| In this section we discuss computational procedures for obtaining the output of the nonlinear decorrelating multiuser detectors [i.e., the solution to (4.15)]. Assume that the penalty function r ( x ) in (4.14) has a bounded second-order derivative: Equation 4.57 
for some a > 0. Then (4.15) can be solved iteratively by the following modified residual method [203]. Let q l be the estimate at the l th step of the iteration; then it is updated according to Equation 4.58 
Equation 4.59 
where m  a is a step-size parameter. Denote the cost function in (4.14) by Equation 4.60 
We have the following result regarding the convergence behavior of the iterative procedure above. The proof is given in the Appendix (Section 4.10.1). Proposition 4.1: If  , the iterative procedure defined by (4.58) and (4.59) satisfies Equation 4.61 
where  is assumed to be positive definite and  . Furthermore, if r ( x ) is convex and bounded from below, then with probability 1, q l  q * as l  , where q * is the unique minimum point of the cost function C ( q ) [ i.e., the unique solution to (4.15) ]. Notice that for the minimax robust decorrelating detector, the Huber penalty function r H ( x ) does not have second-order derivatives at the two "corner" points (i.e., x = ± g n 2 ). In principle, this can be resolved by defining a smoothed version of the Huber penalty function, for example, as follows : Equation 4.62 
where h is a small number. The first- and second-order derivatives of this smoothed Huber penalty function are given, respectively, by Equation 4.63 
Equation 4.64 
We can then apply the iterative procedure (4.58) “(4.59) using this smoothed penalty function and the step size 1/ m = u 2 . In practice, however, convergence is always observed even if the nonsmooth nonlinearity y H ( x ) is used. Notice that the matrix (1/ m ) ( S T S ) “1 S T in (4.59) can be computed off-line, and the major computation involved at each iteration is the product of this K x K matrix with a K -vector z l . For the initial estimate q we can take the least-squares solution: Equation 4.65 
The iteration is stopped if q l “ q l “1  d for some small number d . Simulations show that on average it takes fewer than 10 iterations for the algorithm to converge. Finally, we summarize the robust multiuser detection algorithm as follows. Algorithm 4.1: [Robust multiuser detector ”synchronous CDMA] Equation 4.69 
The operations of the M -decorrelating multiuser detector are depicted in Fig. 4.5. It is evident that it is essentially a robust version of the linear decorrelating detector. At each iteration, the residual signal, which is the difference between the received signal r and the remodulated signal S q l , is passed through the nonlinearity y ( · ). Then the modified residual z l is passed through the linear decorrelating filter to get the modification on the previous estimate. Figure 4.5. M -decorrelating multiuser detector, a robust version of the linear decorrelating multiuser detector.  Simulation Examples In this section we provide some simulation examples to demonstrate the performance of the nonlinear robust multiuser detectors against multiple-access interference and non-Gaussian additive noise. We consider a synchronous system with K = 6 users. The spreading sequence of each user is a shifted version of an m -sequence of length N = 31. We first demonstrate the performance degradation of the linear multiuser detectors in non-Gaussian ambient noise. Two popular linear multiuser detectors are the linear decorrelating and linear MMSE detectors. The performance of the linear decorrelating detector in several different  -mixture channels is depicted in Fig. 4.6. In this figure we plot the BER versus the SNR (defined as  ) corresponding to user 1, assuming that all users have the same amplitudes. The performance of the linear MMSE multiuser detector is indistinguishable in this case from that of the linear decorrelating detector. It is seen that the impulsive character of the ambient noise can substantially degrade the performance of both linear multiuser detectors. Similar situations have been observed for the conventional matched filter receiver in [1]. In [383] it is observed that non-Gaussian-based optimal detection can achieve significant performance gain (more than 10 dB in some cases) over Gaussian-based optimal detection in multiple-access channels when the ambient noise is impulsive. However, this gain is obtained with a significant penalty on complexity. The robust techniques discussed in this chapter constitute some low-complexity multiuser detectors that account for non-Gaussian ambient noise. We next demonstrate the performance gain afforded by this non-Gaussian-based suboptimal detection technique over its Gaussian-based counterpart (i.e., the linear decorrelator). Figure 4.6. BER performance of a linear decorrelating detector for user 1 in a synchronous CDMA channel with Gaussian and -mixture ambient noise. N = 31, K = 6. All users have the same amplitudes.  The next example demonstrates the performance gains achieved by the minimax robust decorrelating detector over the linear decorrelator in impulsive noise. The noise distribution parameters are = 0.01 and k = 100. The BER performance of the two detectors is plotted in Fig. 4.7. Also shown in this figure is the performance of an "approximate" minimax decorrelating detector, in which the nonlinearity y ( ·) is taken as Figure 4.7. BER performance of user 1 for an exact minimax decorrelating detector, approximate minimax decorrelating detector, and linear decorrelating detector in a synchronous CDMA channel with impulsive noise. N = 31, K = 6, = 0.01, k = 100. The powers of the interferers are 10 dB above the power of user 1 (i.e.,  = 10 for k  1).  Equation 4.70 
where the parameter g is taken as Equation 4.71 
and the step-size parameter m in the modified residual method (4.59) is set as Equation 4.72 
The reason for studying such an approximate robust detector is that in practice, it is unlikely that the exact parameters and n in the noise model (4.3) are known to the receiver. However, the total noise variance s 2 can be estimated from the received signal (as discussed in the next section). Hence if we could set some simple rule for choosing the nonlinearity y ( ·) and m , this approximate robust detector is much easier to implement than the exact one. It is seen from Fig. 4.7 that the robust decorrelating multiuser detector offers significant performance gains over the linear decorrelating detector. Moreover, this performance gain increases as the SNR increases . Another important observation is that the performance of the robust multiuser detector is insensitive to the parameters and k in the noise model, which is evidenced by the fact that the performance of the approximate robust detector is very close to that of the exact robust detector. We next consider a synchronous system with 20 users ( K = 20). The spreading sequence of each user is still a shifted version of an m -sequence of length N = 31. The performance of the approximate robust decorrelator and that of the linear decorrelator is shown in Fig. 4.8. Again it is seen that the robust detector offers a substantial performance gain over the linear detector. Figure 4.8. BER performance of user 1 for an approximate minimax decorrelating detector and linear decorrelating detector in a synchronous CDMA channel with impulsive noise. N = 31, K = 20, = 0.01 k = 100. All users have the same amplitudes.  In the third example we consider the performance of the approximate robust decorrelator in Gaussian noise. Shown in Fig. 4.9 are the BER curves for the robust decorrelator and the linear decorrelator in a six-user system ( K = 6). It is seen that there is only a very slight performance degradation by the robust decorrelator in Gaussian channels, relative to the linear decorrelator, which is the optimal decorrelating detector in Gaussian noise. By comparing the BER curves of the robust decorrelator in Figs. 4.7 and 4.9, it is seen that the robust detector performs better in impulsive noise than in Gaussian noise with the same noise variance. This is because in an impulsive environment, a portion of the total noise variance is due to impulses, which have large amplitudes. Such impulses are clipped by the nonlinearity in the detector. Therefore, the effective noise variance at the output of the robust detector is smaller than the input total noise variance. In fact, the asymptotic performance gain by the robust detector in impulsive noise over Gaussian noise is quantified by the asymptotic variance u 2 in (4.47) [cf. Figs. 4.2, 4.3, and 4.4]. Figure 4.9. BER performance of user 1 for a robust decorrelating detector and linear decorrelating detector in a synchronous CDMA channel with Gaussian noise. N = 31, K = 6. The powers of the interferers are 10 dB above the power of user 1.  In summary, we have seen that the performance of the linear decorrelating detector degrades substantially when the distribution of the ambient channel noise deviates even slightly from Gaussian. By using the robust decorrelating detector, such performance loss is prevented and this detector thus offers significant performance gains over the linear detectors, which translates into an increase in capacity in multiple-access channels. On the other hand, even when the ambient noise distribution is indeed Gaussian, the robust detector incurs only negligible performance loss relative to the linear detectors. A number of other techniques have been proposed in the literature to combat impulsive ambient noise in multiple-access channels. These include adaptive receivers with certain nonlinearities [27, 28], a neural network approach [81], maximum- likelihood methods based on the expectation-maximization (EM) algorithm [47, 236, 607], a Bayesian approach based on the Markov chain Monte Carlo technique [540], and extensions to fading channels [382, 384]. | 
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