4.8 Extension to Multipath Channels

In this section we extend the robust group -blind multiuser detection techniques developed in previous sections to general asynchronous CDMA channels with multipath distortion. Let the impulse response of the k th user 's multipath channel be given by

Equation 4.147

where L is the total number of paths in the channel, and where a _l,k and t _l,k are, respectively, the complex path gain and the delay of the k th user's l th path. It is assumed that t _{1, k} < t _{2, k} < . . . < t _L,k . The received continuous-time signal is then given by

Equation 4.148

where * denotes convolution.

As discussed 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 4.149

By stacking m successive sample vectors, we further define the following quantities :

where and where l is the maximum delay spread in terms of symbol intervals. We can then have the following matrix form of the discrete-time signal model:

Equation 4.150

and as before, we write the eigendecomposition of the autocorrelation matrix of the received signal as

Equation 4.151

Equation 4.152

where the signal subspace U _s has r columns .

We next discuss robust blind multiuser detection and robust group-blind multiuser detection in multipath channels.

4.8.1 Robust Blind Multiuser Detection in Multipath Channels

Suppose that user 1 is the user of interest. Then we can rewrite (4.150) as

Equation 4.153

Equation 4.154

where denotes the ( K _I + 1)th column of H (corresponding to the bit b ₁ [ i ]), H denotes the submatrix of H obtained by striking out the ( K _I +1)th column, and b [ i ] denotes the subvector of b [ i ] obtained by striking out the ( K _I +1)th element. As before, the basic idea behind robust blind multiuser detection is first to obtain a robust estimate of z [ i ] using the identified signal subspace U _s . On the other hand, as discussed in Section 2.7.3, given the spreading waveform s ₁ of the desired user, by exploiting the orthogonality between the signal subspace and noise subspace, the composite signature waveform of this user can be estimated (up to a complex scaling factor). Once an estimate of is available, the robust estimate of z [ i ] can then be translated into a robust estimate of b ₁ [ i ] (upto a complex scaling factor) by Proposition 4.2, as

Equation 4.155

Finally, differential detection is performed according to

Equation 4.156

The algorithm is summarized as follows .

Algorithm 4.6: [Robust blind multiuser detector ”multipath CDMA]

• Compute the sample autocorrelation matrix of the received augmented signal r [ i ] and its eigendecomposition.

• Compute the robust estimate of z [ i ] following a procedure similar to (4.128)-(4.132).

• Compute an blind estimate of according to (2.202)-(2.203).

• Compute the output of the robust blind detector according to (4.155).

• Perform differential detection according to (4.156).

4.8.2 Robust Group-Blind Multiuser Detection in Multipath Channels

We now turn to the group-blind version of the robust multiuser detector for the multipath channel. As before, we can rewrite (4.150) as

Equation 4.157

Equation 4.158

where and [ i ] contain the data bits in b [ i ] corresponding to sets of desired users and undesired users, respectively; and contain columns of H corresponding to desired users and undesired users, respectively. As discussed in Section 2.7.3, based on the knowledge of the spreading waveforms of the desired users, by exploiting the orthogonality between the signal subspace and the noise subspace, we can blindly estimate up to a scale and phase ambiguity for each user. With such an estimate, we can write

Equation 4.159

where the term contains the signal carrying the current bits of the desired users; and the term contains the signal carrying the previous and future bits (i.e., the intersymbol interference). Note that in (4.159) the term represents the estimated channel for the desired users' current bits, and is a diagonal matrix containing the complex scalars of ambiguities ; the term represents the estimated channel for the desired users' past and future bits, and q _I [ i ] contains the products of those bits and the complex ambiguities of the corresponding channels. Following the method outlined in Section 4.7, we first obtain a robust estimate of z [ i ] and then translate it into the estimate of [ i ] by again applying Proposition 4.2:

Equation 4.160

Next, we obtain a robust estimate of the sum of the undesired users' signals based on the relationship

Equation 4.161

Equation 4.162

where represents the signal subspace obtained from the eigendecomposition of the autocorrelation matrix of [ i ]. Finally, we subtract the estimated undesired users' signals and the intersymbol interference from r [ i ] to obtain

Equation 4.163

Equation 4.164

Note that the complex ambiguities in can be estimated based on the estimate of [ i ], as discussed in Section 4.7. Note also that (4.164) has the same form as (4.141), and hence similarly to (4.143) “(4.146), the slowest-descent-search method can then be employed to obtain a robust estimate of from (4.164). The algorithm is summarized below.

Algorithm 4.7: [Robust group-blind multiuser detector ”multipath CDMA]

• Compute the sample autocorrelation matrix of the received augmented signal r [ i ] and its eigendecomposition.

• Compute the robust estimate of z [ i ] following a procedure similar to (4.128)-(4.132).

• Compute a blind estimate of according to (3.162)-(3.163).

• Compute the output of the robust blind detector according to (4.160).

• Compute the sum of the undesired users' signals [ i ] according to (4.161); compute the sample autocorrelation matrix of the signal [ i ] and its eigendecomposition.

• Compute the robust estimate of [ i ] in (4.162) following a procedure similar to (4.128)-(4.132).

• Compute the sum of the desired users' signals [ i ] according to (4.163).

• Estimate the complex amplitudes of ambiguities introduced by the blind estimator based on the robust estimate of [ i ] using (3.127)-(3.129) [cf. (3.134)-(3.140)].

• Form the Huber penalty function and apply the slowest-descent search of , similar to (4.143) “(4.146).

• Perform differential decoding.

Simulation Examples

In the following simulation, the number of users is K = 8 and the spreading gain is N = 15. Each user's channel is assumed to have L = 3 paths and a delay spread of up to one symbol. The complex gains and the delays of each user's channel are generated randomly . The chip pulse is a raised cosine pulse with roll-off factor 0.5. The path gains are normalized so that each user's signal arrives at the receiver with unit power. The channel is normalized in such a way that the composite of the multipath channel and the spreading waveform has unit power. The noise parameters are = 0.01 and k = 100. The smoothing factor is m = 2 and the oversampling factor is p = 2. Shown in Fig. 4.14 is the BER performance of the robust blind multiuser detector and that of the robust group-blind multiuser detector ( = 4). It is seen that in the presence of both non-Gaussian noise and multipath channel distortion, the group-blind robust detector substantially improves the performance of the blind robust detector. Furthermore, most of the performance gain offered by the slowest-descent search is obtained by searching along only one direction.

Figure 4.14. BER performance of a group-blind robust multiuser detector in non-Gaussian noise: multipath channel. N = 15, K = 8, = 4, = 0.01, k = 100. 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 the robust blind detector (Algorithm 4.2) and robust group-blind detector (Algorithm 4.7) with one and two search directions are shown.