may be written as

Hence, substituting for the noise statistics, we obtain

(13) Since we have bit interleaving, we may assume that the probabilities of the bits that compose the symbol are independent, we ; hence have

(10) as the log-likelihood for the bit . It is possible to simplify the log-likelihood computations for the bit , by using the following approximation (11) where

(14) Finally, since given we obtain

are conditionally independent and taking the noise statistics into account,

.

C. Iterative Demodulation–Decoding In this section, we present an improved version of the decoding algorithm. In the derivation of the log-likelihoods, we obtained (8), assuming that all constellation points are equally likely. This is a reasonable assumption considering that the a priori probabilities of the transmitted symbols are difficult to compute prior to the decoding process. However, due to the use of a soft-input soft-output (SISO) decoder, we can obtain an estimate of the probabilities of the transmitted symbols and use them in the decoding process. That way we obtain an iterative demodulation–decoding algorithm [28]–[30], [12]. The block diagram of the receiver with iterative demodulation–decoding is given in Fig. 4.

(15) as the log-likelihood for the bit

.

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In order to obtain the probabilities of the systematic and parity bits, we use the soft output turbo decoding algorithm to obtain log-likelihoods for both the systematic, as well as parity bits. In particular, the probabilities are obtained from the log-likelihoods of the extrinsic bit information, as described in [31]. Those probabilities are then fed-back to the demodulator and used as a priori information in the demodulation process, as given in (15). Using the iterative demodulation–decoding algorithm, it is possible to obtain performance improvements over the iterative decoding algorithm but at the expense of increased complexity. The complexity increase is due to the additional computations in the demodulation process, as well as the computation of the log-likelihoods for the parity bits. In order to reduce the complexity at the receiver, we mainly focus on the iterative decoding algorithm presented in the previous subsection, and we only present examples with the iterative demodulation–decoding algorithm to illustrate the potential improvements. Examples of the iterative demodulation–decoding algorithm in the case of serially concatenated codes and multiple antennas are given in [12].

of perfect channel state information at the receiver we can form the following matrix

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..

..

.

.. .

.

The rank of the above matrix will be less then or equal to the number of columns, i.e., (17) On the other hand, we have that (18) [32]. From (17) and (18), we where is the null space of . Hence, if we have have that dim receive antennas, we can suppress the interference from antennas. There are efficient algorithms that can compute a set in . As in of orthonormal vectors denote the matrix whose th [23], let , we have row is . After multiplying both sides of (2) by (19)

IV. LARGE NUMBER OF TRANSMIT AND RECEIVE ANTENNAS It is easy to see from (10) and (15) that the number of computations required to obtain the log-likelihood for each bit grows exponentially with the number of transmit antennas and the constellation size. Hence, the demodulation becomes infeasible if the number of transmit antennas is large. To alleviate the computational burden in the demodulation process, we propose the following method, which is based on the group interference suppression idea introduced in [23]. Let us partition the antennas at the transmitter into groups antennas per group, where with , . Instead of using component codes, one for each group of antennas, as is done in [23], we have a single binary turbo code. At the receiver, we need to compute the log-likelihoods of the transmitted bits. First, we obtain the log-likeliantennas by suphoods corresponding to the first group of pressing (nulling) the interference from the other antennas , . Let the set of log-likelihoods that corresponds to antennas be denoted by . We then prothe first group of ceed by repeating the same procedure for the second group of antennas and we obtain the corresponding set of log-likelihoods, , by suppressing (nulling) the interference from the other an. The interference suppression (nulling) tennas , method is repeated times, once for each set of antennas. Once we obtain the log-likelihoods for all of the transmitted bits, we apply the iterative turbo decoding algorithm. Notice that this method is easily applicable to any other code; however, the decoding algorithm may not be optimal. We describe the group interference suppression method for since the extension to the other cases the case of decoding is straightforward. The description follows the approach taken interfering in [23]. In the case of decoding , we have signals from the other transmit antennas. With the assumption

(16)

, we will have that Due to the construction of , the all zero matrix. Hence, we suppress the signals from an, . Finally, we obtain the following tennas equation for the received signal from antennas (20) where

,

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, and

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We are now able to efficiently compute the log-likelihoods of the transmitted bits, since the computational complexity is rather than , as in the standard approach. Of exponential in course, the penalty for the reduced demodulation complexity is the performance loss due to the decreased diversity, as will be quantified in the numerical results section. V. ESTIMATION OF THE CHANNEL STATE INFORMATION We next consider the effect of imperfect channel state information on the performance of the turbo-coded modulation system for multiple antennas. In order to estimate the channel state information, we follow the approach of [22] and use a training sequence of length . , , We denote the pilot signals for training by where

and are elements of the signal constellation for and . We need , to be orthogwhenever . onal sequences, i.e.,

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During the training period, the received signal is denoted by

(21) , Our goal is to use the received signal to estimate , . Since the training sequences are orthogonal, it is easily observable that (22) thus (23) Hence, the estimated path gain receive antenna , is

from transmit antenna to

(24) We can now use the estimated channel state information [22], to obtain the log-likelihoods of the received bits and proceed with the iterative decoding algorithm. VI. EFFECTS OF CORRELATED FADING AMONG SUBCHANNELS So far, we have assumed that the path gains along different paths (subchannels) are independent of each other. If all the transmit and receive antennas have sufficient separation, this is a reasonable assumption. However, in practice, the subchannels will not undergo perfectly independent fading. Let us now consider the effects of correlation among different subchannels. Here, we only focus on transmit antenna correlation for the case of two transmit antennas and assume that we have sufficient separation between the receive antennas. Under the assumption that we have two transmit antennas, the correlation is given by [33]

where , and denotes the correlation coefficient. At the receiver, we obtain the log-likelihoods of the received bits and we use the iterative decoding algorithm, as described in Section III. VII. SIMULATION RESULTS In this section, we present the performance of the proposed scheme involving turbo codes, and compare it to the performance of space–time block and trellis codes for several examples. The component codes of the turbo code are two recursive , where systematic convolutional codes, described by and are the feedforward and feedback generating polynoand to be 5 and 7 , respectively. mials. We chose The turbo code employs a random interleaver, and it has a rate , obtained by periodically puncturing the parity bits. The interleaver that scrambles the turbo-coded bits consists of two pseudorandom interleavers of length . We use two interleavers, one for the systematic and one for the parity bits, in order to ensure a mapping of one systematic and one parity bit

Fig. 5. BER for several turbo codes, a space–time block code and the 32-state space–time trellis code, and two transmit and one receive antennas.

per constellation point. We present examples where we use the four-PSK constellation at each transmit antenna. It is also pos. Similar perforsible to use a single interleaver of length mance results may be observed with higher order constellation [19], such as eight-PSK or 16-quadrature amplitude modulation (QAM). In all of the numerical results, we compute the exact log-likelihoods, and we use the iterative turbo decoding algorithm employing maximum a posteriori probability (MAP) constituent decoders with ten iterations. Note that the results for the space–time trellis codes are obtained by introducing a random interleaver in order to provide time diversity. Due to the superposition of the transmitted signals at the receive antennas, this interleaver operates on groups of symbols—which are transmitted simultaneously—at the transmitter. At the receiver, we employ the corresponding symbol deinterleaver at each of the receive antennas. Thus, the decoding algorithm is still optimal. In Fig. 5, we present the bit-error rate (BER) for the turbo-coded system for several interleaver lengths, for a 32-state space–time trellis code [20] and a space–time block code [21]. We assume block fading. The path gain is constant for a period of 130 transmissions, which corresponds to 260 information bits. We assume that we have two transmit and one receive antennas. At a BER of 10 , the turbo codes of interleaver size 1300, 2600, and 5200 information bits, provide gains of approximately 3.25, 5, and 6 dB, respectively, over the space–time trellis code. The gains with respect to the space–time block code are even higher. In the case of two transmit and one receive antennas to achieve capacity of two bits/s/Hz, we require an SNR of around 5.5 dB, computed using the result in Section II-A. Hence, at a BER of 10 , the is around seven dB turbo code with interleaver size away from the channel capacity. Note, however that this is the capacity assuming no delay constraints. Fig. 6 presents the BER for the turbo-coded modulation system for several interleaver lengths, and we compare it with the performance of the 32-state space–time trellis code [20]. Again, we assume that we have a block fading channel where

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Fig. 6. BER for several turbo codes and the 32-state space–time trellis code, and two transmit and two receive antennas.

Fig. 7. BER performance of the turbo code versus the number of iterations.

the path gains are constant for a period of 130 transmissions. There are two transmit and two receive antennas. The turbo codes of interleaver size 1300, 2600, and 5200 information bits, provide gains of approximately three, four, and five dB, respectively, over the 32-state space–time trellis code at a BER of 10 . For the case of two transmit and two receive antennas to obtain capacity of two bits/s/Hz, we need an SNR of around one dB. Hence, at a BER of 10 , the turbo code is around four dB away from with interleaver size the capacity. Again, note that this is the capacity assuming no delay constraints. Due to the relatively high complexity of the turbo decoding, it is of interest to consider the convergence of the iterative decoding algorithm. This is demonstrated in Fig. 7, for the case of two transmit and two receive antennas. The frame size of the turbo code is 2600 bits. We assume that we have a block fading channel where the path gains are constant for a period of 130 transmissions. Similar convergence properties were observed in the other examples as well. We observe that at a BER

Fig. 8. Eight-FER performance comparison between the turbo code with iterative decoding and iterative demodulation–decoding for the quasi-static fading channel example.

of 10 , the difference in performance between the decoding algorithm with three and ten iterations is only about 0.5 dB. Furthermore, the difference in performance with four and ten iterations is negligible, and the returns with performing higher number of iterations are diminishing. This is important, since by reducing the number of iterations, we could significantly reduce the decoding complexity. Further reductions in complexity may be achieved by using the Max-Log-MAP decoding algorithm, instead of the regular MAP decoding algorithm, with very small loss in performance. Furthermore, it has been shown by Fossorier et al. [34], that the soft output Viterbi algorithm (SOVA) may be modified in a simple manner, such that it becomes equivalent to the Max-Log-MAP decoding algorithm. This allows for further reductions in complexity without sacrificing performance. These convergence properties and decoding modifications would allow for a significant reduction in complexity and an efficient implementation of the iterative turbo decoding algorithm for the case of multiple antenna systems. A. Iterative Demodulation–Decoding In Fig. 8, we present the frame error rate (FER) comparison between the turbo-coded modulation system with standard iterative decoding and with iterative demodulation–decoding. The . turbo code has an -random interleaver with size, The channel is a quasi-static Rayleigh fading channel. We have two transmit and one or two receive antennas. We see that at a FER of 10 , the iterative demodulation–decoding improves the performance by around 2 and 1 dB, for the case with one and two receive antennas, respectively. This clearly demonstrates the importance of the iterative demodulation–decoding algorithm for turbo codes in the presence of quasi-static Rayleigh fading. At a FER of 10 , the turbo code with iterative demodulation–decoding performs within 2.5 and 1.5 dB of the outage capacity [3], [4] for the case of one and two receive antennas, respectively. Similarly, Fig. 9 presents the FER comparison between the turbo-coded modulation system with standard iterative

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Fig. 9. FER performance comparison between the turbo code with iterative decoding and iterative demodulation–decoding for the block fading channel example.

Fig. 11. BER for the turbo codes with array processing at the receiver and direct computation of the log-likelihoods for the block fading channel example. Four transmit and four receive antennas, 4 bits/s/Hz.

Fig. 10. BER for the turbo codes and the multilayered space–time trellis coded modulation for the block fading channel example. Four transmit and four receive antennas, 4 bits/s/Hz.

both component codes being 32-state space–time codes [20]. We assume that for the block fading channel, the path gains are constant for a period of 130 transmissions, which corresponds to 520 information bits. We also assume that we have perfect channel state information at the receiver. The decoding of the turbo code is performed by partitioning the 4 transmit antennas and into two groups of two antennas. Hence, we have . This scheme achieves a spectral efficiency of 4 bits/s/Hz. For the space–time trellis codes, the average powers radiated from antennas 1 and 2 are equal but each is twice as much as the average power radiated from antennas 3 and 4, as . For described in [23]. Hence, we have the turbo-coded scheme, the total power transmitted from the four transmit antennas is the same as for the space–time trellis codes but we distribute the power equally among all four antennas. At a BER of 10 , the turbo codes of interleaver size 2600 and 5200 information bits, provide gains of approximately 2 and 3.25 dB, respectively, over the multilayered space–time trellis coded modulation scheme. Fig. 11 presents the performance comparison in terms of the BER for the combined array processing and turbo-coded modulation scheme, and the standard turbo-coded modulation scheme with no group interference suppression. We assume that there are four transmit and four receive antennas. Obviously, the complexity of the latter scheme is very high. Nevertheless, the results are useful to quantify the performance loss when array processing/interference suppression technique is used. For this example, the path gains are constant for a period of 130 transmissions, which corresponds to 520 information bits, and we assume that we have perfect channel state information at the receiver. The decoding of the turbo code for the array processing scheme is performed as described in the previous example. The decoding of the standard turbo-coded modulation scheme is performed by direct computation of the log-likelihoods, as described in (10). At a BER of 10 , the turbo codes of interleaver size 2600 and 5200 information bits with direct computation of the log-likelihoods, provide

decoding and with iterative demodulation–decoding for the block fading case. The channel is a block Rayleigh fading channel where the path gains are constant for a period of 65 . transmissions. The turbo code interleaver size is We have two transmit and one or two receive antennas. We see that at a FER of 10 , the iterative demodulation–decoding improves the performance by around two dB for the case with one receive antenna and one dB for the case with two receive antennas. B. Large Number of Transmit and Receive Antennas We next consider the case when we have a large number of transmit and receive antennas. In Fig. 10, we present the BER for the case of four transmit and four receive antennas for the combined array processing and turbo-coded modulation scheme, and the multilayered space–time coding scheme with

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Fig. 12. BER for the turbo code and the multi-layered space–time trellis coded modulation for the block fading channel example. Eight transmit and eight receive antennas, 8 bits/s/Hz.

gains of approximately four and five dB, respectively, over the turbo-coded modulation with array processing. Hence, we see that the reduction in computational complexity results in a substantial performance degradation as compared to the turbo-coded modulation scheme for multiple antenna systems. We consider the case with eight transmit and eight receive antennas in Fig. 12. For this case, we assume that the path gains are constant for a period of 130 transmissions, which corresponds to 1040 information bits. We assume that we have perfect channel state information at the receiver. The decoding of the turbo code is performed by partitioning the 8 transmit antennas into three , such that and . This groups, scheme achieves a spectral efficiency of 8 bits/sec/Hz. For the space–time trellis codes, we assume that the average powers radiated from antennas 1 and 2 are equal, but each is twice as much as the average power radiated from antennas 3 and 4, four times as much as the average power radiated from antennas 5 and 6 and eight times as much as the average power radiated from antennas 7 and 8, i.e., . For the turbo-coded scheme, the total power transmitted from the eight transmit antennas is the same as for the space–time trellis codes; however, we distribute the power equally among all eight antennas. We observe that, at a BER of 10 , the turbo code of interleaver size 4160 information bits, provides a gain of approximately 3.5 dB over the multilayered space–time trellis coded modulation scheme [23]. C. Estimated Channel State Information We next present BER results for the case when we have estimated channel state information (CSI) at the receiver and compare them with the perfect CSI examples. We assume that the path gains along different paths are independent of each other. for the perfect CSI The turbo code interleaver size is symcase. Since the length of the training sequence is bols, which corresponds to 16 bits, and since we use the training

Fig. 13. BER performance comparison between the turbo code and the 32-state space–time trellis code, with perfect CSI and estimated CSI, two transmit and one receive antennas.

Fig. 14. BER performance comparison between the turbo code and the 32-state space–time trellis code, with perfect CSI and estimated CSI, two transmit and two receive antennas.

sequence once per each fading block, the turbo code interleaver . We consider size for the case of estimated CSI is the case with two transmit and one receive antennas in Fig. 13. We assume that the average powers radiated from antennas 1 . We observe that at a BER of and 2 are equal, 10 the loss in performance due to estimation of the channel state information is around 1.5 dB. The gain with respect to the 32-state space–time trellis code with estimated channel state information is still about 4.5 dB at a BER of 10 . Similarly, in Fig. 14, we observe that for the case of two transmit and two receive antennas, the performance loss with respect to the perfect CSI case, is also around 1.5 dB at BER of 10 . The gain with respect to the 32-state space–time trellis code with estimated channel state information is still about 3.5 dB at a BER of 10 .

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the correlation coefficient , the turbo code performance degrades by around two dB, just as the performance of the 32-state space–time trellis code. Hence, the overall gain is preserved even for high correlations and is still around four dB. In both cases, we observe that the performance of the turbo codes degrades gracefully as the correlation between the transmit antennas increases. VIII. CONCLUSION

Fig. 15. BER performance comparison between the turbo codes and the 32-state space–time trellis code for different values of the correlation coefficient, two transmit and one receive antennas.

We studied the performance of turbo-coded modulation schemes for systems with transmit and receive antenna diversity. We showed that turbo-coded modulation provides a significant performance improvement over the space–time codes over block Rayleigh fading channels. Although the performance gains are much higher for the longer interleaver lengths which are suitable for data communications, very short block length turbo codes, suitable for speech applications, also perform very well. We found that the performance of the turbo-coded modulation scheme may be further improved by using the iterative demodulation–decoding algorithm. We also studied a number of practical considerations, such as the case when we have a large number of transmit and receive antennas, the effects of estimated channel state information and correlation among subchannels. The turbo-coded modulation scheme for multiple antenna systems continued to perform well in all these scenarios and outperformed the space–time trellis codes considerably. Hence, turbo-coded modulation represents a viable alternative for use in communication systems with transmit and receive antenna diversity over block fading channels. REFERENCES

Fig. 16. BER performance comparison between the turbo codes and the 32-state space–time trellis code for different values of the correlation coefficient, two transmit and two receive antennas.

D. Correlation Among Subchannels We present examples for the turbo code with interleaver for several values of the correlation coeffilength cient , and compare them to the performance of the 32-state space–time trellis code. Fig. 15 presents results for the case of two transmit and one receive antennas. We observe that the performance loss of the turbo codes, with respect to the case, for the case when the correlation coefficient is around two dB at a BER correlation coefficient of 10 . Similar loss in performance of around two dB, for , is also the case when the correlation coefficient is observed for the 32-state space–time trellis code. Hence, the overall gain that the turbo code provides over the space–time trellis code is preserved and is around five dB. The performance is similar for the case of two transmit and two receive antennas, as depicted in Fig. 16. Again we observe that for the case when

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Andrej Stefanov (S’00) received the B.S. degree in electrical engineering from the University of Cyril and Methodius, Skopje, Macedonia and the M.S. degree in electrical engineering from Arizona State University, Tempe, in 1996 and 1998, respectively. Currently, he is working toward the Ph.D. degree in electrical engineering at Arizona State University. His current research interests are wireless and mobile communications, space–time coding, and turbo codes. During the summer 2000, he held an internship with the Advanced Development Group, Hughes Network Systems, Germantown, MD, where he was involved in research on space–time coding. Mr. Stefanov is the co-recipient of the Best Paper Award from IEEE VTC-Fall 1999, Amsterdam, the Netherlands, for his work on turbo-coded modulation for wireless communication systems with antenna diversity.

Tolga M. Duman (S’97–M’98) received the B.S. degree from Bilkent University, Ankara, Turkey, in 1993, and the M.S. and Ph.D. degrees from Northeastern University, Boston, MA, in 1995 and 1998, respectively, all in electrical engineering. He joined the Electrical Engineering Faculty, Arizona State University, Tempe, as an Assistant Professor in August 1998. His current research interests are in digital communications, wireless and mobile communications, channel coding, turbo codes, coding for recording channels, and coding for wireless communications. Dr. Duman is the recipient of the National Science Foundation CAREER Award, IEEE Third Millennium medal, and IEEE Benelux Joint Chapter Best Paper Award (1999).