The frequency spectrum is heavily regulated internationally, so a common strategy among current cellular systems is to allow multiple users to share the same frequency band at the same time. Therefore, the signal of each desired user is corrupted by the interference originated from the other users in the system. This interference is known as multiple access interference (MAI) and has to be properly mitigated at the receiver to ensure a reliable transmission. In addition to the MAI, the receivers also have to cope with many sources of noise. Today's commercial wide area receivers treat noise and MAI as an indivisible category. The worsening of the MAI is highly expected because of the trend of moving from voice-oriented to high-rate multimedia networks, which require a high occupancy of the channel. Hence, receivers of future networks have to be able to treat the MAI with efficient signal processing algorithms. An ideal receiver, i.e., a receiver that achieves the minimum probability of bit error when MAI is present, has been widely studied in the past 20 years. Unfortunately, it has not been implemented in real systems because of the high complexity of the algorithm, among other problems. Computationally complex algorithms are especially troublesome in mobile receivers owing to the severe power limitations of such devices. Thus, there is a strong demand for low-complexity receivers that provide better performance than those of current commercial systems. Low-complexity receivers have been the topic of my research. In particular, I have studied receivers based on linear filters, which are very attractive from a complexity point of view. In these receivers, as the name implies, MAI is reduced without unduly amplifying the noise by filtering the input signal, and the data can be recovered from the interference-reduced filter output. The desired filter, also called optimal filter in the following, is defined as the one that minimizes or maximizes a tractable cost function closely related to the bit error rate such as the mean-square error (MSE). (Filters reducing the MSE decrease the bit error rate in general.) Given the time-varying characteristics of wireless channels, the optimal filter changes rapidly, so we often use adaptive algorithms to update the filters. We can divide these algorithms into two families: nonblind algorithms and blind algorithms. (The receivers are also termed blind or nonblind according to the algorithm being used.) Nonblind algorithms require a short period that both the transmitter and receiver know the data being transmitted (a.k.a. training sequence); no useful information is sent during this period. They are usually able to provide fairly good estimates of the optimal filter with low complexity, but the throughput is compromised. In contrast, blind algorithms are able to update the filter by processing the useful data information, so no time is wasted with training sequences. They usually trade convergence rate (how fast a reliable estimate of the optimal filter can be found), steady-state performance (how close the estimates are from the optimal filter), and/or computational complexity for increased throughput. To achieve fast convergence rate and good steady-state performance, we have investigated blind receivers based on the adaptive projected subgradient method (APSM), an efficient method for minimizing asymptotically a sequence of convex cost functions over a closed convex set [1],[2]. In the proposed method, the APSM has been used to approximate the filter minimizing the MSE iteratively as a point in the intersection of a time-varying family of closed convex sets. These sets restrict the possible candidates of the desired filter and are defined with measurements of the MAI-contaminated input signal and with (technology-dependent) a priori knowledge. We have considered two technologies: direct sequence-code division multiple access (DS-CDMA) systems [3-6] and multi-user multiple-input multiple-output (MIMO) systems [7]. In the initial study in [3] for DS-CDMA systems, the sets are defined with a priori knowledge of the spreading code (which is used to distinguish each user in the channel), multiple measures of the input signal, and an estimate of the desired user's signal amplitude. All this information required to build the sets of candidate estimates of the optimal filter are readily available at the receiver side. The blind algorithm in [3] is able to achieve similar convergence rate and steady-state performance than those of more computationally complex nonblind schemes. Compared with existing blind algorithms, the proposed algorithm in [3] has both better convergence rate and steady-state performance. By exploiting geometrical properties of the proposed cost function in [3] and by modifying the sets based on the measurements of the input signal, we showed in the IEICE award-winning paper [4] that achieving remarkably good convergence speed with a moderate number of measurements of the input signal is also possible. The results in [3] and [4] were extended in [5] to cope with a variety of scenarios. The main contribution of the work in [5] is that the receiver can achieve good bit error rate performance when the users move with high speed or when a very large number of users are present in the system. The algorithms in [3-5] require that the designer of the system choose a good value for a parameter called "step size", which modifies the convergence characteristics of the algorithm. Usually, increasing the step size improves the convergence speed, but decreases the steady-state performance. ?Hence, to study this trade-off, we derived a closed-form steady-state equation in [6]. The analysis has also proved that an estimate of the desired user's amplitude greatly improves the steady-state performance. More recently, the results in [3-6] have been extended in [7] to MIMO systems, one of the most promising technologies for future wireless systems. In this study, users sharing the same frequency at the same time are distinguished by their channels, which is different for each user owing to their different locations. Practical algorithms cannot estimate the channels precisely, so one of the main contributions of [7] is the development of algorithms robust against channel estimation errors.

References
[1] I. Yamada, “Adaptive projected subgradient method -- a unified view for projection based adaptive algorithms,” J. IEICE, vol. 86, no. 8, pp. 654-658, Aug. 2003.
[2] I. Yamada and N. Ogura, “Adaptive projected subgradient method for asymptotic minimization of sequence of nonnegative convex functions,” Numerical Functional Analysis and Optimization, vol. 25, no. 7/8, pp. 593-617, May 2004.
[3] R. L. G. Cavalcante, I. Yamada, and K. Sakaniwa, "A fast blind multiple access interference reduction in DS/CDMA systems based on adaptive projected subgradient method,"' IEICE Trans. Fundamentals, vol. E87-A, no. 8, pp. 1973-1980, Aug. 2004.
[4] M. Yukawa, R. L. G. Cavalcante, and I. Yamada, "Efficient blind MAI suppression in DS/CDMA systems by embedded constraint parallel projection techniques," IEICE Trans. Fundamentals, vol. E88-A, no. 8, pp. 2062--2071, Aug. 2005.
[5] R. L. G. Cavalcante, M. Yukawa, and I. Yamada, "Set-theoretic DS/CDMA receivers for fading channels by adaptive projected subgradient method," in Proc. IEEE GLOBECOM, Dec. 2005
[6] R. L. G. Cavalcante and I. Yamada, "Steady-state analysis of constrained normalized adaptive filters for MAI reduction by energy conservation arguments," Signal Processing, Elsevier, accepted for publication
[7] R. L. G. Cavalcante and I. Yamada, "Multiaccess interference suppression in orthogonal space-time block coded MIMO systems by adaptive projected subgradient method," IEEE Trans. Signal Processing, in press.

IEICE Trans. Fundamentals 2004/08, 2005/08