3.5 Group -Blind Multiuser Detection in Multipath Channels In this section we extend the linear and nonlinear group-blind multiuser detection methods developed in previous sections to general asynchronous CDMA systems with multipath channel distortion. The signal model for multipath CDMA systems is developed in Section 2.7.1. At the receiver, the received signal is filtered by a chip-matched filter and sampled at a multiple ( p ) of the chip rate. Denote r q [ i ] as the q th signal sample during the i th symbol [cf. (2.167)]. Recall that by denoting  we have the following discrete-time signal model: Equation 3.147 
By stacking m successive sample vectors, we further define the following quantities : 
where the smoothing factor m is chosen according to 
such that the matrix H is a "tall" matrix [i.e., Pm  K ( m + I )]. We can then write (3.147) in matrix form as Equation 3.148 
Assume that H has full column rank; that is,  The autocorrelation matrix of the signal r [ i ] and its eigendecomposition are given, respectively, by Equation 3.149 
Equation 3.150 
where L s = diag ( l 1 , . . ., l r ) contains the r largest eigenvalues of C r . In what follows it is assumed that the receiver has knowledge of the first  (  ) users' signature waveforms,  , whereas the signature waveforms of the remaining (  ) users are unknown to the receiver. Denote by  [ m ] and  the submatrics of  [ m ] and H , respectively, corresponding to the desired users:  It is assumed that has full column rank:  As in the synchronous case, the following projection matrix is needed in the definition of the form I group-blind linear MMSE detector: Equation 3.151 
Note that  projects any signal onto the subspace null (  ). It is then easily seen that the matrix  has an eigenstructure of the form Equation 3.152 
where  , with  , and the columns of  form an orthogonal basis of the subspace range ( H )  null ( ). 3.5.1 Linear Group-Blind Detectors As before, the basic idea behind group-blind linear detectors is to suppress the interference from known users based on the spreading sequences of these users and to suppress the interference from other, unknown users using subspace-based blind methods. Analogous to the synchronous case, we have the following three types of linear group-blind detectors. (In this section, e k denotes an  -vector with all elements zeros except for the k th, which is 1.) Definition 3.4: [Group-blind linear decorrelating detector ”multipath CDMA] The weight vector of the group-blind linear decorrelating detector for user k is given by the solution to the following constrained optimization problem: Equation 3.153 
Definition 3.5: [Group-blind linear hybrid detector ”multipath CDMA] The weight vector of the group-blind linear hybrid detector for user k is given by the solution to the following constrained optimization problem: Equation 3.154 
Definition 3.6: [Group-blind linear MMSE detector ”multipath CDMA] Let  be the components of the received signal r [i] in (3.148) consisting of signals from known users plus noise . (  [ i ] is the subvector of b [ i ] containing bits of the desired users.) The weight vector of the group-blind linear MMSE detector for user k is given by  , where   range ( ) and range ( ) [ note that is given in (3.152)], such that Equation 3.155 
Equation 3.156 
The following results give expressions for the three group-blind linear detectors defined above in terms of the known users' channel matrix and the unknown users' signal subspace components  and defined in (3.152). The proofs of these results are similar to those corresponding to the synchronous case. Proposition 3.7: [Group-blind linear decorrelating detector (form I) ”multipath CDMA] The weight vector of the group-blind linear decorrelating detector for the k th user is given by Equation 3.157 
Proposition 3.8: [Group-blind linear hybrid detector (form I) ”multipath CDMA] The weight vector of the group-blind linear hybrid detector for the k th user is given by Equation 3.158 
Proposition 3.9: [Group-blind linear MMSE detector (form I) ”multipath CDMA] The weight vector of the group-blind linear MMSE detector for the k th user is given by Equation 3.159 
Note that to implement these group-blind linear detectors, the matrix must be estimated first. The blind channel estimation procedure is discussed in Section 2.7.3. The channel estimator discussed there can be used to estimate the channel for each desired user. Once the desired users' channels are estimated, the matrix can be formed . As before, the blind channel estimator has an arbitrary phase ambiguity, which necessitates the use of differential encoding and decoding of the data bits. We next summarize the group-blind linear hybrid multiuser detection algorithm in multipath channels. Algorithm 3.5: [Group-blind linear hybrid detector (form I) ”multipath CDMA] -
Compute the signal subspace: Equation 3.160 
Equation 3.161 
-
Estimate the desired users' channels (cf. Section 2.7.3): Equation 3.162 
Equation 3.163 
Form  using  . -
Compute the unknown users' subspace: Equation 3.164 
Equation 3.165 
-
Form the detectors: Equation 3.166 
-
Perform differential detection: Equation 3.167 
Equation 3.168 
Note that the group-blind linear decorrelating detector and the group-blind linear MMSE detector can be implemented similarly. Both require an estimate of s 2 , which can be obtained simply as the mean of the noise subspace eigenvalues  . Alternatively, the group-blind linear detectors can be expressed in terms of the signal subspace components L s and U s of all users defined in (3.150), as given by the following three results. The proofs are again similar to their counterparts in the synchronous case. Proposition 3.10: [Group-blind linear decorrelating detector (form II) ”multipath CDMA] The group-blind linear decorrelating detector for the k th user is given by Equation 3.169 
Proposition 3.11: [Group-blind linear hybrid detector (form II) ”multipath CDMA] The group-blind linear hybrid detector for the k th user is given by Equation 3.170 
Proposition 3.12: [Group-blind linear MMSE detector (form II) ”multipath CDMA] Let the (rank- deficient ) QR factorization of the Pm x r matrix U s be Equation 3.171 
where Q s is a  matrix, R s is a  nonsingular upper triangular matrix, and P is a permutation matrix. The group-blind linear MMSE detector for the k th user is given by Equation 3.172 
Finally, we summarize the form II group-blind linear hybrid multiuser detection algorithm in multipath channels as follows. Algorithm 3.6: [Group-blind linear hybrid detector (form II) ”multipath CDMA] -
Compute the signal subspace: Equation 3.173 
Equation 3.174 
-
Estimate the desired users' channels (cf. Section 2.7.3): Equation 3.175 
Equation 3.176 
Form using . -
Form the detectors: Equation 3.177 
-
Perform difierential detection: Equation 3.178 
Equation 3.179 
It is seen that form I group-blind detectors are based on an estimate of the signal subspace of the matrix  , whereas form II group-blind detectors are based on an estimate of the signal subspace of the matrix C r . If the signal subspace dimension of is less than that of C r , which is , form I implementation in general gives a more accurate estimation of group-blind detectors. On the other hand, for multipath channels, estimation of the given users' channels is based on eigendecomposition of C r . Hence form II group-blind detectors are more efficient in terms of implementation since they do not require the eigendecomposition (3.152), which is required by form I group-blind detectors. If however, the channels are estimated by some other means not involving the eigendecomposition of C r , form I detectors can be computationally less complex than form II detectors, since the dimension of the estimated signal subspace of the former is less than that of the latter. (That is, of course, if computationally efficient subspace tracking algorithms [98] are used instead of the conventional eigendecomposition.) Simulation Examples Next, we provide computer simulation results to demonstrate the performance of the proposed blind and group-blind linear multiuser detectors under a number of channel conditions. The simulated system is an asynchronous CDMA system with processing gain N = 15. Employed as the user spreading sequences are m -sequences of length 15 and their shifted versions. The chip pulse is a raised cosine pulse with roll-off factor 0.5. Each user's channel has L = 3 paths. The delay of each path is uniform on [0, 10 T c ]. Hence the maximum delay spread is one symbol interval (i.e., i = 1). The fading gain of each path in each user's channel is generated from a complex Gaussian distribution and fixed for all simulations. The path gains in each user's channel are normalized so that all users' signals arrive at the receiver with the same power. The oversampling factor is p = 2. The smoothing factor is m = 2. Hence this system can accommodate up to  = 10 users. The number of users in the simulation is 10, with seven known users (i.e., K = 10 and = 7). The length of each user's signal frame is M = 200. In each simulation, an eigendecomposition is performed on the sample autocorrelation matrix of the received signals. The signal subspace consists of the eigenvectors corresponding to the largest r eigenvalues. [Recall that  is the dimension of the signal subspace.] The remaining eigenvectors constitute the noise subspace. An estimate of the noise variance s 2 is given by the average of the Pm “ r smallest eigenvalues. We first compare the performance of four exact detectors (i.e., assuming that H and s 2 are known): -
The linear MMSE detector -
The linear zero-forcing detector -
The group-blind linear hybrid detector -
The group-blind linear MMSE detector For each of these detectors, and for each SNR value, the minimum and maximum bit-error rate (BER) among the seven known users is plotted in Fig. 3.13. It is seen from this figure that, as expected, the closer the detector is to the true linear MMSE detector, the better its performance is. Figure 3.13. Comparison of the performance of four exact linear detectors in white noise. K = 10, = 7.  Next, the performance of the various estimated group-blind detectors (i.e., the detectors are estimated based on the M received signal vectors) is shown in Fig. 3.14. It is seen that at low SNR, the group-blind MMSE detectors perform best, whereas at high SNR, the group-blind hybrid detectors perform best. This is because the hybrid detector zero-forces the known users' signals and enhances the noise level, whereas the group-blind linear MMSE detector suppresses both the interference and the noise. At high SNR, the group-blind hybrid and group-blind MMSE detectors tend to become the same. However, implementation of the latter requires an estimate of the noise level. When the noise level is low, this estimate is noisy , which causes the performance of the group-blind MMSE detector to deteriorate. It is also seen that the performance of form I detectors is only slightly better than that of the corresponding form II detectors, at the expense of higher computational complexity. Figure 3.14. Comparison of the performance of various estimated group-blind detectors in white noise. K = 10 = 7. (Top: minimum BER; bottom: maximum BER.)  Comparing Fig. 3.13 with Fig. 3.14, it is seen that the performance of the estimated detectors is substantially diffierent from that of the corresponding exact detectors for the block size considered here (i.e., M = 200). It is known that the subspace detectors converge to the exact detectors at a rate of  (  ). It is also seen from Fig. 3.14 that the form II hybrid detector performs very well compared with other forms of group-blind detectors, even though it has the lowest computational complexity. Hence in subsequent simulation studies, we compare the performance of the form II hybrid detector with that of some multiuser detectors proposed previously. We next compare the performance of the group-blind hybrid detector with that of the blind detector for the same system. The result is shown in Fig. 3.15, where the BER curves for the blind linear MMSE detector, the form II group-blind linear hybrid detector, and a partial MMSE detector are plotted. The partial MMSE detector ignores the unknown users and forms the linear MMSE detector for the known users using the estimated matrix . It is seen that the group-blind detector significantly outperforms the blind MMSE detector and the partial MMSE detector. Indeed, the blind MMSE detector exhibits an error floor at high SNR values. This is due to the finite length of the received signal frame, from which the detector is estimated. The group-blind hybrid MMSE detector does not show an error floor in the BER range considered here. Of course, due to the finite frame length, the group-blind detector also has an error floor. But such a floor is much lower than that of the blind linear MMSE detector. Figure 3.15. Comparison of the performance of blind and group-blind linear detectors in white noise. K = 10, = 7.  Theoretically, both the blind and group-blind detectors converge to the true linear MMSE detector (at a high signal-to-noise ratio) as the signal frame size M   . Hence the asymptotic performance of the two detectors is the same at high signal-to-noise ratio. However, for a finite frame length M , the group-blind detector performs significantly better than the blind detector, as seen from the simulation results above. An intuitive explanation for such performance improvement is that more information about the multiuser environment is incorporated in forming the group-blind detector. For example, the computations for subspace decomposition and channel estimation involved in the two detectors are exactly the same. However, the blind detector is formed based solely on the composite channel of the desired user, whereas the group-blind detector is formed based on the composite channels of all known users. By incorporating more information about the multiuser channel, the estimated group-blind detector is more accurate than the estimated blind detector (i.e., the former is "closer" than the latter to the exact detector). It is seen from Fig. 3.13 that when the spreading waveforms and the channels of all users are known, all three forms of the exact group-blind detectors perform worse than the linear MMSE detector, which is the exact blind detector. This is because the zero-forcing and hybrid group-blind detectors zero-force all or some users' signals and enhance the noise level, whereas the group-blind MMSE detector is defined in terms of a specific constrained form which in general is different from the true MMSE detector. However, with imperfect channel information, the roles are reversed and the group-blind detectors outperform the blind detector. Of course, both the blind and group-blind detectors are developed based on the assumption that the multiuser channel is not perfectly known, and a study of the performance of the exact detectors is only of theoretical interest. Nevertheless, it is interesting to observe that by changing the assumption on prior knowledge about the channel, the relative performance of two detectors can be different. 3.5.2 Adaptive Group-Blind Linear Multiuser Detection As for the blind linear multiuser detector discussed in Chapter 2, the group-blind linear multiuser detectors can also be implemented adaptively. Specifically, for example, since the form II linear hybrid detector can be written in closed form as a function of the signal subspace components, we can use a suitable subspace tracking algorithm in conjunction with this detector and a channel estimator to form an adaptive detector that is able to track changes in the number of users and their composite signature waveforms [412]. Figure 3.16 contains a block diagram of such a receiver. The received signal r [ i ] is fed into a subspace tracker which sequentially estimates the signal subspace components ( U s , L s ). The received signal r [ i ] is then projected onto the noise subspace to obtain z [ i ], which is in turn passed through a bank of parallel linear filters, each determined by the signature waveform of a desired user. The output of each filter is fed into a channel tracker which estimates the channel state of that particular user. Finally, the linear hybrid group-blind detector is constructed in closed form based on the estimated signal subspace components and the channel states of the desired users. This adaptive algorithm is summarized as follows. Suppose that at time i “ 1, the estimated signal subspace rank is r [ i “ 1] and the signal subspace components are U s [ i “ 1], L s [ i “1], and s 2 [ i “ 1]. The estimated channel states for the desired users are f k [ i “ 1], 1  k . Then at time i , the adaptive detector performs the following steps to update the detector and detect the data. Figure 3.16. Adaptive group-blind receiver structure.  Algorithm 3.7: [Adaptive group-blind linear hybrid multiuser detector ”multipath CDMA] -
Update the signal subspace: Using a particular signal subspace tracking algorithm, update the signal subspace rank r [ i ] and the signal subspace components U s [ i ], L s [ i ], and s 2 [ i ]. -
Estimate the desired users' channels (cf. Section 2.7.4): Equation 3.180 
Equation 3.181 
Equation 3.182 
Equation 3.183 
Form  using  . -
Form the detectors: Equation 3.184 
-
Perform differential detection: Equation 3.185 
Equation 3.186 
Simulation Examples We next illustrate the performance of the adaptive receiver in an asynchronous CDMA system. The processing gain N = 15 and the spreading codes are Gold codes of length 15. The chip pulse waveform is a raised cosine pulse with a roll-off factor of 0.5. Each user's channel has L = 3 paths. The delay of each path is distributed uniformly on [0, 10 T c ]. Hence, the maximum delay spread is one symbol interval (i.e., I = 1). The channel gain of each path in each user's channel is generated from a complex Gaussian distribution and is fixed for all simulations. The path gains in each user's channel are normalized so that all users' signals arrive at the receiver with the same power. The oversampling factor is p = 2 and the smoothing factor is m = 2. The performance measures are the SINR and the BER. Figure 3.17 is a comparison of the adaptive performance of the MMSE and hybrid group-blind detectors using the NAHJ subspace tracking algorithm discussed in Section 2.6.3. During the first 1000 iterations there are eight total users, six of which are known by the group-blind detector. At iteration 1000, four new users are added to the system. At iteration 2000, one additional known user is added and three unknown users vanish . We see that there is a substantial performance gain using the group-blind detector at each stage and that convergence occurs in less than 500 iterations. Figure 3.17. Performance of an adaptive group-blind receiver employing the NAHJ subspace tracking algorithm.  Figure 3.18 is created with parameters identical to Fig. 3.17 except that the tracking algorithm used is an exact rank-one SVD update. Again we see a significant improvement in performance using the group-blind detector. More important, when we compare Figs. 3.17 and 3.18 we see very little difference between the performance we obtain using the NAHJ subspace tracking and that we obtain using an exact SVD update. Figure 3.18. Performance of an adaptive group-blind receiver employing the exact SVD update.  Figure 3.19 shows the steady-state BER performance of our receiver using NAHJ subspace tracking and the exact SVD update for both blind and group-blind multiuser detection. The number of users is eight and the number of known users is six. At SNR above about 11 dB we see that the group-blind detectors provide a substantial improvement in BER. At lower SNR, the group-blind detectors seem to suffer from the noise enhancement problems that often accompany zero-forcing detectors. Recall that the hybrid group-blind detector zero-forces interference from known users and suppresses interference from unknown users via the MMSE criterion. Once again, note the relatively small difference between the performance of NAHJ and that of exact SVD, especially at high SNR. Figure 3.19. Steady-state performance of adaptive group-blind receivers.  3.5.3 Linear Group-Blind Detection in Correlated Noise The problem of blind linear multiuser detection in unknown correlated noise is discussed in Section 2.7.5. In this section we consider the problem of group-blind linear multiuser detection in the same environment, which was first treated in [551]. Recall that in this case it is assumed that the signal is received by two well-separated antennas, so that the noise is spatially uncorrelated. The two augmented received signal vectors at the two antennas are given, respectively, by Equation 3.187 
Equation 3.188 
where H 1 and H 2 contain the channel information corresponding to the respective antennas; n 1 [ i ] and n 2 [ i ] are the Gaussian noise vectors at the two antennas with the following correlations : Equation 3.189 
Equation 3.190 
Equation 3.191 
Define Equation 3.192 
Equation 3.193 
Equation 3.194 
The canonical correlation decomposition (CCD) of the matrix C 12 is given by Equation 3.195 
Equation 3.196 
The Pm x Pm matrix G has the form G = diag ( g 1 , . . . , g r , 0, . . . , 0), with g 1 . . . g r > 0. Define L j,s and L j,n as, respectively, the first r columns and the last Pm “ r columns of L j , j = 1, 2. It is known then that Equation 3.197 
As discussed in Section 2.7.5, the composite signature waveform  of the desired user k , 1 k , can be estimated based on the orthogonality relationship  . We next consider the group-blind linear detector in correlated ambient noise based on the CCD method. Since the signal subspace cannot be identified directly in the CCD, we will not consider the group-blind linear zero-forcing or MMSE detectors, which require the identification of some signal subspace. Nevertheless, the form II group-blind linear hybrid detector can easily be constructed for correlated noise, as given by the following result. Proposition 3.13: [Group-blind linear hybrid detector in correlated noise (form II)] The weight vector of the group-blind linear hybrid detector for the k th user at the jth antenna in correlated noise is given by Equation 3.198 
Proof: By definition, the group-blind linear hybrid detector is given by the following constrained optimization problem: Equation 3.199 
Using the method of Lagrange multipliers to solve (3.199), we obtain Equation 3.200 
where  is the Lagrange multiplier , and  . Substituting (3.200) into the constraint that  , we obtain  Hence Equation 3.201 
Moreover, by definition, Equation 3.202 
Substituting (3.202) into (3.201), and using the fact that  , we obtain (3.198). The group-blind linear multiuser detection algorithm in multipath channels with correlated noise is summarized as follows. Algorithm 3.8: [Group-blind linear hybrid detector ”multipath CDMA and correlated noise] Simulation Example The simulated system is the same as that described in Section 3.5.1. The noise at each antenna j is modeled by a second-order autoregressive (AR) model with coefficients [ a j,1 , a j,2 ]; that is, the noise field is generated according to Equation 3.209 
where v j [ n ] is the noise at antenna j and sample n , and w j [ n ] is a complex white Gaussian noise sample. The AR coefficients at the two antennas are chosen as [ a 1,1 , a 1,2 ] = [1, “0.2] and [ a 2,1 , a 2,2 ] = [1.2, “0.3]. The performance of the group-blind linear hybrid detector is compared with that of the blind linear MMSE detector. The result is shown in Fig. 3.20. It is seen that similar to the white noise case, the proposed group-blind linear detector offers substantial performance gain over the blind linear detector. 3.5.4 Nonlinear Group-Blind Detection In this section we extend the nonlinear multiuser detection methods discussed in Section 3.4 to asynchronous CDMA systems with multipath [456]. The idea is essentially the same as in the synchronous case. We first estimate the decorrelating detectors of the desired users, given by Equation 3.210 
Note that can be estimated only up to a phase ambiguity. Denote the output of the decorrelating detector as Equation 3.211 
Figure 3.20. Comparison of the performance of blind and group-blind linear detectors in correlated noise. K = 10, = 7.  where  is the phase ambiguity induced by the channel estimator which can be estimated using (3.129). Denote  Then (3.211) can be written as Equation 3.212 
Note that the covariance of v is given by Equation 3.213 
with Equation 3.214 
Based on (3.212), the slowest-descent-search method for estimating  is given by the same procedure as (3.121) “(3.126), with the covariance matrix given by (3.214). The algorithm is summarized as follows. Algorithm 3.9: [Nonlinear group-blind detector ”multipath CDMA] -
Compute the signal subspace: Equation 3.215 
Equation 3.216 
-
Estimate the desired users' channels (cf. Section 2.7.3): Equation 3.217 
Equation 3.218 
Form using . -
Form the decorrelating detectors using (3.210). -
Estimate the complex amplitudes  : Equation 3.219 
Equation 3.220 
Equation 3.221 
Equation 3.222 
Equation 3.223 
Equation 3.224 
-
Compute the Hessian: Equation 3.225 
and the Q smallest eigenvectors  . -
Detect each symbol by solving the following discrete optimization problem using an exhaustive search [over ( Q + 1) points]: Equation 3.226 
Equation 3.227 
Equation 3.228 
Equation 3.229 
-
Perform differential decoding: Equation 3.230 
Simulation Examples The simulation set is the same as that in Section 3.5.2. Figure 3.21 shows that similar to the synchronous case, in multipath channels, the nonlinear group-blind multiuser detector outperforms the linear group-blind detector by a significant margin. Furthermore, most of the performance gain offered by the slowest-descent method is obtained by searching along only one direction. Figure 3.21. Performance of a slowest-descent-based group-blind multiuser detector in a multipath channel. N = 15, K = 8, = 4. Each user's channel consists of three paths with randomly generated complex gains and delays. Only the spreading waveforms of the desired users are assumed known to the receiver. The BER curves of a linear group-blind detector and slowest-descent (nonlinear) group-blind detector with Q = 1 and Q = 2 are shown.   | 
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