Generally speaking, space-time processing involves the exploitation of spatial diversity using multiple transmit and/or receive antennas and, perhaps, some form of coding. In previous sections we have focused on systems that employ one transmit antenna and multiple receive antennas. Recently, however, much of the work in this area has focused on transmit diversity schemes that use multiple transmit antennas. They include delay schemes [444, 572, 573] in which copies of the same symbol are transmitted through multiple antennas at different times, the space-time trellis coding algorithm in [477], and the simple space-time block coding (STBC) scheme developed in [12], which has been adopted in third-generation (3G) wideband CDMA (WCDMA) standards [294, 479]. A generalization of this simple space-time block coding concept is developed in [475, 476]. It has been shown that these techniques can significantly increase capacity [122, 478]. In this section we discuss adaptive receiver structures for synchronous CDMA systems with multiple transmit antennas and multiple receive antennas. Specifically, we focus on three configurations: (1) one transmit antenna, two receive antennas; (2) two transmit antennas, one receive antenna; and (3) two transmit antennas, two receive antennas. It is assumed that the orthogonal space-time block code [12] is employed in systems with two transmit antennas. For each of these configurations, we discuss two possible linear receiver structures and compare their performance in terms of diversity gain and signal separation capability. We also describe blind adaptive receiver structures for such multiple-antenna CDMA systems. The methods discussed in this section are generalized in the next section to mutipath CDMA systems. The materials discussed in this and the following sections first appeared in [415]. 5.5.1 One Transmit Antenna, Two Receive AntennasConsider the following discrete-time K - user synchronous CDMA channel with one transmit antenna and two receive antennas. The received baseband signal at the p th antenna can be modeled as Equation 5.95
where s k is the N -vector of the discrete-time signature waveform of the k th user with unit norm (i.e., s k = 1), b k {+1, “1} is the data bit of the k th user, g p,k is the complex channel response of the p th receive antenna element to the k th user's signal, and n p ~ N c ( , s 2 I N ) is the ambient noise vector at antenna p . It is assumed that n 1 and n 2 are independent. Linear Diversity Multiuser DetectorDenote
Suppose that user 1 is the user of interest. We first consider the linear diversity multiuser detection scheme, which first applies a linear multiuser detector to the received signal r p in (5.95) at each antenna p = 1, 2, and then combines the outputs of these linear detectors to make a decision. For example, a linear decorrelating detector for user 1 based on the signal in (5.95) is simply Equation 5.96
where e 1 denotes the first unit vector in . This detector is applied to the received signal at each antenna p = 1, 2, to obtain z = [ z 1 z 2 ] T , where Equation 5.97
with Equation 5.98
where w 1 2 = [ R “1 ] 1,1 . Denote Equation 5.99
and . Since the noise vectors from different antennas are independent, we can write Equation 5.100
with Equation 5.101
The maximum- likelihood (ML) decision rule for b 1 based on z in (5.100) is then Equation 5.102
Let be the total received desired user's signal energy. The decision statistic in (5.102) can be expressed as Equation 5.103
with Equation 5.104
The probability of detection error is computed as Equation 5.105
Linear Space-Time Multiuser DetectorDenote
Then, by augmenting the received signals at two antennas, (5.95) can be written as Equation 5.106
with . A linear space-time multiuser detector operates on the augmented received signal directly. For example, the linear decorrelating detector for user 1 in this case is given by Equation 5.107
This detector is applied to the augmented received signal to obtain Equation 5.108
with Equation 5.109
where . Denote Equation 5.110
An expression for can be found as follows . Note that Equation 5.111
Equation 5.112
where (5.111) and (5.112) follow, respectively, from the following two matrix identities: Equation 5.113
Equation 5.114
Hence Equation 5.115
where ° denotes the Schur matrix product (i.e., elementwise product). The ML decision rule for b 1 based on in (5.108) is then Equation 5.116
The probability of detection error is computed as Equation 5.117
Performance ComparisonFrom the discussion above it is seen that the linear space-time multiuser detector exploits the signal structure in both the time domain (i.e., induced by the signature waveform s k ) and the spatial domain (i.e., induced by the channel response g k ) for interference rejection; whereas for the linear diversity multiuser detector, interference rejection is performed only in the time domain, and the spatial domain is used only for diversity combining. The next result, which first appeared in [324], shows that the linear space-time multiuser detector always outperforms the linear diversity multiuser detector. Proposition 5.6: Let ( e ) given by (5.105) and ( e ) given by (5.117) be, respectively, the probability of detection error of the linear diversity detector and the linear space-time detector. Then
Proof: By (5.105) and (5.117) it suffices to show that
We make use of the following facts. Denote by A i,j the submatrix of A obtained by striking out the i th row and the j th column. Then it is known that Equation 5.118
It is also known that Equation 5.119
Assuming that and , and using the two results above, we have Equation 5.120
Equation 5.121
Equation 5.122
where (5.120) follows from the fact that and ; (5.121) follows from the matrix identity Equation 5.123
and (5.122) follows from Equation 5.124
Hence we have Equation 5.125
We next consider a simple example to demonstrate the performance difference between the two receivers discussed above. Consider a two-user system with
where r is the correlation of the signature waveforms of the two users and q 1 and q 2 are the directions of arrival of the two users' signals. Define . Then we have E 1 = E 2 = 1 and Equation 5.126
Equation 5.127
Equation 5.128
Equation 5.129
These expressions are plotted in Fig. 5.12. It is seen that while the multiuser space-time receiver can exploit both the temporal signal separation (along the r -axis) and the spatial signal separation (along the a -axis), the multiuser diversity receiver can exploit only the temporal signal separation. For example, for large r , the performance of the multiuser diversity receiver is poor, no matter what value a takes; but the performance of the multiuser space-time receiver can be quite good as long as a is large. Figure 5.12. Performance comparison between a multiuser diversity receiver (top) and multiuser space-time receiver (bottom).
5.5.2 Two Transmit Antennas, One Receive AntennaWhen two antennas are employed at the transmitter, we must first specify how the information bits are transmitted across the two antennas. Here we adopt the well-known orthogonal space-time block coding scheme [12, 475]. Specifically, for user k , two information symbols, b k ,1 and b k ,2 , are transmitted over two symbol intervals. At the first time interval, the symbol pair ( b k ,1 , b k ,2 ) is transmitted across the two transmit antennas; and at the second time interval, the symbol pair ( “ b k ,2 , b k ,1 ) is transmitted. The received signals corresponding to these two time intervals are given by Equation 5.130
Equation 5.131
where g 1, k ( g 2, k ,) is the complex channel response between the first (second) transmit antenna and the receive antenna; n 1 and n 2 are independent received N c ( 0, I N ) noise vectors at the two time intervals. Linear Diversity Multiuser DetectorWe first consider the linear diversity multiuser detection scheme, which first applies the linear multiuser detector w 1 in (5.96) to the received signals r 1 and r 2 during the two time intervals, and then performs a space-time decoding. Specifically, denote Equation 5.132
Equation 5.133
with Equation 5.134
where w 1 2 = [ R “1 ] 1,1 . Denote
It is easily seen that . Then (5.132) “(5.134) can be written as Equation 5.135
with Equation 5.136
As before, denote . Note that Equation 5.137
The ML decision rule for b 1,1 and b 2,1 based on z in (5.135) is then given by Equation 5.138
Using (5.135), it is easily seen that the decision statistic in (5.138) is distributed according to Equation 5.139
Equation 5.140
Hence the probability of error is given by Equation 5.141
This is the same expression as (5.117) for the linear diversity receiver with one transmit antenna and two receive antennas. Linear Space-Time Multiuser DetectorDenote and . Then (5.130) and (5.131) can be written as Equation 5.142
On denoting
the decorrelating detector for detecting the bit b 1,1 based on in (5.142) is given by Equation 5.143
where is the first unit vector in . We have the following result. Proposition 5.7: The decorrelating detector in (5.143) is given by Equation 5.144
where w 1 is given by (5.96). Proof: We need to verify that Equation 5.145
We have Equation 5.146
Equation 5.147
Equation 5.148
Equation 5.149
This verifies (5.145), so that (5.144) is indeed the decorrelating detector given by (5.143). Thus the output of the linear space-time detector in this case is given by Equation 5.150
with Equation 5.151
where using (5.99) and (5.144), we have Equation 5.152
Therefore, the probability of detection error is given by Equation 5.153
On comparing (5.141) with (5.153) we see that for the case of two transmit antennas and one receive antenna, the linear diversity receiver and the linear space-time receiver have the same performance. Hence the multiple transmit antennas with space-time block coding provide only diversity gain and no signal separation capability. 5.5.3 Two Transmit and Two Receive AntennasWe combine the results from the two preceding sections to investigate an environment in which we use two transmit antennas and two receive antennas. We adopt the space-time block coding scheme used in the preceding section. The received signals at antenna 1 during the two symbol intervals are Equation 5.154
Equation 5.155
and the corresponding signals received at antenna 2 are Equation 5.156
Equation 5.157
where is the complex channel response between transmit antenna i and receive antenna j for user k . The noise vectors , and are independent and identically distributed with distribution N c ( , s 2 I N ). Linear Diversity Multiuser DetectorAs before, we first consider the linear diversity multiuser detection scheme for user 1, which applies the linear multiuser detector w 1 in (5.96) to each of the four received signals , and and then performs a space-time decoding. Specifically, denote the filter outputs as Equation 5.158
Equation 5.159
Equation 5.160
Equation 5.161
with Equation 5.162
where, as before, . We define the following quantities :
Then (5.158) “(5.162) can be written as Equation 5.163
with Equation 5.164
It is readily verified that Equation 5.165
with Equation 5.166
To form the ML decision statistic, we premultiply z by G 1 and obtain Equation 5.167
with Equation 5.168
The corresponding bit estimates are given by Equation 5.169
The bit error probability is then given by Equation 5.170
Linear Space-Time Multiuser DetectorWe denote
Then (5.154) “(5.157) may be written as Equation 5.171
Equation 5.172
where
Since and (5.171) has the same form as (5.142), it is easy to show that the decorrelating detector for detecting the bit b 1,1 based on is given by Equation 5.173
Hence the output of the linear space-time detector in this case is given by Equation 5.174
with Equation 5.175
where Equation 5.176
Therefore, the probability of detection error is given by Equation 5.177
Comparing (5.177) with (5.170), it is seen that when two transmit antennas and two receive antennas are employed and the signals are transmitted in the form of a space-time block code, the linear diversity receiver and the linear space-time receiver have identical performance. RemarksWe have seen that the performance of space-time multiuser detection (STMUD) and linear diversity multiuser detection (LDMUD) are similar for two transmit/one receive and two transmit/two receive antenna configurations. What, then, are the benefits of the space-time detection technique? They are as follows:
5.5.4 Blind Adaptive ImplementationsWe next develop both batch and sequential blind adaptive implementations of the linear space-time receiver. These implementations are blind in the sense that they require only knowledge of the signature waveform of the user of interest. Instead of the decorrelating detector used in previous sections, we will use a linear MMSE detector for the adaptive implementations because the MMSE detector is more suitable for adaptation and its performance is comparable to that of the decorrelating detector. We consider only the environment in which we have two transmit antennas and two receive antennas. The other cases can be derived in a similar manner. Note that inherent to any blind receiver in multiple transmit antenna systems is an ambiguity issue. That is, if the same spreading waveform is used for a user at both transmit antennas, the blind receiver cannot distinguish which bit is from which antenna. To resolve such an ambiguity, here we use two different spreading waveforms for each user (i.e., s j,k , j {1, 2} is the spreading code for user k for the transmission of bit b j, k ). There are two bits, b 1, k [ i ] and b 2, k [ i ], associated with each user at each time slot i , and the difference in time between slots is 2 T , where T is the symbol interval. The received signal at antenna 1 during the two symbol periods for time slot i is Equation 5.178
Equation 5.179
and the corresponding signals received at antenna 2 are Equation 5.180
Equation 5.181
We stack these received signal vectors and denote
Then we may write Equation 5.182
where
The autocorrelation matrix of the stacked signal [ i ], C , and its eigendecomposition are given by Equation 5.183
Equation 5.184
where L s = diag { l 1 , l 2 , . . . , l 2 K } contains the largest (2 K ) eigenvalues of C , the columns of U s are the corresponding eigenvectors, and the columns of U n are the 4 N “ 2 K eigenvectors corresponding to the smallest eigenvalue s 2 . The blind linear MMSE detector for detecting [ b [ i ]] 1 = b 1,1 [ i ] is given by the solution to the optimization problem Equation 5.185
From Chapter 2, a scaled version of the solution can be written in terms of the signal subspace components as Equation 5.186
and the decision is made according to Equation 5.187
and Equation 5.188
or Equation 5.189
Before we address specific batch and sequential adaptive algorithms, we note that these algorithms can also be implemented using linear group -blind multiuser detectors instead of blind MMSE detectors. This would be appropriate, for example, in uplink environments in which the base station has knowledge of the signature waveforms of all of the users in the cell, but not those of users outside the cell . Specifically, we may rewrite (5.182) as Equation 5.190
where we have separated the users into two groups. The composite signature sequences of the known users are the columns of . The unknown users' composite sequences are the columns of . Then, from Chapter 3, the group-blind linear hybrid detector for bit b 1,1 [ i ] is given by Equation 5.191
This detector offers a significant performance improvement over (5.186) for environments in which the signature sequences of some of the interfering users are known. Batch Blind Linear Space-Time Multiuser DetectionTo obtain an estimate of g 1 , we make use of the orthogonality between the signal and noise subspaces [i.e., the fact that . In particular, we have Equation 5.192
Equation 5.193
In (5.193), specifies g 1 up to an arbitrary complex scale factor a (i.e., ). The following is a summary of a batch blind space-time multiuser detection algorithm for the two transmit antenna/two receive antenna configuration. Algorithm 5.4: [Batch blind linear space-time multiuser detector ”synchronous CDMA, two transmit antennas, and two receive antennas]
A batch group-blind space-time multiuser detector algorithm can be implemented with simple modifications to (5.200) and (5.201). Adaptive Blind Linear Space-Time Multiuser DetectionTo form a sequential blind adaptive receiver, we need adaptive algorithms for sequentially estimating the channel and the signal subspace components U s and L s . First, we address sequential adaptive channel estimation. Denote by z [ i ] the projection of the stacked signal [ i ] onto the noise subspace: Equation 5.206
Equation 5.207
Since z [ i ] lies in the noise subspace, it is orthogonal to any signal in the signal subspace, and in particular, it is orthogonal to ( ). Hence g 1 is the solution to the following constrained optimization problem: Equation 5.208
To obtain a sequential algorithm to solve the optimization problem above, we write it in the following (trivial) state space form:
The standard Kalman filter can then be applied to the system above as follows. Denote . We have Equation 5.209
Equation 5.210
Equation 5.211
Once we have obtained channel estimates at time slot i , we can combine them with estimates of the signal subspace components to form the detector in (5.186). Since we are stacking received signal vectors, and subspace tracking complexity increases at least linearly with signal subspace dimension, it is imperative that we choose an algorithm with minimal complexity. The best existing low-complexity algorithm for this purpose appears to be the NAHJ subspace tracking algorithm discussed in Section 2.6.3. This algorithm has the lowest complexity of any algorithm used for similar purposes and has performed well when used for signal subspace tracking in multipath fading environments. Since the size of U s is 4 N x 2 K , the complexity is 40 · 4 N · 2 K + 3 · 4 N + 7.5(2 K ) 2 + 7 · 2 K floating point operations per iteration. Algorithm 5.5: [Blind adaptive linear space-time multiuser detector ”synchronous CDMA, two transmit antennas, and two receive antennas]
Equation 5.223
Equation 5.224
Equation 5.225
Equation 5.226
A group-blind sequential adaptive space-time multiuser detector can be implemented similarly. The adaptive receiver structure is illustrated in Fig. 5.13. Figure 5.13. Adaptive receiver structure for linear space-time multiuser detectors.
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