6.7 Turbo Multiuser Detection in Space-Time Block-Coded Systems

The recently developed space-time coding (STC) techniques [350] integrate the methods of transmitter diversity and channel coding and provide significant capacity gains over traditional communication systems in fading wireless channels. STC has been developed along two major directions: space-time block coding (STBC) and space-time trellis coding (STTC). The common features of STBC and STTC lie in their realizations of spatial diversity (i.e., both methods transmit a vector of complex code symbols simultaneously from multiple transmitter antennas). Their differences, on the other hand, lie in their realizations of temporal diversity: In STBC, the temporal constraint is represented in matrix form; whereas in STTC, the temporal constraint is represented in the form of a trellis tree, which is akin to the trellis-coded modulation (TCM) code.

From the coding perspective, the single- user performance of STBC and STTC has been studied in [476, 477], and some code design criteria have been developed. However, in wireless communication systems, sharing the limited radio resources among multiple users is inevitable. Indeed, the emerging wireless systems with multiple transmitter and receiver antennas enable a new dimension for multiple accessing: space-division multiple access (SDMA) [492], which when employed with the more conventional TDMA or CDMA techniques, can substantially increase system capacity. However, if not properly ameliorated, the presence of multiuser interference can significantly degrade receiver performance as well as system capacity. Therefore, the development of efficient detection and decoding techniques for multiuser STC systems ( illustrated in Fig. 6.21) is the key to bringing STC techniques into the practical arena of wireless communications. Research results along this direction first appeared in [349, 476], where some techniques for combined array processing, interference cancellation, and space-time decoding for multiuser STC systems were proposed.

Figure 6.21. Multiuser wireless communication system employing multiple transmitter and receiver antennas. There are K users in the system, each user employing N transmitter antennas. At the receiver side, there are M receiver antennas.

In this and the following sections we discuss turbo receiver structures for joint detection and decoding in multiuser STC systems, based on the techniques developed in previous sections. Such iterative receivers and their variants, which were first developed in [288], are described for both STBC and STTC systems. During iterations, extrinsic information is computed and exchanged between a soft multiuser demodulator and a bank of MAP decoders to achieve successively refined estimates of the users' signals. Further developments of low-complexity turbo structures for multiuser STTC systems are given in [217].

6.7.1 Multiuser STBC System

The transmitter end of a multiuser STBC system is shown in Fig. 6.22. The information bit stream for the k th user, { d _k [ n ]} _n , is encoded by a convolutional encoder; the resulting convolutional code-bit stream { b _k [ i ]} _i is then interleaved by a code-bit interleaver. After interleaving, the interleaved code-bit stream is then fed to an M -PSK modulator , which maps the binary bits into complex symbols { c _k [ l ]} _l , where , and W _C is the M -PSK symbol constellation set ( M = W _C ). The symbol stream { c _k [ l ]} _l , is partitioned into blocks, with each block consisting of N symbols. Due to the existence of the interleaver, we can ignore the temporal constraint induced by the outer convolutional encoder and assume that the set { c _k [ l ]} _l contains independent symbols. Hence, from the STBC decoder's perspective, we need only consider one block of symbols in the code symbol stream: the code vector .

Figure 6.22. Transmitter structure for a multiuser STBC system.

STBC was first proposed in [12] and was later generalized systematically in [476]. Following [476], the k th user's STBC is defined by a P x N code matrix , where N denotes the number of transmitter antennas or the spatial transmitter diversity order, and P denotes the number of time slots for transmitting an STBC code word or the temporal transmitter diversity order. Each row of is a permuted and transformed (i.e., negated and/or conjugated) form of the code vector c _k . An STBC encoder takes as input the code vector c _k and transmits each row of symbols in at P consecutive time slots. At each time slot, the symbols contained in an N -dimensional row vector of are transmitted through N transmitter antennas simultaneously.

As a simple example, we consider a particular user employing a 2 x 2 STBC (i.e., P = 2, N = 2). Its code matrix is defined by

Equation 6.184

The input to this STBC is the code vector c = [ c [1] c [2]] ^t . During the first time slot, the two symbols in the first row of (i.e., c [1] and c [2]) are transmitted simultaneously at the two transmitter antennas; during the second time slot, the symbols in the second row of (i.e., “ c [2] ^* and c [1]*) are transmitted.

We assume a flat-fading channel between each transmitter “receiver pair. Specifically, denote a _m,n as the complex fading gain from the n th transmitter antenna to the m th receiver antenna, where a _m,n ~ N _c (0, 1) is assumed to be a zero-mean circularly symmetric complex Gaussian random variable with unit variance. It is also assumed that the fading gains remain constant over an entire signal frame, but they may vary from one frame to another.

In general, we consider an STBC system with K users, each employing N transmitter antennas. At the receiver side, there are M receiver antennas. In this case, the received signal can be written as

Equation 6.185

In (6.185), m = 1,2, ..., M , consists of the received signal from time slots 1 to P , at the m th receiver antenna; H _k , k = 1, 2, ..., K, is the channel response matrix for the k th user, as explained below; is the code vector for the k th user; and contains the additive Gaussian noise samples from time slots 1 to P at the m th receiver antenna.

As a simple example, consider a single-user ( K = 1) STBC system with two ( N = 2) transmitter antennas and M receiver antennas, employing the code matrix g ₁ in (6.184), the received signal at the m th receiver antenna for this single user can be written as

Equation 6.186

For notational convenience, we write (6.186) in an alternative form by conjugating r ^m [1]:

Equation 6.187

We can see that contains information not only of the channel response related to the m th receiver antenna, but also the code constraint of the STBC . Finally, by stacking all the r ^m in (6.187), we obtain

Equation 6.188

The signal model in (6.188) can easily be extended to the general model (6.185) of a K -user P x N STBC system, in which each user employs the code defined in [476]. The analogy between this multiuser STBC signal model and the synchronous CDMA signal model (6.74) is evident. Note that to effectively suppress the interfering signals in model (6.185), the size of the receiver signal r should be larger than the number of symbol to be decoded (i.e., MP NK ).

6.7.2 Turbo Multiuser Receiver for STBC System

The iterative receiver structure for a multiuser STBC system is illustrated in Fig. 6.23. It consists of a soft multiuser demodulator, followed by K parallel MAP convolutional decoders. The two stages are separated by interleavers and deinterleavers. The soft multiuser demodulator takes as input the signals received from the M receiver antennas and the interleaved extrinsic log- likelihood ratios (LLRs) of the code bits of all users { l ₂ ( b _k [ i ])} _i;k (which are fed back by the K single-user MAP decoders), and computes as output the a posteriori LLRs of the code bits of all users, { L ₁ ( b _k [ i ])} _i;k . The MAP decoder of the k th user takes as input the deinterleaved extrinsic LLRs of the code bits { l ₁ ( [ i ])} _i from the soft multiuser demodulator and computes as output the a posteriori LLRs of the code bits { L ₂ ( [ i ])} _i , as well as the LLRs of the information bits { D ₂ ( [ n ])} _n . We next describe each component of the receiver in Fig. 6.23.

Figure 6.23. Iterative receiver structure for a multiuser STBC system.

Soft Multiuser Demodulator

The soft multiuser demodulator is based on the technique described in Section 6.3.3. First, a soft estimate [ l ] of the k th user's l th code symbol c _k ( l ) is formed by

Equation 6.189

where W _c is the set of all code symbols. At the first iteration, no prior information about code symbols is available; thus code symbols are assumed to be equiprobable [i.e., P (c _k [ l ] = C _i ) = 1/ W _c ]. In subsequent iterations, the probability P (c _k [ i ] = C _i ) is computed from the extrinsic information delivered by the MAP decoder, as will be explained later [cf. (6.204)].

For the K -user STBC system (6.185), define an NK -dimensional soft code vector

The basic idea is to treat every element in as a virtual user , and therefore there are a total of NK virtual users in the system (6.185). Viewing it this way, the model (6.185) is similar to the synchronous CDMA signal model treated in Section 6.3.3. Henceforth in this section, the notation _k [ l ] is used to index a virtual user. Define

Equation 6.190

In (6.190), e _k [ l ] is an NK -vector of all zeros, except for the "1" element in the corresponding entry of the _k [ l ]th virtual user. That is, [ l ] is obtained from by setting the _k [ l ]th element to zero.

Subtracting the soft estimate of the interfering signals of other virtual users from the received signal r in (6.185), we get

Equation 6.191

Equation 6.192

As before, to further suppress the residual interference in , we apply an instantaneous linear minimum mean-square error (MMSE) filter to . The linear MMSE weight vector w _k [ l ] is chosen to minimize the MSE between the transmitted symbol c _k [ l ] and the filter output:

Equation 6.193

Using (6.191) and assuming that the M -PSK symbol c _k [ l ] is of unit energy (i.e., c _k [ l ] ² = 1) and E{ n n ^H } = s ² I _{M P} , we have

Equation 6.194

Equation 6.195

with

Equation 6.196

Using (6.193) “(6.196), the instantaneous MMSE estimate for c _k [ l ] is then given by

Equation 6.197

The instantaneous MMSE filter output is modeled by an equivalent additive white Gaussian noise channel having c _k [ l ] as its input symbol. The output of this filter can then be written as

Equation 6.198

with

Equation 6.199

Equation 6.200

Note that m _k [ l ] and [ l ] are real numbers . Equations (6.198) “(6.200) give the probability distribution of the code symbol [ l ], based on which the a posteriori probability of the code bits are computed, as discussed below.

From the discussion above, the major computation involved in the soft multiuser demodulator is the MP x MP matrix inversion, . As before, this can be done recursively by making use of the matrix inversion lemma. As a result, the computational complexity of the proposed soft multiuser demodulator per user per symbol is [( MP ) ³ / NK ]. (Recall that M is the number of receiver antennas, N is the number of transmitter antennas, P is the number of time slots in an STBC code word, and K is the number of users.)

Computing a posteriori Code-Bit LLRs

The convolutional code is chosen as the outer channel code in the system considered here (Fig. 6.22). First we need to compute the a posteriori LLRs of the code bits based on the estimated code symbols given by the soft multiuser demodulator. Since each user decodes its convolutional code independently, henceforth we drop the subscript k , the user index, to simplify the notation.

Every complex symbol c [ l ] can be represented by a J -dimensional binary bit vector, [ b [ l , 1], . . . b [ l,J ]], where and b [ l,j ] {+1, -1} denotes the j th binary bit of the l th complex code symbol. By (6.177),

Equation 6.201

with

Due to the existence of the interleaver, we can assume that b [ l,j ] is independent of c [ l' ], l' l ; and b [ l,j ] is independent of b [ l,j' ], j j' . Based on the Gaussian model (6.198), we have

Equation 6.202

where C _i W _c and with a binary representation, C _i [ B [ i , 1] · · · B [ i,J ]]. Then the a posteriori LLR of b [ l, j ] at the output of soft multiuser demodulator can be computed as

Equation 6.203

where is the set of the complex code symbols whose j th binary bit equals +1; and is defined similarly. The last equality in (6.203) follows from (6.201) and (6.202). The term l ₂ ( b [ l, j ]) in (6.203) is the interleaved extrinsic LLR of the j th code bit for the l th complex code symbol in the previous iteration, and is computed by bitwise-subtracting the code-bit LLR at the input of the decoder from the corresponding code-bit LLR at the output (cf. Fig. 6.23). At the first iteration, no prior information about code bits is available; thus l ₂ ( b [ l, j ]) = 0. It is seen from (6.203) that the output of the soft multiuser demodulator is the sum of the a priori information l ₂ ( b [ l, j ]) provided by the MAP convolutional decoder in the previous iteration and the extrinsic information l ₁ ( b [ l, j ]). Finally, the extrinsic LLRs of the code bits calculated in (6.203) are deinterleaved and then fed to the MAP convolutional decoder.

MAP Decoding Algorithm for Convolutional Code

Consider a rate- k / n binary convolutional code of overall constraint length k n . At each time t , the input to the encoder is the k -dimensional binary vector d _t = ( ) and the corresponding output is the n -dimensional binary vector b _t = ( ).

As shown in Fig. 6.23, the deinterleaved extrinsic LLRs of the code bits { l ₁ ( b ^p [ i ])} _i are fed as input to the MAP convolutional decoder. We partition this stream into n - sized blocks, each block consisting of n code bit LLRs, corresponding to the n output code bits at one time instant. Denote each block by a vector , t = 1, 2, . . ., t , where t blocks of code bits are transmitted in each signal frame. The partitioned code-bit LLR stream is then denoted as .

The extrinsic LLRs of the code bits { l ₂ ( b ^p [ i ])} _i are computed based on and the convolutional code structure by the MAP decoding algorithm described in Section 6.2. Based on the interleaved extrinsic LLRs of the code bits in the previous iteration, l ₂ ( b [ l, j ]), j = 1, 2, ... , J , the soft estimate [ l ] cf. (6.189)] of the code symbol c [ l ] can then be computed as

Equation 6.204

where (6.204) follows from (6.175). At the first iteration, no prior information is available, and thus l ₂ ( b [ l, j ]) = 0. At the last iteration, the information bits are recovered by the MAP decoding algorithm.

6.7.3 Projection-Based Turbo Multiuser Detection

In Section 6.7.2 we considered the problem of decoding the information of all users in the system. In some cases, however, we are interested in decoding the information of specific users only and are not willing to pay extra receiver complexity for decoding the information of undesired users. One approach to addressing this problem is to null out the signals of the K _u undesired users at the front end and then to apply the iterative soft multiuser demodulation algorithm on the rest of K _d = K “ K _u users' signals [439]. In [476], a projection-based technique was proposed for interference cancellation in STC multiuser systems. Here, we discuss applying soft multiuser demodulation and iterative processing on the projected signal to further enhance the receiver performance.

Consider again the signal model (6.185). Divide the users into two groups, desired users and undesired users. Rewrite (6.185) as

Equation 6.205

where the subscript d denotes desired users and u denotes undesired users. Define , , and (where is the Moore “Penrose generalized inverse of H _u [189]). It is assumed that the matrix H _u is "tall" (i.e., MP > NK _u ) and has full column rank. It is easily seen that

Equation 6.206

(i.e., the undesired users' signals are nulled out by this projection operation). Moreover, before projection, the number of linearly independent rows of H is MP , whereas after projection, the number of linearly independent rows of becomes MP “ NK _u , which implies that in the projected system the effective number of receiver antennas (the effective receiver diversity order) is reduced to M' ( MP “ NK _u )/ P . Hence, the projection operation incurs a diversity loss. Following the same derivations as in Section 6.7.2, we can apply the soft multiuser demodulator and MAP decoder on the projected signal in (6.206) to iteratively detect and decode the information bits of the desired users.

Since we assume that the fading channel remains static within an entire signal frame, which normally contains hundreds of code blocks (of N symbols), we need only compute the projection matrix once per signal frame. Therefore, the dominant computation of the projection-based soft multiuser demodulator is the same as before. However, the overall computational complexity of the multiuser receiver is reduced, since now we need only decode K _d users' information [i.e., only K _d (instead of K ) MAP decoders are needed].

Simulation Examples

Next we provide computer simulation results to illustrate the performance of the turbo receivers in multiuser STBC systems. It is assumed that the fading processes are uncorrelated among all transmitter “receiver antenna pairs of all users; and for each user, the fading processes are uncorrelated from frame to frame but remain static within each frame. It is also assumed that the channel response matrix H in (6.185) is known perfectly . All users employ the same STBC code, but each user uses a different random interleaver. Furthermore, all users transmit M -PSK symbols with equal powers, a scenario in space-division multiple-access (SDMA) systems. Such an equal-power setup is also the worst-case scenario from the interference mitigation point of view.

We consider a four-user ( K = 4) STBC system, as shown in Fig. 6.22. Each user employs the STBC defined in (6.184) and two transmitter antennas ( N = 2). An 8-PSK signal constellation is used in the modulator. The outer convolutional code, which is the same for all users, is a four-state, rate- ½ code with generator (5,7) in octal notation. The encoder is forced to the all-zero state at the end of every signal frame. Each signal frame contains 128 8-PSK symbols. At the receiver side, four antennas ( M = 4) are used.

Assume that all K users' signals are to be decoded ( K = 4). We first demonstrate the performance of the iterative receiver discussed in Section 6.7.2. The frame-error rate (FER) and the bit-error rate (BER) are shown in Fig. 6.24. For the purpose of comparison, we also include the performance of the single-user STBC system with iterative decoding. The dotted lines (denoted as SU1-1 ) in Fig. 6.24 represent the performance of the single-user system with two transmitter antennas ( N = 2) and four receiver antennas ( M = 4) after its first iteration [i.e., the conventional (noniterative) single-user performance]. The dash-dotted lines (denoted as SU1-6 ) in Figs. 6.24 and 6.25 represent the single-user performance after six iterations using the same iterative structure discussed in Section 6.7.2.

Figure 6.24. Frame-error rate (FER) and bit-error rate (BER) for a four-user STBC system. K = 4, N = 2, M = 4. All four users are iteratively detected and decoded. SU1-1 and SU1-6 denote the iterative decoding performance of the single-user system with K = 1, N = 2, M = 4.

Since a single user transmits different STC code symbols from its N transmitter antennas, it could be viewed as a virtual N -user system as discussed in Section 6.7.2. Then the iterative receiver structure for the multiuser STC systems can also be applied to single-user STC systems. Note that the optimal receiver of the proposed multiuser STBC system involves joint decoding of the multiuser STBC and outer convolutional codes, which has a prohibitive complexity [ W _c ^NK 2 ⁿ ]. However, the STBC signal model in (6.188), either single-user or multiuser, is analogous to the synchronous CDMA multiuser system model. As seen from previous sections, at high SNR, the iterative technique for interference suppression and decoding in a multiuser system can approach the performance of a single-user system (which is a lower bound for optimal performance). Hence it is reasonable to view the performance of the iterative single-user STBC system as an approximate lower bound on optimal joint decoding performance. It is seen from Fig. 6.24 that after six iterations the performance, in terms of both FER and BER, of both the single-user and multiuser STBC systems is significantly improved compared with that of the noniterative receivers (i.e., the performance after the first iteration). More impressively, the performance of the iterative receiver in a multiuser system approaches that of the iterative single-user receiver at high SNR.

We next demonstrate the performance of the projection-based turbo receiver. Assume that K _d out of all K users' signals are to be decoded ( K = 4, K _d = 2). In this scenario, two users are first nulled out by a projection operation, and the other two users are iteratively detected and decoded, as discussed in Section 6.7.3. The performance is shown in Fig. 6.25. It is shown in [476] that due to the projection operation, the equivalent receiver antenna number (the receiver diversity) is reduced from MP to MP “ N K _u . So, for a fair comparison, in Fig. 6.25 we also present the iterative decoding performance after the first iteration (denoted by SU2-1 ) and after the sixth iteration (denoted by SU2-6 ) of the single-user system with two transmitter antennas ( N = 2) and two receiver antennas ( M' = 2), where M' denotes the effective number of receiver antennas for the projected system, with M' ( MP “ NK _u )/ P . It is seen that the projection-based turbo receiver still significantly outperforms the projection-based noniterative receiver. However, compared with the turbo receiver discussed above, the projection operation incurs a substantial performance loss. The reason for such a performance loss is twofold. First, the projection operation causes a diversity loss by suppressing the interference from other K _u users; second, the projection operation enhances the background ambient noise. It is therefore preferable to use turbo receiver operating on all users' signals in STBC systems.

Figure 6.25. Frame-error rate (FER) and bit-error rate (BER) for a four-user STBC system. K = 4, N = 2, M = 4, M' = 2. Two users are first nulled out, the other two users are iteratively detected and decoded. SU2-1 and SU2-6 denote the iterative decoding performance of the single-user system with K = 1, N = 2, M' = 2. The gap between SU1-6 and SU2-6 constitutes the diversity loss caused by the projection operation.