1736 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 16, NO. 9, DECEMBER 1998 Recursive Structures and Finite Impulse Response Implementations of Linear Multiuser Detectors for an Asynchronous CDMA System Sathyadev V. Uppala and John D. Sahr Abstract— We present order-recursive structures for imple- menting the linear decorrelating and linear minimum mean square error (MMSE) detectors in a computationally efficient manner. These infinite memory length, linear time invariant, noncausal systems can be approximated by implementing them as -input -output finite impulse response (FIR) filters. We developed a linear multiuser receiver architecture called a recur- sive linear multiuser detector which has lower computational and memory requirements then an equivalent tap delay line FIR filter implementation. We establish the tradeoff between window length and bit error rate and find that relatively small window size yields performance indistinguishable from the ideal decorrelating detector and the ideal MMSE detector. Index Terms—Cyclic reduction, FIR filters, iterative methods, linear decorrelating detector, MMSE detector, multiuser detec- tion, recursive algorithms. I. INTRODUCTION W HEN several users asynchronously transmit messages in a code division multiple access (CDMA) system, the signals of the users interfere with one another. This resulting interference is called multiple access interference (MAI). Of the several multiuser detectors known [1], we examine implementation issues for two, the linear decorre- lating detector and the MMSE detector. Both these detectors belong to the class of linear multiuser detectors. The linear decorrelating detector (LDD) proposed by Lupas and Verdu [2] eliminates the MAI and is the optimal linear detector in the generalized maximum likelihood sense. The minimum mean square error detector (MMSE) was introduced by Xie et al. [3] and minimizes the variance of the MAI and noise together. The MMSE detector reduces to the LDD when the noise variance is zero and has better performance than the LDD in the presence of noise [4]. Neither of these receivers can be implemented in practice because all the users’ messages must be decoded en masse—leading to a huge computational burden, and the decoding cannot be started until all the messages are complete—resulting in a large delay. Manuscript received August 29, 1997; revised April 10, 1998. This paper was presented in part in the 35th Annual Allerton Conference on Communi- cation, Control and Computing, September 1997. S. V. Uppala is with Bell Laboratories, Lucent Technologies, Murray Hill, NJ 07974 USA. J. D. Sahr is with the Department of Electrical Engineering, University of Washington, Seattle, WA 98195-2500 USA. Publisher Item Identifier S 0733-8716(98)08643-0. Xie et al. [3] have exploited the structure inherent in the implied equations and have reduced the computational burden. Nevertheless, the time delay and computational burden of the LDD and the MMSE detectors are still very large, motivating the development of several suboptimal linear receiver archi- tectures. Moshavi et al. [5] have proposed the “polynomial expansion detector” which reduces the computational burden in inverting the correlation matrix by approximating it as a polynomial of the correlation matrix. This approach cannot be extended for decoding infinite length messages. Another approach used involves subdividing the matched filter outputs into manageable blocks and processing the blocks while ac- counting for the edge effects in various ways [6], [3]. These approaches, while applicable for infinite length messages, lead to nonlinear suboptimal receiver structures. Another approach requires all the users to modify their transmitted bit sequences by introducing an isolation bit periodically so that the re- ceiver can exploit the structure introduced [7]. This requires a modification of the transmission protocol and a decrease in bandwidth efficiency. The “one shot decorrelating detector” was proposed in [8] and [9] as an approximation to the decorrelating detector and requires the correlation matrix to be strongly diagonal. When the bit sequences are infinite length, Lupas and Verdu [2] have shown that the LDD may be implemented as a -input, -output, noncausal, linear time invariant, infinite memory length system. This system has a doubly infinite length impulse response. Since for a stable system the impulse response as , truncating the impulse response leads to an finite impulse response (FIR) filter approximation to the LDD. The MMSE detector is also amenable to a similar analysis and an FIR implementation. This idea was used by Juntti and Aazhang [10] and they have proposed the truncated LDD and MMSE detector with a tap delay line, finite memory length implementation. In this paper we introduce an efficient receiver structure which also truncates the impulse response for infinite length messages. The proposed receiver structure relies upon an iterative method for solving block tridiagonal systems of equations known as the “incomplete cyclic odd–even re- duction technique” [11]. We call the result the recursive linear multiuser detector (RLMD). The RLMD has three principal strengths: it is computationally efficient compared to an equivalent tap delay line FIR filter implementation; it has an order-recursive structure; and it provides a conve- 0733–8716/98$10.00  1998 IEEE