8 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 3, NO. 1, JANUARY 2004

The Application of Nonlinear Programming for Multiuser Detection in CDMALiu Hongwei, Wang Xinhui, and Liu Sanyang

Abstract—In this paper, a heuristic algorithm based on anonlinear nonconvex programming relaxation of the CDMAmaximum likelihood (ML) problem is presented. Simulationresults have shown that the BER performances of a detectionstrategy based on the heuristic algorithm are similar to thatof the detection strategy based on the semidefinite relaxation.Furthermore, average CPU time of the heuristic algorithm is sig-nificantly lower than that of the randomized rounding algorithmbased on a semidefinite relaxation. This approach provides goodapproximations to the ML performance.

Index Terms—Code division multiple access, heuristic algo-rithm, multiuser detection.

I. INTRODUCTION

I N A code-division multiple-access (CDMA) system, usersare assigned unique signature waveforms that are used to

modulate their transmitted symbols. It is, however, not possibleto ensure orthogonality among received signature waveforms ina mobile environment, and thus, multiple access interferencearises. Multiuser detection [1] plays an important role in sup-pressing the performance degrading effect of multiuser inter-ference. Consider a users synchronous CDMA system withadditive white Gaussian noise (AWGN) of variance

Each user transmits data using BPSK signaling and spreading.Without loss of generality, we assume that all signature wave-forms have unit energy. A minimal set of sufficient statistics ofdimension is obtained through matched filtering of the re-ceived spreading code of the desired user , where isthe matched filter output vector, is the spreading code, is thecorrelation matrix and is the zero-mean Gaussian noise vectorwith autocorrelation matrix . The optimum ML detector se-lects the maximum likelihood hypothesis given the matchedfilter output. Since we are considering an AWGN channel, thenegative log-likelihood function based on is described as

. The binary constrained maximum likeli-hood (ML) problem is then described as [2]

(1)

The problem (1) can be solved by an exhaustive search,however, the exhaustive search is prohibitive for large numberof users because of its exponentially increasing computational

Manuscript received January 6, 2002; revised November 25, 2002; acceptedJanuary 8, 2003. The editor coordinating the review of this paper and approvingit for publication is W.-Y. Kuo. This work was supported by the National Sci-ence Foundation under Grant 69972036 and by the Shaanxi Province NationalScience Foundation under Grant 2001SL05.

The authors are with the Department of Applied Mathematics, XidianUniversity, Xi’an 710071, China. (e-mail: xdliuhongwei@hotmail.com;xinhu_wang@hotmail.com)

Digital Object Identifier 10.1109/TWC.2003.821183

complexity. It is known that the polynomial-time algorithms ofthe problem (1) exist if the autocorrelation matrix exhibits somespecial structure. However, in general case, it is an NP-hardproblem [1].

Because of intrinsic difficulty in solving the detectionproblem (1), there has been much interest in the developmentof suboptimal but computationally efficient ML detector. A treesearch method [3] has been proposed to perform an incompletesearch for a solution to the problem (1) with limited complexity.The coordinate ascent algorithm [4] has also been proposed tosolve this problem. But the performance of coordinate ascentalgorithm strongly depends on the initialization. In [2], [5],a detection strategy based on a semidefinite relaxation of theCDMA maximum likelihood (ML) problem is investigated.The simulated bit error rate performance demonstrates thatthe semidefinite relaxation approach provides a good approx-imation to the ML performance. However, the semidefiniterelaxation encounters difficulty in practice because the cost ofsolving semidefinite programming goes up quickly as the sizeof the problem increases.

In this paper, we propose a method to seek a suboptimal so-lution to the ML detection problem by using a nonlinear non-convex program. The paper is organized as follows. In Sec-tion II, a nonlinear programming relaxation for ML detectionproblem is presented. In Section III, a heuristic algorithm forthe ML detection problem is developed. The simulation resultsand conclusion are found in Sections IV and V, respectively.

II. NONLINEAR PROGRAMMING RELAXATION FOR MLDETECTION PROBLEM

The ML detection problem (1) can be reformulated as in [2]

where and

Let . Let, and be a function

defined as

When or for every

Then we obtain the following relaxation for the ML detectionproblem

(2)

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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 3, NO. 1, JANUARY 2004 9

This is an unconstrained optimization problem with a non-convex objective function [6].

The derivatives of the function can be easily computed.Indeed, the first-order partial derivatives of the function aregiven by

The second-order partial derivatives of the function aregiven by

if

if .

When or

Let , and

Then is a stationary point of the function . However, thefollowing theorem shows that only minimum points of the MLdetection problem (1) possibly be local minimum of .(Apoint called local minimal point if the objective function is lowerthan 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

(1). Otherwise, and an eigenvectorcorresponding to the minimal eigenvalue is a decentdirection of at . (The detailed proof is found in theAppendix)

Since nonminimum feasible point of the ML detectionproblem (1) cannot be local minimum point of , a goodminimization algorithm would not be attracted to stationarypoints corresponding to nonminimum feasible point of the MLdetection problem (1). According to this fact, we construct analgorithm as follows.

III. A HEURISTIC ALGORITHM FOR THE ML DETECTION

PROBLEM

In order to produce a suboptimal solution to the ML detec-tion problem, we first minimize the function and give asuboptimal solution to the ML detection problem by the fol-lowing Algorithm-1. Using periodicity, we may easily assumethat for each . Without loss of gener-ality, we assume that

after a reordering if necessary.Algorithm-1 (Input , Output ): Let , , .

Let be the smallest index such that if there is one;otherwise let . Set .

While1. Generate feasible point of (1) by

ifotherwise.

for every , and compute2. If , then , and .3. If , let and increase by1; otherwise let and increase by1.EndWhile4. Compute

5. Let andEnd

We know that, for all ,

is a stationary point—most likely a saddle point—of the func-tion , but not a local minimal point unless it is already aminimum point of the ML detection problem. Based on the the-orem, we can restart the minimization from a new initialpoint and continue this process until further improvement seemsunlikely. We state this heuristic algorithm as following :

Algorithm-2 (Input , Output ): Given letand . Let be a very small positive number.

While1. Starting from , minimize to get .2. By Algorithm 1 compute associatedwith .3. Let . Compute

, and eigenvectorcorresponding to .

If , let , ;If and , let ,and ,

compute

Let . Otherwise .End.While4. Compute

5. Let , and .End

IV. SIMULATION RESULTS

In this section, we first report numerical results on the bit-error-rate (BER) performance of the detector based on heuristicalgorithms with and randomized rounding algorithm

10 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 3, NO. 1, JANUARY 2004

Fig. 1. Frame structure for DS-CDMA and CIBS-CDMA systems.

Fig. 2. Transceiver model of DS-CDMA using chip equalizers.

based on the semidefinite relaxation [7] in Fig. 1. A synchronousCDMA system with length-63 Gold codes is used. Two differentscenarios with , 50 are considered. The simulation re-sults show that the BER of heuristic algorithm is approachingthat of the single user for , 50 and no appreciable per-formance difference between the detection strategy based on theheuristic algorithm and that on the semidefinite relaxation.

Secondly, we illustrate near-far resistance. The case is con-sidered 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 thefirst user with dB is shown against the ratio ofthe strength of the interfering user’s signals to the first user’ssignals strength (SNR(i)-SNR(1) in dB). The simulation resultsshow that the detection strategy based on the heuristic algorithmand that on the semidefinite relaxation have the same BER per-formance also.

Finally, we use simulations to evaluate the computationaltime of heuristic algorithms and randomized rounding algorithmbased on the semidefinite relaxation. The simulation is run inthe MATLAB 5.3 environment on a 450-MHz Pentium personalcomputer with 128 Mb of Ram. We use interior point algorithm tosolve the SDP relaxation problem. The result is shown in Fig. 3.Clearly, for large , the CPU time of the heuristic algorithms is

Fig. 3. Transceiver model of CIBS-CDMA.

significantly lower than that of randomized rounding algorithmsbased on the semidefinite relaxation.

V. CONCLUSION

In this paper, the nonlinear programming relaxation methodapproximately is applied to solve the NP-hard multiuser detec-tion problem. Simulation results have shown that the approachprovides a good approximation to the ML performance. Further-more, for large , the CPU time of the heuristic algorithms issignificantly lower than that of randomized rounding algorithmbased on the semidefinite relaxation.

APPENDIX

The proof of theorem in paper is described as the following.Proof: For all ,

let for every

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Where is the Hessian of the function. If is a positive semidefinite matrices, and hence

Noting that for all , theabove inequality holds, then . In particular, the isa local minimum point of , the is a positive semidefinitematrix, then is a minimum point of the ML detection problem(1).

If the is not a positive semidefinite matrix, sinceand or 2 (for all

), takes the place of if the num-bers of nonzero components of the are larger than , wehave

Hence, . Now, we consider thesecond order Tailor’s formula of .

Where is real, and is infintesmall real numberthan of . Let be an eigenvector corresponding to the minimaleigenvalue

Since when the absolute value of is very small,we have

Then an eigenvector corresponding to the minimal eigenvalueis 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 pro-gramming 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-searchdetection schemes for bit-synchronous multiuser CDMA systemover 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 receiverusing 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 re-laxation,” IEEE Trans. Signal Processing, vol. 50, pp. 912–922, Apr.2002.

[6] S. Burer, R. D. C. Monterio, and Y, Zhang, “Rank-two relaxationheuristic 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 semidefiniteprogramming,” SIAM J. Optimization, vol. 6, no. 2, pp. 342–361, 1996.