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International Journal of Antennas and Propagation

Volume 2015 (2015), Article ID 873890, 13 pages

http://dx.doi.org/10.1155/2015/873890

## On the Efficient Generation of --** and **-- White Samples with Applications

^{1}National Institute of Telecommunications (Inatel), P. O. Box 05, 37540-000 Santa Rita do Sapucaí, MG, Brazil^{2}Department of Communications, School of Electrical and Computation Engineering, University of Campinas (DECOM/FEEC/UNICAMP), 13083-852 Campinas, SP, Brazil

Received 8 May 2014; Revised 18 July 2014; Accepted 1 August 2014

Academic Editor: Jose F. Paris

Copyright © 2015 Rausley Adriano Amaral de Souza et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper is concerned with a simple and highly efficient random sequence generator for uncorrelated and variates. The algorithm may yield an efficiency of almost 100%, and this high efficiency can be reached for all special cases such as , , , Nakagami-*m*, Nakagami-*q*, Weibull, Hoyt, Rayleigh, Rice, Exponential, and the One-Sided Gaussian. This generator is implemented via the rejection technique and allows for arbitrary fading parameters. The goodness-of-fit is measured using the Kolmogorov-Smirnov and Anderson-Darling tests. The maximum likelihood parameter estimation for the distribution is proposed and verified against true values of the parameters chosen in the generator. We also provide two important applications for the random sequence generator, the first one dealing with the performance assessment of a digital communication system over the and fading channels and the second one dealing with the performance assessment of the spectrum sensing with energy detection over special cases of these channels. Theoretical and simulation results are compared, validating again the accuracy of the generators.

#### 1. Introduction

In nearly all fields of science, simulation is a strikingly powerful tool widely adopted to help develop a better understanding of some phenomenon under investigation. Particularly in engineering, it is used, for instance, to successfully test equipment, algorithms, and techniques, and, to some extent and whenever applicable, to avoid or minimize time-consuming, costly, and inexhaustible field trials. Wireless communications are no exception and in this challenging, lively, and unkind area, with systems becoming increasingly more complex, both industry and academy engage themselves in developing simulators. Such simulators for wireless communications almost certainly include a block for the fading channel.

The fading channel can be described by a number of models. Among them, the general models, namely, --, -- [1], and some particular cases such as - [2], - [2], and *α*- [3], have been gaining wide acceptance [4–25]. Their flexibility renders them adaptable to situations in which none of the traditional distributions yield good fit [2, 3]. In addition, their applicability has been recognized in practical and real scenarios. Field measurements carried out in diverse propagation environments have shown that, in many situations, these models better accommodate the statistical variations of the propagated signal [1, Section VII], [2, 7–10, 26]. In this sense, developing and ameliorating methods in order to simulate the -- and -- fading models and their special cases for* arbitrary* values of their parameters are of paramount importance. One first step in such a direction is to generate uncorrelated samples and then, if required, correlate them.

This paper is concerned with the generation of uncorrelated samples of -- and -- fading models for arbitrary values of their parameters. Two largely applied methods in this case are the inversion method and the rejection method. The former involves the knowledge of the inverse of the cumulative distribution function (cdf) of the variate, which is either not always available or cannot be easily implemented, but, on the other hand, is highly efficient. The latter is general and applies to any variate but can be rather inefficient.

A useful method for generating independent -, -, and - sequences with an arbitrary fading parameter was recently investigated in [27]. The method is reported to achieve an efficiency higher than for - and for -. More interestingly, a transformation was proposed in which, from an - sequence, a new - sequence can be obtained with an almost efficiency.

In this paper, we extend the applicability of the approach in [28] to provide an easy-to-implement and highly efficient algorithm that generates -- and -- uncorrelated sequences for arbitrary values of their parameters. A simple transformation is also proposed, in which, from an -- or -- sequence, a new -- or -- sequence can be obtained with an almost efficiency. To the best of the authors' knowledge, the results reported here are new.

With the aim of quantifying the performance of the random sequence generators, we compare empirical cdfs to hypothesized ones by carrying out goodness-of-fit Kolmogorov-Smirnov (KS) and Anderson-Darling (AD) tests. We also generate a large number of - random variables, as a particular case of the -- generator, and perform the maximum likelihood (ML) estimation of the parameters and . We then verify these estimates against true values of and defined in the generator. In this context, we use the maximum likelihood technique as its estimators have notable properties, mainly for large sample size [29]. In fact, under regularity conditions, for large sample size ML estimators are consistent and have normal distribution with variance attaining the Cramér-Rao lower bound (CRLB) [29].

In order to demonstrate the usefulness of the proposed method, we provide theoretical and simulated bit error rates of a coherent binary phase-shift keying (BPSK) modulation over the -- and -- fading channels. We also provide the performance assessment of the spectrum sensing with energy detection over special cases of these channels, namely, the - and - channels.

The remaining of the paper is organized as follows. Section 2 presents the preliminary proposed algorithm and briefly describes the general distributions that are the focus of this paper. Numerical results, including the goodness-of-fit test, and their interpretations are presented in Section 3. In Section 4 a near- efficient and definitive algorithm for generating -- and -- variates is discussed in detail. Section 5 verifies the -- generator performance by checking ML parameter estimates from - random samples against true values of the distribution parameters. In Section 6 the average error probability of the BPSK modulation over the -- and -- channels and the performance of the spectrum sensing over the - and - channels are presented. Some conclusions are drawn in Section 7.

#### 2. Proposed Algorithm: Preliminary Results

In this section we present a preliminary proposed algorithm. However, in Section 4 the definitive and more efficient algorithm will be presented.

The majorizing hat function used here is given as [30]where is the majorizing density, , , and are coefficients to be obtained for the specific fading model so that can majorize for all , and is the desired probability density function (pdf) given in terms of the normalized envelope , with standing for -- or --. The parameter is given in an exact form aswhere is the error function. The coefficient is obtained as the solution of . In all cases, the parameter can be easily found numerically using well-known software tools such as Mathematica or MATLAB. The coefficient is found as the mode of the pdf; that is, . Finally, the coefficient is found asAlgorithm 1 summarizes the steps for generating the desired sequences. The probability of acceptance in step 7 is . is the uniform distribution over the unit interval . The rejection method is well known and it is described in detail in [31]. Notice that the function has the form of a truncated-Gaussian density. Random variables with pdf can be generated in a fast and accurate way by truncated-Gaussian random variables generation methods (e.g., [32]).