886 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 3, MARCH 2010 BER Optimal Linear Combiner for Signal Detection in Symmetric Alpha-Stable Noise: Small Values of Alpha S. Niranjayan, Student Member, IEEE, and Norman C. Beaulieu, Fellow, IEEE Abstract—The maximum likelihood optimal combiner for sig- nal detection in alpha-stable noise is not known in general, except for some special values of the characteristic exponent . A linear Rake combiner receiver is simple and easy to realize. The optimal linear Rake receiver, in the sense of minimizing the bit error rate, for the detection of signals contaminated by symmetric alpha stable noise is derived for values of , 0 < ≤ 1. Interestingly, for this range of , the optimal combiner is found to be a selection combiner which selects the channel (finger) with the largest signal amplitude and suppresses all other channels (fingers). This interesting result is valid over the range 0 < ≤ 1 and allows one to implement effective signal detection without having to know the actual value of the parameter . Therefore, the result yields a very simple form of diversity combiner for signal detection in symmetric alpha-stable noise for 0 < ≤ 1. Comparisons with the widely used maximal ratio combining and equal gain combining schemes are made in terms of signal-to- noise ratio advantage defined in the bit error rate sense. Index Terms—Alpha-stable noise, diversity, fading channels, linear Rake receiver. I. I NTRODUCTION I N many communication systems, the signal of interest is corrupted by additive impulsive noise. Some examples are shot noise, radar clutter [1] and multiple access interference (MAI) in ultra-wide bandwidth (UWB) systems [2], [3]. The symmetric alpha-stable distribution () is one of the mod- els suggested for impulsive non-Gaussian noise distributions [4] - [11]. Also, in a recent example, the model is used to model the MAI distribution in designing improved receivers for UWB multiple access communication [12]. In this paper, the problem of signal detection using a set of independent samples which are corrupted by additive noise is considered. These samples may be obtained from multipaths, or antennas, or frequency multiplexed channels. Closed-form expressions for the PDFs of -stable processes do not exist for every value of . Known closed- form expressions exist only for the cases =0.5 (L´ evy distribution), = 1 (Cauchy distribution) and = 2 (Gaussian distribution) [10] (among these distributions, the L´ evy distribution is skewed and therefore not a symmetric - stable distribution.). Therefore, the maximal likelihood (ML) detector for is not known except for the aforementioned Manuscript received October 13, 2008; revised June 15, 2009; accepted August 3, 2009. The associate editor coordinating the review of this letter and approving it for publication was K. B. Lee. This work is funded by the Alberta Ingenuity Fund and the Informatics Circle of Research Excellence (iCORE), Alberta, Canada. The authors are with the Department of Electrical and Computer Engineer- ing, University of Alberta, Edmonton, Alberta, Canada, T6G 2V4 (e-mail: {niranjan, beaulieu}@ece.ualberta.ca). Digital Object Identifier 10.1109/TWC.2010.03.081362 Fig. 1. A block diagram of the linear Rake combiner with fingers. special cases. Several suboptimal techniques have been pro- posed in the literature [4] - [7], [9], [11], [12]. All these techniques ensure better performance than a conventional linear receiver but have higher complexity. A linear Rake combiner (as illustrated in Fig. 1) is a simple and easy to implement form of suboptimal receiver. The optimal linear combiner for 1 < ≤ 2 was previously derived in [13]. However, in our derivations in the sequel, it will be shown that the structure of the optimal linear combiner is very different for 0 < ≤ 1. The optimal linear Rake combiner derived here reduces to a selection combiner which selects the strongest channel and disregards the rest. Therefore, the implementation of this combiner has no requirement to estimate the alpha- stable distribution (noise distribution) parameters; instead one needs only to compare the signal strengths in the channels. It is also found that the optimal linear combiner can yield significant performance enhancement in terms of SNR over the conventionally used maximal ratio combiner (MRC) and equal gain combiner (EGC). II. SYSTEM MODEL AND THE OPTIMAL COMBINER The approach used in [13] is used here to model the system and formulate the problem. Fig. 1 depicts the linear Rake combiner considered in this letter. The received corrupted signal on the ℎ channel is given by = + where is the signal component, denotes the noise and ∈ {−1, 1} denotes the binary symbol. The noise is distributed according to an distribution, (,). The characteristic function 1536-1276/10$25.00 c ⃝ 2010 IEEE