- Email: [email protected]
Multiuser Detection in Heavy Tailed Noise
Digital Signal Processing 12, 262–273 (2002) doi:10.1006/dspr.2002.0457
Multiuser Detection in Heavy Tailed Noise 1 Abdelhak M. Zoubir and Ramon F. Brcich Australian Telecommunications Research Institute (ATRI) and School of Electrical and Computer Engineering, Curtin University of Technology, GPO Box U1987, Perth, Western Australia 6845, Australia E-mail: [email protected], [email protected] Zoubir, A. M., and Brcich, R. F., Multiuser Detection in Heavy Tailed Noise, Digital Signal Processing 12 (2002) 262–273. We consider the problem of multiuser detection in impulsive noise channels. Multiuser detection methods have been shown to effectively combat multiple access interference in Gaussian noise, but are highly vulnerable to impulsive noise common in urban and indoor areas. Many multiuser detectors proposed for impulsive noise are based on a specific parametric noise model. While such a detector will perform well near the chosen model, performance is uncertain under larger deviations from the model. Robust detectors seek to minimize this loss, though they still rely on a static, albeit broader, model. We propose a nonparametric detector which makes minimal a priori assumptions on the noise model, requiring only a symmetric density. The detector is based on a nonparametric estimate of the noise density, obtained from the observations without the need for training data. Simulations show the nonparametric detector offers improved performance over existing methods when the noise is highly impulsive. 2002 Elsevier Science (USA) Key Words: multiuser detection; M-estimation; robust statistics; impulsive noise; density estimation.
1. INTRODUCTION The problem of detection in multiple access channels has received much attention. Multiuser detection (MUD) techniques are being intensively investigated because they have the ability to combat the multiple access interference (MAI) caused by the presence of more than one user in the channel. At best MUD can achieve single user performance by removing MAI altogether. Most MUD schemes are designed under the assumption of Gaussian noise and tend 1 This work was in part supported by the Australian Telecommunications Cooperative Research Centre (AT-CRC).
262 1051-2004/02 $35.00 2002 Elsevier Science (USA) All rights reserved.
Zoubir and Brcich: Multiuser Detection in Heavy Tailed Noise
263
to be linear in nature; the decorrelating detector which removes MAI for high signal-to-noise ratios (SNRs) is simply a linear transformation [13]. Impulsive noise has been shown to be the more accurate description of reality for many communications channels including urban and indoor wireless channels [8, 9]. Impulsive noise generally arises from transients such as electromagnetic discharge, automobile ignition, and the operation of common appliances such as microwaves and fluorescent lighting. Unfortunately, impulsive noise can severely degrade the performance of detectors based on the Gaussian assumption, warranting the design of MUD schemes for impulsive noise [1–3]. One way to proceed is to assume a parametric form for the noise distribution, chosen from one of the many impulsive noise models, and then to design an optimal or suboptimal detector. This assumes the chosen distribution is a good model; of course it can only be an approximation to physical reality. How far the model deviates from reality and what effect these deviations have on the estimator become problematic and we can only hope that the estimator is robust with respect to deviations in the assumed model. To decrease the sensitivity of the estimator to the underlying distribution one may turn to the theory of robust estimation and use M-estimators to implement a suboptimal nonlinear detector which is robust to changes in distribution [14]. We propose a nonparametric procedure based on the M-estimation concept but with the noise density estimated from the observations. This enables the detector to adapt to various noise models with minimal a priori information; at most we assume a unimodal symmetric density. To estimate the noise density we use a nonparametric kernel estimator modified to yield good estimates for impulsive noise. This paper is organized as follows. In Section 2 we present the MUD problem. The proposed nonparametric detector is presented in Section 3 after which we discuss nonparametric density estimation in Section 4. Some results are presented in Section 5 before conclusions are drawn in Section 6.
2. SIGNAL MODEL We consider a DSSS MA communications channel where K users transmit synchronously. The received signal is xn =
K
(S)nk Ak bk + vn
n = 1, . . . , N.
(1)
k=1
(S)nk denotes element n, k of the matrix S whose columns consist of the normalized spreading codes of the K users, S = (s1 , . . . , sk ), the codes being of length N . Ak > 0 is the amplitude of user k and bk ∈ {−1, 1} is the bit sent by user k. vn is additive i.i.d. zero mean noise. We will use the vector form of the above model, x = Sθ + v, where θ = Ab, A = diag(A1 , . . . , AK ), b = (b1 , . . . , bk )T , and θk = Ak bk denotes element k of θ , so that all the unknowns are collected into θ .
Digital Signal Processing Vol. 12, Nos. 2,3, April and July 2002
264
Classical least squares (LS) gives θˆLS as θˆLS = argmin θ
N n=1
K xn − (S)nk θk
2 (2)
,
k=1
for which the solution is θˆLS = (S T S)−1 S T x.
(3)
Existence of (S T S)−1 is ensured as the columns of the code matrix are linearly independent spreading codes. The users’ amplitudes and data bits are easily recovered as Aˆ k = |θˆk |, bˆk = sgn(θˆk ). This gives the so-called decorrelating detector, aptly named since it decorrelates the users, removing the MAI [13]. For Gaussian vn the LS solution for θ is equivalent to ML. Now consider the general case where vn has a density fV (v). The ML estimates are obtained as θˆML = argmin θ
N
− log fV xn −
n=1
K
(S)nk θk .
(4)
k=1
If the log-likelihood function has a single minimum we can obtain a unique solution by solving the K coupled equations K K (S)nk ϕ xn − (S)nk θk = 0, k=1
k = 1, . . . , K,
(5)
k =1
where ϕ(v) = −fV (v)/fV (v) is the location score function of fV (v). It is clear that without a priori knowledge of fV (v) estimation of θ cannot be optimal. In impulsive noise of unknown distribution both these methods have severe disadvantages. First, the decorrelating detector is sensitive to impulsive noise, and its performance rapidly degrades as the level of impulsive behavior increases. Second, even for known fV (v), the computational cost of ML is often too high. To alleviate the general problem of estimation in the presence of outliers or impulsive noise, Huber proposed the concept of M-estimators [5, 7]. In an Mestimator the function − log fV (v) is replaced with a similarly behaved penalty function, ρ(v). The penalty function is chosen to confer robustness on the estimator under deviations from the assumed density. One then estimates θ by solving N K (S)nk ψ xn − (S)nk θk = 0, k = 1, . . . , K, (6) n=1
k =1
where ψ(v) = ρ (v). ψ(v) is commonly known as the influence function; when ψ(v) = v the equations reduce to LS. When fV (v) is unknown one cannot be sure whether ψ(v) is close to ϕ(v). Huber considered estimation of location in a nominal Gaussian distribution
Zoubir and Brcich: Multiuser Detection in Heavy Tailed Noise
265
contaminated by an unknown symmetric distribution [7]. This ε contaminated mixture model is defined as fε (v) = (1 − ε)fG (v; 0, γ 2 ) + εH,
(7)
where fG (v; 0, γ 2 ) is a Gaussian distribution with variance γ 2 and H is an unknown symmetric density. By taking a minimax approach, that is, minimizing the maximum asymptotic variance in the estimation of location, Huber arrived at the soft limiter as the minimax solution for ψ(v), for |v| ≤ kγ 2 v/γ 2 , (8) ψ(v) = k sgn(v), for |v| > kγ 2 . The principal of the soft limiter is intuitively obvious. When outliers are present they will be clipped by the limiter, reducing their deleterious effect on the estimator. Solutions to (6) can be obtained by iteratively reweighted least squares or the iterative modified residual method proposed by Huber, which is used here. To achieve a consistent solution it is required that E[ψ(v)] = 0. Here we make the assumption that the noise density is symmetric, so that this condition is met if ψ(v) is antisymmetric. M-estimates using Huber’s penalty function have been considered in the context of MUD; see [14] for more details. Next we consider motivations for and suggest a variation on M-estimation where the penalty function is not set in advance but is estimated from the observations.
3. A NONPARAMETRIC DETECTOR Although the soft limiter influence function is well motivated, like any minimax solution it may be far from optimal for many distributions in the class for which it was designed. Also, its asymptotic optimality property is not indicative of its behavior for small samples. If the penalty function could adapt itself to the observations the estimates may exhibit robust behavior over a wider class of distributions. This is what is proposed here; instead of fixing the penalty function we base it on the observations. To achieve this a nonparametric density estimator is utilized to estimate the score function, ψ(v), in (6). Score function estimation is incorporated into the iterative parameter estimation procedure used to obtain the M-estimates. At each step the parameters are updated and the residuals found, these residuals are used to estimate the score function which in the next step is utilized to update the parameters and so forth. The procedure is summarized as Algorithm 1. A LGORITHM 1 (Iterative procedure for the nonparametric detector). 1. Initialization Set i = 0. Obtain an initial estimate of θ from the decorrelator, θˆ 0 = θˆLS .
Digital Signal Processing Vol. 12, Nos. 2,3, April and July 2002
266
2. Determine the residuals vˆ = x − S θˆ i 3. Estimate the score function From v, ˆ estimate the density, fˆV (v), and evaluate the score function estimate as ψ(v) = −
fˆV (v) . fˆV (v)
4. Update the parameter estimates Evaluate z = ψ(v) ˆ and update the parameters, θˆ i+1 = θˆ i + µ(S T S)−1 S T z, where µ is the step size. 5. Check for convergence If θˆ i+1 − θˆ i < stop, otherwise set i → i + 1 and go to step 2. The use of nonparametric density estimates in the context of detection was considered in [15, 16]. There, a training sequence was used to estimate the locally optimum nonlinearity for detection of a signal in impulsive noise. A training sequence gives a priori information about fV (v), necessary to estimate the distribution of the observations under the null hypothesis that no signal is present. Without such knowledge one cannot implement the Neyman– Pearson detector. The proposed method requires no training sequence as it relies on an iterative scheme where the density and parameters are estimated together. Such a procedure was considered in [11] where a set of basis functions was used to approximate ϕ(v). Here, we approach the problem from a more nonparametric viewpoint. Instead of using linear combinations of basis functions to approximate ϕ(v), we use nonparametric density estimates. Next we address the problem of density estimation.
4. DENSITY ESTIMATION The problem of density estimation is to estimate f (x) from N i.i.d. observations, xn , n = 1, . . . , N . A survey reveals many methods including kernel and nearest neighbor methods, series estimators, and maximum penalized likelihood estimators. Kernel methods are popular because of their wide applicability and the properties of the estimates [4, 10, 12]. Here an adaptive kernel estimator is used, which we briefly review before considering several extensions to cater for specific properties of f (x) such as heavy tails, symmetry, unimodality, and multimodality.
Zoubir and Brcich: Multiuser Detection in Heavy Tailed Noise
267
4.1. Adaptive Kernel Estimator The fundamental basis of kernel methods is to smooth the empirical density by locating a kernel at each observation, their sum giving an estimate of f (x). The adaptive kernel estimator (AKE) is defined as N 1 1 x − xn ˆ , f (x) = K N hλn hλn
(9)
n=1
where K(x) is the kernel function, h the global bandwidth, and λn the local bandwidths. As kernel methods inherit the properties of K(x) it is recommended to ensure K(x) is differentiable, nonnegative, and integrates to one; this guarantees a valid density. We used a Gaussian kernel since it fulfills all these requirements and gives good results over a wide range of distributions. The λn are used to adjust for local structure in f (x) such as heavy tails, while h alters fˆ(x) globally and may be used to control the smoothness. Bandwidth selection. To determine h and λn we proceed as follows. First, find a pilot estimate fˆ0 (x) by setting λn = 1 and for a suitable h, as will 1/N )−β where 0 ≤ β ≤ 1 ˆ be discussed. Second, evaluate λn = (fˆ0 (xn )/ N n=1 f0 (xn ) ˆ controls the sensitivity to f0 (x). A recommended choice is β = 1/2 since asymptotically the resulting estimator will have a bias of lower order than when β = 0. Finally, evaluate fˆ(x) at the required x. For fixed λn , an optimal h can be obtained by minimizing a measure of distance between fˆ(x) and f (x) with respect ∞to h. An oft-used measure is the mean integrated squared error, MISE = E[ −∞ (fˆ(x) − f (x))2 dx]. For the special case of a Gaussian distribution and kernel one obtains hopt = 0.79RN −1/5 , where R is the interquartile range. In general hopt is a reasonable choice even for nonGaussian K(x) and f (x); however, should f (x) be multimodal this choice gives large errors. In this case, or merely to refine h, one must turn to more intelligent methods of bandwidth selection such as cross-validation or resampling schemes such as the bootstrap. In cross-validation one estimates the MISE from the observations and then minimizes this with respect to h. More details can be found in [6, 10]. Bandwidth corrections for heavy tailed distributions. Even with local bandwidth adjustments the AKE may undersmooth and produce spurious peaks in the tails for heavy tailed distributions. Modes in the tails are undesirable as they produce a score function which oscillates in the extremities. This can lead to inaccurate estimates as the iterative scheme used in the parameter estimation algorithm will only converge to a unique minimum if the step size µ ≤ 1/|ψ (x)| [14]. To avoid this problem one can adaptively set the step size, µ = 1/δ max |ψ (xn )|, where δ ≥ 1 allows for a margin of error when estimating ψ (xn ); we used δ = 1.25. However, rapid oscillations result in small step sizes which slow convergence. Possibly the simplest way to remove these peaks is to selectively increase the local bandwidths. We can then correct the tails while not distorting the estimate near the mode. First we define the tail regions as the lower and upper 100ζ % of the observations; we used ζ = 0.25. For each tail we move from the
268
Digital Signal Processing Vol. 12, Nos. 2,3, April and July 2002
extremal observations inward, adjusting local bandwidths for successive pairs of observations. If fˆ(x) at the midpoint between the observations is less than that at both observations a local minima is deemed present and the two local bandwidths are increased. The correction is based on the assumption of a Gaussian kernel. If two Gaussian kernels are separated by a distance of 2" their sum will be unimodal if their local bandwidths are at least "/ h. The component with the smaller local bandwidth is assigned this critical value, while the other is assigned the critical value scaled by the ratio of the original bandwidths to preserve their original proportionality. Should a correction reduce the local bandwidth, it is not applied. This simple scheme produced good results when the tails were heavy. Symmetry. Recall that for (6) to give consistent estimates f (x) should be symmetric and ψ(x) antisymmetric. We ensure ψ(x) is antisymmetric by imposing symmetry on fˆ(x). Let fˆ(x) be the raw asymmetric estimate; then ˆ = a symmetric estimate, fˆs (x), is obtained by taking the even part of fˆ(x), fˆs (x) (fˆ(x) + fˆ(−x))/2. Unimodality. If f (x) is known to be unimodal, incorporating this information into the estimator will reduce error. To impose unimodality on kernel estimators Silverman suggests increasing h; this follows from a result for Gaussian kernels which states that the number of modes is a decreasing function of h [10]. One can check for unimodality by determining the number of peaks in fˆ(x), should there be more than a single peak h is increased by a factor η. The process is then repeated until unimodality is attained, with η = 1.05 a few iterations was usually enough. Consistency of the estimates. Pointwise and uniform consistency using the kernel estimators can be achieved under some mild regularity conditions on f (x) and K(x), and some further conditions on h, both of which are usually fulfilled. Uniform consistency implies that Pr{supx |fˆ(x) − f (x)| → 0} = 1 as n → ∞, more details and other consistency results can be found in [4, 10, 12].
5. SIMULATIONS We compare the decorrelating detector and a robust MUD [14] which uses Mestimates with Huber’s penalty function to the proposed scheme for a variety of noise models. We refer to the method of [14] as the robust detector and the proposed scheme as the nonparametric detector. Unless otherwise stated all noise models are symmetric and unimodal, this being incorporated into the density estimator. Bit error rates (BER) versus SNR are shown for K = 6 users. The spreading codes are shifted maximal length sequences of length N = 31, meaning density estimation is performed with a sample size of 31. The amplitude of the first user is at −10 dB relative to the others; BERs are shown for this user. For Gaussian noise in Fig. 1 the decorrelator and robust detectors perform similarly. This is expected as for Gaussian noise the influence function for the robust detector clips only a minority of the incoming observations, but
Zoubir and Brcich: Multiuser Detection in Heavy Tailed Noise
269
FIG. 1. Bit error rate vs SNR in Gaussian noise (left) and generalized Gaussian noise with p = 0.5 (right).
appears linear, the optimum shape in Gaussian noise, for the majority. Since the nonparametric detector estimates the score function it will never be perfectly linear for Gaussian noise, leading to a decrease in performance of approximately 1 dB. The generalized Gaussian distribution has a density proportional to exp(−|x|p ), 0 < p ≤ 2. A p less than 2 gives a heavy tailed distribution suitable as an impulsive noise model. In Fig. 1 we have shown results for p = 0.5. The nonparametric detector has the lowest BER up to an SNR of 9 dB. There appears to be a range of p and SNR which favor use of the nonparametric detector, namely for low p and SNR, a region where accurate detection is difficult. This suggests that the nonparametric detector is more robust in highly impulsive noise. Next we consider a special case of the ε contaminated mixture model where the contaminant has a Gaussian distribution with variance κγ 2 . This is often used as a model for impulsive noise as it approximates Middleton’s class A model [9]. Results for ε = 0.01, κ = 100 and ε = 0.1, κ = 100 are shown in Fig. 2. Both the robust and the nonparametric detectors outperform the decorrelator,
FIG. 2. Bit error rate vs SNR in ε mixture noise with ε = 0.01, κ = 100 (left) and ε = 0.1, κ = 100 (right).
270
Digital Signal Processing Vol. 12, Nos. 2,3, April and July 2002
FIG. 3. Bit error rate vs SNR in SαS noise for α = 1.5 (left) and α = 1.0 (right).
while the robust detector outperforms the nonparametric detector in the first case but not the second. Which detector is best obviously depends on the model parameters, but we suggest a possible explanation. In the first case there is a small proportion of very large noise values, making tail estimation difficult as we have so few observations in the tail regions. In the second case large noise values occur with a greater probability, aiding in tail estimation and hence estimation of the score function, leading to smaller BERs. The set of (κ, ε) for which one detector outperforms the other has yet to be determined. Next we consider two cases where the noise is symmetric alpha stable (SαS). SαS distributions are a popular model for impulsive noise for which there has been shown to be a physical motivation in multiuser environments [8]. SαS distributions are parameterized by their characteristic exponent, 0 < α ≤ 2, and a scale parameter σ > 0. The level of impulsive behavior increases as α decreases. For α = 2 the distribution is Gaussian, while for α = 1, we have the highly impulsive Cauchy distribution. For α < 2 they have infinite variance and for α ≤ 1 an infinite mean. Results for characteristic exponents of 1.5 and 1.0 are shown in Fig. 3. In these cases the nonparametric detector outperforms both the robust and the decorrelating detectors. Further experiments have shown this to be true for smaller values of α. This gives more evidence to the suggestion that the nonparametric detector is more robust to highly impulsive noise. Finally we show an example where the noise distribution is symmetric but bimodal. This is incorporated into the density estimation procedure as outlined in Section 4. The bimodal density is composed of two Gaussian densities with means of ±1 and a variance of σ 2 = 0.01. Figure 4 shows results for this scenario. The nonparametric detector is able to adapt itself to the bimodal nature of the noise and outperforms the other methods at low SNR. Although asynchronous users and fading multipath channels were not considered, experiments show a similar relative performance between the detectors. To briefly comment on computational complexity, we found the nonparametric detector took on average four iterations to converge while the robust detector took two.
Zoubir and Brcich: Multiuser Detection in Heavy Tailed Noise
271
FIG. 4. Bit error rate vs SNR in symmetric bimodal noise generated from a two component Gaussian mixture model where the components have a mean of ±1 and a common variance of 0.12 .
6. CONCLUSION We proposed a nonparametric detection scheme for the problem of MUD in non-Gaussian noise. The proposed method is conceptually similar to Mestimation in that we use an influence function to approximate the score function. Instead of using a static influence function, we adaptively estimate the score function of the noise using nonparametric density estimation. This results in an iterative procedure where estimates of the signal parameters are used to refine the score function estimate. Simulations show the nonparametric detector outperforms a recently developed robust detector for highly impulsive noise models. When the noise is slightly impulsive performance is improved compared to the decorrelating detector. When the noise models are more complicated, such as for multimodal densities, the nonparametric detector has the advantage as it can adapt itself to the noise distribution. There are many possible avenues for improvement of this basic scheme. The kernel used here for density estimation was chosen to be Gaussian as it gave the best results over a wide range of noise models. It has been suggested that the kernel should match the noise model, so that one should use a Cauchy kernel when the noise is Cauchy. However, as the Cauchy kernel will give tail estimates that are too heavy for distributions with light or only moderately heavy tails this is not recommended. What is needed is a method to automatically select the best kernel. Given a repertoire of possible kernels a scheme can be envisaged where some goodness-of-fit criterion is minimized with respect to the kernel, possibly using cross-validation or resampling methods. The set of kernels need not be very large but must capture the essential characteristics of the distributions we expect, so that we should include a kernel with heavy tails to cater for heavy tailed distributions. As it stands, automatic bandwidth selection using cross-validation gives equal weight across the entire density. In goodness-of-fit tests where one is concerned with deciding whether a set of observations arose from a specific density it is
272
Digital Signal Processing Vol. 12, Nos. 2,3, April and July 2002
common to use a weighting function so that different regions of the density contribute more or less to the final decision. If we are testing for heavy tailed distributions one would want to assign more importance to the tail regions by using a weighting function which is concentrated away from the mode. This idea can be incorporated into cross-validation, producing bandwidths which favor good estimation of the tails, possibly leading to better estimates for heavy tailed distributions. As a final comment we note that although the motivation for developing this nonparametric detection scheme was MUD in non-Gaussian noise, it can be applied whenever we want to estimate the parameters of a signal in additive noise with an unknown distribution.
REFERENCES 1. Aazhang, B. and Poor, H., Performance of DS/SSMA communications in impulsive channels, I: Linear correlation receivers. IEEE Trans. Comm. 35 (1987), 1179–1188. 2. Aazhang, B. and Poor, H., Performance of DS/SSMA communications in impulsive channels, II: Hard-limiting correlation receivers. IEEE Trans. Comm. 36 (1988), 88–97. 3. Aazhang, B. and Poor, H., An analysis of nonlinear direct-sequence correlators. IEEE Trans. Comm. 37 (1989), 23–31. 4. Devroye, L., A Course in Density Estimation. Birkhäuser, Basel, 1987. 5. Rousseeuw, P., Hampel, F., Ronchetti, E., and Stahel, W., Robust Statistics, The Approach Based on Influence Functions. Wiley, New York, 1986. 6. Faraway, J. and Jhun, M., Bootstrap choice of bandwidth selection for density estimation. J. Amer. Statist. Assoc. 85 (1990), 1119–1122. 7. Huber, P., Robust Statistics. Wiley, New York, 1981. 8. Ilow, J. and Hatzinakos, D., Analytic alpha-stable noise modeling in a poisson field of interferers or scatterers, IEEE Trans. Signal Process. 46 (1998), 1601–1611. 9. Middleton, D., Statistical-physical models of electromagnetic interference. IEEE Trans. Electromag. Compat. 19 (1977), 106–127. 10. Silverman, B., Density Estimation for Statistics and Data Analysis. Chapman and Hall, London, 1986. 11. Taleb, A., Brcich, R., and Green, M., Suboptimal robust estimation for signal plus noise models. In Proceedings of the 34th Asilomar Conference on Signals, Systems and Computing. Pacific Grove, CA, October 2000. 12. Thompson, J. and Tapia, R., Nonparametric Function Estimation, Modeling, and Simulation. Society for Industrial and Applied Mathematics, Philadelphia, 1990. 13. Verdu, S., Multiuser Detection. Cambridge University Press, Cambridge, UK, 1998. 14. Wang, X. and Poor, H., Robust multiuser detection in non-Gaussian channels. IEEE Trans. Signal Process. 47 (1999), 289–304. 15. Zabin, S. and Wright, G., Nonparametric density estimation and detection in impulsive interference channels, I: Estimators. IEEE Trans. Comm. 42 (1994), 1684–1697. 16. Zabin, S. and Wright, G., Nonparametric density estimation and detection in impulsive interference channels, II: Detectors. IEEE Trans. Comm. 42 (1994), 1698–1711.
ABDELHAK ZOUBIR received the Dipl.-Ing. degree (BSc/BEng) from Fachhochschule Niederrhein, Germany, in 1983, the Dipl.-Ing, (MSc/MEng) and the Dr.-Ing. (PhD) degree from Ruhr University Bochum, Germany, in 1987 and 1992, all in Electrical Engineering. He is currently Professor of Telecommunications at Curtin University of Technology and Head of the School of Electrical and
Zoubir and Brcich: Multiuser Detection in Heavy Tailed Noise
273
Computer Engineering. His general interest lies in statistical methods for signal processing with applications to communications, sonar, radar, biomedical engineering and vibration analysis. He is a Senior Member of the IEEE, serves on the IEEE SPS Technical Committee on Signal Processing Theory and Methods and is an Associate Editor of IEEE Transactions on Signal Processing. RAMON BRCICH received the B.Eng. degree in Aerospace Avionics (Hons 1) in 1998 from Queensland University of Technology, Brisbane, Australia. At present he is pursuing the Ph.D. degree in signal processing at Curtin University of Technology, Perth, Australia, where he is also a Research Fellow in the School of Electrical and Computer Engineering. His current research interests include irregular sampling and applications of heavy tailed modelling, particularly to telecommunications.
Multiuser Detection in Heavy Tailed Noise 1 Abdelhak M. Zoubir and Ramon F. Brcich Australian Telecommunications Research Institute (ATRI) and School of Electrical and Computer Engineering, Curtin University of Technology, GPO Box U1987, Perth, Western Australia 6845, Australia E-mail: [email protected], [email protected] Zoubir, A. M., and Brcich, R. F., Multiuser Detection in Heavy Tailed Noise, Digital Signal Processing 12 (2002) 262–273. We consider the problem of multiuser detection in impulsive noise channels. Multiuser detection methods have been shown to effectively combat multiple access interference in Gaussian noise, but are highly vulnerable to impulsive noise common in urban and indoor areas. Many multiuser detectors proposed for impulsive noise are based on a specific parametric noise model. While such a detector will perform well near the chosen model, performance is uncertain under larger deviations from the model. Robust detectors seek to minimize this loss, though they still rely on a static, albeit broader, model. We propose a nonparametric detector which makes minimal a priori assumptions on the noise model, requiring only a symmetric density. The detector is based on a nonparametric estimate of the noise density, obtained from the observations without the need for training data. Simulations show the nonparametric detector offers improved performance over existing methods when the noise is highly impulsive. 2002 Elsevier Science (USA) Key Words: multiuser detection; M-estimation; robust statistics; impulsive noise; density estimation.
1. INTRODUCTION The problem of detection in multiple access channels has received much attention. Multiuser detection (MUD) techniques are being intensively investigated because they have the ability to combat the multiple access interference (MAI) caused by the presence of more than one user in the channel. At best MUD can achieve single user performance by removing MAI altogether. Most MUD schemes are designed under the assumption of Gaussian noise and tend 1 This work was in part supported by the Australian Telecommunications Cooperative Research Centre (AT-CRC).
262 1051-2004/02 $35.00 2002 Elsevier Science (USA) All rights reserved.
Zoubir and Brcich: Multiuser Detection in Heavy Tailed Noise
263
to be linear in nature; the decorrelating detector which removes MAI for high signal-to-noise ratios (SNRs) is simply a linear transformation [13]. Impulsive noise has been shown to be the more accurate description of reality for many communications channels including urban and indoor wireless channels [8, 9]. Impulsive noise generally arises from transients such as electromagnetic discharge, automobile ignition, and the operation of common appliances such as microwaves and fluorescent lighting. Unfortunately, impulsive noise can severely degrade the performance of detectors based on the Gaussian assumption, warranting the design of MUD schemes for impulsive noise [1–3]. One way to proceed is to assume a parametric form for the noise distribution, chosen from one of the many impulsive noise models, and then to design an optimal or suboptimal detector. This assumes the chosen distribution is a good model; of course it can only be an approximation to physical reality. How far the model deviates from reality and what effect these deviations have on the estimator become problematic and we can only hope that the estimator is robust with respect to deviations in the assumed model. To decrease the sensitivity of the estimator to the underlying distribution one may turn to the theory of robust estimation and use M-estimators to implement a suboptimal nonlinear detector which is robust to changes in distribution [14]. We propose a nonparametric procedure based on the M-estimation concept but with the noise density estimated from the observations. This enables the detector to adapt to various noise models with minimal a priori information; at most we assume a unimodal symmetric density. To estimate the noise density we use a nonparametric kernel estimator modified to yield good estimates for impulsive noise. This paper is organized as follows. In Section 2 we present the MUD problem. The proposed nonparametric detector is presented in Section 3 after which we discuss nonparametric density estimation in Section 4. Some results are presented in Section 5 before conclusions are drawn in Section 6.
2. SIGNAL MODEL We consider a DSSS MA communications channel where K users transmit synchronously. The received signal is xn =
K
(S)nk Ak bk + vn
n = 1, . . . , N.
(1)
k=1
(S)nk denotes element n, k of the matrix S whose columns consist of the normalized spreading codes of the K users, S = (s1 , . . . , sk ), the codes being of length N . Ak > 0 is the amplitude of user k and bk ∈ {−1, 1} is the bit sent by user k. vn is additive i.i.d. zero mean noise. We will use the vector form of the above model, x = Sθ + v, where θ = Ab, A = diag(A1 , . . . , AK ), b = (b1 , . . . , bk )T , and θk = Ak bk denotes element k of θ , so that all the unknowns are collected into θ .
Digital Signal Processing Vol. 12, Nos. 2,3, April and July 2002
264
Classical least squares (LS) gives θˆLS as θˆLS = argmin θ
N n=1
K xn − (S)nk θk
2 (2)
,
k=1
for which the solution is θˆLS = (S T S)−1 S T x.
(3)
Existence of (S T S)−1 is ensured as the columns of the code matrix are linearly independent spreading codes. The users’ amplitudes and data bits are easily recovered as Aˆ k = |θˆk |, bˆk = sgn(θˆk ). This gives the so-called decorrelating detector, aptly named since it decorrelates the users, removing the MAI [13]. For Gaussian vn the LS solution for θ is equivalent to ML. Now consider the general case where vn has a density fV (v). The ML estimates are obtained as θˆML = argmin θ
N
− log fV xn −
n=1
K
(S)nk θk .
(4)
k=1
If the log-likelihood function has a single minimum we can obtain a unique solution by solving the K coupled equations K K (S)nk ϕ xn − (S)nk θk = 0, k=1
k = 1, . . . , K,
(5)
k =1
where ϕ(v) = −fV (v)/fV (v) is the location score function of fV (v). It is clear that without a priori knowledge of fV (v) estimation of θ cannot be optimal. In impulsive noise of unknown distribution both these methods have severe disadvantages. First, the decorrelating detector is sensitive to impulsive noise, and its performance rapidly degrades as the level of impulsive behavior increases. Second, even for known fV (v), the computational cost of ML is often too high. To alleviate the general problem of estimation in the presence of outliers or impulsive noise, Huber proposed the concept of M-estimators [5, 7]. In an Mestimator the function − log fV (v) is replaced with a similarly behaved penalty function, ρ(v). The penalty function is chosen to confer robustness on the estimator under deviations from the assumed density. One then estimates θ by solving N K (S)nk ψ xn − (S)nk θk = 0, k = 1, . . . , K, (6) n=1
k =1
where ψ(v) = ρ (v). ψ(v) is commonly known as the influence function; when ψ(v) = v the equations reduce to LS. When fV (v) is unknown one cannot be sure whether ψ(v) is close to ϕ(v). Huber considered estimation of location in a nominal Gaussian distribution
Zoubir and Brcich: Multiuser Detection in Heavy Tailed Noise
265
contaminated by an unknown symmetric distribution [7]. This ε contaminated mixture model is defined as fε (v) = (1 − ε)fG (v; 0, γ 2 ) + εH,
(7)
where fG (v; 0, γ 2 ) is a Gaussian distribution with variance γ 2 and H is an unknown symmetric density. By taking a minimax approach, that is, minimizing the maximum asymptotic variance in the estimation of location, Huber arrived at the soft limiter as the minimax solution for ψ(v), for |v| ≤ kγ 2 v/γ 2 , (8) ψ(v) = k sgn(v), for |v| > kγ 2 . The principal of the soft limiter is intuitively obvious. When outliers are present they will be clipped by the limiter, reducing their deleterious effect on the estimator. Solutions to (6) can be obtained by iteratively reweighted least squares or the iterative modified residual method proposed by Huber, which is used here. To achieve a consistent solution it is required that E[ψ(v)] = 0. Here we make the assumption that the noise density is symmetric, so that this condition is met if ψ(v) is antisymmetric. M-estimates using Huber’s penalty function have been considered in the context of MUD; see [14] for more details. Next we consider motivations for and suggest a variation on M-estimation where the penalty function is not set in advance but is estimated from the observations.
3. A NONPARAMETRIC DETECTOR Although the soft limiter influence function is well motivated, like any minimax solution it may be far from optimal for many distributions in the class for which it was designed. Also, its asymptotic optimality property is not indicative of its behavior for small samples. If the penalty function could adapt itself to the observations the estimates may exhibit robust behavior over a wider class of distributions. This is what is proposed here; instead of fixing the penalty function we base it on the observations. To achieve this a nonparametric density estimator is utilized to estimate the score function, ψ(v), in (6). Score function estimation is incorporated into the iterative parameter estimation procedure used to obtain the M-estimates. At each step the parameters are updated and the residuals found, these residuals are used to estimate the score function which in the next step is utilized to update the parameters and so forth. The procedure is summarized as Algorithm 1. A LGORITHM 1 (Iterative procedure for the nonparametric detector). 1. Initialization Set i = 0. Obtain an initial estimate of θ from the decorrelator, θˆ 0 = θˆLS .
Digital Signal Processing Vol. 12, Nos. 2,3, April and July 2002
266
2. Determine the residuals vˆ = x − S θˆ i 3. Estimate the score function From v, ˆ estimate the density, fˆV (v), and evaluate the score function estimate as ψ(v) = −
fˆV (v) . fˆV (v)
4. Update the parameter estimates Evaluate z = ψ(v) ˆ and update the parameters, θˆ i+1 = θˆ i + µ(S T S)−1 S T z, where µ is the step size. 5. Check for convergence If θˆ i+1 − θˆ i < stop, otherwise set i → i + 1 and go to step 2. The use of nonparametric density estimates in the context of detection was considered in [15, 16]. There, a training sequence was used to estimate the locally optimum nonlinearity for detection of a signal in impulsive noise. A training sequence gives a priori information about fV (v), necessary to estimate the distribution of the observations under the null hypothesis that no signal is present. Without such knowledge one cannot implement the Neyman– Pearson detector. The proposed method requires no training sequence as it relies on an iterative scheme where the density and parameters are estimated together. Such a procedure was considered in [11] where a set of basis functions was used to approximate ϕ(v). Here, we approach the problem from a more nonparametric viewpoint. Instead of using linear combinations of basis functions to approximate ϕ(v), we use nonparametric density estimates. Next we address the problem of density estimation.
4. DENSITY ESTIMATION The problem of density estimation is to estimate f (x) from N i.i.d. observations, xn , n = 1, . . . , N . A survey reveals many methods including kernel and nearest neighbor methods, series estimators, and maximum penalized likelihood estimators. Kernel methods are popular because of their wide applicability and the properties of the estimates [4, 10, 12]. Here an adaptive kernel estimator is used, which we briefly review before considering several extensions to cater for specific properties of f (x) such as heavy tails, symmetry, unimodality, and multimodality.
Zoubir and Brcich: Multiuser Detection in Heavy Tailed Noise
267
4.1. Adaptive Kernel Estimator The fundamental basis of kernel methods is to smooth the empirical density by locating a kernel at each observation, their sum giving an estimate of f (x). The adaptive kernel estimator (AKE) is defined as N 1 1 x − xn ˆ , f (x) = K N hλn hλn
(9)
n=1
where K(x) is the kernel function, h the global bandwidth, and λn the local bandwidths. As kernel methods inherit the properties of K(x) it is recommended to ensure K(x) is differentiable, nonnegative, and integrates to one; this guarantees a valid density. We used a Gaussian kernel since it fulfills all these requirements and gives good results over a wide range of distributions. The λn are used to adjust for local structure in f (x) such as heavy tails, while h alters fˆ(x) globally and may be used to control the smoothness. Bandwidth selection. To determine h and λn we proceed as follows. First, find a pilot estimate fˆ0 (x) by setting λn = 1 and for a suitable h, as will 1/N )−β where 0 ≤ β ≤ 1 ˆ be discussed. Second, evaluate λn = (fˆ0 (xn )/ N n=1 f0 (xn ) ˆ controls the sensitivity to f0 (x). A recommended choice is β = 1/2 since asymptotically the resulting estimator will have a bias of lower order than when β = 0. Finally, evaluate fˆ(x) at the required x. For fixed λn , an optimal h can be obtained by minimizing a measure of distance between fˆ(x) and f (x) with respect ∞to h. An oft-used measure is the mean integrated squared error, MISE = E[ −∞ (fˆ(x) − f (x))2 dx]. For the special case of a Gaussian distribution and kernel one obtains hopt = 0.79RN −1/5 , where R is the interquartile range. In general hopt is a reasonable choice even for nonGaussian K(x) and f (x); however, should f (x) be multimodal this choice gives large errors. In this case, or merely to refine h, one must turn to more intelligent methods of bandwidth selection such as cross-validation or resampling schemes such as the bootstrap. In cross-validation one estimates the MISE from the observations and then minimizes this with respect to h. More details can be found in [6, 10]. Bandwidth corrections for heavy tailed distributions. Even with local bandwidth adjustments the AKE may undersmooth and produce spurious peaks in the tails for heavy tailed distributions. Modes in the tails are undesirable as they produce a score function which oscillates in the extremities. This can lead to inaccurate estimates as the iterative scheme used in the parameter estimation algorithm will only converge to a unique minimum if the step size µ ≤ 1/|ψ (x)| [14]. To avoid this problem one can adaptively set the step size, µ = 1/δ max |ψ (xn )|, where δ ≥ 1 allows for a margin of error when estimating ψ (xn ); we used δ = 1.25. However, rapid oscillations result in small step sizes which slow convergence. Possibly the simplest way to remove these peaks is to selectively increase the local bandwidths. We can then correct the tails while not distorting the estimate near the mode. First we define the tail regions as the lower and upper 100ζ % of the observations; we used ζ = 0.25. For each tail we move from the
268
Digital Signal Processing Vol. 12, Nos. 2,3, April and July 2002
extremal observations inward, adjusting local bandwidths for successive pairs of observations. If fˆ(x) at the midpoint between the observations is less than that at both observations a local minima is deemed present and the two local bandwidths are increased. The correction is based on the assumption of a Gaussian kernel. If two Gaussian kernels are separated by a distance of 2" their sum will be unimodal if their local bandwidths are at least "/ h. The component with the smaller local bandwidth is assigned this critical value, while the other is assigned the critical value scaled by the ratio of the original bandwidths to preserve their original proportionality. Should a correction reduce the local bandwidth, it is not applied. This simple scheme produced good results when the tails were heavy. Symmetry. Recall that for (6) to give consistent estimates f (x) should be symmetric and ψ(x) antisymmetric. We ensure ψ(x) is antisymmetric by imposing symmetry on fˆ(x). Let fˆ(x) be the raw asymmetric estimate; then ˆ = a symmetric estimate, fˆs (x), is obtained by taking the even part of fˆ(x), fˆs (x) (fˆ(x) + fˆ(−x))/2. Unimodality. If f (x) is known to be unimodal, incorporating this information into the estimator will reduce error. To impose unimodality on kernel estimators Silverman suggests increasing h; this follows from a result for Gaussian kernels which states that the number of modes is a decreasing function of h [10]. One can check for unimodality by determining the number of peaks in fˆ(x), should there be more than a single peak h is increased by a factor η. The process is then repeated until unimodality is attained, with η = 1.05 a few iterations was usually enough. Consistency of the estimates. Pointwise and uniform consistency using the kernel estimators can be achieved under some mild regularity conditions on f (x) and K(x), and some further conditions on h, both of which are usually fulfilled. Uniform consistency implies that Pr{supx |fˆ(x) − f (x)| → 0} = 1 as n → ∞, more details and other consistency results can be found in [4, 10, 12].
5. SIMULATIONS We compare the decorrelating detector and a robust MUD [14] which uses Mestimates with Huber’s penalty function to the proposed scheme for a variety of noise models. We refer to the method of [14] as the robust detector and the proposed scheme as the nonparametric detector. Unless otherwise stated all noise models are symmetric and unimodal, this being incorporated into the density estimator. Bit error rates (BER) versus SNR are shown for K = 6 users. The spreading codes are shifted maximal length sequences of length N = 31, meaning density estimation is performed with a sample size of 31. The amplitude of the first user is at −10 dB relative to the others; BERs are shown for this user. For Gaussian noise in Fig. 1 the decorrelator and robust detectors perform similarly. This is expected as for Gaussian noise the influence function for the robust detector clips only a minority of the incoming observations, but
Zoubir and Brcich: Multiuser Detection in Heavy Tailed Noise
269
FIG. 1. Bit error rate vs SNR in Gaussian noise (left) and generalized Gaussian noise with p = 0.5 (right).
appears linear, the optimum shape in Gaussian noise, for the majority. Since the nonparametric detector estimates the score function it will never be perfectly linear for Gaussian noise, leading to a decrease in performance of approximately 1 dB. The generalized Gaussian distribution has a density proportional to exp(−|x|p ), 0 < p ≤ 2. A p less than 2 gives a heavy tailed distribution suitable as an impulsive noise model. In Fig. 1 we have shown results for p = 0.5. The nonparametric detector has the lowest BER up to an SNR of 9 dB. There appears to be a range of p and SNR which favor use of the nonparametric detector, namely for low p and SNR, a region where accurate detection is difficult. This suggests that the nonparametric detector is more robust in highly impulsive noise. Next we consider a special case of the ε contaminated mixture model where the contaminant has a Gaussian distribution with variance κγ 2 . This is often used as a model for impulsive noise as it approximates Middleton’s class A model [9]. Results for ε = 0.01, κ = 100 and ε = 0.1, κ = 100 are shown in Fig. 2. Both the robust and the nonparametric detectors outperform the decorrelator,
FIG. 2. Bit error rate vs SNR in ε mixture noise with ε = 0.01, κ = 100 (left) and ε = 0.1, κ = 100 (right).
270
Digital Signal Processing Vol. 12, Nos. 2,3, April and July 2002
FIG. 3. Bit error rate vs SNR in SαS noise for α = 1.5 (left) and α = 1.0 (right).
while the robust detector outperforms the nonparametric detector in the first case but not the second. Which detector is best obviously depends on the model parameters, but we suggest a possible explanation. In the first case there is a small proportion of very large noise values, making tail estimation difficult as we have so few observations in the tail regions. In the second case large noise values occur with a greater probability, aiding in tail estimation and hence estimation of the score function, leading to smaller BERs. The set of (κ, ε) for which one detector outperforms the other has yet to be determined. Next we consider two cases where the noise is symmetric alpha stable (SαS). SαS distributions are a popular model for impulsive noise for which there has been shown to be a physical motivation in multiuser environments [8]. SαS distributions are parameterized by their characteristic exponent, 0 < α ≤ 2, and a scale parameter σ > 0. The level of impulsive behavior increases as α decreases. For α = 2 the distribution is Gaussian, while for α = 1, we have the highly impulsive Cauchy distribution. For α < 2 they have infinite variance and for α ≤ 1 an infinite mean. Results for characteristic exponents of 1.5 and 1.0 are shown in Fig. 3. In these cases the nonparametric detector outperforms both the robust and the decorrelating detectors. Further experiments have shown this to be true for smaller values of α. This gives more evidence to the suggestion that the nonparametric detector is more robust to highly impulsive noise. Finally we show an example where the noise distribution is symmetric but bimodal. This is incorporated into the density estimation procedure as outlined in Section 4. The bimodal density is composed of two Gaussian densities with means of ±1 and a variance of σ 2 = 0.01. Figure 4 shows results for this scenario. The nonparametric detector is able to adapt itself to the bimodal nature of the noise and outperforms the other methods at low SNR. Although asynchronous users and fading multipath channels were not considered, experiments show a similar relative performance between the detectors. To briefly comment on computational complexity, we found the nonparametric detector took on average four iterations to converge while the robust detector took two.
Zoubir and Brcich: Multiuser Detection in Heavy Tailed Noise
271
FIG. 4. Bit error rate vs SNR in symmetric bimodal noise generated from a two component Gaussian mixture model where the components have a mean of ±1 and a common variance of 0.12 .
6. CONCLUSION We proposed a nonparametric detection scheme for the problem of MUD in non-Gaussian noise. The proposed method is conceptually similar to Mestimation in that we use an influence function to approximate the score function. Instead of using a static influence function, we adaptively estimate the score function of the noise using nonparametric density estimation. This results in an iterative procedure where estimates of the signal parameters are used to refine the score function estimate. Simulations show the nonparametric detector outperforms a recently developed robust detector for highly impulsive noise models. When the noise is slightly impulsive performance is improved compared to the decorrelating detector. When the noise models are more complicated, such as for multimodal densities, the nonparametric detector has the advantage as it can adapt itself to the noise distribution. There are many possible avenues for improvement of this basic scheme. The kernel used here for density estimation was chosen to be Gaussian as it gave the best results over a wide range of noise models. It has been suggested that the kernel should match the noise model, so that one should use a Cauchy kernel when the noise is Cauchy. However, as the Cauchy kernel will give tail estimates that are too heavy for distributions with light or only moderately heavy tails this is not recommended. What is needed is a method to automatically select the best kernel. Given a repertoire of possible kernels a scheme can be envisaged where some goodness-of-fit criterion is minimized with respect to the kernel, possibly using cross-validation or resampling methods. The set of kernels need not be very large but must capture the essential characteristics of the distributions we expect, so that we should include a kernel with heavy tails to cater for heavy tailed distributions. As it stands, automatic bandwidth selection using cross-validation gives equal weight across the entire density. In goodness-of-fit tests where one is concerned with deciding whether a set of observations arose from a specific density it is
272
Digital Signal Processing Vol. 12, Nos. 2,3, April and July 2002
common to use a weighting function so that different regions of the density contribute more or less to the final decision. If we are testing for heavy tailed distributions one would want to assign more importance to the tail regions by using a weighting function which is concentrated away from the mode. This idea can be incorporated into cross-validation, producing bandwidths which favor good estimation of the tails, possibly leading to better estimates for heavy tailed distributions. As a final comment we note that although the motivation for developing this nonparametric detection scheme was MUD in non-Gaussian noise, it can be applied whenever we want to estimate the parameters of a signal in additive noise with an unknown distribution.
REFERENCES 1. Aazhang, B. and Poor, H., Performance of DS/SSMA communications in impulsive channels, I: Linear correlation receivers. IEEE Trans. Comm. 35 (1987), 1179–1188. 2. Aazhang, B. and Poor, H., Performance of DS/SSMA communications in impulsive channels, II: Hard-limiting correlation receivers. IEEE Trans. Comm. 36 (1988), 88–97. 3. Aazhang, B. and Poor, H., An analysis of nonlinear direct-sequence correlators. IEEE Trans. Comm. 37 (1989), 23–31. 4. Devroye, L., A Course in Density Estimation. Birkhäuser, Basel, 1987. 5. Rousseeuw, P., Hampel, F., Ronchetti, E., and Stahel, W., Robust Statistics, The Approach Based on Influence Functions. Wiley, New York, 1986. 6. Faraway, J. and Jhun, M., Bootstrap choice of bandwidth selection for density estimation. J. Amer. Statist. Assoc. 85 (1990), 1119–1122. 7. Huber, P., Robust Statistics. Wiley, New York, 1981. 8. Ilow, J. and Hatzinakos, D., Analytic alpha-stable noise modeling in a poisson field of interferers or scatterers, IEEE Trans. Signal Process. 46 (1998), 1601–1611. 9. Middleton, D., Statistical-physical models of electromagnetic interference. IEEE Trans. Electromag. Compat. 19 (1977), 106–127. 10. Silverman, B., Density Estimation for Statistics and Data Analysis. Chapman and Hall, London, 1986. 11. Taleb, A., Brcich, R., and Green, M., Suboptimal robust estimation for signal plus noise models. In Proceedings of the 34th Asilomar Conference on Signals, Systems and Computing. Pacific Grove, CA, October 2000. 12. Thompson, J. and Tapia, R., Nonparametric Function Estimation, Modeling, and Simulation. Society for Industrial and Applied Mathematics, Philadelphia, 1990. 13. Verdu, S., Multiuser Detection. Cambridge University Press, Cambridge, UK, 1998. 14. Wang, X. and Poor, H., Robust multiuser detection in non-Gaussian channels. IEEE Trans. Signal Process. 47 (1999), 289–304. 15. Zabin, S. and Wright, G., Nonparametric density estimation and detection in impulsive interference channels, I: Estimators. IEEE Trans. Comm. 42 (1994), 1684–1697. 16. Zabin, S. and Wright, G., Nonparametric density estimation and detection in impulsive interference channels, II: Detectors. IEEE Trans. Comm. 42 (1994), 1698–1711.
ABDELHAK ZOUBIR received the Dipl.-Ing. degree (BSc/BEng) from Fachhochschule Niederrhein, Germany, in 1983, the Dipl.-Ing, (MSc/MEng) and the Dr.-Ing. (PhD) degree from Ruhr University Bochum, Germany, in 1987 and 1992, all in Electrical Engineering. He is currently Professor of Telecommunications at Curtin University of Technology and Head of the School of Electrical and
Zoubir and Brcich: Multiuser Detection in Heavy Tailed Noise
273
Computer Engineering. His general interest lies in statistical methods for signal processing with applications to communications, sonar, radar, biomedical engineering and vibration analysis. He is a Senior Member of the IEEE, serves on the IEEE SPS Technical Committee on Signal Processing Theory and Methods and is an Associate Editor of IEEE Transactions on Signal Processing. RAMON BRCICH received the B.Eng. degree in Aerospace Avionics (Hons 1) in 1998 from Queensland University of Technology, Brisbane, Australia. At present he is pursuing the Ph.D. degree in signal processing at Curtin University of Technology, Perth, Australia, where he is also a Research Fellow in the School of Electrical and Computer Engineering. His current research interests include irregular sampling and applications of heavy tailed modelling, particularly to telecommunications.