A new approach to greedy multiuser detection - Semantic Scholar
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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 50, NO. 8, AUGUST 2002
A New Approach to Greedy Multiuser Detection Amina AlRustamani, Member, IEEE, and Branimir R. Vojcic, Senior Member, IEEE
Abstract—In this paper, we propose a new suboptimum multiuser detector for synchronous and asynchronous multiuser communications. In this approach, a greedy strategy is used to maximize the cost function, the maximum-likelihood (ML) metric. The coefficients of the ML metric are utilized as weights indicating in which order bits can be estimated. The complexity of per bit, where is the algorithm is approximately 2 log the number of users. We analyze the performance of the greedy multiuser detection in the additive white Gaussian noise channel as well as in the frequency-nonselective Rayleigh fading channel, and compare it with the optimum detector and several suboptimum schemes such as conventional, successive interference cancellation, decorrelator, sequential, and multistage detectors. The proposed greedy approach considerably outperforms these suboptimum schemes, especially for moderate and high loads in low and moderate signal-to-noise ratio regions. The results show that when there is a significant imbalance in the values of the coefficients of the ML metric due to moderate to high noise, fading, and asynchronous transmission, near-optimum performance is achieved by the greedy detection. Index Terms—Code-division multiaccess, interference suppression, multiuser channels.
I. INTRODUCTION
T
HE exponential complexity and thus, the impracticality [1], [2] of joint optimum multiuser detection in code-division multiple-access (CDMA) systems has driven researchers to explore suboptimum solutions with practical significance in terms of performance and complexity. This resulted in discovering a wide range of possible complexity/performance tradeoffs. A detailed analysis of the most well known suboptimum multiuser detectors, and an extensive list of references, are given in [3]. Although each proposed improving the practicality by substantially reducing the complexity, the performance of these suboptimum schemes is far from the optimum performance, especially for systems with large numbers of users. In this paper, we propose a new suboptimum multiuser scheme that is based on the greedy principle. We show that the performance of the proposed detector is close to the optimum performance, yet with significantly lower complexity. Indeed, the complexity is, at most, polynomial in the number of users.
Paper approved by S. L. Miller, the Editor for Spread Spectrum of the IEEE Communications Society. Manuscript received October 10, 2000; revised June 20, 2001 and November 19, 2001. This paper was presented in part at IEEE 6th International Symposium on Spread Spectrum Techniques and Applications (ISSSTA), NJ, September 2000, and at the 2000 Conference on Information Sciences and Systems (CISS), Princeton, NJ, March 2000. A. AlRustamani is with Dubai Internet City, Dubai (e-mail: [email protected]). B. R. Vojcic is with the Department of Electrical and Computer Engineering, George Washington University, Washington, DC 20052 USA (e-mail: [email protected]). Publisher Item Identifier 10.1109/TCOMM.2002.801493.
Also, we demonstrate comparatively that the performance is significantly better than that of most of the existing suboptimum schemes for a wide range of operating conditions, and specifically for systems with moderate to large numbers of users. The greedy principle is applied to a wide variety of combinatorial optimization problems, such as the knapsack problem, Hoffman coding, and minimum spanning tree problem [4], [5]. Most of these problems have inputs, such as the objects in the knapsack problem, and require obtaining a subset that satisfies certain constraints, for example, the capacity of the knapsack. The greedy method suggests that one can think of an algorithm which works in stages, considering one input at a time. At each stage, the algorithm examines one input and decides on a partially constructed solution or on a complete solution. The main distinctive feature of the greedy principle is that the inputs are considered in an order determined by some selection procedure that is based on some optimization measure, such as maximizing the increase in the profit in every stage. For example, some of the criteria that are used to order the objects in the knapsack problem are profit, weight, or density (profit/weight). In the multiuser detection problem, the maximum-likelihood (ML) metric is a quadratic function of variables (the transmitted bits) that take discrete values. The task of the optimum detector is to find the values of the transmitted bits that maximize the ML metric. The quadratic nature of the problem and the discrete values of the variables give rise to the exponential complexity of the optimum solution. To apply the greedy concept to reduce the complexity of the solution, we need to define the inputs and the optimization measure that will determine the order of these inputs. Some of the suboptimum detectors that are present in the literature can be viewed as algorithms that are based on the greedy principle [6]–[10]. The inputs of these detectors are the received signals of users in the system. The signals are ordered based on their received powers, as in successive interference cancellation [6] and the decorrelating decision-feedback detector [7]. In every stage, the bit corresponding to the user in that stage is estimated. In [8]–[10] the authors adapt the same algorithms given in [6] and [7], and show that if more than one partially constructed solution is passed from one stage to the other, the performance of such detectors can be significantly improved. Moreover, in [8], the authors show that for certain scenarios, the way users are ordered has no significant effect on the performance of such detectors. Instead of ordering the users, the algorithm proposed in this paper views the coefficients of the bits in the ML metric as weights (inputs) that indicate the order in which bits could be estimated [11]–[13]. The dominating coefficients, the large values, will have more impact on the value of the ML metric
0090-6778/02$17.00 © 2002 IEEE
ALRUSTAMANI AND VOJCIC: A NEW APPROACH TO GREEDY MULTIUSER DETECTION
and thus, carry more weight in determining the values of the bits associated with them than smaller terms. The complexity , and we show that its of the algorithm is performance is significantly better than that of most of the existing suboptimum detectors for systems with moderate to high loads. Specifically, the results show that when there is a significant imbalance in the values of the coefficients of the ML metric due to moderate to high noise, fading, and asynchronous transmission, near-optimum performance is achieved by the proposed greedy detection. The rest of the paper is organized as follows. In Section II, the system model and the greedy multiuser detection (MUD) algorithm for synchronous CDMA is presented. The proposed algorithm for the asynchronous CDMA system is given in Section III as well. In Section IV, simulation results for the proposed greedy detector and other considered detectors for the additive white Gaussian noise (AWGN) channel and for the nonselective Rayleigh fading channel are presented. Concluding remarks are given in Section V. II. SYNCHRONOUS CDMA We consider a frequency-nonselective fading channel for which the multipath components are not resolvable [14]. In this case, the received signal is the transmitted signal multiplied by a complex-valued random process representing the time-variant characteristics of the channel. Furthermore, we assume that the signal duration is significantly smaller than the coherence time of the channel and thus, the channel is slowly fading and the attenuation and phase shift are essentially constant for the duration of at least one bit interval [14]. Therefore, the equivalent low-pass received signal in the th interval for a synchronous CDMA system over such a channel is
(1) is the number of users, and is the bit interval; , and represent energy per bit, unit-energy signature waveform, and bit value of the th user , are in the th bit interval, respectively; independent zero-mean complex-valued Gaussian random, , and such that is the space-time correlation function of the channel associated with th user. denotes complex conjugate is mathematical expectation. is a complex and zero-mean Gaussian random process. The receiver consists of a bank of matched filters and a multiuser detector. The output of the filter matched to the signature waveform of user and is sampled at where
(2)
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where denotes the cross correlation of the signature waveforms of users and , and denotes the noise at the output of the th matched filter
The AWGN channel can be viewed as a special case of the for , model represented above, where is AWGN of one-sided power spectral density . and The outputs of the matched filters are sufficient statistics for optimum multiuser detection and can be expressed in vector form as [3], [15]
where is the normalized cross-correlation matrix of the signature waveforms, , and is the noise vector with . The superscript autocorrelation matrix denotes the complex conjugate transpose. The ML receiver selects the bits that maximize the metric [1], [15], where is the real component of (3) Hence, the optimum receiver for the synchronous case consists single-user matched filters followed by a detector that of possible transmitted informacomputes the metrics for the tion bit vectors, and selects the vector that gives the largest metric value. In combinatorial optimization, (3) is known as the unconstrained bivalent quadratic programming problem, and it belongs to a class with many different applications [16]–[18]. The , such as density (the ratio characteristics of the matrix of the number of nonzero entries to the total number of entries) , and and sparsity, the noise level affecting the elements in the dimensionality of the problem mainly determine the complexity of the problem and the performance of suboptimum detectors [16], [17]. When the matrix has low density, or when it is diagonally dominant indicating low cross correlations, or when the signal-to-noise ratio (SNR) is low, then finding a near-optimum solution becomes much easier. In this case, most of the proposed algorithms in the literature provide a significant improvement in performance compared to the conventional detector. Furthermore, it has been shown that a careful design of the signature waveforms and thus, the cross-correlation matrix , results in significant reduction in the complexity of the jointly optimum detection [19]–[24]. Indeed, polynomial complexity is achievable if stringent requirements are applied on the design of the signature waveforms [3], [19]–[24]. However, these requirements limit the system capacity [24]–[26], assume synchronous transmission, and cannot be maintained in multipath channels.
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The above discussion suggests that the elements in the maplay a crucial role in determining the optimal bit trix and the AWGN channel, the values. For example, for ML metric is
The largest two coefficients will determine the optimum bits [3]. If or then the outputs of the matched filters determine the optimum . However, if bits, that is or or
or
then the largest absolute value of the matched filter outputs multiplied by the amplitudes and the cross-correlation terms determine the optimum bit decisions. Accordingly, we can apply the . However, most of the existing subopsame idea for timum detectors do not explicitly utilize the elements of the mato decide on the bit values. These values are, rather, trix used to reconstruct and then suppress interference due to other , we utilize users. Therefore, in addition to the elements in as well in the detection process. the elements in The greedy MUD algorithm is based on the ML metric in (3), or, equivalently
(4) where
By examining the above equation, we can view the ML metric real numbers, where deciding on the as a sum of directly affects terms sign of one bit in the summation. Based on the greedy principle, the proposed and as weights that algorithm views the coefficients indicate in which order the bits of users could be sequentially estimated. By ordering the absolute values of the coefficients in descending order, and examining the incremental effect of each coefficient on the whole ML metric value, we choose the values of the bits that, in combination with the corresponding coefficient, make as large as possible positive contributions to the ML metric. When the coefficient is considered, the effect or is examined in of choosing the corresponding bit as coefficients in the conjunction to its impact on the other , then the metric. Similarly, when considering the coefficient effect of the corresponding bits and on the other
terms is examined. Thus, to maximize the ML metric (4), we always try to resolve the information bits in the order of their contribution to the ML metric. and Initially, the absolute value of the coefficients, and , are sorted in descending order. The subscripts of these coefficients indicate in which order the bits must be estimated. Since for every bit interval the values of depend on the outputs of the matched the coefficients filters, and all the coefficients depend on the amplitudes of the users, the order of detection is dynamic on a bit-interval basis. The algorithm consists of, at most, stages, which is the total number of coefficients in (4), and times. The th stage correeach bit is estimated up to sponds to the th coefficient after sorting. In some cases, stages the number of stages might be less than for reasons that will be evident later. Let the elements of the be the absolute values of the vector be coefficients after sorting. Let in the th stage and the tentative estimate of the vector . Fig. 1 illustrates the main steps in the let , then the ML greedy algorithm. In stage , if metric, in (4), is calculated twice for the two possible vectors and in , obtained by substituting and i.e.,\break . The bit vector that results in the largest metric value is chosen as the new estimate of the vector in the th stage if and only if and On the other hand, if the coefficient is of the form , then the ML metric, in (4), is calculated four times for the four possible vectors obtained by substituting in , i.e., , . The bit and vector that results in the largest metric value is chosen as the new estimate of the vector in the th stage if and only if and Therefore, in the th stage, one or two bits in , depending on the coefficient corresponding to the th stage, are examined and updated based on the decisions on the bits made in the previous stages, corresponding to the largest increment of the ML metric. One should note that calculating the metric in (4) in every stage is equivalent to subtracting the interference of users estimated in the previous stages and either jointly detecting the bits , or deciding on of two users, if the respective coefficient is the bit of one user if the respective coefficient is . Moreover, estimates of all the bits might not be available, at most, until . This can occur if all the coefficients of stage
ALRUSTAMANI AND VOJCIC: A NEW APPROACH TO GREEDY MULTIUSER DETECTION
Fig. 1.
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Flow diagram of the greedy algorithm for synchronous CDMA.
a specific user are examined in the last stages in the greedy algorithm. Once estimates of all bits are available or estimates of the bits examined in the stage already exist, then one of the possible bit combinations that should be examined in the stage is the one chosen in the previous stage. Therefore, to reduce the number of computations, the algorithm must forward the vector and the metric value associated with this vector to the next stage. in In addition, the number of stages might be less than some cases. In Lemma 1 we summarize the cases in which the number of stages can be reduced. Lemma 1: In the greedy MUD algorithm, if two consecutive or , or if three consecustages examine or all the other five possible tive stages examine and , then deleting the stages correpermutations of and will not affect the result. sponding to Proof: The vectors examined in the consecutive stages or , or all other possible permutations, have the same elements in all other positions besides and , i.e., same bit estimates for all other users except users and . (and Moreover, the two examined bit vectors in the stages ) are subsets of the four examined vectors in the stage . (and ) will not have an effect Therefore, deleting stages on the results. When discussing the convergence of the algorithm to the global maximum, one needs to think of the algorithm as consisting of two phases. For example, let us assume that estimates of all the bits are available at stage . In the first phase, stages one to , estimates of all the transmitted bits, the
elements in , are obtained. This first phase suggests that and thus, the global maximum is in the neighborhood of it determines the region that must be examined by the to , the second phase. In the second phase, stages neighboring points, vectors differing in one or two elements compared to the vector chosen in every stage, are examined. Therefore, the second phase of the algorithm is a search phase for a maximum point in the region indicated by the first phase. Since the algorithm might not converge to the global maximum region, and also the number of the examined vectors in the second phase is determined by the number of stages, the final estimate of might not be the global maximum. One way to increase the possibility of converging to the global largest metrics and the maximum is to forward the corresponding bit vectors from the current stage to the next will be estimates of one. Therefore, the input to stage and the corresponding . metric values , two or four metric In stage , for every or metric values, the values are calculated. Then among largest values and the corresponding bit vectors are chosen as the output of the stage. As the value of is increased, the algorithm explores different regions, and thus, the possibility of converging to the global maximum also increases. Whether the algorithm converges to the global maximum and the speed of convergence is determined by the order of the weights after sorting, and the variability in the values of the weights which is a measure of the increase in the value of the metric in every stage.
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The greedy algorithm is summarized with the following steps1 : Step 1) Sort the absolute values of the coefficients , , and in a descending order.
EXAMPLE
OF
TABLE I SYNCHRONOUS GREEDY MUD ALGORITHM FOR
K = 4 AND L = 1
sort descending
where . Step 2) Based on Lemma 1, delete stages that are redundant . and find Step 3) Set and . , then find all the possible Step 4) If metric values, corresponding to all the possible in bit vectors obtained by substituting . Set , where is the minimum of the set of values . Choose the largest metric values and the . corresponding vectors , find all the possible metric Else, values, corresponding to all the possible bit vecin tors obtained by substituting . Set . Choose the largest metric values and the corresponding . vectors , then and go back to Step Step 5) If 4, else stop and the bit vector corresponding to the largest metric value is the final estimate of the bits. and An example for the greedy MUD algorithm for is illustrated in Table I. Although the complexity of the greedy algorithm for this specific example is higher than that of the optimum detector, this example is chosen just for illustration purposes. The three gray rows indicate the deleted stages. The calculations in these stages are also given to demonstrate the reasoning behind Lemma 1. Estimates of all bits are available in the sixth stage. The complexity of the algorithm can be calculated as follows. terms that are sorted and examined. The There are . The complexity of sorting is and four ML metric, , is calculated twice for the terms , if and/or , else it is caltimes for the term culated once and three times, respectively. Thus, in total, if all terms are considered, the metric is calculated times. Therefore, the complexity of the algorithm is , and for large it is , which is significantly values of smaller than the computational complexity of the optimum de. tector for , the algorithm is optimal, since in the stage where For is examined, all possible combinations are considthe term ered and the performance in this case is known [3]. Moreover, 1In the beginning of the algorithm the number of examined vectors might be less than L; thus, is the number of metric values and the corresponding bit vectors that can be passed from one stage to the other.
q
the suboptimum detectors presented in [6]–[9] can be viewed as special cases of the greedy algorithm, in which only the , are considered to determine the coefficients and only are used for sorting order of detection. If and selection, the greedy algorithm is equivalent to successive interference cancellation [6], while it is equivalent to the decorrelating decision-feedback detector [7] if a whitening filter is and only used prior to the greedy detection. For are used, the greedy algorithm is equivalent to the tree search algorithms presented in [8]–[10], with and without a whitening filter. III. ASYNCHRONOUS CDMA SYSTEM For the same model presented in the synchronous case for the frequency-nonselective fading channel, the equivalent low-pass received waveform for asynchronous CDMA system over such a channel can be expressed as [3], [12], [15]
(5) is the frame length and is the time delay of the where th user. As in the synchronous case, for the AWGN channel, for , and and is AWGN of one-sided power spectral density . Without loss
ALRUSTAMANI AND VOJCIC: A NEW APPROACH TO GREEDY MULTIUSER DETECTION
of generality, we can assume that The outputs of the matched filters are
.
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due to previous and future intervals. Therefore, we define the following metric for the th interval, which is part of the whole metric in (8):
(6) terms, in which The ML metric can be written as a sum of each term depends on components of and any consecutive components [3] as shown in (7) at the bottom terms share of the page, where
, and for , and represents the modulo- remainder of . Consequently, dynamic programming can be used to find the longest path in a layered di; thus, the comrected graph that maximizes the ML metric . plexity of jointly optimum detection is independent of The ML metric in (7) can be rewritten as (8) where
The ML detector chooses that maximizes (8). As in the synchronous case, the greedy algorithm can be applied on the whole metric, however, the complexity will depend on the frame length . As a simpler approach, the greedy MUD algorithm is applied sequentially on -bit intervals to maximize parts of the ML metric, i.e., the algorithm is applied intervals of the frame, where indicates on the smallest integer greater than or equal to . Therefore, the suboptimality is due to the partitioning of the ML metric and the greedy maximization of every partition. In the th interval, , the bits where are estimated. Final estimates of the bits be, are longing to the th interval, bits in the obtained, and tentative estimates of the first , are obtained future interval, to account for their interference. Therefore, tentative estimates in the th interval, of the first , are already available from the th interval, and are used as initial estimates in the th interval. The metric defined for an interval should be closely related to the ML metric (8), and should take into account the interference
(9) This metric consists of four terms (9). The first term represents and , and final the shared coefficients between intervals , bits , are estimates of the bits belonging to interval already available and are used in the th interval. The second term (second, third, and fourth summations) corresponds to all the coefficients of the bits belonging to the th interval. The third term (fifth summation) corresponds to the shared coeffi. As mentioned earlier, cients between interval and tentative estimates of the bits belonging to the future interval are obtained to account for their interference. To increase the reliability of these tentative estimates, the last summation is included in the metric, which consists of all the terms in the that are nonzero2 if the first bits in ininterval are estimated. Unlike the sequential detector [27], terval which uses the conventional detector to get tentative estimates of future bits to account for their interference, in the greedy approach, better estimates are obtained by including all the terms , that are affected when the bits are estimated. By substituting the final estimates of the bits belonging to the , we can write the metric in (9) as previous interval
(10) 2This is because we initialize the vector b to be zero before we start the algorithm.
(7)
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where, as shown in (11) at the bottom of the page, are the final estimates obtained from the previous interval. This is equivalent to subtracting the interference due to the bits in the previous interval from the outputs of the matched filters (11). To determine the order of detection, the greedy algorithm sorts, in descending order, the absolute values of the coefficients in
(12) which corresponds to the first three lines in (9). As for the synchronous case, the subscripts of these coefficients indicate in consists of all which order the bits must be estimated. the terms in the likelihood metric that contain the bits , while consists of all the terms that are affected when the bits are estimated. stages, which The algorithm consists of, at most, is the total number of coefficients in (12), and each bit is estitimes. Based on Lemma 1, in some cases the mated up to . Therefore, the same number of stages may be less than algorithm presented for the synchronous case can be used for every interval in the asynchronous case. The greedy MUD algorithm for the asynchronous case is described as follows. and . Step 1) Set Step 2) Find all the coefficients (12) that belong to the th interval
Step 3)
Step 4) Step 5) Step 6)
or Sort the absolute values of the coefficients in in . If , then descending order, . If and , then . If and , then . Based on Lemma 1, delete the stages that are redun. dant and find Set and . , then find all the possible metric If values (10), corresponding to all the possible in bit vectors obtained by substituting . Set . Choose the largest metric values and the corresponding . Else, , find vectors all the possible metric values (10), corresponding
if otherwise
to all the possible bit vectors obtained by substituting in . Set . Choose the largest metric values . and the corresponding vectors , then and go back to Step 6, Step 7) If else proceed. Step 8) The final estimates of , are the estimates in the bit vector corresponding to the largest metric value. , then , and go back to Step 2, Step 9) If else stop. The complexity of the algorithm can be calculated as follows. The number of terms that are sorted in one -bits in; the complexity of sorting is . terval is is calculated times for the terms and The metric times for the term , if and/or , else, it times, respectively. Overall, the metric is calculated and times. Therefore, the is calculated complexity of the algorithm per-bit interval is on the order of per users, , which is significantly and for large it is smaller than the computational complexity of the optimum de, if and . tector for IV. SIMULATION RESULTS Analytical examination of the performance of the algorithm is intractable, since the detector has a nonfor arbitrary linear structure. Therefore, the performance of the algorithm is evaluated using computer simulations. A. Synchronous CDMA System In order to show the performance of the greedy MUD algorithm, we conducted several simulations that are used to evaluate and compare its performance with other detectors. In all of the following simulations, we consider a bit-synchronous CDMA system with binary random signatures and with equal to 31. New signature waveforms are spreading factor generated every 100 bits. The obtained results can be viewed as an upper bound for a system with a good set of signature waveforms. 1) AWGN Channel: In the first set of simulations, Figs. 2–4, we assumed perfect power control, i.e., all users have the same . Fig. 2 shows the perreceived energies formance of the greedy MUD, the optimum detector, and several . Specifically, we consider suboptimum detectors, for the following suboptimum detectors: conventional detector, successive interference cancellation (SIC) [6], decorrelator [28], and two-stage detector with conventional or decorrelator detectors in the first stage [29]. The greedy MUD achieves almost and outperthe same performance as the optimum for forms the considered suboptimum detectors. The performance
and
(11)
ALRUSTAMANI AND VOJCIC: A NEW APPROACH TO GREEDY MULTIUSER DETECTION
Fig. 2. Bit error probability for K = 10; N = 31, synchronous transmission, random spreading sequences, and equal amplitudes over AWGN channel.
Fig. 3. Bit error probability for K = 20; N = 31, synchronous transmission, random spreading sequences, and equal amplitudes over AWGN channel.
of the system for is shown in Fig. 3. The algorithm outperforms the two-stage detector with the decorwith dB for both relator in the first stage for and . This is because, unlike the decorrelator detector, error-free demodulation in the absence of noise is not achievable by the algorithm, since the algorithm considers a subset of the possible bit combinations and might not converge to the global maximum. However, the performance of the algorithm , since in the high SNR region improves significantly for a larger number of bit combinations are examined. It should be mentioned that when forward error correction is used, we are interested in low to moderate SNR, where the greedy multiuser detector performance is very close to the optimum performance [12]. One aspect that needs to be examined in the performance of the greedy algorithm is the presence of the error floor in the region. The joint multiuser detector partitions the high
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Fig. 4. Bit error probability as a function of the number of users for N = 31, synchronous transmission, random spreading sequences, and equal amplitudes over AWGN channel.
observation space into decision regions , where each region corresponds to a certain hypothesis or possible vector . Any suboptimum approach simplifies the boundaries of joint optimum detection and thus, reduces the complexity of the detection process. In some cases, however, is transmitted, denoted by the noiseless observations when , might be included in the decision region of a neighboring hypothesis, due to the simplification of these boundaries. As the noise level approaches zero, the vector , where given is transmitted, approaches , and consequently, an is transmitted. This will error will always occur whenever result in an error floor in the performance of the algorithm in the high SNR region, and the number of noiseless observations that mitigate from their decision region determines the level of the error floor. However, as the value of increases, the boundaries defined by the greedy algorithm approaches the optimum boundaries, and the number of such observations is reduced as well, resulting in lower levels of error floor. Fig. 4 shows the bit-error rate (BER) as a function of the dB. For a BER of , number of users, , for the capacity in terms of the number of users for the suboptimum detectors are 3, 5, 8, 14, and 18 users for conventional, SIC [6], two-stage detector with conventional in the first stage [29], decorrelator [28] and two-stage detector with decorrelator in the first stage [29], respectively. On the other hand, the capacity of 1, 2, 4, 6, and 8 is 19, 24, 29, 31, the greedy MUD for and 32 users, respectively. Therefore, a significant increase in , compared to the other examined capacity is observed for suboptimum detectors. and algorithms, with and In [10] the results for the without a whitening filter, are obtained for the same system. The algorithm, with three the maximum allowcapacity of the able number of surviving paths and without a whitening filter, is 13 users, while with the whitening filter it is 25 users, which is greatly improved and it is similar to the one for the greedy . Therefore, the greedy algorithm, with no algorithm for
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Fig. 5. Bit error probability for K = 20; N = 31, synchronous transmission, random spreading sequences and frequency-nonselective Rayleigh-fading channel.
Fig. 6. Bit error probability for K = 30; N = 31, synchronous transmission, random spreading sequences and frequency-nonselective Rayleigh-fading channel.
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Bit-error probability as a function of the number of users for N = 31; M = 50, asynchronous transmission, random sequences and delays and
Fig. 7.
equal amplitudes, over AWGN channel.
greedy MUD and the two-stage detector with the decorrelator . For SNR dB, the greedy in the first stage for significantly outperforms all other examined suboptimum detectors. For low SNR, the greedy detector performance is close to the single-user bound, whereas for the high SNR region dB), the performance of the greedy algorithm is (SNR improved by increasing the value of to overcome the error floor effect. Comparing the results obtained for the AWGN channel and the frequency-nonselective fading channel, we observe that the greedy MUD algorithm performs better in the fading channel. This is specifically evident in Figs. 3 and 5. This is because the greedy MUD significantly exploits the variability and sparsity of the coefficients of the ML metric, which, in turn, depend on the average received powers of the users. When there are large differences between the values of the coefficients in the ML metric, it is more likely that the greedy detector approaches the optimum performance. B. Asynchronous CDMA System
whitening filter and with , can achieve the same perforalgorithm with a whitening filter with . mance as the This demonstrates the importance of including the cross-correlation coefficients to improve the performance of a detector employing the greedy principle. 2) Frequency-Nonselective Fading Channel: Figs. 5 and 6 show the performance of the considered detectors over the frequency-nonselective Rayleigh fading channel for and , respectively. The statistical characteristics of the , Fig. 5, fading channel for each user are the same. For is the performance of the greedy MUD algorithm with dB. Furthermore, close to the single-user bound for SNR among all the other considered suboptimum detectors, only the two-stage detector with the decorrelator in the first stage gives the same performance as the greedy MUD algorithm. However, there is a substantial difference in performance between the
In the simulations for the asynchronous case, the signature waveforms and time delays for every user are generated randomly for every frame. 1) AWGN Channel: Fig. 7 shows the BER as a function of dB, for the conventhe number of users, , for tional, sequential [27], and greedy detectors, and the single-user bound. Here, the sequential detector is chosen for comparison because it also attempts to approximate the ML sequence detection. The greedy algorithm significantly outperforms the other examined suboptimum detectors. The capacity of the sequential detector [27], in terms of the number of users at is 14 users, whereas the conventional detector cannot achieve this performance even for four users. On the other hand, for the is greater than greedy MUD detector, the capacity at 36 users for all the examined values of and . It should be noted that the estimated capacities of all the examined detectors
ALRUSTAMANI AND VOJCIC: A NEW APPROACH TO GREEDY MULTIUSER DETECTION
Fig. 8. Bit-error probability for K = 20; N = 31; M = 50, asynchronous transmission, random spreading sequences and delays, and equal amplitudes over AWGN channel.
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Fig. 10. Bit error probability for K = 20; N = 15; M = 50, asynchronous transmission, random spreading sequences and delays, and frequencynonselective Rayleigh fading channel.
detector is close to the single-user bound for SNR dB. Unlike the results for the AWGN channel, the performance dB improves by increasing of the algorithm for SNR the window size , while it is not affected by the value of . Fig. 10 shows the performance of the greedy algorithm and users and the examined suboptimum detectors for . The results show that even for a high load , the greedy algorithm performance is close to the single-user dB. For SNR dB, the performance bound for SNR of the greedy algorithm is improved by increasing the value of the window size and number of the largest values examined in every stage . V. CONCLUSION
Fig. 9. Bit-error probability for K = 20; N = 31; M = 50, asynchronous transmission, random spreading sequences and delays, and frequencynonselective Rayleigh fading channel.
would approximately double had we included the phase asynchronism in the model. The performance of the considered deand for is shown in Fig. 8. tectors versus The results for the greedy algorithm for the AWGN channel show that the performance is more sensitive to the value of , than the window size . The performance of the algorithm for different values of and the same is almost the same. However for the same , significant gain is achieved by increasing the value of . This suggests that the performance of the greedy MUD for AWGN channel is more sensitive to the greedy optimization of the metric in every interval than the interval size . 2) Frequency-Nonselective Fading Channel: The performance of the greedy MUD algorithm and the conventional detector in the frequency-nonselective Rayleigh fading channel is shown in Fig. 9. The performance of the greedy for
In this paper, we proposed a new suboptimum multiuser detector for synchronous and asynchronous CDMA systems, which utilizes a greedy algorithm to decide on the transmitted bits. The algorithm considers both the outputs of the matched filters and cross correlations of the signature waveforms to decide on the order in which users are detected to give a performance close to the optimum one. The performance of the proposed scheme is compared with different suboptimum multiuser detectors proposed in the literature for the AWGN channel and frequency-nonselective Rayleigh fading channel. The complexity of the proposed detector is approximately , and it is evident that its performance is significantly better and more robust compared to other examined suboptimum schemes. The proposed greedy approach considerably outperforms these suboptimum schemes, especially for moderate and high loads in low and moderate SNR regions. The results show that the greedy MUD algorithm increases the capacity and improves the performance in the presence of fading. For many cases that we could check, the proposed detector has identical or almost identical performance to that of the optimum detector in the range of SNR of most practical interest.
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REFERENCES [1] S. Verdu, “Minimum probability of error for asynchronous Gaussian multiple-access channels,” IEEE Trans. Inform. Theory, vol. IT-32, pp. 85–95, Jan. 1986. , “Computational complexity of optimum multiuser detection,” Al[2] gorithmica, vol. 4, pp. 303–312, 1989. [3] , Multiuser Detection. Cambridge, U.K.: Cambridge Univ. Press, 1998. [4] T. C. Hu, Combinatorial Algorithms. Reading, MA: Addison-Wesley, 1982. [5] K. P. Bogart, Introductory Combinatorics. New York: Pitman, 1983. [6] J. Holtzman, “DS/CDMA successive interference cancellation,” in Code Division Multiple Access Communications, S. Glisic and P. Leppanen, Eds, Dordrecht, The Netherlands: Kluwer, 1995, pp. 161–182. [7] A. Duel-Hellen, “Decorrelating decision-feedback multiuser detector for synchronous code-division multiple-access channel,” IEEE Trans. Commun., vol. 41, pp. 285–290, Feb. 1993. [8] L. Wei and C. Schlegel, “Synchronous DS-CDMA systems with improved decorrelating decision-feedback multiuser detection,” IEEE Trans. Veh. Technol., vol. 43, pp. 767–772, Aug. 1994. [9] L. K. Rasmussen, T. J. Lim, and T. M. Aulin, “Breadth-first maximum likelihood detection in multiuser CDMA,” IEEE Trans. Commun., vol. 45, pp. 1176–1178, Oct. 1997. [10] 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. [11] A. AlRustamani and B. R. Vojcic, “Greedy-based multiuser detection for CDMA systems,” in Proc. 2000 Conf. on Information Sciences and Systems, vol. II, Princeton, NJ, Mar. 2000, pp. FA8-11–FA8-14. [12] B. Vojcic, A. AlRustamani, and A. Damnjanovic, “Greedy iterative multiuser detection for turbo coded multiuser communications,” presented at the ICT 2000, Mexico, May 2000. [13] A. AlRustamani and B. Vojcic, “Greedy multiuser detection over single-path fading channel,” in Proc. IEEE Sixth Int. Symp. on Spread Spectrum Techniques & Applications (ISSSTA), NJ, Sept. 2000, pp. 708–712. [14] J. G. Proakis, Digital Communications, 3rd ed. New York: McGrawHill, 1995. [15] Z. Zvonar and D. Brady, “Multiuser detection in single-path fading channels,” IEEE Trans. Commun., vol. 42, pp. 1729–1739, Feb.-Apr. 1994. [16] P. M. Pardalos, “Construction of test problems in quadratic bivalent programming,” ACM Trans. Math. Software, vol. 17, pp. 74–87, 1991. [17] M. W. Carter, “The indefinite zero-one quadratic problem,” Discrete Appl. Math., vol. 7, pp. 23–44, 1984. [18] V. P. Gulati, S. K. Gupta, and A. K. Mittal, “Unconstrained quadratic bivalent programming problem,” Euro. J. Oper. Res., vol. 15, pp. 121–125, 1984. [19] F. Barahona, “A solvable case for quadratic 0–1 programming,” Discrete Appl. Math., vol. 13, pp. 127–137, 1986. [20] P. Kempf, “A nonorthogonal synchronous DS-CDMA case, where successive cancellation and maximum-likelihood multiuser detectors are equivalent,” in Proc. 1995 IEEE Int. Symp. on Information Theory, Sept. 1995, p. 321. [21] J. C. Picard and M. Queyranne, “Selected applications of min cut in networks,” INFOR, vol. 20, no. 4, pp. 395–422, 1982.
[22] R. Learned, A. Willsky, and D. Boroson, “Low complexity optimal joint detection for oversaturated multiple access communications,” IEEE Trans. Signal Processing, vol. 45, pp. 113–123, Jan. 1997. [23] J. A. F. Ross and D. P. Taylor, “Vector assignment scheme for users in -dimensional global additive channel,” Electron. Lett., vol. 28, no. 17, pp. 1634–1636, Aug. 1992. [24] C. Sankaran and A. Ephremides, “Solving a class of optimum multiuser detection problems with polynomial complexity,” IEEE Trans. Inform. Theory, vol. 44, pp. 1958–1961, Sept. 1998. [25] L. Welch, “Lower bounds on the maximum crosscorrelation of signals,” IEEE Trans. Inform. Theory, vol. 20, pp. 397–399, May 1974. [26] J. L. Massey and T. Mittelholzer, “Welch’s bound and sequence sets for code-division multiple-access systems,” in Sequence II, Methods in Communication, Security, and Computer Science, R. Capocelli, A. De Santis, and U. Vaccaro, Eds. New York: Springer-Verlag, 1993. [27] Z. Xie and C. K. Rushforth, “Multiuser signal detection using sequential decoding,” IEEE Trans. Commun., vol. 38, pp. 578–583, May 1990. [28] R. Lupas and S. Verdu, “Linear multiuser detectors for synchronous code-division multiple-access channels,” IEEE Trans. Inform. Theory, vol. 35, pp. 123–136, Jan. 1989. [29] M. Varanasi and B. Aazhang, “Multistage detection in asynchronous code-division multiple-access communications,” IEEE Trans. Commun., vol. 38, pp. 509–519, Apr. 1990.
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Amina AlRustamani received the B.S., M.S., and D.Sc. degrees in electrical engineering from George Washington University, Washington, DC, in 1993, 1996, and 2001, respectively. She is currently with Dubai Internet City, Dubai, UAE. Her areas of interest are spread spectrum, multiuser detection, wireless/mobile networks, and satellite communications.
Branimir R. Vojcic (M’86-SM’96) received the Dipl. and D.Sc. degrees in electrical engineering from the Faculty of Electrical Engineering, University of Belgrade, Yugoslavia, in 1980, 1986, and 1989, respectively. Since 1991, he has been a Professor with the Department of Electrical and Computer Engineering, George Washington University, Washington, DC, where he is also the Department Chairman. His current research interests are in the areas of communication theory, cellular and satellite networks, code-division multiple-access, multiuser detection, adaptive antenna arrays and space-time coding, and ad hoc networks. Dr. Vojcic was an Associate Editor for IEEE COMMUNICATIONS LETTERS. He was also a recipient of 1995 National Science Foundation CAREER Award.
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 50, NO. 8, AUGUST 2002
A New Approach to Greedy Multiuser Detection Amina AlRustamani, Member, IEEE, and Branimir R. Vojcic, Senior Member, IEEE
Abstract—In this paper, we propose a new suboptimum multiuser detector for synchronous and asynchronous multiuser communications. In this approach, a greedy strategy is used to maximize the cost function, the maximum-likelihood (ML) metric. The coefficients of the ML metric are utilized as weights indicating in which order bits can be estimated. The complexity of per bit, where is the algorithm is approximately 2 log the number of users. We analyze the performance of the greedy multiuser detection in the additive white Gaussian noise channel as well as in the frequency-nonselective Rayleigh fading channel, and compare it with the optimum detector and several suboptimum schemes such as conventional, successive interference cancellation, decorrelator, sequential, and multistage detectors. The proposed greedy approach considerably outperforms these suboptimum schemes, especially for moderate and high loads in low and moderate signal-to-noise ratio regions. The results show that when there is a significant imbalance in the values of the coefficients of the ML metric due to moderate to high noise, fading, and asynchronous transmission, near-optimum performance is achieved by the greedy detection. Index Terms—Code-division multiaccess, interference suppression, multiuser channels.
I. INTRODUCTION
T
HE exponential complexity and thus, the impracticality [1], [2] of joint optimum multiuser detection in code-division multiple-access (CDMA) systems has driven researchers to explore suboptimum solutions with practical significance in terms of performance and complexity. This resulted in discovering a wide range of possible complexity/performance tradeoffs. A detailed analysis of the most well known suboptimum multiuser detectors, and an extensive list of references, are given in [3]. Although each proposed improving the practicality by substantially reducing the complexity, the performance of these suboptimum schemes is far from the optimum performance, especially for systems with large numbers of users. In this paper, we propose a new suboptimum multiuser scheme that is based on the greedy principle. We show that the performance of the proposed detector is close to the optimum performance, yet with significantly lower complexity. Indeed, the complexity is, at most, polynomial in the number of users.
Paper approved by S. L. Miller, the Editor for Spread Spectrum of the IEEE Communications Society. Manuscript received October 10, 2000; revised June 20, 2001 and November 19, 2001. This paper was presented in part at IEEE 6th International Symposium on Spread Spectrum Techniques and Applications (ISSSTA), NJ, September 2000, and at the 2000 Conference on Information Sciences and Systems (CISS), Princeton, NJ, March 2000. A. AlRustamani is with Dubai Internet City, Dubai (e-mail: [email protected]). B. R. Vojcic is with the Department of Electrical and Computer Engineering, George Washington University, Washington, DC 20052 USA (e-mail: [email protected]). Publisher Item Identifier 10.1109/TCOMM.2002.801493.
Also, we demonstrate comparatively that the performance is significantly better than that of most of the existing suboptimum schemes for a wide range of operating conditions, and specifically for systems with moderate to large numbers of users. The greedy principle is applied to a wide variety of combinatorial optimization problems, such as the knapsack problem, Hoffman coding, and minimum spanning tree problem [4], [5]. Most of these problems have inputs, such as the objects in the knapsack problem, and require obtaining a subset that satisfies certain constraints, for example, the capacity of the knapsack. The greedy method suggests that one can think of an algorithm which works in stages, considering one input at a time. At each stage, the algorithm examines one input and decides on a partially constructed solution or on a complete solution. The main distinctive feature of the greedy principle is that the inputs are considered in an order determined by some selection procedure that is based on some optimization measure, such as maximizing the increase in the profit in every stage. For example, some of the criteria that are used to order the objects in the knapsack problem are profit, weight, or density (profit/weight). In the multiuser detection problem, the maximum-likelihood (ML) metric is a quadratic function of variables (the transmitted bits) that take discrete values. The task of the optimum detector is to find the values of the transmitted bits that maximize the ML metric. The quadratic nature of the problem and the discrete values of the variables give rise to the exponential complexity of the optimum solution. To apply the greedy concept to reduce the complexity of the solution, we need to define the inputs and the optimization measure that will determine the order of these inputs. Some of the suboptimum detectors that are present in the literature can be viewed as algorithms that are based on the greedy principle [6]–[10]. The inputs of these detectors are the received signals of users in the system. The signals are ordered based on their received powers, as in successive interference cancellation [6] and the decorrelating decision-feedback detector [7]. In every stage, the bit corresponding to the user in that stage is estimated. In [8]–[10] the authors adapt the same algorithms given in [6] and [7], and show that if more than one partially constructed solution is passed from one stage to the other, the performance of such detectors can be significantly improved. Moreover, in [8], the authors show that for certain scenarios, the way users are ordered has no significant effect on the performance of such detectors. Instead of ordering the users, the algorithm proposed in this paper views the coefficients of the bits in the ML metric as weights (inputs) that indicate the order in which bits could be estimated [11]–[13]. The dominating coefficients, the large values, will have more impact on the value of the ML metric
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ALRUSTAMANI AND VOJCIC: A NEW APPROACH TO GREEDY MULTIUSER DETECTION
and thus, carry more weight in determining the values of the bits associated with them than smaller terms. The complexity , and we show that its of the algorithm is performance is significantly better than that of most of the existing suboptimum detectors for systems with moderate to high loads. Specifically, the results show that when there is a significant imbalance in the values of the coefficients of the ML metric due to moderate to high noise, fading, and asynchronous transmission, near-optimum performance is achieved by the proposed greedy detection. The rest of the paper is organized as follows. In Section II, the system model and the greedy multiuser detection (MUD) algorithm for synchronous CDMA is presented. The proposed algorithm for the asynchronous CDMA system is given in Section III as well. In Section IV, simulation results for the proposed greedy detector and other considered detectors for the additive white Gaussian noise (AWGN) channel and for the nonselective Rayleigh fading channel are presented. Concluding remarks are given in Section V. II. SYNCHRONOUS CDMA We consider a frequency-nonselective fading channel for which the multipath components are not resolvable [14]. In this case, the received signal is the transmitted signal multiplied by a complex-valued random process representing the time-variant characteristics of the channel. Furthermore, we assume that the signal duration is significantly smaller than the coherence time of the channel and thus, the channel is slowly fading and the attenuation and phase shift are essentially constant for the duration of at least one bit interval [14]. Therefore, the equivalent low-pass received signal in the th interval for a synchronous CDMA system over such a channel is
(1) is the number of users, and is the bit interval; , and represent energy per bit, unit-energy signature waveform, and bit value of the th user , are in the th bit interval, respectively; independent zero-mean complex-valued Gaussian random, , and such that is the space-time correlation function of the channel associated with th user. denotes complex conjugate is mathematical expectation. is a complex and zero-mean Gaussian random process. The receiver consists of a bank of matched filters and a multiuser detector. The output of the filter matched to the signature waveform of user and is sampled at where
(2)
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where denotes the cross correlation of the signature waveforms of users and , and denotes the noise at the output of the th matched filter
The AWGN channel can be viewed as a special case of the for , model represented above, where is AWGN of one-sided power spectral density . and The outputs of the matched filters are sufficient statistics for optimum multiuser detection and can be expressed in vector form as [3], [15]
where is the normalized cross-correlation matrix of the signature waveforms, , and is the noise vector with . The superscript autocorrelation matrix denotes the complex conjugate transpose. The ML receiver selects the bits that maximize the metric [1], [15], where is the real component of (3) Hence, the optimum receiver for the synchronous case consists single-user matched filters followed by a detector that of possible transmitted informacomputes the metrics for the tion bit vectors, and selects the vector that gives the largest metric value. In combinatorial optimization, (3) is known as the unconstrained bivalent quadratic programming problem, and it belongs to a class with many different applications [16]–[18]. The , such as density (the ratio characteristics of the matrix of the number of nonzero entries to the total number of entries) , and and sparsity, the noise level affecting the elements in the dimensionality of the problem mainly determine the complexity of the problem and the performance of suboptimum detectors [16], [17]. When the matrix has low density, or when it is diagonally dominant indicating low cross correlations, or when the signal-to-noise ratio (SNR) is low, then finding a near-optimum solution becomes much easier. In this case, most of the proposed algorithms in the literature provide a significant improvement in performance compared to the conventional detector. Furthermore, it has been shown that a careful design of the signature waveforms and thus, the cross-correlation matrix , results in significant reduction in the complexity of the jointly optimum detection [19]–[24]. Indeed, polynomial complexity is achievable if stringent requirements are applied on the design of the signature waveforms [3], [19]–[24]. However, these requirements limit the system capacity [24]–[26], assume synchronous transmission, and cannot be maintained in multipath channels.
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The above discussion suggests that the elements in the maplay a crucial role in determining the optimal bit trix and the AWGN channel, the values. For example, for ML metric is
The largest two coefficients will determine the optimum bits [3]. If or then the outputs of the matched filters determine the optimum . However, if bits, that is or or
or
then the largest absolute value of the matched filter outputs multiplied by the amplitudes and the cross-correlation terms determine the optimum bit decisions. Accordingly, we can apply the . However, most of the existing subopsame idea for timum detectors do not explicitly utilize the elements of the mato decide on the bit values. These values are, rather, trix used to reconstruct and then suppress interference due to other , we utilize users. Therefore, in addition to the elements in as well in the detection process. the elements in The greedy MUD algorithm is based on the ML metric in (3), or, equivalently
(4) where
By examining the above equation, we can view the ML metric real numbers, where deciding on the as a sum of directly affects terms sign of one bit in the summation. Based on the greedy principle, the proposed and as weights that algorithm views the coefficients indicate in which order the bits of users could be sequentially estimated. By ordering the absolute values of the coefficients in descending order, and examining the incremental effect of each coefficient on the whole ML metric value, we choose the values of the bits that, in combination with the corresponding coefficient, make as large as possible positive contributions to the ML metric. When the coefficient is considered, the effect or is examined in of choosing the corresponding bit as coefficients in the conjunction to its impact on the other , then the metric. Similarly, when considering the coefficient effect of the corresponding bits and on the other
terms is examined. Thus, to maximize the ML metric (4), we always try to resolve the information bits in the order of their contribution to the ML metric. and Initially, the absolute value of the coefficients, and , are sorted in descending order. The subscripts of these coefficients indicate in which order the bits must be estimated. Since for every bit interval the values of depend on the outputs of the matched the coefficients filters, and all the coefficients depend on the amplitudes of the users, the order of detection is dynamic on a bit-interval basis. The algorithm consists of, at most, stages, which is the total number of coefficients in (4), and times. The th stage correeach bit is estimated up to sponds to the th coefficient after sorting. In some cases, stages the number of stages might be less than for reasons that will be evident later. Let the elements of the be the absolute values of the vector be coefficients after sorting. Let in the th stage and the tentative estimate of the vector . Fig. 1 illustrates the main steps in the let , then the ML greedy algorithm. In stage , if metric, in (4), is calculated twice for the two possible vectors and in , obtained by substituting and i.e.,\break . The bit vector that results in the largest metric value is chosen as the new estimate of the vector in the th stage if and only if and On the other hand, if the coefficient is of the form , then the ML metric, in (4), is calculated four times for the four possible vectors obtained by substituting in , i.e., , . The bit and vector that results in the largest metric value is chosen as the new estimate of the vector in the th stage if and only if and Therefore, in the th stage, one or two bits in , depending on the coefficient corresponding to the th stage, are examined and updated based on the decisions on the bits made in the previous stages, corresponding to the largest increment of the ML metric. One should note that calculating the metric in (4) in every stage is equivalent to subtracting the interference of users estimated in the previous stages and either jointly detecting the bits , or deciding on of two users, if the respective coefficient is the bit of one user if the respective coefficient is . Moreover, estimates of all the bits might not be available, at most, until . This can occur if all the coefficients of stage
ALRUSTAMANI AND VOJCIC: A NEW APPROACH TO GREEDY MULTIUSER DETECTION
Fig. 1.
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Flow diagram of the greedy algorithm for synchronous CDMA.
a specific user are examined in the last stages in the greedy algorithm. Once estimates of all bits are available or estimates of the bits examined in the stage already exist, then one of the possible bit combinations that should be examined in the stage is the one chosen in the previous stage. Therefore, to reduce the number of computations, the algorithm must forward the vector and the metric value associated with this vector to the next stage. in In addition, the number of stages might be less than some cases. In Lemma 1 we summarize the cases in which the number of stages can be reduced. Lemma 1: In the greedy MUD algorithm, if two consecutive or , or if three consecustages examine or all the other five possible tive stages examine and , then deleting the stages correpermutations of and will not affect the result. sponding to Proof: The vectors examined in the consecutive stages or , or all other possible permutations, have the same elements in all other positions besides and , i.e., same bit estimates for all other users except users and . (and Moreover, the two examined bit vectors in the stages ) are subsets of the four examined vectors in the stage . (and ) will not have an effect Therefore, deleting stages on the results. When discussing the convergence of the algorithm to the global maximum, one needs to think of the algorithm as consisting of two phases. For example, let us assume that estimates of all the bits are available at stage . In the first phase, stages one to , estimates of all the transmitted bits, the
elements in , are obtained. This first phase suggests that and thus, the global maximum is in the neighborhood of it determines the region that must be examined by the to , the second phase. In the second phase, stages neighboring points, vectors differing in one or two elements compared to the vector chosen in every stage, are examined. Therefore, the second phase of the algorithm is a search phase for a maximum point in the region indicated by the first phase. Since the algorithm might not converge to the global maximum region, and also the number of the examined vectors in the second phase is determined by the number of stages, the final estimate of might not be the global maximum. One way to increase the possibility of converging to the global largest metrics and the maximum is to forward the corresponding bit vectors from the current stage to the next will be estimates of one. Therefore, the input to stage and the corresponding . metric values , two or four metric In stage , for every or metric values, the values are calculated. Then among largest values and the corresponding bit vectors are chosen as the output of the stage. As the value of is increased, the algorithm explores different regions, and thus, the possibility of converging to the global maximum also increases. Whether the algorithm converges to the global maximum and the speed of convergence is determined by the order of the weights after sorting, and the variability in the values of the weights which is a measure of the increase in the value of the metric in every stage.
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The greedy algorithm is summarized with the following steps1 : Step 1) Sort the absolute values of the coefficients , , and in a descending order.
EXAMPLE
OF
TABLE I SYNCHRONOUS GREEDY MUD ALGORITHM FOR
K = 4 AND L = 1
sort descending
where . Step 2) Based on Lemma 1, delete stages that are redundant . and find Step 3) Set and . , then find all the possible Step 4) If metric values, corresponding to all the possible in bit vectors obtained by substituting . Set , where is the minimum of the set of values . Choose the largest metric values and the . corresponding vectors , find all the possible metric Else, values, corresponding to all the possible bit vecin tors obtained by substituting . Set . Choose the largest metric values and the corresponding . vectors , then and go back to Step Step 5) If 4, else stop and the bit vector corresponding to the largest metric value is the final estimate of the bits. and An example for the greedy MUD algorithm for is illustrated in Table I. Although the complexity of the greedy algorithm for this specific example is higher than that of the optimum detector, this example is chosen just for illustration purposes. The three gray rows indicate the deleted stages. The calculations in these stages are also given to demonstrate the reasoning behind Lemma 1. Estimates of all bits are available in the sixth stage. The complexity of the algorithm can be calculated as follows. terms that are sorted and examined. The There are . The complexity of sorting is and four ML metric, , is calculated twice for the terms , if and/or , else it is caltimes for the term culated once and three times, respectively. Thus, in total, if all terms are considered, the metric is calculated times. Therefore, the complexity of the algorithm is , and for large it is , which is significantly values of smaller than the computational complexity of the optimum de. tector for , the algorithm is optimal, since in the stage where For is examined, all possible combinations are considthe term ered and the performance in this case is known [3]. Moreover, 1In the beginning of the algorithm the number of examined vectors might be less than L; thus, is the number of metric values and the corresponding bit vectors that can be passed from one stage to the other.
q
the suboptimum detectors presented in [6]–[9] can be viewed as special cases of the greedy algorithm, in which only the , are considered to determine the coefficients and only are used for sorting order of detection. If and selection, the greedy algorithm is equivalent to successive interference cancellation [6], while it is equivalent to the decorrelating decision-feedback detector [7] if a whitening filter is and only used prior to the greedy detection. For are used, the greedy algorithm is equivalent to the tree search algorithms presented in [8]–[10], with and without a whitening filter. III. ASYNCHRONOUS CDMA SYSTEM For the same model presented in the synchronous case for the frequency-nonselective fading channel, the equivalent low-pass received waveform for asynchronous CDMA system over such a channel can be expressed as [3], [12], [15]
(5) is the frame length and is the time delay of the where th user. As in the synchronous case, for the AWGN channel, for , and and is AWGN of one-sided power spectral density . Without loss
ALRUSTAMANI AND VOJCIC: A NEW APPROACH TO GREEDY MULTIUSER DETECTION
of generality, we can assume that The outputs of the matched filters are
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due to previous and future intervals. Therefore, we define the following metric for the th interval, which is part of the whole metric in (8):
(6) terms, in which The ML metric can be written as a sum of each term depends on components of and any consecutive components [3] as shown in (7) at the bottom terms share of the page, where
, and for , and represents the modulo- remainder of . Consequently, dynamic programming can be used to find the longest path in a layered di; thus, the comrected graph that maximizes the ML metric . plexity of jointly optimum detection is independent of The ML metric in (7) can be rewritten as (8) where
The ML detector chooses that maximizes (8). As in the synchronous case, the greedy algorithm can be applied on the whole metric, however, the complexity will depend on the frame length . As a simpler approach, the greedy MUD algorithm is applied sequentially on -bit intervals to maximize parts of the ML metric, i.e., the algorithm is applied intervals of the frame, where indicates on the smallest integer greater than or equal to . Therefore, the suboptimality is due to the partitioning of the ML metric and the greedy maximization of every partition. In the th interval, , the bits where are estimated. Final estimates of the bits be, are longing to the th interval, bits in the obtained, and tentative estimates of the first , are obtained future interval, to account for their interference. Therefore, tentative estimates in the th interval, of the first , are already available from the th interval, and are used as initial estimates in the th interval. The metric defined for an interval should be closely related to the ML metric (8), and should take into account the interference
(9) This metric consists of four terms (9). The first term represents and , and final the shared coefficients between intervals , bits , are estimates of the bits belonging to interval already available and are used in the th interval. The second term (second, third, and fourth summations) corresponds to all the coefficients of the bits belonging to the th interval. The third term (fifth summation) corresponds to the shared coeffi. As mentioned earlier, cients between interval and tentative estimates of the bits belonging to the future interval are obtained to account for their interference. To increase the reliability of these tentative estimates, the last summation is included in the metric, which consists of all the terms in the that are nonzero2 if the first bits in ininterval are estimated. Unlike the sequential detector [27], terval which uses the conventional detector to get tentative estimates of future bits to account for their interference, in the greedy approach, better estimates are obtained by including all the terms , that are affected when the bits are estimated. By substituting the final estimates of the bits belonging to the , we can write the metric in (9) as previous interval
(10) 2This is because we initialize the vector b to be zero before we start the algorithm.
(7)
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where, as shown in (11) at the bottom of the page, are the final estimates obtained from the previous interval. This is equivalent to subtracting the interference due to the bits in the previous interval from the outputs of the matched filters (11). To determine the order of detection, the greedy algorithm sorts, in descending order, the absolute values of the coefficients in
(12) which corresponds to the first three lines in (9). As for the synchronous case, the subscripts of these coefficients indicate in consists of all which order the bits must be estimated. the terms in the likelihood metric that contain the bits , while consists of all the terms that are affected when the bits are estimated. stages, which The algorithm consists of, at most, is the total number of coefficients in (12), and each bit is estitimes. Based on Lemma 1, in some cases the mated up to . Therefore, the same number of stages may be less than algorithm presented for the synchronous case can be used for every interval in the asynchronous case. The greedy MUD algorithm for the asynchronous case is described as follows. and . Step 1) Set Step 2) Find all the coefficients (12) that belong to the th interval
Step 3)
Step 4) Step 5) Step 6)
or Sort the absolute values of the coefficients in in . If , then descending order, . If and , then . If and , then . Based on Lemma 1, delete the stages that are redun. dant and find Set and . , then find all the possible metric If values (10), corresponding to all the possible in bit vectors obtained by substituting . Set . Choose the largest metric values and the corresponding . Else, , find vectors all the possible metric values (10), corresponding
if otherwise
to all the possible bit vectors obtained by substituting in . Set . Choose the largest metric values . and the corresponding vectors , then and go back to Step 6, Step 7) If else proceed. Step 8) The final estimates of , are the estimates in the bit vector corresponding to the largest metric value. , then , and go back to Step 2, Step 9) If else stop. The complexity of the algorithm can be calculated as follows. The number of terms that are sorted in one -bits in; the complexity of sorting is . terval is is calculated times for the terms and The metric times for the term , if and/or , else, it times, respectively. Overall, the metric is calculated and times. Therefore, the is calculated complexity of the algorithm per-bit interval is on the order of per users, , which is significantly and for large it is smaller than the computational complexity of the optimum de, if and . tector for IV. SIMULATION RESULTS Analytical examination of the performance of the algorithm is intractable, since the detector has a nonfor arbitrary linear structure. Therefore, the performance of the algorithm is evaluated using computer simulations. A. Synchronous CDMA System In order to show the performance of the greedy MUD algorithm, we conducted several simulations that are used to evaluate and compare its performance with other detectors. In all of the following simulations, we consider a bit-synchronous CDMA system with binary random signatures and with equal to 31. New signature waveforms are spreading factor generated every 100 bits. The obtained results can be viewed as an upper bound for a system with a good set of signature waveforms. 1) AWGN Channel: In the first set of simulations, Figs. 2–4, we assumed perfect power control, i.e., all users have the same . Fig. 2 shows the perreceived energies formance of the greedy MUD, the optimum detector, and several . Specifically, we consider suboptimum detectors, for the following suboptimum detectors: conventional detector, successive interference cancellation (SIC) [6], decorrelator [28], and two-stage detector with conventional or decorrelator detectors in the first stage [29]. The greedy MUD achieves almost and outperthe same performance as the optimum for forms the considered suboptimum detectors. The performance
and
(11)
ALRUSTAMANI AND VOJCIC: A NEW APPROACH TO GREEDY MULTIUSER DETECTION
Fig. 2. Bit error probability for K = 10; N = 31, synchronous transmission, random spreading sequences, and equal amplitudes over AWGN channel.
Fig. 3. Bit error probability for K = 20; N = 31, synchronous transmission, random spreading sequences, and equal amplitudes over AWGN channel.
of the system for is shown in Fig. 3. The algorithm outperforms the two-stage detector with the decorwith dB for both relator in the first stage for and . This is because, unlike the decorrelator detector, error-free demodulation in the absence of noise is not achievable by the algorithm, since the algorithm considers a subset of the possible bit combinations and might not converge to the global maximum. However, the performance of the algorithm , since in the high SNR region improves significantly for a larger number of bit combinations are examined. It should be mentioned that when forward error correction is used, we are interested in low to moderate SNR, where the greedy multiuser detector performance is very close to the optimum performance [12]. One aspect that needs to be examined in the performance of the greedy algorithm is the presence of the error floor in the region. The joint multiuser detector partitions the high
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Fig. 4. Bit error probability as a function of the number of users for N = 31, synchronous transmission, random spreading sequences, and equal amplitudes over AWGN channel.
observation space into decision regions , where each region corresponds to a certain hypothesis or possible vector . Any suboptimum approach simplifies the boundaries of joint optimum detection and thus, reduces the complexity of the detection process. In some cases, however, is transmitted, denoted by the noiseless observations when , might be included in the decision region of a neighboring hypothesis, due to the simplification of these boundaries. As the noise level approaches zero, the vector , where given is transmitted, approaches , and consequently, an is transmitted. This will error will always occur whenever result in an error floor in the performance of the algorithm in the high SNR region, and the number of noiseless observations that mitigate from their decision region determines the level of the error floor. However, as the value of increases, the boundaries defined by the greedy algorithm approaches the optimum boundaries, and the number of such observations is reduced as well, resulting in lower levels of error floor. Fig. 4 shows the bit-error rate (BER) as a function of the dB. For a BER of , number of users, , for the capacity in terms of the number of users for the suboptimum detectors are 3, 5, 8, 14, and 18 users for conventional, SIC [6], two-stage detector with conventional in the first stage [29], decorrelator [28] and two-stage detector with decorrelator in the first stage [29], respectively. On the other hand, the capacity of 1, 2, 4, 6, and 8 is 19, 24, 29, 31, the greedy MUD for and 32 users, respectively. Therefore, a significant increase in , compared to the other examined capacity is observed for suboptimum detectors. and algorithms, with and In [10] the results for the without a whitening filter, are obtained for the same system. The algorithm, with three the maximum allowcapacity of the able number of surviving paths and without a whitening filter, is 13 users, while with the whitening filter it is 25 users, which is greatly improved and it is similar to the one for the greedy . Therefore, the greedy algorithm, with no algorithm for
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Fig. 5. Bit error probability for K = 20; N = 31, synchronous transmission, random spreading sequences and frequency-nonselective Rayleigh-fading channel.
Fig. 6. Bit error probability for K = 30; N = 31, synchronous transmission, random spreading sequences and frequency-nonselective Rayleigh-fading channel.
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 50, NO. 8, AUGUST 2002
Bit-error probability as a function of the number of users for N = 31; M = 50, asynchronous transmission, random sequences and delays and
Fig. 7.
equal amplitudes, over AWGN channel.
greedy MUD and the two-stage detector with the decorrelator . For SNR dB, the greedy in the first stage for significantly outperforms all other examined suboptimum detectors. For low SNR, the greedy detector performance is close to the single-user bound, whereas for the high SNR region dB), the performance of the greedy algorithm is (SNR improved by increasing the value of to overcome the error floor effect. Comparing the results obtained for the AWGN channel and the frequency-nonselective fading channel, we observe that the greedy MUD algorithm performs better in the fading channel. This is specifically evident in Figs. 3 and 5. This is because the greedy MUD significantly exploits the variability and sparsity of the coefficients of the ML metric, which, in turn, depend on the average received powers of the users. When there are large differences between the values of the coefficients in the ML metric, it is more likely that the greedy detector approaches the optimum performance. B. Asynchronous CDMA System
whitening filter and with , can achieve the same perforalgorithm with a whitening filter with . mance as the This demonstrates the importance of including the cross-correlation coefficients to improve the performance of a detector employing the greedy principle. 2) Frequency-Nonselective Fading Channel: Figs. 5 and 6 show the performance of the considered detectors over the frequency-nonselective Rayleigh fading channel for and , respectively. The statistical characteristics of the , Fig. 5, fading channel for each user are the same. For is the performance of the greedy MUD algorithm with dB. Furthermore, close to the single-user bound for SNR among all the other considered suboptimum detectors, only the two-stage detector with the decorrelator in the first stage gives the same performance as the greedy MUD algorithm. However, there is a substantial difference in performance between the
In the simulations for the asynchronous case, the signature waveforms and time delays for every user are generated randomly for every frame. 1) AWGN Channel: Fig. 7 shows the BER as a function of dB, for the conventhe number of users, , for tional, sequential [27], and greedy detectors, and the single-user bound. Here, the sequential detector is chosen for comparison because it also attempts to approximate the ML sequence detection. The greedy algorithm significantly outperforms the other examined suboptimum detectors. The capacity of the sequential detector [27], in terms of the number of users at is 14 users, whereas the conventional detector cannot achieve this performance even for four users. On the other hand, for the is greater than greedy MUD detector, the capacity at 36 users for all the examined values of and . It should be noted that the estimated capacities of all the examined detectors
ALRUSTAMANI AND VOJCIC: A NEW APPROACH TO GREEDY MULTIUSER DETECTION
Fig. 8. Bit-error probability for K = 20; N = 31; M = 50, asynchronous transmission, random spreading sequences and delays, and equal amplitudes over AWGN channel.
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Fig. 10. Bit error probability for K = 20; N = 15; M = 50, asynchronous transmission, random spreading sequences and delays, and frequencynonselective Rayleigh fading channel.
detector is close to the single-user bound for SNR dB. Unlike the results for the AWGN channel, the performance dB improves by increasing of the algorithm for SNR the window size , while it is not affected by the value of . Fig. 10 shows the performance of the greedy algorithm and users and the examined suboptimum detectors for . The results show that even for a high load , the greedy algorithm performance is close to the single-user dB. For SNR dB, the performance bound for SNR of the greedy algorithm is improved by increasing the value of the window size and number of the largest values examined in every stage . V. CONCLUSION
Fig. 9. Bit-error probability for K = 20; N = 31; M = 50, asynchronous transmission, random spreading sequences and delays, and frequencynonselective Rayleigh fading channel.
would approximately double had we included the phase asynchronism in the model. The performance of the considered deand for is shown in Fig. 8. tectors versus The results for the greedy algorithm for the AWGN channel show that the performance is more sensitive to the value of , than the window size . The performance of the algorithm for different values of and the same is almost the same. However for the same , significant gain is achieved by increasing the value of . This suggests that the performance of the greedy MUD for AWGN channel is more sensitive to the greedy optimization of the metric in every interval than the interval size . 2) Frequency-Nonselective Fading Channel: The performance of the greedy MUD algorithm and the conventional detector in the frequency-nonselective Rayleigh fading channel is shown in Fig. 9. The performance of the greedy for
In this paper, we proposed a new suboptimum multiuser detector for synchronous and asynchronous CDMA systems, which utilizes a greedy algorithm to decide on the transmitted bits. The algorithm considers both the outputs of the matched filters and cross correlations of the signature waveforms to decide on the order in which users are detected to give a performance close to the optimum one. The performance of the proposed scheme is compared with different suboptimum multiuser detectors proposed in the literature for the AWGN channel and frequency-nonselective Rayleigh fading channel. The complexity of the proposed detector is approximately , and it is evident that its performance is significantly better and more robust compared to other examined suboptimum schemes. The proposed greedy approach considerably outperforms these suboptimum schemes, especially for moderate and high loads in low and moderate SNR regions. The results show that the greedy MUD algorithm increases the capacity and improves the performance in the presence of fading. For many cases that we could check, the proposed detector has identical or almost identical performance to that of the optimum detector in the range of SNR of most practical interest.
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Amina AlRustamani received the B.S., M.S., and D.Sc. degrees in electrical engineering from George Washington University, Washington, DC, in 1993, 1996, and 2001, respectively. She is currently with Dubai Internet City, Dubai, UAE. Her areas of interest are spread spectrum, multiuser detection, wireless/mobile networks, and satellite communications.
Branimir R. Vojcic (M’86-SM’96) received the Dipl. and D.Sc. degrees in electrical engineering from the Faculty of Electrical Engineering, University of Belgrade, Yugoslavia, in 1980, 1986, and 1989, respectively. Since 1991, he has been a Professor with the Department of Electrical and Computer Engineering, George Washington University, Washington, DC, where he is also the Department Chairman. His current research interests are in the areas of communication theory, cellular and satellite networks, code-division multiple-access, multiuser detection, adaptive antenna arrays and space-time coding, and ad hoc networks. Dr. Vojcic was an Associate Editor for IEEE COMMUNICATIONS LETTERS. He was also a recipient of 1995 National Science Foundation CAREER Award.
