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Semidefinite Relaxation Based Multiuser Detection for -Ary PSK Multiuser Systems
M
Wing-Kin Ma, Member, IEEE, Pak-Chung Ching, Senior Member, IEEE, and Zhi Ding, Fellow, IEEE
Abstract—Because of the powerful symbol error performance of multiuser maximum-likelihood (ML) detection, recently, there has been much interest in seeking effective ways of approximating multiuser ML detection (MLD) with affordable computational costs. It has been illustrated that for the synchronous code division multiple access (CDMA) scenario, the so-called semidefinite relaxation (SDR) algorithm can accurately and efficiently approximate multiuser MLD. This SDR-MLD algorithm, however, can only handle binary and quadratic phase shift keying (PSK) symbol constellations. In this sequel, we propose an extended SDR algorithm -ary PSK (MPSK) constellations. For the synfor MLD with chronous CDMA scenario, the proposed SDR algorithm provides an attractive polynomial-time complexity order of 3 5 , where is the number of users. Simulation results indicate that the proposed detector provides improved symbol error performance compared with several commonly used multiuser detectors. Index Terms—M-ary phase shift keying, maximum likelihood detection, multiuser detection, relaxation methods, semidefinite programming.
I. INTRODUCTION
I
N MULTIUSER communication systems such as code division multiple access (CDMA), multiuser detection plays an important role in suppressing the multiuser interference effects induced by multiple access. Extensive studies have indicated that multiuser detection shows good potential in enhancing system performance as well as increasing system capacity. There are various multiuser detection methods [1] providing different tradeoffs between symbol error performance and computational cost. In this work, our emphasis is placed on the maximum-likelihood (ML) multiuser detector [1]. The ML detector is optimal in the sense that it achieves the minimum error probability under the assumption of independent and identically distributed (i.i.d.) data symbols. The major obstacle of implementing the ML detector is that multiuser ML detection (MLD) is a computationally prohibitive optimization problem [2] in most cases. To deal with this computational difficulty, various computationally efficient approximation techniques for MLD have been studied [3]–[17]. Recently, it has been shown
[14]–[17] that for the synchronous CDMA scenario with binary phase shift keying (BPSK) or quadratic (QPSK), MLD can be accurately and efficiently approximated using the so-called semidefinite relaxation (SDR) algorithm [18], [19]. Extensive simulation results [14]–[17] have illustrated that the SDR-ML detector yields bit error performance close to that of exact MLD. Motivated by the potential of SDR, recently, there has been much interest in applying the SDR detection technique to more challenging communication scenarios. The extension of the SDR method to soft ML symbol decisions has been considered in [20]. Moreover, the application of SDR in the asynchronous CDMA case has been investigated in [21] and [22]. In this paper, our emphasis is placed on -ary PSK (MPSK) multiuser MLD, which is a problem that the previously developed SDR-MLD algorithm cannot handle (except for the special cases of 2-ary and 4-ary PSK symbol constellations). We will propose an extended SDR algorithm for general MPSK symbol constellations. An approximation advantage of the proposed detector over several existing suboptimal detectors, such as the decorrelator, will be discussed. For the synchronous CDMA scenario, the SDR-ML detector provides a polynomial-time , where represents the number complexity order of of users. It will be illustrated that for large number of users, the complexity order of the SDR-ML detector is substantially lower than that of the exact ML detector via exhaustive search. Simulation results will show that the SDR-ML detector exhibits improved symbol error performance compared with several commonly used suboptimal multiuser detectors including the decorrelator, the linear minimum mean squared error (LMMSE) detector, and the multistage interference-canceling detectors. II. PROBLEM STATEMENT In this section, we review the formulation of the ML multiuser detection problem. For ease of exposition of the ideas behind the proposed detector, the synchronous CDMA scenario will be focused. We will first describe the received CDMA signal model. Then, the ML detection problem for the synchronous CDMA scenario will be presented. A. Signal Model
Manuscript received February 14, 2003; revised October 7, 2003. This work was supported in part by a research grant from the Hong Kong Research Grant Council. The associate editor coordinating the review of this paper and approving it for publication was Dr. Martin Haardt. W.-K. Ma and P.-C. Ching are with the Department of Electronic Engineering, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong (e-mail:
[email protected]). Z. Ding is with the Department of Electrical and Computer Engineering, University of California, Davis, CA 95616 USA (e-mail:
[email protected]). Digital Object Identifier 10.1109/TSP.2004.834267
We consider a standard -user CDMA system [1] in which noise is assumed additive white Gaussian, and multipath fading is negligible. For such a system, the received signal over a length- data frame can be represented by
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(1)
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where
and . Here symbol sequence transmitted by th user; alphabet set from which are drawn; th-user spreading code waveform with unit energy; symbol interval; th-user transmission delay which equals 0 for the synchronous CDMA scenario; received signal energy of the th user; carrier phase of the th user; circular additive white Gaussian noise with zero mean and spectral density . In this paper, we focus on MPSK symbol constellations, i.e., (2)
denotes the alphabet size. In particular, we are interwhere , meaning that the special cases of ested in cases where and , respectively) will BPSK and QPSK (i.e., not be considered. Moreover, we assume that the spreading code waveforms are constructed using a rectangular chip function, are limited within . For the synchronous and thus, , the spreading CDMA scenario where code waveforms transmitted at different time instants do not have time overlap, i.e., there is no overlap between and for and for any . Hence, the received signal in (1) over any individual symbol interval can simply be expressed as
(3)
B. ML Multiuser Detection for Synchronous CDMA
The ML detection (MLD) problem in (5) is a computationally difficult combinatorial optimization problem [1], [2]. A straightforward way of finding the globally optimal MLD solution is to . evaluate the objective function in (5) for all possible However, such a “brute-force” search method requires operations, which is prohibitive for large number of users. Instead of considering finding the (globally) optimal MLD solution, we focus on using the SDR technique to efficiently approximate the MLD solution. This will be elaborated upon in the next section. III. RELAXED ML DETECTION BY SEMIDEFINITE PROGRAMMING SDR [18], [19] is an efficient approximation tool for quadratic programming problems, and it has been shown to provide good approximation accuracy in the application of MLD with BPSK [14]–[17] and QPSK [21] constellations. In this section, we will develop an extended SDR algorithm for the case of MLD with general MPSK constellations. Like most relaxation methods, SDR consists of three steps. 1) Relax the feasible set of the original problem in such a way that the relaxed problem is easier to solve. 2) Solve the relaxed problem. 3) Convert the relaxation solution to an approximate solution of the original problem. All these steps will be considered in the first two subsections. Then, in the last subsection, we will describe how the proposed SDR algorithm is applied to the MLD of synchronous CDMA multiuser signals. An important property of the resulting SDR-ML detector will also be examined. Table I gives a summarized procedure of implementing the proposed SDR-ML detector. A. SDR for a Combinatorial Quadratic Program
In multiuser detection, we consider detecting all user symbols and . Among simultaneously, given knowledge of the various multiuser detectors, the ML multiuser detector is optimal in the sense that it achieves the minimum error probability under the common assumption of i.i.d. data symbols [1]. To describe the ML multiuser detector for the synchronous CDMA . Define the received signal in (3), let th-user sampled matched filter output to be
Consider the following homogeneous quadratic program: (7) where , and the feasible set is the MPSK alphabet set in (2). It will be illustrated in Section III-C that the MPSK MLD problem in (5) can be reformulated to a form reminiscent of (7). To describe SDR, define a vector (8)
(4) where is the interval on which the signals are defined, and . The ML detector for the received let signal in (3) is shown to be [1] Re where Diag signal correlation matrix with
(5) , and
is the
th entry (6)
, where denotes for some arbitrary vector the 2-norm. It is easy to verify that (7) is equivalent to (9a) s.t.
(9b) (9c) (9d)
Both (7) and (9) are difficult to solve. For (9), the difficulty lies are restricted to be spanned by the same column in that all vector , with the coefficients lying in the alphabet set. Since
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TABLE I SUMMARY OF THE SDR-ML RANDOMIZATION DETECTOR
TABLE II SDR APPROXIMATION FOR MLD USING DOMINANT EIGENVECTOR APPROXIMATION
TABLE III SDR APPROXIMATION FOR MLD USING RANK-1 APPROXIMATION
and
in (9), a feasible for (9) has unit . If we relax (9) by replacing (9b) to (9d) norm, i.e., for all , then a relatively simpler with constraints that optimization problem is obtained
(10a) s.t.
(10b)
The relaxed problem in (10) is nonconvex, but it can be turned to a convex optimization problem, the advantages of which will , and define a matrix be discussed soon. Let . As , the objective in (10a) can be , where tr is the trace opequivalently expressed as tr if and only if is Hererator. Using the fact that mitian positive semidefinite (PSD), (10) can be reformulated as tr s.t.
(11a) (11b) (11c)
means that is Hermitian PSD, and (11c) is from where (10b). Problem (11) is referred to as an SDR of the homogeneous quadratic program in (7). The SDR problem is a semidefinite program (SDP) [23], which i) is convex and thus does not suffer from local maxima and ii) can be efficiently solved using the interior-point optimization technique [23], [24]. In particular, an interior-point algorithm that is tailor-made for the SDR
problem will be developed in the next section. As we will elaborate in the next section, the proposed interior-point algorithm . yields a polynomial-time complexity of B. SDR Approximation by Randomization Once the SDR problem is solved, we use the SDR solution to approximate the solution of the original problem in (7). There are several ways of performing this approximation task, such as randomization [15], [18], [19], dominant eigenvector approximation [14], [17], and rank-1 approximation [16]. Here, we focus on the randomization method. It will be shown by simulations in Section V that the proposed randomization method provides a better approximation than the other two approximation methods. The procedures of the dominant eigenvector and rank-1 approximation methods for MPSK MLD are given in Tables II and III, respectively; see [14], [16], and [17] for the details of these two approximation methods. To present the MPSK randomization method, let be the solution of the SDR problem in (11). Let , where is a square-root factor of , e.g., the Cholesky factor. Then, is the solution of the equivalent SDR problem be the solution of the homoin (10). On the other hand, let geneous quadratic program in (7), and let be the solution of the equivalent homogeneous quadratic program in (9). The and is that relationship between
for any
(12)
. Since SDR drops the constraints that lead and for may not satisfy to the structure in (12), the SDR solution
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the structure in (12). For this reason, we consider approximating by performing a least square fitting
where denotes the optimal objective function value of MLD, has to be reformulated to a homogeneous form as in (7). Problem (17) is equivalent to
(13) (18) Minimizing (13) over both solving (13) for a given
and
is difficult. Let us consider
Re
(14)
Lemma 1: Re where dec i.e., dec
dec
(15)
is the MPSK threshold decision function, , where is such that , and is the phase angle
of . The proof of Lemma 1 is analogous to that of the single-user MPSK correlator receiver [25]. Applying Lemma 1 to (14), it is shown that dec
(16)
is the element-wise MPSK threshold where dec decision function. To determine in (16), we use randomization. The idea behind randomization is to use random search to suboptimally over .1 maximize the objective function Let define the SDR approximate solution using randomization. Given the number of randomizations, which is denoted by , the procedure of the randomization method is as follows: for Randomly generate a complex vector uniformly distributed on an -dimensional unit sphere. end; Choose the SDR approximate solution to be where .
,
As will be illustrated by the simulation results in Section V, this MPSK randomization method provides good approximation accuracy with modest . C. SDR-ML Detector for Synchronous CDMA Systems To apply the above developed SDR algorithm to the MLD of the synchronous CDMA signals, the MLD problem [in (5)] Re
(17)
approximation of x could be done by applying randomization to the least square fitting cost function in (13). However, it is more desirable to consider randomization for the original problem in (7) as the objective here is to maximize (7). 1The
is the solution of (18), then It is easy to verify that if is the solution of the MLD problem in (17). Since , we can use (18) is in the same form as (7) with the SDR algorithm to approximate the equivalent MLD problem in (18), followed by using the relationship mentioned above to yield an approximate MLD solution. Putting all the components together, the SDR-ML detector is summarized in Table I. It should be pointed out that the complexity of the randomiza) is much smaller than that of solving tion (with modest ]. Thus, the comthe SDR problem in (11) [i.e., putational cost of the whole SDR-ML detection process is of the . order of An important result shown in [15] is that for the BPSK case, the SDR-ML detector provides a tighter relaxation than two other relaxation-based suboptimal detection methods, namely, the unconstrained relaxation and bound relaxation [12], [13]. In the MPSK case, these two relaxation methods are represented, respectively, by Re
(19)
Re
(20)
where and denote the maximal objective function values of the unconstrained and bound relaxation methods, respectively. The unconstrained and bound relaxation methods are simpler than the SDR in that the former replaces the MPSK constellation constraints with simpler convex constraints. The unconstrained and bound relaxation methods are either equivalent or closely related to several linear and interference-canceling detectors [12], [13], [15]. For example, the unconstrained relaxation method is equivalent to the commonly used decorrelator [1]. The tighter approximation advantage of the SDR-ML detector over the unconstrained and bound relaxation detectors is also true in the general MPSK case, as described in the following theorem. Theorem 1: Define the maximal objective function value of SDR-MLD to be tr
(21)
The maximal objective function value provides a tighter upper bound on the optimal MLD objective function value than that of the unconstrained and bound relaxation detectors, i.e., (22) The proof of Theorem 1 may be obtained by following the development in [15] and is not shown here for brevity. Equation (22) shows that among the various relaxation methods, SDR has its maximal objective function value closest to the optimal MLD
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objective function value. Hence, the SDR-ML detector is expected to give a better approximation than the unconstrained and bound relaxation based detectors. In Section V, we will use simulations to compare the symbol error performance of the SDR-ML detector and various suboptimal detectors. The subsequent section will focus on the technique of numerically finding the SDR solution in (11).
TABLE IV INTERIOR-POINT ALGORITHM FOR SDR
IV. NUMERICAL COMPUTATION OF THE SDR SOLUTION As we have examined in the previous section, the process of solving the SDR optimization problem in (11) is integral to the proposed SDR-ML detector. The SDR problem is a SDP for which the most promising optimization technique at present is the interior-point (IP) methods [23], [26]. There are readily available IP algorithms for general SDPs, e.g., [23], [24], [27]. However, it is computationally more efficient to use an IP algorithm designed specially for the SDR problem. In this section, such an algorithm will be developed. The proposed algorithm is based on the primal-dual IP method by Helmberg [24], [28]. To gain better insights into the IP method, several key ideas of IP optimization will be reviewed in the first subsection. Then, the IP algorithm for SDR will be presented in the second subsection. Table IV shows the summarized procedure of the proposed IP algorithm. A. Some Basic Concepts of Interior-Point Methods for General SDPs We consider an SDP in the following standard form: tr s.t. (23) is a linear where , a general form of which is function of tr tr tr represents the number of linear equality constraints, and . Based on the theory of Lagrangian duality [29], the dual optimization problem associated with (23) is given by
s.t. (24) where is given by . Problems (23) and (24) are commonly referred to as the primal and dual SDPs, respectively. An important lemma regarding the optimality conditions for the SDPs is presented as follows. Lemma 2 [28]: Suppose that the Slater condition,2 which is a condition that rarely violates in optimization, is satisfied. In is optimal for (23) and (24) if and only if this case, is feasible for (23), is feasible for (24), and . in As will be shown, the complementarity condition Lemma 2 plays a key role in finding a close-to-optimal primal 2The
Slater condition is said to be satisfied if there exist strictly feasible
X; ; Z), i.e., there are (X; ; Z) such that X and Z are positive definite, F (X) = g, and Z = F~() 0 Q. (
SDP solution. Now, let us turn our attention to solving the dual SDP. How dual SDP optimization leads to the primal SDP solution will soon become apparent. A popular approach of dealing with the dual SDP is to consider the following so-called barrier dual SDP:
s.t.
(25)
where the prespecified parameter is called the barrier parameter. Comparing (24) and (25), it is clear that the barrier
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dual SDP drops the PSD constraint and adds a logain the objective function. rithmetic barrier function Simply speaking, the barrier function serves as a penalty function to enforce to lie within the set of positive definite (PD) matrices. There are several advantages of choosing the function as the barrier function, an important one of which is that (25) is a convex differentiable optimization problem; see [29] for details and for other advantages. The Lagrangian of (25) is tr
it is necessary to convert (31) to a real-valued SDP following Re and the standard formulation in (23). Let Im define the real and imaginary parts of . It can be verified that (31) is equivalent to the following real-valued problem:
tr
(32a)
s.t.
(32b)
(26)
Here, is defined to be the Lagrangian multiplier for . Based on the the equality constraint is optimal for Karush-Kuhn-Tucker (KKT) condition, (25) if and only if the following equations are simultaneously satisfied:
(32c) (32d) (32e) where (33)
(27) (28) (29) (30) where is the gradient operator, and means that is PD. define the solution of (27) to (30). It is verified Let from (27), (29), and (30) that is feasible for the primal SDP . By noting that (29) can be rewritten as [in (23)] for any and by Lemma 2, the following lemma is shown. approaches the opLemma 3: When timal solution of the primal-dual SDP pair in (23) and (24). Lemma 3 illustrates that the optimal SDP solution, both primal and dual, can be accurately approximated by solving the barrier dual SDP with a small enough . However, directly is numerically hard solving the barrier dual SDP with because the corresponding Hessian matrix usually changes rapidly near the boundary of the feasible set [29]. In the IP approach, this difficulty is circumvented by sequentially solving (27) to (29), with reducing at each iteration. At each iteration, the current iterate, which is denoted by , is in modified to form a new iterate a way that the new iterate solves (or well approximates) the optimality conditions in (27) to (29) for the new . The iteration continues until is small enough or some stopping criteria are satisfied (e.g., the number of iterations being greater some prespecified number). The next subsection will describe the sequential IP optimization process for finding the optimal SDR solution.
and and are the real and imaginary parts of , respectively. To have (32) amenable to the standard (primal) SDP formulation in (23), a number of linear equality constraints are required to construct the structure in (32b) and the skew-symmetry in (32c). A simpler way of solving (32) without considof ering those structural constraints is suggested by the following theorem. Theorem 2: Consider the following SDP: tr
(34a) (34b)
s.t.
(34c) Let
denote the optimal solution of (34), and partition (35)
where for all . Problems (32) and (34) are equivalent in that the optimal objective function values of (32) and (34) are equal and that the optimal solution of (32), which , , is the same as is denoted by (36) and denote the optimal objective funcProof: Let tion values of (32) and (34), respectively. We show Theorem 2 and . Since by proving that (37)
B. Interior-Point Algorithm for SDR To develop an IP algorithm for the complex-valued SDR problem in (11), i.e.,
is feasible for (34), we have that , we seek to prove that trate
tr
. To illus(38)
tr s.t. (31)
are feasible for (32). The matrix in (38) satisfies the constraints in (32e), due to (34c). Using the fact that for any , it is verified that the and in (38) satisfy
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the symmetry and skew-symmetry constraints in (32c), respectively. Furthermore, we substitute (38) into (32b) to obtain the following expression: (39)
such that solves (43) to (45) approximately. By applying Newton’s method to (43) to (45) and by noting that satisfies (43) and (45), it is shown [24] that the set of Newton is the solution of the following three equasearch directions tions: Diag
where
is the optimal solution of (34) defined in (35), and
Diag
(40) is PSD as long as is PSD. It folIt can be verified that lows that the left-hand side of (39) is PSD, and thus, the PSD constraint in (32d) is satisfied. These results conclude that (38) are feasible for (32). By putting (38) into (32a), it is shown that . , it is straightforward that (38) is an optimal Since solution of (32). The equivalent SDR problem in (34) in Theorem 2 is clearly structurally simpler than the SDR problem in (32). To describe the IP optimization process for (34), we follow the standard SDP formulation in (23) to rewrite (34) as tr
It is interesting to note that the search direction equations in (46) to (48) can be alternatively and more easily obtained by diinto (43) to (45) and then assuming rectly substituting . We show from (46) to (48) that the Newton search directions are given by
Diag
Diag
(41)
Diag is an where identity matrix, and is a length- all one column vector. Note that the generic linear operator in (23) is now replaced . The associated dual by the special linear structure Diag SDR problem is shown to be
Diag (42)
Diag
(50) (51)
represents the Hadamard product, i.e., . In Step 3 of Table IV, we adopt a more efficient way of computing the Newton search directions in (49) to (51), where is exploited. Several remarks the simple structure on the IP algorithm are as follows. in (51) is not symmetric. To 1) The search direction , the anti-symmetric part ensure the symmetry of is discarded; see Step 3 of Table IV. of 2) To safeguard the new iterate from leaving the strictly feasible set, step sizes are used to modify the primal and dual search directions; see Steps 4 and 5 of Table IV. as a 3) The IP algorithm requires a strictly feasible is suggested starting point. An initialization for as follows:
where Diag is a diagonal matrix with elements of on the main diagonal. In the SDR case, the barrier dual SDP optimality conditions in (27) to (29) can be simplified as Diag Diag
(49)
where
s.t.
s.t.
(46) (47) (48)
(43) (44) (45)
Table IV shows the proposed IP algorithm for (41), which is developed based on Helmberg’s framework [24]. As studied in the previous subsection, the IP algorithm iteratively solves (43) to (45), with the barrier parameter reducing at each iteration (see Steps 1 and 2 of Table IV). The major difference between the proposed IP algorithm and the general-purpose IP algorithm in [24] is Step 3 of Table IV, where the SDR problem structure is exploited to make the IP optimization process more efficient. Specifically, the problem addressed in Step 3 of Table IV is as that follows: Given the current iterate satisfies the feasibility conditions in (43) and (45) but violates the complementarity condition in (45) for the new (Step 2 of Table IV), find a set of search directions
(52) abs
abs
(53)
Diag (54) Diag where abs is the elementwise absolute function, i.e., abs . It is straightforward that (52) is strictly primal feasible. The strict dual feasibility of (53) and (54) can be shown using the standard property of dominant diagonal matrices [24], [30]. The computational complexity of the above developed SDR IP algorithm is determined by two factors: i) the number of iterations required to achieve the stopping criterion and ii) the operational cost at one iteration. It has been shown [24] that given a solution accuracy , the worst-case number of iterations required to satisfy the stopping criterion is of the order of . On the other hand, the computational cost at one iteration is mainly contributed by the search direction computations (Step 3 of Table IV) and the line search process (Step 4 of Table IV). It is
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A. Symbol Error Performance
Fig. 1. Near–far performance in a four-user, 8-ary PSK system. The energy of = 17 dB. (a) Various user 3 is E = 0 dB, and the SNR is fixed at E = SDR-ML detection algorithms. (b) Various detectors. Note that the multistage IC detectors are initialized by the decorrelator decision and that their number of stages is 4.
N
inspected from Table IV that the computations of the search dioperations. Moreover, the line search rections require process, which uses Cholesky factorization to check the positive and for given definiteness of and [28], requires operations. Hence, the compu. It then follows tational complexity per iteration is that given a solution accuracy, the computational complexity of . the IP algorithm is of the order of
V. SIMULATION RESULTS
The following three simulation examples illustrate the SERs of various multiuser detectors when the MPSK constellation . size is Example 1: This example compares the SER performance of the SDR-ML detector and the true ML detector. To do so, we such that direct implemenset the number of users to be tation of the true ML detector (via exhaustive search) is computationally affordable. The simulation settings are as follows. The spreading sequences are the length-7 Gold codes. All user are set to be 0. We simulated a near–far carrier phase values scenario where the SERs of a particular user are evaluated under various interfering users’ power. User 3 is chosen as the user of dB, whereas the other users are ininterest with energy terferers with their energies being equal. The SNR of user 3 is dB. Fig. 1(a) plots the SER performance of fixed at the proposed SDR-ML randomization algorithm against the interfering user energy. Note that the signal-to-interference ratios3 associated with the figure range from 13 to 3 dB. For comparison, the SERs of the other two SDR-based approximation algorithms, namely, the dominant eigenvector approximation [14], [17] and rank-1 approximation [16] (the procedures of which are given in Tables II and III, respectively), were plotted in the same in the figure represents the figure. Recall that the symbol number of randomizations of the SDR-ML randomization algorithm. We see that the SDR-ML randomization algorithm has its and that SER performance improved significantly with it performs much better than the other two SDR-ML algorithms . In addition, the performance gap between the for SDR-ML randomization algorithm and the true ML detector be. comes small when In Fig. 1(b), we compare the SER performance of the SDR-ML detector (with randomization) and four commonly used suboptimal multiuser detectors, namely i) decorrelator; ii) LMMSE detector; iii) multistage parallel interference canceling (IC) detector [4]; iv) multistage serial IC detector4 [5], [31], [32]. It is observed that the SDR-ML detector provides better SER performance than the other detectors, except at dB, where the SER of the SDR-ML detector is only slightly worse than that of the multistage IC detectors. Example 2: In this example, we evaluate the SER performance of the various detectors when the number of active users increases but the spreading factor remains fixed. The length-31 Gold spreading-code sequences are used. All user SNRs are dB, and all user carrier phase values fixed at are set to be 0. Fig. 2(a) shows the SER performance of the SDR-ML detectors. Like the results in the previous example, the SDR-ML randomization detector with sufficiently large (i.e., ) provides better SERs than the other 3Here,
In the following simulation examples, we compare the symbol error rate (SER) performance and computational complexity of the proposed SDR-ML detector and several standard multiuser detectors.
the signal-to-interference ratio is defined to be SIR = E ), where the index k denotes the user of interest. 4It is interesting to note that the multistage serial IC detector is equivalent to the space alternating generalized expectation-maximization (SAGE) detector [5] and the coordinate ascent detector [31], both of which are suboptimal MLD methods.
E =(
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Fig. 3. Near–far performance of the various detectors in a 24-user, 8-ary PSK system. The energies of the desired users equal E = E = 1 1 1 = E = 0 dB. The SNRs of the desired users are fixed at E =N = 17 dB. Note that the multistage IC detectors are initialized by the decorrelator decision and that their number of stages is 4.
We should point out that the SER performance results for were found to be similar to those MPSK constellations of the above 8-ary PSK examples, given the same SER ranges. B. Computational Complexity
Fig. 2. Symbol error performance versus the number of users in an 8-ary PSK = 17 dB. (a) Various SDR-ML system. All user SNRs are fixed at E = detection algorithms. (b) Various detectors. Note that the multistage IC detectors are initialized by the decorrelator decision and that their number of stages is 4.
N
two SDR-ML approximation algorithms. Fig. 2(b) plots the SER performance of the SDR-ML randomization detector and the various suboptimal detectors. It is observed that the SDR-ML detector yields improved SER performance compared with the other suboptimal detectors, particularly when the system is loaded with more users. Example 3: This example examines the near–far performance of the various detectors in a 24-user system. The spreading sequences are the length-31 Gold codes. All user are set to be 0. Users 13 to 24 are the carrier phase values desired users with dB, whereas users 1 to 12 are the interfers with their energies being equal. dB The SNRs of the desired users are fixed at . Fig. 3 shows the averaged SER of the for desired users versus the interfering user energy. Note that the SIRs associated with the figure range from 20 dB to 10 dB. Fig. 3 demonstrates that the SER performance of the SDR-ML detector is better than that of the other suboptimal detectors.
The previous simulation examples have indicated that the SDR-ML detector offers better SER performance than the other suboptimal multiuser detector. The following simulation example considers the computational complexity of the SDR-ML detector and the other detectors. Example 4: In this example, we measure the average numbers of floating point operations (FLOPs) of the following detectors: operai) true ML detector via exhaustive search [ tions]; operations]; ii) SDR-ML randomization detector [ operations]; iii) decorrelator [ operations without iniv) serial IC detector [ represents cluding the initialization process, where the number of stages]. The computational costs of the LMMSE detector and the parallel IC detector are not considered because they are similar, respectively, to those of the decorrelator and the serial IC detector. Likewise, the FLOP counts of the SDR-ML dominant eigenvector approximation method and the SDR-ML rank-1 approximation method are not shown as they were found to be similar to that of the SDR-ML randomization method. The con. The numbers of FLOPs stellation size is set to be of the various detectors are shown in Fig. 4. We see that the SDR-ML detector, which is inherently iterative in nature, yields a higher computational cost than the one-shot decorrelator and the low-complexity serial IC detectors. However, the number of FLOPs of the SDR-ML detector is much lower than that of the exhaustive-search based ML detector when the number of users
MA et al.: SEMIDEFINITE RELAXATION-BASED MULTIUSER DETECTION
Fig. 4. Computational costs of the various multiuser detectors. The number of stages of the serial IC detector is fixed at 4.
is large, i.e., when . Moreover, by comparing the complexity curves of the SDR-ML detector and the decorrelator, it is seen that the complexity order of the SDR-ML detector is similar to that of the decorrelator. VI. CONCLUSION AND DISCUSSION An efficient approximate ML detector based on semidefinite relaxation (SDR) has been proposed for -ary PSK CDMA multiuser communication systems. For the synchronous CDMA scenario, the proposed SDR-ML detector yields an attractive , where stands for polynomial-time complexity order of the number of users. Simulation results have indicated that the SDR-ML detector provides improved symbol error performance compared with several standard suboptimal detectors. In this work, it has been discussed that the computational complexity of the SDR-ML detector is dominated by that of the interior-point iterative optimization algorithm (cf., Section IV). It will be interesting to consider more efficient SDR-ML detection by further modifying the interior-point algorithm. Likewise, some other optimization algorithms, such as [17], [33], [34], could be used in place of the present interior-point algorithm for reduction in complexity. Such reduced-complexity possibilities may lead to different solution accuracy, and their impacts on the symbol error performance and computational efficiency remain explored. REFERENCES [1] S. Verdu, Multiuser Detection. Cambridge, U.K.: Cambridge Univ. Press, 1998. [2] , “Computational complexity of optimum multiuser detection,” Algorithmica, vol. 4, pp. 303–312, 1989. [3] L. Wei, L. K. Rasmussen, and R. Wyrwas, “Near optimum tree-search detection schemes for bit-synchronous multiuser CDMA systems over Gaussian and two-path Rayleigh-fading channels,” IEEE Trans. Commun., vol. 45, pp. 691–700, June 1997. [4] M. K. Varansi and B. Aazhang, “Near-optimum detection in synchronous code-division multiple-access systems,” IEEE Trans Commun., vol. 39, pp. 725–736, May 1991.
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[5] L. B. Nelson and H. V. Poor, “Iterative multiuser receivers for CDMA channels: An EM-based approach,” IEEE Trans. Commun., vol. 44, pp. 1700–1710, Dec. 1996. [6] T. Miyajima, T. Hasegawa, and M. Haneishi, “On the multiuser detection using a neural network in code-division multiple-access communications,” IEICE Trans. Commun., vol. E76-B, no. 8, pp. 961–968, 1993. [7] W. G. Teich and M. Seidl, “Code division multiple access communications: Multiuser detection based on a recurrent neural network structure,” in Proc. IEEE 4th Int. Symp. Spread Spectrum Techn. Applications, vol. 3, 1996, pp. 22–25. [8] C. Ergün and K. Hacioglu, “Multiuser detection using a genetic algorithm in CDMA communication systems,” IEEE Trans. Commun., vol. 48, pp. 1374–1383, Aug. 2000. [9] K. Yen and L. Hanzo, “Genetic algorithm assisted joint multiuser symbol detection and fading channel estimation for synchronous CDMA systems,” IEEE J. Select. Areas Commun., vol. 19, pp. 985–998, Aug. 2001. [10] P. Spasojeviæ and C. N. Georghiades, “The slowest descent method and its application to sequence estimation,” IEEE Trans. Commun., vol. 49, pp. 1592–1604, Sept. 2001. [11] A. Alrustamani and B. R. Vojcic, “A new approach to greedy multiuser detection,” IEEE Trans. Commun., vol. 50, pp. 1326–1336, Aug. 2002. [12] P. H. Tan, L. K. Rasmussen, and T. J. Lim, “Constrained maximumlikelihood detection in CDMA,” IEEE Trans. Commun., vol. 49, pp. 142–153, Jan. 2001. [13] A. Yener, R. D. Yates, and S. Ulukus, “CDMA multiuser detection: A nonlinear programming approach,” IEEE Trans. Commun., vol. 50, pp. 1016–1024, June 2002. [14] P. H. Tan and L. K. Rasmussen, “The application of semidefinite programming for detection in CDMA,” IEEE J. Select. Areas Commun., vol. 19, pp. 1442–1449, Oct. 2001. [15] W.-K. Ma, T. N. Davidson, K. M. Wong, Z.-Q. Luo, and P. C. Ching, “Quasimaximum-likelihood multiuser detection using semi-definite relaxation with applications to synchronous CDMA,” IEEE Trans. Signal Processing, vol. 50, pp. 912–922, Apr. 2002. [16] X. M. Wang, W.-S. Lu, and A. Antoniou, “A near-optimal multiuser detector for CDMA systems using semidefinite programming relaxation,” IEEE Trans. Signal Processing, vol. 51, pp. 2446–2450, Sept. 2003. [17] M. Abdi, H. El Nahas, A. Jard, and E. Moulines, “Semidefinite positive relaxation of the maximum-likelihood criterion applied to multiuser detection in a CDMA context,” IEEE Signal Processing Lett., vol. 9, pp. 165–167, June 2002. [18] M. X. Goemans and D. P. Williamson, “Improved approximation algorithms for maximum cut and satisfiability problem using semi-definite programming,” J. ACM, vol. 42, pp. 1115–1145, 1995. [19] Y. E. Nesterov, “Semidefinite relaxation and nonconvex quadratic optimization,” Optim. Methods Software, vol. 9, pp. 140–160, 1998. [20] B. Steingrimsson, Z.-Q. Luo, and K. M. Wong, “Soft quasi-maximum-likelihood detection for multiple-antenna wireless channels,” IEEE Trans. Signal Processing, vol. 51, pp. 2710–2719, Nov 2003. [21] W.-K. Ma, T. N. Davidson, K. M. Wong, and P. C. Ching, “A block alternating likelihood maximization approach to multiuser detection,” IEEE Trans. Signal Processing, vol. 52, pp. 2600–2611, Sept. 2004. [22] , “Multiuser detection for asynchronous CDMA using block coordinate ascent and semi-definite relaxation,” in Proc. Int. Conf. Acoust., Speech Signal Process., vol. 3, 2002, pp. 2309–2312. [23] L. Vandenberghe and S. Boyd, “Semidefinite programming,” SIAM Rev., vol. 38, pp. 49–95, 1996. [24] C. Helmberg, F. Rendl, R. J. Vanderbei, and H. Wolkowicz, “An interiorpoint method for semidefinite programming,” SIAM J. Optim., vol. 6, no. 2, pp. 342–361, 1996. [25] J. G. Proakis, Digital Communications, Third ed. New York: McGrawHill, 1995. [26] R. M. Freund and S. Mizuno et al., “Interior point methods: Current status and future directions,” in High Performance Optimization, H. Frenk et al., Eds. Boston, MA: Kluwer, 2000, pp. 441–446. [27] J. F. Sturm. (1999) Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optim. Methods Software [Online], pp. 625–653. Available: http://fewcal.kub.nl/sturm/software/sedumi.html [28] F. Rendl, “Semidefinite programming and combinatorial optimization,” Appl. Numer. Math., vol. 29, pp. 255–281, 1999. [29] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge, U.K.: Cambridge Univ. Press, 2004. [30] F. A. Graybill, Matrices with Applications in Statistics, Second ed. Belmont, CA: Wadsworth, 1983. [31] H. 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.
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[32] K. Jamal and E. Dahlman, “Multi-stage serial interference cancellation for DS-CDMA,” in Proc. IEEE Veh. Tech. Conf., vol. 2, 1996, pp. 671–675. [33] S. Burer and R. D. C. Monteiro, “A projected gradient algorithm for solving the maxcut SDP relaxation,” Optim. Methods Software, vol. 15, pp. 175–200, 2001. [34] S. Burer, R. D. C. Monteiro, and Y. Zhang, “Rank-2 relaxation heuristics for MAX-CUT and other binary quadratic programs,” SIAM J. Optim., vol. 12, no. 2, pp. 503–521, 2001.
Wing-Kin Ma (M’01) received the B.Eng. (Hons.) degree in electrical and electronic engineering from the University of Portsmouth, Portsmouth, U.K., in 1995 and the M.Phil. and Ph.D. degrees, both in electronic engineering, from the Chinese University of Hong Kong in 1997 and 2001, respectively. In 2000, he was with the Department of Electrical and Computer Engineering, McMaster University, Hamilton, ON, Canada, as a Visiting Scholar. Since 2001, he has been a Research Associate with the Department of Electronic Engineering, the Chinese University of Hong Kong. Since November 2003, he has been with the Department of Electrical and Electronic Engineering, the University of Melbourne, Parkville, Australia, as a Research Fellow. His research interests are in signal processing for communications and statistical signal processing. Dr. Ma’s Ph.D. dissertation was commended to be “of very high quality and well deserved honorary mentioning” by the Faculty of Engineering, the Chinese University of Hong Kong, in 2001.
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 52, NO. 10, OCTOBER 2004
Pak-Chung Ching (SM’90) received the B.Eng. (Hons.) and Ph.D. degrees in electrical engineering and electronics from the University of Liverpool, Liverpool, U.K., in 1977 and 1981, respectively. From 1981 to 1982, he was a research officer at the School of Electrical Engineering, University of Bath, Bath, U.K. From 1982 to 1984, he was a Lecturer with the Department of Electronic Engineering of the then Hong Kong Polytechnic University. Since 1984, he has been with the Chinese University of Hong Kong, where he is presently Dean of Engineering and a full professor at the Department of Electronic Engineering. He has taught courses in digital signal processing, stochastic processes, speech processing, and communication systems. His research interests include adaptive filtering, time delay estimation, statistical signal processing, and hands-free speech communication. Dr. Ching has actively participated in many professional activities. He was the Chairman of the IEEE Hong Kong Section from 1993 to 1994. He is a member of the Signal Processing Theory and Methods Technical Committee of the IEEE Signal Processing Society. He has been serving as an associate editor for the IEEE SIGNAL PROCESSING LETTERS since January 2002 and was also an associate editor for the IEEE TRANSACTIONS ON SIGNAL PROCESSING from 1997 to 2000. He is a member of the Accreditation Board and Fellowship Committee of the Hong Kong Institution of Engineers and a Council member of IEE, U.K. He has been involved in organizing many international conferences, including the 1997 IEEE International Symposium on Circuits and Systems, where he was the Vice-Chairman, and the 2003 IEEE International Conference on Acoustics, Speech, and Signal Processing, where he served as the Technical Program Co-chair. He is a Fellow of IEE and HKIE.
Zhi Ding (F’03) received the Ph.D. degree in electrical engineering from Cornell University, Ithaca, NY, in 1990. He is currently is a Professor at the University of California, Davis. From 1990 to 2000, he was a faculty member of Auburn University, Auburn, AL, and later, at the University of Iowa, Iowa City. He has held visiting positions with the Australian National University, Canberra, Australia; Hong Kong University of Science and Technology; the NASA Lewis Research Center, Cleveland, OH; and USAF Wright Laboratory, Wright-Patterson AFB, OH. Dr. Ding has been an active member of IEEE Signal Processing Society, serving on technical programs of several workshops and conferences. He was associate editor for IEEE TRANSACTIONS ON SIGNAL PROCESSING from 1994 to 1997. He is currently associate editor of IEEE TRANSACTIONS ON SIGNAL PROCESSING and the IEEE SIGNAL PROCESSING LETTERS. He was a member of technical committee on statistical signal and array processing. He is currently a member of technical committee on signal processing for communications.
