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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 3, NO. 1, JANUARY 2004
The Application of Nonlinear Programming for Multiuser Detection in CDMA Liu Hongwei, Wang Xinhui, and Liu Sanyang
Abstract—In this paper, a heuristic algorithm based on a nonlinear nonconvex programming relaxation of the CDMA maximum likelihood (ML) problem is presented. Simulation results have shown that the BER performances of a detection strategy based on the heuristic algorithm are similar to that of the detection strategy based on the semidefinite relaxation. Furthermore, average CPU time of the heuristic algorithm is significantly lower than that of the randomized rounding algorithm based on a semidefinite relaxation. This approach provides good approximations to the ML performance. Index Terms—Code division multiple access, heuristic algorithm, multiuser detection.
I. INTRODUCTION
I
N A code-division multiple-access (CDMA) system, users are assigned unique signature waveforms that are used to modulate their transmitted symbols. It is, however, not possible to ensure orthogonality among received signature waveforms in a mobile environment, and thus, multiple access interference arises. Multiuser detection [1] plays an important role in suppressing the performance degrading effect of multiuser interusers synchronous CDMA system with ference. Consider a additive white Gaussian noise (AWGN) of variance
Each user transmits data using BPSK signaling and spreading. Without loss of generality, we assume that all signature waveforms have unit energy. A minimal set of sufficient statistics of is obtained through matched filtering of the redimension ceived spreading code of the desired user , where is the matched filter output vector, is the spreading code, is the correlation matrix and is the zero-mean Gaussian noise vector . The optimum ML detector sewith autocorrelation matrix lects the maximum likelihood hypothesis given the matched filter output. Since we are considering an AWGN channel, the is described as negative log-likelihood function based on . The binary constrained maximum likelihood (ML) problem is then described as [2]
complexity. It is known that the polynomial-time algorithms of the problem (1) exist if the autocorrelation matrix exhibits some special structure. However, in general case, it is an NP-hard problem [1]. Because of intrinsic difficulty in solving the detection problem (1), there has been much interest in the development of suboptimal but computationally efficient ML detector. A tree search method [3] has been proposed to perform an incomplete search for a solution to the problem (1) with limited complexity. The coordinate ascent algorithm [4] has also been proposed to solve this problem. But the performance of coordinate ascent algorithm strongly depends on the initialization. In [2], [5], a detection strategy based on a semidefinite relaxation of the CDMA maximum likelihood (ML) problem is investigated. The simulated bit error rate performance demonstrates that the semidefinite relaxation approach provides a good approximation to the ML performance. However, the semidefinite relaxation encounters difficulty in practice because the cost of solving semidefinite programming goes up quickly as the size of the problem increases. In this paper, we propose a method to seek a suboptimal solution to the ML detection problem by using a nonlinear nonconvex program. The paper is organized as follows. In Section II, a nonlinear programming relaxation for ML detection problem is presented. In Section III, a heuristic algorithm for the ML detection problem is developed. The simulation results and conclusion are found in Sections IV and V, respectively. II. NONLINEAR PROGRAMMING RELAXATION FOR ML DETECTION PROBLEM The ML detection problem (1) can be reformulated as in [2]
where
and
Let , and
(1)
. Let be a function
defined as The problem (1) can be solved by an exhaustive search, however, the exhaustive search is prohibitive for large number of users because of its exponentially increasing computational Manuscript received January 6, 2002; revised November 25, 2002; accepted January 8, 2003. The editor coordinating the review of this paper and approving it for publication is W.-Y. Kuo. This work was supported by the National Science Foundation under Grant 69972036 and by the Shaanxi Province National Science Foundation under Grant 2001SL05. The authors are with the Department of Applied Mathematics, Xidian University, Xi’an 710071, China. (e-mail:
[email protected];
[email protected]) Digital Object Identifier 10.1109/TWC.2003.821183
When
or
for every
Then we obtain the following relaxation for the ML detection problem
1536-1276/04$20.00 © 2004 IEEE
(2)
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 3, NO. 1, JANUARY 2004
This is an unconstrained optimization problem with a nonconvex objective function [6]. can be easily computed. The derivatives of the function Indeed, the first-order partial derivatives of the function are given by
The second-order partial derivatives of the function given by
are
if if When
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While 1. Generate feasible point
of (1) by
if otherwise. for every , and compute 2. If , then , and . , let and increase 3. If and increase 1; otherwise let 1. End While 4. Compute
. 5. Let End
or
and
,
We know that, for all Let
by by
, and
Then is a stationary point of the function . However, the following theorem shows that only minimum points of the ML .(A detection problem (1) possibly be local minimum of point called local minimal point if the objective function is lower than that of other in a region of this point.) Theorem: Let and . If is a positive semidefinite matrix, then , in particular, If is a local minimum point of , then is a minimum point of the ML detection problem and an eigenvector (1). Otherwise, is a decent corresponding to the minimal eigenvalue at . (The detailed proof is found in the direction of Appendix) Since nonminimum feasible point of the ML detection , a good problem (1) cannot be local minimum point of minimization algorithm would not be attracted to stationary points corresponding to nonminimum feasible point of the ML detection problem (1). According to this fact, we construct an algorithm as follows. III. A HEURISTIC ALGORITHM FOR THE ML DETECTION PROBLEM In order to produce a suboptimal solution to the ML detecand give a tion problem, we first minimize the function suboptimal solution to the ML detection problem by the following Algorithm-1. Using periodicity, we may easily assume for each . Without loss of generthat ality, we assume that
after a reordering if necessary. Algorithm-1 (Input , Output ): Let Let be the smallest index such that . Set . otherwise let
is a stationary point—most likely a saddle point—of the func, but not a local minimal point unless it is already a tion minimum point of the ML detection problem. Based on the thefrom a new initial orem, we can restart the minimization point and continue this process until further improvement seems unlikely. We state this heuristic algorithm as following : let Algorithm-2 (Input , Output ): Given and . Let be a very small positive number. While 1. Starting from , minimize to get . 2. By Algorithm 1 compute associated with . . Compute 3. Let , and eigenvector corresponding to . , let , ; If and , let , If and , compute
Let End. While 4. Compute
5. Let End
. Otherwise
, and
.
.
IV. SIMULATION RESULTS ,
, . if there is one;
In this section, we first report numerical results on the biterror-rate (BER) performance of the detector based on heuristic and randomized rounding algorithm algorithms with
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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 3, NO. 1, JANUARY 2004
Fig. 1. Frame structure for DS-CDMA and CIBS-CDMA systems.
Fig. 3. Transceiver model of CIBS-CDMA.
significantly lower than that of randomized rounding algorithms based on the semidefinite relaxation. V. CONCLUSION In this paper, the nonlinear programming relaxation method approximately is applied to solve the NP-hard multiuser detection problem. Simulation results have shown that the approach provides a good approximation to the ML performance. Furthermore, for large , the CPU time of the heuristic algorithms is significantly lower than that of randomized rounding algorithm based on the semidefinite relaxation. APPENDIX Fig. 2.
Transceiver model of DS-CDMA using chip equalizers.
based on the semidefinite relaxation [7] in Fig. 1. A synchronous CDMA system with length-63 Gold codes is used. Two different , 50 are considered. The simulation rescenarios with sults show that the BER of heuristic algorithm is approaching , 50 and no appreciable perthat of the single user for formance difference between the detection strategy based on the heuristic algorithm and that on the semidefinite relaxation. Secondly, we illustrate near-far resistance. The case is considered that all but one user have the same signal-to-noise ratio (SNR). The user stays at a fixed SNR. In Fig. 2, the BER of the dB is shown against the ratio of first user with the strength of the interfering user’s signals to the first user’s signals strength (SNR(i)-SNR(1) in dB). The simulation results show that the detection strategy based on the heuristic algorithm and that on the semidefinite relaxation have the same BER performance also. Finally, we use simulations to evaluate the computational time of heuristic algorithms and randomized rounding algorithm based on the semidefinite relaxation. The simulation is run in the MATLAB 5.3 environment on a 450-MHz Pentium personal computer with 128 Mb of Ram. We use interior point algorithm to solve the SDP relaxation problem. The result is shown in Fig. 3. Clearly, for large , the CPU time of the heuristic algorithms is
The proof of theorem in paper is described as the following. , Proof: For all for every let
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 3, NO. 1, JANUARY 2004
Where . If
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is the Hessian of the function is a positive semidefinite matrices, and hence Since we have
Noting that for all , the . In particular, the is above inequality holds, then , the is a positive semidefinite a local minimum point of matrix, then is a minimum point of the ML detection problem (1). is not a positive semidefinite matrix, since If the and or 2 (for all ), takes the place of if the numbers of nonzero components of the are larger than , we have
Hence, second order Tailor’s formula of
. Now, we consider the .
Where is real, and is infintesmall real number than of . Let be an eigenvector corresponding to the minimal eigenvalue
when the absolute value of
is very small,
Then an eigenvector corresponding to the minimal eigenvalue is a decent direction of at .
REFERENCES [1] S. Verdu, Multiuser Detection. Cambridge, MA: Cambridge Univ. Press, 1998. [2] P. Huitian and L. K. Rasmussen, “The application of semidefinite programming for detection CDMA,” IEEE Select. Areas Commun., vol. 19, pp. 1442–1449, Aug. 2001. [3] L. Wei, L. K. Rasmussen, and R. Wyrwas, “Near optimum tree-search detection schemes for bit-synchronous multiuser CDMA system over Gaussian and two-path Rayleigh-fading channels,” IEEE Trans. Commun., vol. 39, pp. 725–736, May 1991. [4] Sharfer and A. O. Hero III, “A maximum likelihood digital receiver using coordinate ascent and the discrete wavelet transform,” IEEE Trans. Signal Processing, vol. 47, pp. : 813–825, Mar. 1999. [5] W. K. Ma, T. N. Davidson, K. M. Wong, Z. Q. Luo, and P. C. Ching, “Quasi-maximum-likelihood multiuser detection using semidefinite relaxation,” IEEE Trans. Signal Processing, vol. 50, pp. 912–922, Apr. 2002. [6] S. Burer, R. D. C. Monterio, and Y, Zhang, “Rank-two relaxation heuristic for max-cut and other binary quadratic programs,” SIAM J. Optimization, vol. 12, pp. 503–521, 2002. [7] C. Helmberg and F. Rendl, “An interior-point method for semidefinite programming,” SIAM J. Optimization, vol. 6, no. 2, pp. 342–361, 1996.
