This work was supported in part by Telstra Australia under Grant 7368, and by the Commonwealth of Australia under International. S&T Grant 56. C. Schlegel is ...
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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 45, NO. 5, MAY 1997
A Simple Way to Compute the Minimum Distance in Multiuser CDMA Systems Christian Schlegel and Lei Wei
Abstract—A simple method to compute the minimum distance in multiuser code division multiple access (CDMA) systems based on the Cholesky decomposition of the positive-definite symmetric correlation matrix is proposed. Although finding min is known to be NP-hard, this decomposition allows the computation of the minimum distance and the asymptotic efficiency of optimum multiuser detection to be performed very efficiently in almost all cases of practical interest. Numerical results for synchronous CDMA with binary random signature waveforms of length 31 are used to illustrate the method.
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I. INTRODUCTION
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ECENTLY joint multiuser detection in which the multiuser interference is treated as a part of the information, rather than noise, has attracted much attention [1]–[8]. The optimal detector proposed by Verd´u achieves optimum near–far resistance and a significant performance improvement over the conventional detector [1], [2]. These improvements, however, are obtained at the expense of a dramatic increase in computational complexity, which grows exponentially with the number of users. Similarly, reliable performance measures like the bit-error rate, the minimum Euclidean distance, or the asymptotic efficiency of optimum detection are equally difficult to calculate, and are essentially not known for large numbers of users. Currently, only the single-user bound is readily computed. This quandary makes it hard to gauge the performance of different suboptimum detectors. In this letter, we present a simple way to calculate the minimum distance for multiuser CDMA systems. This, in turn, then directly gives the asymptotic efficiency of optimum detection, which can be used as a benchmark for performance. The computation of the asymptotic efficiency of optimum multiuser detection is known to be an NP-hard problem [2]. However, Verd´u’s work, which, like most early work on multiuser detection, can be viewed as being based on the matched filter receiver. It is known, for the case where the receiver knows perfect channel state information, however, that suboptimum detectors can significantly improve performance if they are based on the whitening matched filter (WMF) instead [7], [8]. Our method to compute the minimum distance of the optimum multiuser detection is based on the Cholesky Paper approved by B. Aazhang, the Editor for Spread Spectrum Networks of the IEEE Communications Society. Manuscript received September 8, 1994; revised May 27, 1996. This work was supported in part by Telstra Australia under Grant 7368, and by the Commonwealth of Australia under International S&T Grant 56. C. Schlegel is with the Department of Electrical Engineering, University of Utah, Salt Lake City, UT 84112 USA. L. Wei is with the Department of Engineering, Faculty of Engineering and Information Technology, Australian National University, Canberra, ACT 0200, Australia. Publisher Item Identifier S 0090-6778(97)03722-7.
factorization of the positive-definite symmetric correlation matrix, also needed to derive the WMF. We demonstrate that the computation of the minimum distance (and hence the asymptotic efficiency) of the optimum multiuser detection, albeit NP-hard in general, can be performed very efficiently in almost all cases of practical interest. (Pathological cases with a very strong correlation between all users or very different power levels can be constructed for which the method fails to be significantly less complex than an exhaustive search.) In this letter, we will concentrate on synchronous CDMA; however, the method presented here can be generalized easily for asynchronous CDMA systems. In Section II, we present the system of synchronous CDMA with multiuser detection, the relationship between the minimum distance and the asymptotic efficiency, and give our method to calculate the minimum distance of optimum multiuser detection. We also show the new problem to be NP-hard. In Section III, we present numerical results, and in Section IV, we conclude with a summary. II. THE MINIMUM DISTANCE CALCULATION Consider a synchronous CDMA system with users and a set of unity energy preassigned signature waveforms of duration as shown in Fig. 1. At the receiver, a bank of filters matched to the set of preassigned waveforms is sampled at times The sample values
(1) form a sufficient statistic for optimum multiuser detection is vector [2], where of information bits, is the nonnegative-definite symmetric correlation matrix with entries and where is the energy per bit of user The superscript denotes matrix transposition. The noise vector in (1) is a Gaussian noise vector of dimension with the autocorrelation matrix where is the one-sided noise power spectral density of the zero-mean additive white Gaussian noise (AWGN) source and In a synchronous CDMA system, the relationship between and the matrix of discrete signature sequences is given by where
is the discrete signature sequence of the th user with and is the length of the sequences.
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Fig. 1. Baseband equivalent model of system.
To find the Euclidean minimum distance of user to search and minimize the quadratic form [2]:
we need (2)
From (2), the asymptotic efficiency of user i is readily found as [2] (3) If is a positive-definite symmetric matrix (nonsingular), there exists a unique lower triangular, nonsingular matrix such that (Cholesky decomposition [9]). Using this, we obtain (4)
Let be rewritten as
for
Then, (4) can
3) deactivate nodes with i.e., drop these paths from future consideration; 4) set if stop; otherwise, go to 2). Since the paths with are dropped, the number of tree branches searched by the above algorithm is significantly smaller than the number of branches of the full tree Proposition: The modified minimum distance calculation in (4) is NP-hard. Proof: In [2], it is shown that the original minimum distance calculation (2) is NP-hard by transforming the KNAPSACK problem (a known NP-hard problem) into the problem (2). The Cholesky decomposition can be performed in steps, and hence we can transform the original problem (2) into the modified problem (4) in polynomial time. Hence, we can transform the KNAPSACK problem into (4) in polynomial time. III. NUMERICAL RESULTS
(5) where
and since
is lower triangular, (6)
depends only on we can perform Since a tree search to evaluate the minimum value of expression (6). This tree branches into two new nodes at level and the nodes are labeled with The branch is labeled with which connects the two nodes and The node is labeled with the The key accumulated weight observation now is that only a small part of this tree needs to be searched in general since, due to distances will quickly accumulate and most nodes can be discarded from further consideration. From the single-user bound, we know that which can be used as a discard threshold. Now, we present the following simple algorithm to find the minimum distance of user : 1) initialize activate the root node; 2) compute for all active nodes;
In this section, we use the above algorithm to evaluate the minimum distance of synchronous CDMA with binary random signature waveforms of length 31. The procedure is as follows. First, a set of random signature waveforms of length 31 is generated for every bit duration and the correlation matrix is computed. Then we check whether the matrix is singular. If for any is singular and the WMF does not exist; hence, if the set is discarded and a new set of random waveforms is generated. The signature waveforms are used to spread the signal with at the transmitter and to detect the signal at the receiver. Let for i.e., we look at an equal power system. In Fig. 2, we show the widths of the active tree as a function of the search depth for the worst case found. Among the first five sets of random signature waveforms generated, the maximum width of the active tree is 48 479 and 147 for 31user and 20-user systems, respectively, which is significantly less than the total number of tree branches for and for In Figs. 3 and 4, we show a histogram of the minimum distances of the binary random signature waveform sets which have been tested for a system with 20 users and 31 users,
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Fig. 2.
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 45, NO. 5, MAY 1997
Widths of the survived tree branches as a function of the user index.
Fig. 4. Minimum distances for a 31-user system.
If
Fig. 3. Minimum distance for a 20-user system.
respectively. Interestingly, we found that among 100 sets of random waveforms tested, there were only five discrete values of the minimum distance (see figure). Fig. 4 shows the minimum distances for a full set of users, while Fig. 3 shows the case for a smaller number of users. Not surprisingly, the minimum distances in Fig. 3 concentrate more around the value 1.0. As an example of unequal powers, let us study the following matrix, (7), shown at the bottom of the page, particular which is generated by binary random signature waveforms of length 10.
for
for respectively. The maximum width of the search tree is 81. If and for the maximum width is 339 if we drop nodes with The maximum width is 51 if we drop nodes with For both cases, we have for respectively. If and for the maximum width is 75 if we drop the paths with We have that for respectively (i.e., we do not know the asymptotic efficiency of user 4 with this threshold). The maximum width is 1011 if we raise the threshold to and we find that for respectively. Obviously, when is large, the algorithm needs to go through more nodes. However, the complexity of finding the asymptotic efficiency is still small. IV. DISCUSSION
AND
CONCLUSION
In [2], Verd´u concluded that “ the worst-case complexity measure may be overly pessimistic for specific instances exhibiting low cross-correlations.” The application of the Cholesky decomposition allows us to extend this conclusion to almost all cases, in particular, all cases of practical importance, excluding only such pathological cases with very high cross correlations among all users or very different power levels. Numerical results have been presented for synchronous
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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 45, NO. 5, MAY 1997
CDMA with randomly generated binary signature waveforms of length 31. They show that the maximum width of the active tree of the search algorithm is significantly smaller than the total number of tree branches.
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REFERENCES [1] S. Verd´u, “Minimum probability of error for asynchronous Gaussian multiple-access channels,” IEEE Trans. Inform. Theory, vol. IT-32, pp. 85–96, Jan. 1986. , “Optimum, multiuser asymptotic efficiency,” IEEE Trans. Com[2] mun., vol. COM-34, pp. 890–897, Sept. 1986. [3] R. Lupas and S. Verd´u, “Linear multiuser detectors for synchronous code-division multiple-access channels,” IEEE Trans. Inform. Theory, vol. 35, pp. 123–136, Jan. 1989.
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, “Near-far resistance of multiuser detectors in asynchronous channels,” IEEE Trans. Commun., vol. 38, pp. 496–508, Apr. 1990. M. K. Varanasi and B. Aazhang, “Multistage detection in asynchronous code-division muliple-access communications,” IEEE Trans. Commun., vol. 38, pp. 509–519, Apr. 1990. , “Near-optimum detection in synchronous code-division mulipleaccess systems,” IEEE Trans. Commun., vol. 39, pp. 725–736, May 1991. A. Duel-Hallen, “Decorrelating decision-feedback multiuser detector for synchronous code-division multiple-access channel,” IEEE Trans. Commun., vol. 41, pp. 285–290, Feb. 1993. L. Wei and C. Schlegel, “Synchronous DS-SSMA with improved decorrelating decision-feedback multiuser detector,” IEEE Trans. Veh. Technol. (Special Issue on PCN), pp. 767–772, Aug. 1994. G. W. Stewart, Introduction to Matrix Computations. New York: Academic, 1973.
