ABSTRACT
CONTENTS 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 5.2 α-Stable Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 5.3 Running Myriad Smoothers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 5.4 Optimality of the Sample Myriad in the α-Stable Model . . . . . . . . . . . 163 5.5 Weighted Myriad Smoothers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 5.6 Fast Weighted Myriad Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 5.7 Weighted Myriad Smoother Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 5.8 Weighted Myriad Filters with Real-Valued Weights . . . . . . . . . . . . . . . . 178 5.9 Fast Real-Valued Weighted Myriad Computation . . . . . . . . . . . . . . . . . . 180 5.10 Weighted Myriad Filter Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 5.11 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
In recent years, there has been considerable interest in signal processing based on α-stable distributions. The motivations are simple yet profound. First, good empirical fits are often found through the use of stable distributions on data exhibiting skewness and heavy tails. Second, there is solid theoretical justification that non-Gaussian stable processes emerge in practice, e.g., multiple access interference in a Poisson distributed communication network,1
Internet traffic,2 and numerous other examples as described in Uchaikin and Zolotarev3 and in Feller.4 The third argument for modeling with stable distributions is perhaps the most significant and compelling. Stable distributions satisfy an important generalization of the central limit theorem, which states that the only possible limit of normalized sums of independent and identically distributed terms is stable.5 A wide variety of impulsive processes found in these applications arise as the superposition of many small independent effects. While Gaussian models are clearly inappropriate, stable distributions
Methods, and
model these types of impulsive processes.6,7 Stable models are appealing because the generalization of the central limit theorem explains the apparent contradictions of its “ordinary” version, which could not naturally explain the presence of heavytailed signals.
