This paper is concerned with a simple and highly efficient random sequence generator for uncorrelated
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 timeconsuming, 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,
This paper is concerned with the generation of uncorrelated samples of
A useful method for generating independent
In this paper, we extend the applicability of the approach in [
With the aim of quantifying the performance of the random sequence generators, we compare empirical cdfs to hypothesized ones by carrying out goodnessoffit KolmogorovSmirnov (KS) and AndersonDarling (AD) tests. We also generate a large number of
In order to demonstrate the usefulness of the proposed method, we provide theoretical and simulated bit error rates of a coherent binary phaseshift keying (BPSK) modulation over the
The remaining of the paper is organized as follows. Section
In this section we present a preliminary proposed algorithm. However, in Section
The majorizing hat function
(1) Define the distribution parameters
(2) Find
(3) Find
(4) Find
(5) Generate
(6) Generate
(7)
(8)
(9)
(10)
(11)
For a fading signal with envelope
In particular, the first derivative of the
For a fading signal with envelope
In particular, the first derivative of the
In Figure
Simulated and theoretical
The acceptance proportion, or efficiency, is the performance measure of the acceptancerejection method. It is the ratio between the number of samples accepted by the method and the total number of samples generated from the respective hat function (majorizing function). Figures
Rejection method efficiency for the
Rejection method efficiency for the
The efficiency of the proposed method for the
The efficiency of the proposed method for the
In all the cases, the acceptance ratio does not vary significantly with the variation of
A strikingly interesting result is shown next. Refer to Figure
Figure
Rejection method efficiency for the
Figure
Rejection method efficiency for the
The difference between the theoretical and experimental distributions is minimal as visually perceived in Figure
Parameters 



0.8218 

0.6293 

0.3654 

0.6745 
It is well known that the AndersonDarling test gives more weight to the tails than the KS test. Also, because the AndersonDarling test is specific for the hypothesized distribution, this test is likely to be more powerful than the traditional KS test [
A modified procedure for a highefficient and definitive algorithm can be noticed in this section. Let us consider first the
Considering the
Because
In the same way, one can conclude that the high efficiency can be reached for all the particular cases of
The steps for generating the desired sequences using the definitive algorithm are summarized in the Algorithm
(1) Define the distribution parameters
(2) Find
(3) Find
(4) Find
(5) Generate
(6) Generate
(7)
(8)
(9)
(10)
(11)
(12) Make the transformation
The generator ability in providing random samples following a given distribution can be alternatively verified by generating a large number of random variables and obtaining maximum likelihood (ML) estimates for the distribution parameters. In this section, as a particular case of the
Let
Taking the derivative of (
In general, it is less computationally intensive to evaluate
The variance of an estimator is a measurement of its ability to perform reliably as it gives the degree of certainty in which the parameter is being estimated. In this context, the CramérRao lower bound (CRLB) sets a lower limit for the variance of all unbiased estimators for
Figure
Normalized standard deviation curves for the ML estimators of
Also, it has been shown [
We use Monte Carlo simulations in order to study the performance of the
Taking advantage of notable properties of the ML estimation for large sample sizes, we calculate ML estimates of the parameters
Figure
Sample mean and confidence region (
Similar results of the sample mean for
Sample mean and confidence region (
The unbiasedness of the ML estimators for large sample sizes, depicted in Figures
In this section we give applications of the proposed random variable generators and use theoretical and simulation results for certifying the accuracy of these generators.
Here we analyze the bit error rate (BER) of the BPSK modulation over frequencyflat fading channels modeled by the
One possible analytical method employed for determining the performance of a mobile radio communication system is by evaluating the error probability as a function of a fixed signaltonoise ratio (SNR) and then averaging the result over the probability density function of the SNR variations, which is governed by the particular envelope fading distribution. For instance, the bit error probability of the BPSK modulation over the pure AWGN channel as a function of the received SNR
Now, we must average
Applying a transformation of random variables, from (
The integral in (
Average error probability for BPSK with
Average error probability for BPSK with
It is worth mentioning that the application just described can be used to check the adherence of the generated random numbers to the tails of their probability distributions as follows. The agreement between theoretical and simulation results in the high
Modern wireless communication systems are now facing a huge obstacle, spectrum scarcity. New services and applications appear every day, demanding increased bandwidth, new spectrum bands, or both. However, the currently adopted fixed spectrum allocation policy prevents those services and applications to be deployed in adequate pace. Nevertheless, recent studies have demonstrated that, in fact, the radiofrequency spectrum is quite underutilized in some areas and during some time [
To detect the idle bands, also called spectral holes or whitespaces, the CRs must have some sort of
Several studies consider the problem of spectrum sensing with energy detection over the pure AWGN channel, an approach, that is, by far unrealistic since typical wireless communication channels are also subjected to fading. Then, it is of paramount importance to access the performance of a spectrum sensing technique taking into account the channel fading.
There are several fading channel models available in the literature. Among them, two wellaccepted models deserve attention due to their ability for accurately modelling several real channel conditions in practice. They are the
In this section we apply
The discretetime model for the hypothesis test associated with the spectrum sensing problem is given by
The average signaltonoise ratio (SNR) is defined by
The performance of a spectrum sensing technique is often measured in terms of the probability of detection,
Figure
ROC curves for
From Figure
In this paper, sequences of general distributions
The authors declare that there is no conflict of interests regarding the publication of this paper.
This paper was partially supported by the CNPq Postdoctoral Program (501281/20134).