Multiuser Detections for Optical CDMA Networks ... - Semantic Scholar
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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 52, NO. 4, APRIL 2004
Multiuser Detections for Optical CDMA Networks Based on Expectation-Maximization Algorithm Abolfazl S. Motahari and Masoumeh Nasiri-Kenari, Member, IEEE
Abstract—In this paper, we introduce new unblind and blind multiuser detectors for an optical code-division multiple-access system. The detectors have two soft and hard stages. In the soft stage, a soft estimation of the interference is obtained by solving an unconstrained maximum-likelihood problem via the iterative expectation-maximization (EM) algorithm. Then, the hard stage detects the user information bit by solving a one-dimensional Boolean constrained problem conditioned on knowing the interference. Our results reveal that the proposed detectors have very low complexity, and are robust against changes in parameters. Moreover, the numerical results illustrate that despite of their simplicities, our detectors substantially outperform other well-known suboptimum detectors, such as multistage and decorrelating detectors. Index Terms—Interference cancellation, multiuser and blind detection, optical code-division multiple-access networks.
I. INTRODUCTION
I
N ANY multiple-access system, the available resources are shared in some ways among all active users. In code-division multiple-access (CDMA) systems, all resources (in principle) are available to all users, simultaneously. The users are distinguished from each other by a user-specific signature sequence (PN sequence). In a fiber-optic network, CDMA is considered as a viable multiple-access technique due to its ability to establish an asynchronous and robust multiple-access system for a number of users. In [1], Salehi proposed optical orthogonal codes (OOCs) as spreading codes for intensity modulation/direct detection (IM/DD) optical CDMA networks, which yield low crosscorrelation and out-of-phase autocorrelation, and therefore suitable for high-speed asynchronous networks using conventional correlation detectors [2], [3]. The correlation detector, which has a very low complexity, is optimal for a single-user system. However, as the number of simultaneous users increases, the simple correlation detector performance seriously degrades. This is due to the fact that the correlation detector does not take into account the existence of multiple-access interference (MAI) and treats them as a noise. In order to overcome this deficiency, Verdù [5] proposed an optimal multiuser detector, which is a maximum-likelihood
Paper approved by W. C. Kwong, the Editor for Optical Communications of the IEEE Communications Society. Manuscript received July 18, 2002; revised October 19, 2002. This paper was presented in part at the IEEE 7th International Symposium on Spread Spectrum Technology and Applications, Czech Republic, September 2–5, 2002. The authors are with the Department of Electrical Engineering, Sharif University of Technology, Tehran, Iran (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TCOMM.2003.811410
(ML) detector for an equiprobable channel input data. The optimum detector, which is the solution of a Boolean constrained ML problem, provides a performance comparable to that of a single-user system, but at the expense of a computational complexity that is known to be NP-hard. Therefore, the optimum detector is, in general, too complex for practical optical CDMA systems, even with a moderate number of users. To circumvent the complexity problem, much effort has been devoted to developing suboptimum receivers. In [6], Brandt-Pearce and Aazhang proposed a multistage (iterative) detector. This receiver, in each stage, first estimates the interference using the decisions made at the previous stage, and then based on this estimation derives a new decision for the current stage. The initial estimates used at the first iteration are provided by the outputs of the correlation detectors. It should be noted that the initial estimates of the information bits play an important role in the performance of the multistage detector. The decorrelating receiver, proposed in [8], has good resistance against the near–far problem. This detector requires the knowledge of the signature sequences and delays of the interfering users. However, in contrast to the multistage detector, it does not require the knowledge of the interfering users’ powers. The chip-level detector was proposed in [13]. This detector has a very simple structure, and its performance is good in the absence of dark current. But in the presence of dark current, its performance substantially degrades [14]. In this paper, we propose two suboptimum multiuser detectors for an optical CDMA system. In order to reduce the complexity of the optimum receiver, we first solve an unconstrained 1 in which the symbols can take any positive ML problem on real value (soft-decision stage). We then use these soft decisions as an estimate of the interference and solve a one-dimensional (1-D) Boolean constraint ML problem, conditioned on knowing the interference. This stage is called the hard-decision stage. By applying this approach, new blind and unblind multiuser detectors are introduced. For solving the unconstrained ML problem, we use the expectation-maximization (EM) algorithm. This algorithm provides an iterative approach to likelihood-based parameter estimation where direct maximization of the likelihood function may not be feasible [9]. Previous studies on the applications of the EM algorithm for multiuser detection in CDMA systems, to the best knowledge of the authors, have been in the radio-frequency (RF) domain, see [11] and [12]. In this paper, we use the EM algorithm for the iterative soft-decision stage, as described above. 1R = f[x] j x 2 Rg. Throughout this paper, the notation [x] implies that if x > 0, then [x] = x, else [x] = ", where " is a small positive number near zero.
0090-6778/04$20.00 © 2004 IEEE
MOTAHARI AND NASIRI-KENARI: MULTIUSER DETECTIONS FOR OPTICAL CDMA NETWORKS
Fig. 1.
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Discrete model of an optical CDMA system.
Numerical results show that only a few iterations are required for practical convergence of the EM algorithm in our applications. The rest of this paper is organized as follows. Section II describes the system model for an optical CDMA network. Section III introduces a multiuser detector with two soft and hard stages, in which the soft stage uses the EM algorithm. The blind implementation of the detectors is derived in Section IV. Finally, numerical results are presented in Section V, and concluding remarks are then given in Section VI. Throughout this paper, scalars are lowercase, vectors are boldfaced lowercase, and matrices are boldfaced uppercase. The notation ( ) and ( ) are used for soft and hard estimation operators, respectively. II. SYSTEM MODEL We consider an IM/DD optical CDMA network with users. Each user uses an OOK modulation for transmitting independent and equiprobable binary data over a common optical channel. The signature sequence is OOC with property 2 [4], where is the signature code length (or is the weight of codes. This set of processing gain), and codes are designed such that each pair of codes satisfies the following autocorrelation and crosscorrelation properties, i.e., (1)
are the signal strength and the relative where and transmission delay of user , respectively. The term repreis a rectangular waveform sents the dark current effect. are bit and chip with amplitude one and duration . and durations, respectively. For simplicity of presentation, the system is assumed synchronous. The generalization to the asynchronous case is straightforward, and will not be considered here. For a synchronous system, a more convenient form of (3) is in discrete vector notation as follows: (4) where
, , and represent the signature sequences matrix, users bits vector, and users power matrix, respectively, and denotes a vector of all one. Also, is the received intensity . Note that in a fiber-optic at chip interval , for system, the background noise is weak, and thus it is neglected in (4). Fig. 1 shows an equivalent discrete model of this network. The received signal is passed through a photo detector (p-i-n detector). We usually expect a Poisson process at the output of a photo detector. That is, if the photo counts collected from a s can be modeled as chip position is denoted by , then Poisson random variables, i.e., (5) where is given by (4). The correlation detector for user detects the user’s bit based on the following decision rule:
(2) where and are members of the OOC. be the transmitted bit of the th user at the Let th bit interval and be its signature code. Then the received intensity signal can be written as
(6) where is a threshold which depends on the user power and dark current intensity. This simple detector is optimal only in a single-user system. The ML detector decides based on the following rule [5]: (7) where (8)
(3) 2Even though we use the OOC as signature code for performance evaluation of our proposed detector, any other signature code can be used as well.
is the log-likelihood function (LLF), and denotes the density function of conditioned on . This detector, incidentally, is identical to the maximum a posteriori (MAP) detector
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Fig. 2.
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 52, NO. 4, APRIL 2004
Complete data for the unblind multiuser detector.
for an equiprobable channel input data. The computational complexity of this detector is known to be NP-hard. Several suboptimum multiuser detectors for mitigating the complexity of the optimum receiver have been proposed [6]–[8]. These detectors mostly have the following structure: (9) where ’s are the weighting coefficients. In this structure, those chips observing stronger interference are weighted smaller, compared with the other chips. III. PROPOSED DETECTOR: MULTIUSER DETECTION IMPLEMENTATION In this section, we propose a new EM-based multiuser detector which, as in [7], contains two soft and hard stages. The soft stage is implemented iteratively using the EM algorithm. We describe the soft- and hard-decision stages in parts A and B, respectively. In part C, we summarize some advantages of the new detector. Before describing the detector, we first briefly introduce the EM algorithm. Consider the general ML estimation problem of a parameter from observation data. In many situations for which the ML solution is intractable or has high complexity, the EM algorithm provides an attractive alternative to computing ML estimates. The key element of EM is to replace the ML problem with an iterative maximization of the following objective function: (10)
A. Soft-Decision Stage In this stage, we relax Boolean constrained in (7) and as. With this sume that the transmitted bits vector belongs to assumption, the LLF, i.e., , is concave [7] and bounded below, therefore, it has only a global maximum that can be found via the EM algorithm, and in sequel, the convergence to the global maximum is guaranteed [9]. Fig. 2 shows a new equivalent mathematical model for the network. Note that the models of Figs. 1 and 2 are equivalent if we set (11) where is the dark current assigned to the th user. The photo , plays counts of each user, i.e., represents the photo the role of the complete data, where counts due to the th user collected at the th chip interval. , where The complete data is represented by is a photo-counts matrix with the columns as is the photo-counts described above, and vector due to . The observable data (incomplete data) is related to the complete data with the following many-to-one mapping: (12) As mentioned above, the EM algorithm is implemented in two steps, expectation and maximization, as follows. The E-step is defined by function (13)
where is the expectation operator, is the LLF defined in (8), and is the complete data containing some missing data (unobservable data) that would aid in estimation of param, the th iteration of the EM eter . Given an initial estimate algorithm is described by the following. . Expectation-step: compute . Maximization-step: Where is the estimate of at the th iteration. E-step and M-step iterate until a global or local maximum is obtained [9].
where
MOTAHARI AND NASIRI-KENARI: MULTIUSER DETECTIONS FOR OPTICAL CDMA NETWORKS
Since all densities obey Poisson distribution, we have
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Now the detector is implemented iteratively using (15), (16), (18), and (19). B. Hard-Decision Stage In this stage, the solution of the soft-decision stage, i.e. , is converted onto a valid data point through a suboptimum mapping, as follows. Interference at the th chip of user is estimated using the soft decisions as
By taking the expectation, we simply obtain
(23)
(14) (the symbol “ ” denotes the equivalent up to a constant term which is independent from ), where in Appendix A, we show
where can be replaced with if the dark current is unknown and must be estimated by (19). Then, the user ’s bit is detected by solving the following 1-D Boolean constrained ML problem conditioned on having interference as (23): (24)
(15) (16) Since the ,
where to [7]
. Solving the above problem leads
function in (14) is a separable function of , the following problem in the M-step:
(25)
(17) C. Advantages of Proposed Detector has a closed-form solution for we obtain
’s and
. After some algebra,
(18)
(19) The equations (15), (16), (18), and (19) can be used iteratively to obtain the unconstrained ML solution. We can divide the dark current among the users, i.e., ’s in (11), in two ways, as follows. 1) Case I: Known Dark Current When the dark-current intensity is known, we do the following division: (20) In this case, the detector is implemented iteratively using (15) and (18), and (16) and (19) are not used. Also, (14) is modified as
The proposed detector has some crucial advantages, as follows. — Low complexity: There is no comparison required at the soft-decision stage. Only at the hard-decision stage, comparisons are required, one for each user. Moreover, the numerical results show that two or three iterations are sufficient for the EM algorithm convergence. — Unique convergence: In contrast to the multistage receiver, the performance of this detector does not depend on the initial estimation at first iteration, and global convergence can be achieved for any initial values. — Robustness: In this detector, the knowledge of the dark current intensity is not required, and in fact, the dark current can be estimated simultaneously. — Blindness: There is a very interesting result from (15), (16), and (18), as follows. If we want to decode the information bit of only one user, we do not need to know the powers and bits of the interfering users separately. . By substituting We actually need to estimate in (15), (16), and (18), we obtain (26)
(21) 2) Case II: Unknown Dark Current In this case, information symbols and dark current must be estimated simultaneously, and thus, we set
(27)
(22)
(28)
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Fig. 3. Complete data for the blind detector.
Now, the interference can be estimated as (29) The final hard stage can be implemented by (25). This detector requires the same knowledge as required by the decorrelating detector, even though, as our simulation results indicate, it substantially outperforms the decorrelating detector. IV. PROPOSED DETECTOR: BLIND IMPLEMENTATION In the previous section, we proposed a multiuser detector based on the EM algorithm, in which the spreading codes and delays of all interfering users are assumed to be known. However, when this knowledge does not exist, the previous derivations can no longer be used, and some changes must be made to derive the blind implementation. In this case, we again follow the same approach used in Section III. That is, the blind detector that will be derived in this section has two soft and hard stages, in which the interference is estimated at the soft stage, and the information bit is decoded at the hard stage. Without loss of generality, we assume that the user one is the desired user.
has Gaussian distribution. Note that many papers on optical CDMA in the literature have assumed Gaussian distribution for the interference, for instance, see [2]. Moreover, we have . Therefore, (30) can be simplified as follows: (31) are the mean and variance of the interference where and at th position, respectively. This equation is similar to (8), except for a second term. This objective function is also a concave function, and can be solved via the EM algorithm to achieve the global convergence. Fig. 3 shows the complete data defined for the EM algorithm implementation of the problem in (31). In this case, the complete , where data is denoted by and are photo-counts vectors for user one and the interference at the marked chips, respectively. The function in (13) is now easily computed, as follows:
(32)
A. Soft-Decision Stage In this stage, the desired user bit and interference vector are estimated simultaneously. Although the bit estimate is not used at hard stage, it is required for interference estimation. be the desired parameter Let the vector that must be estimated, where represents the interference at the th marked position in the signature code of user one for . Note that the weight of the signature code is . The MAP detector makes a decision based on the following rule:
where (33) (34) The derivatives of
, with respect to
and
, are
(30) where is defined as in (8). Assume that all components of are statistically independent. Even though this assumption is not very realistic, the derived blind detector performs quite as well, as will be shown in Section V. Thus, we can write . Since we want to capture only the first and second statistics of the interference, we assume that
(35) By setting these derivatives equal to zero, we obtain a solution for as in (18), and a quadratic equation for interference as (36)
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This function has the following solution: (37) Now (18), (33), (34), and (37) can be iteratively used to obtain soft estimation of the interference. In Appendix B, it has been shown that the mean and variance of interference, i.e., and , can be estimated from the mean and variance of , the observed data, as follows: (38) (39) B. Hard-Decision Stage In the final stage, the suboptimal mapper of (24) is used for hard estimation, as follows: (40)
V. NUMERICAL RESULTS In this section, some numerical results are presented to demonstrate the performance of our proposed detectors, namely, EM-multiuser detector and EM-blind detector. For simulation, we consider a chip-synchronous OOK-CDMA with OOC signature sequence (200,3,1,1). The maximum number of users is 33 [4]. The bit-error rates (BERs) of the proposed detectors are obtained by Monte Carlo simulation and then compared with those of various previously proposed detectors, namely, the decorrelating, multistage, and known interference detectors. Note that the performance of the known interference detector is a lower bound for the performance of the optimum detector [6]. Intensity of all users is assumed to be identical, and the dark current intensity is chosen to be 0.1. For the EM-multiuser detector, we have used (25)–(29), which do not require the intensity of the interfering users. As mentioned before, in our proposed detectors, the initial value of the transmitted bits at the soft stage can be chosen arbitrarily. In simulation, we have, however, set the soft initial values of the other user bits equal to half value of the desired user’s (user one) power. It must be noted that although the average performance of the system depends on the number of simultaneous users, the performance at a particular instance is dictated by the interference pattern [10]. So, we consider three interference patterns. First, we consider a weak interference pattern, in which only one of the three marked chips of the desired user signature code collides with the interfering pulses of other users. In the second pattern, we assume only two marked chips collide with interfering pulses, and in last pattern, we consider the case in which all three marked chips will collide with interference pulses. Note that at most one marked chip (among three marked chips) can be hit by each interfering user. The number of iterations in any iterative detector somehow indicates its complexity. Fig. 4 shows the plot of the BER of our proposed detectors versus the number of iterations for three
Fig. 4. Bit-error probability of user one versus the number of iterations for all three cases, where F = 200, w = 3, and M = 8.
Fig. 5. Bit-error probability of user one versus intensity for case 1, where F = 200, w = 3, and M = 8.
interference patterns, as described above. The number of active users is eight, and their intensities are set to six. As can be observed, the proposed detectors converge after a few iterations for all three cases. Figs. 5–7 present the plots of BER versus the users powers for cases 1-3, respectively. As can be observed, for case 3, the EM-multiuser detector substantially outperforms the multistage and decorrelating detectors. Also, our blind detector performs
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Fig. 8. Bit-error probability of user one versus number of users, where F = 200, w = 3, and intensity = 5.
Fig. 6. Bit-error probability of user one versus intensity for case 2, where F = 200, w = 3, and M = 8.
Fig. 9. Bit-error probability of user one versus intensity for EM-multiuser detector in two cases, known dark current and unknown dark current, where F = 200, w = 3, and M = 8.
Fig. 7. Bit-error probability of user one versus intensity for case 3, where F = 200, w = 3, and M = 8.
much better than the decorrelating detector for all three cases, in spite of its simplicity. Note that in the blind detector, only the signature code and related parameters of the desired user are required to be known. The overall system performance can be revealed from Fig. 8, where the BERs of the detectors are plotted versus the number of users. In this simulation, the intensities of all users are set to five. From Fig. 8, it can be realized that our proposed detec-
tors outperform other suboptimum detectors. Furthermore, our unblind implementation performs so close to the known-interference detector that its performance is a lower bound for the optimum detector performance. In the above simulations, we have assumed that the dark current intensity is known. To investigate the robustness of the above receivers, we have evaluated the performance of the detectors when the dark current intensity is not known and must be estimated. In this case, (15), (16), (18), and (19) are used. Figs. 9–10 illustrate the performance of the proposed unblind and blind detectors for two cases, namely, known and unknown dark current intensity, respectively. As expected, the performance for the above two cases are almost the same.
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and the intensity of desired user, user , as (A.3) it can be easily shown that (A.4) (A.5) (A.6) By substituting these equations in (A.1), we obtain (A.7)
Fig. 10. Bit-error probability of user one versus intensity for EM-blind detector in two cases, known dark current and unknown dark current, where F = 200, w = 3, and M = 8.
VI. CONCLUSION
. Thus, conditioned on and has a where binomial distribution with parameter . As a result, the expecconditioned on and is tation of (A.8) and then (15) is derived.
In this paper, we have introduced two blind and unblind multiuser detectors for an optical CDMA system based on the EM algorithm. These detectors have two stages, in which at the soft stage, a soft estimation of the interference is obtained by solving an unconstrained likelihood function through the EM algorithm, and at the hard stage by solving a 1-D constrained Boolean likelihood function conditioned on knowing the interference, the input bit is detected. The derived unblind and blind detectors have a very low computational complexity and are robust against the channel parameters’ variations. Our simulation results have shown that the proposed detectors substantially outperform the conventional, multistage, and decorrelating detectors in the cases considered. Moreover, the numerical results have indicated that at the soft stage, only a few iterations are required for the EM algorithm convergence.
APPENDIX B MEAN AND VARIANCE ESTIMATIONS In this appendix, we present an approach to estimate the mean and variance of the interference, which is required in the blind detector. We can formulate this estimation problem as follows. Suppose that we observe a double Poisson process with intensity , and we want to estimate mean and variance of . We take the following steps. The characteristic function of is given by (B.1) then, we can write (B.2) Since
APPENDIX A
conditioned on
has Poisson distribution, we obtain (B.3)
EXPECTATION COMPUTATION In this appendix, we derive (15). First, the probability density conditioned on and is computed. function (pdf) of From Baye’s formula, we have (A.1)
From (B.2) and (B.3), we conclude (B.4) Since the mean and variance of a random variable are obtained from its characteristic function, it can be easily shown that (B.5) (B.6)
By denoting the intensity at the th chip position as
(A.2)
Thus, we can first estimate the mean and variance of , the observed data, and then from (B.5) and (B.6) and , underlying intensity statistics, are derived.
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ACKNOWLEDGMENT The authors would like to thank Dr. J. A. Salehi, M. Razavi, and A. Keshavarzian for their helpful comments.
[13] H. M. Shalaby, “Chip-level detection in optical code division multipleaccess,” IEEE J. Lightwave Technol., vol. 16, pp. 1077–1087, June 1998. [14] S. Zahedi and J. A. Salehi, “Analytical comparison of various fiber-optic CDMA receivers,” IEEE J. Lightwave Technol., vol. 18, pp. 1718–1727, Dec. 2000.
REFERENCES [1] J. A. Salehi, “Code division multiple-access techniques in optical fiber network-Part I: Fundamental principles,” IEEE Trans. Commun., vol. 37, pp. 824–833, Aug. 1989. [2] J. A. Salehi and C. A. Brackett, “Code division multiple-access techniques in optical fiber network-Part II: System performance analysis,” IEEE Trans. Commun., vol. 37, pp. 834–842, Aug. 1989. [3] M. Azizoglu, J. A. Salehi, and Y. Li, “Optical CDMA via temporal codes,” IEEE Trans. Commun., vol. 40, pp. 1162–1170, July 1992. [4] F. R. K. Chang, J. A. Salehi, and V. K. Wei, “Optical orthogonal codes: design, analysis and applications,” IEEE Trans. Inform. Theory, vol. 35, pp. 866–873, July 1990. [5] S. Verdù, “Multiple-access channels with point process observation: optimum demodulation,” IEEE Trans. Inform. Theory, vol. IT-32, pp. 642–651, Sept. 1986. [6] M. B. Pearce and B. Aazhang, “Multiuser detection for optical code division multiple access,” IEEE Trans. Commun., vol. 42, pp. 1801–1810, Feb.-Apr. 1994. , “Performance analysis of single user and multiuser detectors for [7] optical code division multiple access communication systems,” IEEE Trans. Commun., vol. 43, pp. 435–444, Feb.-Apr. 1995. [8] L. B. Nelson and H. V. Poor, “Performance of multiuser detection for optical CDMA—Part I: Error probabilities,” IEEE Trans. Commun., vol. 43, pp. 2803–2811, Nov. 1995. [9] A. P. Dempster, N. M. Laired, and D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. Roy. Statist. Soc., ser. B, vol. 39, no. 1, pp. 1–38, 1977. [10] J. T. K. Tang and K. B. Letaief, “Optical CDMA communication systems with multiuser and blind detection,” IEEE Trans. Commun., vol. 47, pp. 1211–1217, Aug. 1999. [11] 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. [12] M. J. Borran and M. Nasiri-Kenari, “An efficient detection technique for synchronous CDMA communication systems based on the expectation maximization algorithm,” IEEE Trans. Veh. Technol., vol. 49, pp. 1663–1668, Sept. 2000.
Abolfazl S. Motahari was born in 1977. He received the B.S. degree from the Iran University of Science and Technology (IUST), Tehran, Iran, in 1999 and the M.S. degree from Sharif University of Technology, Tehran, Iran, in 2001, both in electrical engineering. He is currently working toward the Ph.D. degree in the Department of Electrical Engineering, Sharif University of Technology. From August 2000 to August 2001, he was a Member of Research Staff with the Advanced Communication Science Research Laboratory, Iran Telecommunication Research Center (ITRC), Tehran, Iran. His research interests include spread-spectrum systems, optical communication systems, and statistical digital signal processing.
Masoumeh Nasiri-Kenari (S’90–M’94) received the B.S. and M.S. degrees in electrical engineering from Isfahan University of Technology, Isfahan, Iran, in 1986 and 1987, respectively, and the Ph.D. degree in electrical engineering from the University of Utah, Salt Lake City, in 1993. From 1987 to 1988, she was a Technical Instructor and Research Assistant at Isfahan University of Technology. Since 1994, she has been with the Department of Electrical Engineering, Sharif University of Technology, Tehran, Iran, where she is now an Associate Professor. She was a Co-Director of the Advanced CDMA Laboratory at the Iran Telecom Research Center (ITRC), Tehran, Iran, in 1999–2001. Her research interests are in radio and optical communications and error-control coding.
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 52, NO. 4, APRIL 2004
Multiuser Detections for Optical CDMA Networks Based on Expectation-Maximization Algorithm Abolfazl S. Motahari and Masoumeh Nasiri-Kenari, Member, IEEE
Abstract—In this paper, we introduce new unblind and blind multiuser detectors for an optical code-division multiple-access system. The detectors have two soft and hard stages. In the soft stage, a soft estimation of the interference is obtained by solving an unconstrained maximum-likelihood problem via the iterative expectation-maximization (EM) algorithm. Then, the hard stage detects the user information bit by solving a one-dimensional Boolean constrained problem conditioned on knowing the interference. Our results reveal that the proposed detectors have very low complexity, and are robust against changes in parameters. Moreover, the numerical results illustrate that despite of their simplicities, our detectors substantially outperform other well-known suboptimum detectors, such as multistage and decorrelating detectors. Index Terms—Interference cancellation, multiuser and blind detection, optical code-division multiple-access networks.
I. INTRODUCTION
I
N ANY multiple-access system, the available resources are shared in some ways among all active users. In code-division multiple-access (CDMA) systems, all resources (in principle) are available to all users, simultaneously. The users are distinguished from each other by a user-specific signature sequence (PN sequence). In a fiber-optic network, CDMA is considered as a viable multiple-access technique due to its ability to establish an asynchronous and robust multiple-access system for a number of users. In [1], Salehi proposed optical orthogonal codes (OOCs) as spreading codes for intensity modulation/direct detection (IM/DD) optical CDMA networks, which yield low crosscorrelation and out-of-phase autocorrelation, and therefore suitable for high-speed asynchronous networks using conventional correlation detectors [2], [3]. The correlation detector, which has a very low complexity, is optimal for a single-user system. However, as the number of simultaneous users increases, the simple correlation detector performance seriously degrades. This is due to the fact that the correlation detector does not take into account the existence of multiple-access interference (MAI) and treats them as a noise. In order to overcome this deficiency, Verdù [5] proposed an optimal multiuser detector, which is a maximum-likelihood
Paper approved by W. C. Kwong, the Editor for Optical Communications of the IEEE Communications Society. Manuscript received July 18, 2002; revised October 19, 2002. This paper was presented in part at the IEEE 7th International Symposium on Spread Spectrum Technology and Applications, Czech Republic, September 2–5, 2002. The authors are with the Department of Electrical Engineering, Sharif University of Technology, Tehran, Iran (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TCOMM.2003.811410
(ML) detector for an equiprobable channel input data. The optimum detector, which is the solution of a Boolean constrained ML problem, provides a performance comparable to that of a single-user system, but at the expense of a computational complexity that is known to be NP-hard. Therefore, the optimum detector is, in general, too complex for practical optical CDMA systems, even with a moderate number of users. To circumvent the complexity problem, much effort has been devoted to developing suboptimum receivers. In [6], Brandt-Pearce and Aazhang proposed a multistage (iterative) detector. This receiver, in each stage, first estimates the interference using the decisions made at the previous stage, and then based on this estimation derives a new decision for the current stage. The initial estimates used at the first iteration are provided by the outputs of the correlation detectors. It should be noted that the initial estimates of the information bits play an important role in the performance of the multistage detector. The decorrelating receiver, proposed in [8], has good resistance against the near–far problem. This detector requires the knowledge of the signature sequences and delays of the interfering users. However, in contrast to the multistage detector, it does not require the knowledge of the interfering users’ powers. The chip-level detector was proposed in [13]. This detector has a very simple structure, and its performance is good in the absence of dark current. But in the presence of dark current, its performance substantially degrades [14]. In this paper, we propose two suboptimum multiuser detectors for an optical CDMA system. In order to reduce the complexity of the optimum receiver, we first solve an unconstrained 1 in which the symbols can take any positive ML problem on real value (soft-decision stage). We then use these soft decisions as an estimate of the interference and solve a one-dimensional (1-D) Boolean constraint ML problem, conditioned on knowing the interference. This stage is called the hard-decision stage. By applying this approach, new blind and unblind multiuser detectors are introduced. For solving the unconstrained ML problem, we use the expectation-maximization (EM) algorithm. This algorithm provides an iterative approach to likelihood-based parameter estimation where direct maximization of the likelihood function may not be feasible [9]. Previous studies on the applications of the EM algorithm for multiuser detection in CDMA systems, to the best knowledge of the authors, have been in the radio-frequency (RF) domain, see [11] and [12]. In this paper, we use the EM algorithm for the iterative soft-decision stage, as described above. 1R = f[x] j x 2 Rg. Throughout this paper, the notation [x] implies that if x > 0, then [x] = x, else [x] = ", where " is a small positive number near zero.
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Fig. 1.
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Discrete model of an optical CDMA system.
Numerical results show that only a few iterations are required for practical convergence of the EM algorithm in our applications. The rest of this paper is organized as follows. Section II describes the system model for an optical CDMA network. Section III introduces a multiuser detector with two soft and hard stages, in which the soft stage uses the EM algorithm. The blind implementation of the detectors is derived in Section IV. Finally, numerical results are presented in Section V, and concluding remarks are then given in Section VI. Throughout this paper, scalars are lowercase, vectors are boldfaced lowercase, and matrices are boldfaced uppercase. The notation ( ) and ( ) are used for soft and hard estimation operators, respectively. II. SYSTEM MODEL We consider an IM/DD optical CDMA network with users. Each user uses an OOK modulation for transmitting independent and equiprobable binary data over a common optical channel. The signature sequence is OOC with property 2 [4], where is the signature code length (or is the weight of codes. This set of processing gain), and codes are designed such that each pair of codes satisfies the following autocorrelation and crosscorrelation properties, i.e., (1)
are the signal strength and the relative where and transmission delay of user , respectively. The term repreis a rectangular waveform sents the dark current effect. are bit and chip with amplitude one and duration . and durations, respectively. For simplicity of presentation, the system is assumed synchronous. The generalization to the asynchronous case is straightforward, and will not be considered here. For a synchronous system, a more convenient form of (3) is in discrete vector notation as follows: (4) where
, , and represent the signature sequences matrix, users bits vector, and users power matrix, respectively, and denotes a vector of all one. Also, is the received intensity . Note that in a fiber-optic at chip interval , for system, the background noise is weak, and thus it is neglected in (4). Fig. 1 shows an equivalent discrete model of this network. The received signal is passed through a photo detector (p-i-n detector). We usually expect a Poisson process at the output of a photo detector. That is, if the photo counts collected from a s can be modeled as chip position is denoted by , then Poisson random variables, i.e., (5) where is given by (4). The correlation detector for user detects the user’s bit based on the following decision rule:
(2) where and are members of the OOC. be the transmitted bit of the th user at the Let th bit interval and be its signature code. Then the received intensity signal can be written as
(6) where is a threshold which depends on the user power and dark current intensity. This simple detector is optimal only in a single-user system. The ML detector decides based on the following rule [5]: (7) where (8)
(3) 2Even though we use the OOC as signature code for performance evaluation of our proposed detector, any other signature code can be used as well.
is the log-likelihood function (LLF), and denotes the density function of conditioned on . This detector, incidentally, is identical to the maximum a posteriori (MAP) detector
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Fig. 2.
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Complete data for the unblind multiuser detector.
for an equiprobable channel input data. The computational complexity of this detector is known to be NP-hard. Several suboptimum multiuser detectors for mitigating the complexity of the optimum receiver have been proposed [6]–[8]. These detectors mostly have the following structure: (9) where ’s are the weighting coefficients. In this structure, those chips observing stronger interference are weighted smaller, compared with the other chips. III. PROPOSED DETECTOR: MULTIUSER DETECTION IMPLEMENTATION In this section, we propose a new EM-based multiuser detector which, as in [7], contains two soft and hard stages. The soft stage is implemented iteratively using the EM algorithm. We describe the soft- and hard-decision stages in parts A and B, respectively. In part C, we summarize some advantages of the new detector. Before describing the detector, we first briefly introduce the EM algorithm. Consider the general ML estimation problem of a parameter from observation data. In many situations for which the ML solution is intractable or has high complexity, the EM algorithm provides an attractive alternative to computing ML estimates. The key element of EM is to replace the ML problem with an iterative maximization of the following objective function: (10)
A. Soft-Decision Stage In this stage, we relax Boolean constrained in (7) and as. With this sume that the transmitted bits vector belongs to assumption, the LLF, i.e., , is concave [7] and bounded below, therefore, it has only a global maximum that can be found via the EM algorithm, and in sequel, the convergence to the global maximum is guaranteed [9]. Fig. 2 shows a new equivalent mathematical model for the network. Note that the models of Figs. 1 and 2 are equivalent if we set (11) where is the dark current assigned to the th user. The photo , plays counts of each user, i.e., represents the photo the role of the complete data, where counts due to the th user collected at the th chip interval. , where The complete data is represented by is a photo-counts matrix with the columns as is the photo-counts described above, and vector due to . The observable data (incomplete data) is related to the complete data with the following many-to-one mapping: (12) As mentioned above, the EM algorithm is implemented in two steps, expectation and maximization, as follows. The E-step is defined by function (13)
where is the expectation operator, is the LLF defined in (8), and is the complete data containing some missing data (unobservable data) that would aid in estimation of param, the th iteration of the EM eter . Given an initial estimate algorithm is described by the following. . Expectation-step: compute . Maximization-step: Where is the estimate of at the th iteration. E-step and M-step iterate until a global or local maximum is obtained [9].
where
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Since all densities obey Poisson distribution, we have
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Now the detector is implemented iteratively using (15), (16), (18), and (19). B. Hard-Decision Stage In this stage, the solution of the soft-decision stage, i.e. , is converted onto a valid data point through a suboptimum mapping, as follows. Interference at the th chip of user is estimated using the soft decisions as
By taking the expectation, we simply obtain
(23)
(14) (the symbol “ ” denotes the equivalent up to a constant term which is independent from ), where in Appendix A, we show
where can be replaced with if the dark current is unknown and must be estimated by (19). Then, the user ’s bit is detected by solving the following 1-D Boolean constrained ML problem conditioned on having interference as (23): (24)
(15) (16) Since the ,
where to [7]
. Solving the above problem leads
function in (14) is a separable function of , the following problem in the M-step:
(25)
(17) C. Advantages of Proposed Detector has a closed-form solution for we obtain
’s and
. After some algebra,
(18)
(19) The equations (15), (16), (18), and (19) can be used iteratively to obtain the unconstrained ML solution. We can divide the dark current among the users, i.e., ’s in (11), in two ways, as follows. 1) Case I: Known Dark Current When the dark-current intensity is known, we do the following division: (20) In this case, the detector is implemented iteratively using (15) and (18), and (16) and (19) are not used. Also, (14) is modified as
The proposed detector has some crucial advantages, as follows. — Low complexity: There is no comparison required at the soft-decision stage. Only at the hard-decision stage, comparisons are required, one for each user. Moreover, the numerical results show that two or three iterations are sufficient for the EM algorithm convergence. — Unique convergence: In contrast to the multistage receiver, the performance of this detector does not depend on the initial estimation at first iteration, and global convergence can be achieved for any initial values. — Robustness: In this detector, the knowledge of the dark current intensity is not required, and in fact, the dark current can be estimated simultaneously. — Blindness: There is a very interesting result from (15), (16), and (18), as follows. If we want to decode the information bit of only one user, we do not need to know the powers and bits of the interfering users separately. . By substituting We actually need to estimate in (15), (16), and (18), we obtain (26)
(21) 2) Case II: Unknown Dark Current In this case, information symbols and dark current must be estimated simultaneously, and thus, we set
(27)
(22)
(28)
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Fig. 3. Complete data for the blind detector.
Now, the interference can be estimated as (29) The final hard stage can be implemented by (25). This detector requires the same knowledge as required by the decorrelating detector, even though, as our simulation results indicate, it substantially outperforms the decorrelating detector. IV. PROPOSED DETECTOR: BLIND IMPLEMENTATION In the previous section, we proposed a multiuser detector based on the EM algorithm, in which the spreading codes and delays of all interfering users are assumed to be known. However, when this knowledge does not exist, the previous derivations can no longer be used, and some changes must be made to derive the blind implementation. In this case, we again follow the same approach used in Section III. That is, the blind detector that will be derived in this section has two soft and hard stages, in which the interference is estimated at the soft stage, and the information bit is decoded at the hard stage. Without loss of generality, we assume that the user one is the desired user.
has Gaussian distribution. Note that many papers on optical CDMA in the literature have assumed Gaussian distribution for the interference, for instance, see [2]. Moreover, we have . Therefore, (30) can be simplified as follows: (31) are the mean and variance of the interference where and at th position, respectively. This equation is similar to (8), except for a second term. This objective function is also a concave function, and can be solved via the EM algorithm to achieve the global convergence. Fig. 3 shows the complete data defined for the EM algorithm implementation of the problem in (31). In this case, the complete , where data is denoted by and are photo-counts vectors for user one and the interference at the marked chips, respectively. The function in (13) is now easily computed, as follows:
(32)
A. Soft-Decision Stage In this stage, the desired user bit and interference vector are estimated simultaneously. Although the bit estimate is not used at hard stage, it is required for interference estimation. be the desired parameter Let the vector that must be estimated, where represents the interference at the th marked position in the signature code of user one for . Note that the weight of the signature code is . The MAP detector makes a decision based on the following rule:
where (33) (34) The derivatives of
, with respect to
and
, are
(30) where is defined as in (8). Assume that all components of are statistically independent. Even though this assumption is not very realistic, the derived blind detector performs quite as well, as will be shown in Section V. Thus, we can write . Since we want to capture only the first and second statistics of the interference, we assume that
(35) By setting these derivatives equal to zero, we obtain a solution for as in (18), and a quadratic equation for interference as (36)
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This function has the following solution: (37) Now (18), (33), (34), and (37) can be iteratively used to obtain soft estimation of the interference. In Appendix B, it has been shown that the mean and variance of interference, i.e., and , can be estimated from the mean and variance of , the observed data, as follows: (38) (39) B. Hard-Decision Stage In the final stage, the suboptimal mapper of (24) is used for hard estimation, as follows: (40)
V. NUMERICAL RESULTS In this section, some numerical results are presented to demonstrate the performance of our proposed detectors, namely, EM-multiuser detector and EM-blind detector. For simulation, we consider a chip-synchronous OOK-CDMA with OOC signature sequence (200,3,1,1). The maximum number of users is 33 [4]. The bit-error rates (BERs) of the proposed detectors are obtained by Monte Carlo simulation and then compared with those of various previously proposed detectors, namely, the decorrelating, multistage, and known interference detectors. Note that the performance of the known interference detector is a lower bound for the performance of the optimum detector [6]. Intensity of all users is assumed to be identical, and the dark current intensity is chosen to be 0.1. For the EM-multiuser detector, we have used (25)–(29), which do not require the intensity of the interfering users. As mentioned before, in our proposed detectors, the initial value of the transmitted bits at the soft stage can be chosen arbitrarily. In simulation, we have, however, set the soft initial values of the other user bits equal to half value of the desired user’s (user one) power. It must be noted that although the average performance of the system depends on the number of simultaneous users, the performance at a particular instance is dictated by the interference pattern [10]. So, we consider three interference patterns. First, we consider a weak interference pattern, in which only one of the three marked chips of the desired user signature code collides with the interfering pulses of other users. In the second pattern, we assume only two marked chips collide with interfering pulses, and in last pattern, we consider the case in which all three marked chips will collide with interference pulses. Note that at most one marked chip (among three marked chips) can be hit by each interfering user. The number of iterations in any iterative detector somehow indicates its complexity. Fig. 4 shows the plot of the BER of our proposed detectors versus the number of iterations for three
Fig. 4. Bit-error probability of user one versus the number of iterations for all three cases, where F = 200, w = 3, and M = 8.
Fig. 5. Bit-error probability of user one versus intensity for case 1, where F = 200, w = 3, and M = 8.
interference patterns, as described above. The number of active users is eight, and their intensities are set to six. As can be observed, the proposed detectors converge after a few iterations for all three cases. Figs. 5–7 present the plots of BER versus the users powers for cases 1-3, respectively. As can be observed, for case 3, the EM-multiuser detector substantially outperforms the multistage and decorrelating detectors. Also, our blind detector performs
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Fig. 8. Bit-error probability of user one versus number of users, where F = 200, w = 3, and intensity = 5.
Fig. 6. Bit-error probability of user one versus intensity for case 2, where F = 200, w = 3, and M = 8.
Fig. 9. Bit-error probability of user one versus intensity for EM-multiuser detector in two cases, known dark current and unknown dark current, where F = 200, w = 3, and M = 8.
Fig. 7. Bit-error probability of user one versus intensity for case 3, where F = 200, w = 3, and M = 8.
much better than the decorrelating detector for all three cases, in spite of its simplicity. Note that in the blind detector, only the signature code and related parameters of the desired user are required to be known. The overall system performance can be revealed from Fig. 8, where the BERs of the detectors are plotted versus the number of users. In this simulation, the intensities of all users are set to five. From Fig. 8, it can be realized that our proposed detec-
tors outperform other suboptimum detectors. Furthermore, our unblind implementation performs so close to the known-interference detector that its performance is a lower bound for the optimum detector performance. In the above simulations, we have assumed that the dark current intensity is known. To investigate the robustness of the above receivers, we have evaluated the performance of the detectors when the dark current intensity is not known and must be estimated. In this case, (15), (16), (18), and (19) are used. Figs. 9–10 illustrate the performance of the proposed unblind and blind detectors for two cases, namely, known and unknown dark current intensity, respectively. As expected, the performance for the above two cases are almost the same.
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and the intensity of desired user, user , as (A.3) it can be easily shown that (A.4) (A.5) (A.6) By substituting these equations in (A.1), we obtain (A.7)
Fig. 10. Bit-error probability of user one versus intensity for EM-blind detector in two cases, known dark current and unknown dark current, where F = 200, w = 3, and M = 8.
VI. CONCLUSION
. Thus, conditioned on and has a where binomial distribution with parameter . As a result, the expecconditioned on and is tation of (A.8) and then (15) is derived.
In this paper, we have introduced two blind and unblind multiuser detectors for an optical CDMA system based on the EM algorithm. These detectors have two stages, in which at the soft stage, a soft estimation of the interference is obtained by solving an unconstrained likelihood function through the EM algorithm, and at the hard stage by solving a 1-D constrained Boolean likelihood function conditioned on knowing the interference, the input bit is detected. The derived unblind and blind detectors have a very low computational complexity and are robust against the channel parameters’ variations. Our simulation results have shown that the proposed detectors substantially outperform the conventional, multistage, and decorrelating detectors in the cases considered. Moreover, the numerical results have indicated that at the soft stage, only a few iterations are required for the EM algorithm convergence.
APPENDIX B MEAN AND VARIANCE ESTIMATIONS In this appendix, we present an approach to estimate the mean and variance of the interference, which is required in the blind detector. We can formulate this estimation problem as follows. Suppose that we observe a double Poisson process with intensity , and we want to estimate mean and variance of . We take the following steps. The characteristic function of is given by (B.1) then, we can write (B.2) Since
APPENDIX A
conditioned on
has Poisson distribution, we obtain (B.3)
EXPECTATION COMPUTATION In this appendix, we derive (15). First, the probability density conditioned on and is computed. function (pdf) of From Baye’s formula, we have (A.1)
From (B.2) and (B.3), we conclude (B.4) Since the mean and variance of a random variable are obtained from its characteristic function, it can be easily shown that (B.5) (B.6)
By denoting the intensity at the th chip position as
(A.2)
Thus, we can first estimate the mean and variance of , the observed data, and then from (B.5) and (B.6) and , underlying intensity statistics, are derived.
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ACKNOWLEDGMENT The authors would like to thank Dr. J. A. Salehi, M. Razavi, and A. Keshavarzian for their helpful comments.
[13] H. M. Shalaby, “Chip-level detection in optical code division multipleaccess,” IEEE J. Lightwave Technol., vol. 16, pp. 1077–1087, June 1998. [14] S. Zahedi and J. A. Salehi, “Analytical comparison of various fiber-optic CDMA receivers,” IEEE J. Lightwave Technol., vol. 18, pp. 1718–1727, Dec. 2000.
REFERENCES [1] J. A. Salehi, “Code division multiple-access techniques in optical fiber network-Part I: Fundamental principles,” IEEE Trans. Commun., vol. 37, pp. 824–833, Aug. 1989. [2] J. A. Salehi and C. A. Brackett, “Code division multiple-access techniques in optical fiber network-Part II: System performance analysis,” IEEE Trans. Commun., vol. 37, pp. 834–842, Aug. 1989. [3] M. Azizoglu, J. A. Salehi, and Y. Li, “Optical CDMA via temporal codes,” IEEE Trans. Commun., vol. 40, pp. 1162–1170, July 1992. [4] F. R. K. Chang, J. A. Salehi, and V. K. Wei, “Optical orthogonal codes: design, analysis and applications,” IEEE Trans. Inform. Theory, vol. 35, pp. 866–873, July 1990. [5] S. Verdù, “Multiple-access channels with point process observation: optimum demodulation,” IEEE Trans. Inform. Theory, vol. IT-32, pp. 642–651, Sept. 1986. [6] M. B. Pearce and B. Aazhang, “Multiuser detection for optical code division multiple access,” IEEE Trans. Commun., vol. 42, pp. 1801–1810, Feb.-Apr. 1994. , “Performance analysis of single user and multiuser detectors for [7] optical code division multiple access communication systems,” IEEE Trans. Commun., vol. 43, pp. 435–444, Feb.-Apr. 1995. [8] L. B. Nelson and H. V. Poor, “Performance of multiuser detection for optical CDMA—Part I: Error probabilities,” IEEE Trans. Commun., vol. 43, pp. 2803–2811, Nov. 1995. [9] A. P. Dempster, N. M. Laired, and D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. Roy. Statist. Soc., ser. B, vol. 39, no. 1, pp. 1–38, 1977. [10] J. T. K. Tang and K. B. Letaief, “Optical CDMA communication systems with multiuser and blind detection,” IEEE Trans. Commun., vol. 47, pp. 1211–1217, Aug. 1999. [11] 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. [12] M. J. Borran and M. Nasiri-Kenari, “An efficient detection technique for synchronous CDMA communication systems based on the expectation maximization algorithm,” IEEE Trans. Veh. Technol., vol. 49, pp. 1663–1668, Sept. 2000.
Abolfazl S. Motahari was born in 1977. He received the B.S. degree from the Iran University of Science and Technology (IUST), Tehran, Iran, in 1999 and the M.S. degree from Sharif University of Technology, Tehran, Iran, in 2001, both in electrical engineering. He is currently working toward the Ph.D. degree in the Department of Electrical Engineering, Sharif University of Technology. From August 2000 to August 2001, he was a Member of Research Staff with the Advanced Communication Science Research Laboratory, Iran Telecommunication Research Center (ITRC), Tehran, Iran. His research interests include spread-spectrum systems, optical communication systems, and statistical digital signal processing.
Masoumeh Nasiri-Kenari (S’90–M’94) received the B.S. and M.S. degrees in electrical engineering from Isfahan University of Technology, Isfahan, Iran, in 1986 and 1987, respectively, and the Ph.D. degree in electrical engineering from the University of Utah, Salt Lake City, in 1993. From 1987 to 1988, she was a Technical Instructor and Research Assistant at Isfahan University of Technology. Since 1994, she has been with the Department of Electrical Engineering, Sharif University of Technology, Tehran, Iran, where she is now an Associate Professor. She was a Co-Director of the Advanced CDMA Laboratory at the Iran Telecom Research Center (ITRC), Tehran, Iran, in 1999–2001. Her research interests are in radio and optical communications and error-control coding.
