Highly efficient rejection method for generating Nakagami-m sequences

Abstract

A simple and highly efficient rejection method is presented for generating Nakagami-m sequences with arbitrary average power and fading factor. With a novel exponential decay hat function, the proposed method's rejection efficiency is up to 80% and the difference between theoretical and simulated distribution curves are neglectable.Introduction: The Nakagami-m (or Nakagami) distribution has been widely used to model the wireless fading channel for its good fitting to experimental data [1]. It is equivalent to the well-known one-sided Gaussian, Rayleigh and Rice distributions with certain values of fading parameter m. A simple, efficient and accurate way of generating Nakagami sequences is particularly important for large-scale and realtime fading channel simulation such as multiple-input multiple-output (MIMO) systems. Unfortunately, most of the generation methods known are not satisfactory for this. Yip and Ng's method [2] based on the product of square-root beta random variables (RVs) and Rayleigh RVs, is quite accurate but only works for m , 1. The decomposition method [3] is valid for arbitrary m but becomes inaccurate as m is neither integer nor half-integer. Beaulieu and Cheng's method [4] guarantees accuracy but requires a different set of coefficients for each m by some numerical method or curve fitting.The acceptance/rejection technique is a universal and accurate generation method for non-uniform RVs [5] and its effectiveness highly depends on the hat function (or comparison function). Reference [6] suggests the use of a folded-Gaussian (0.5 ≤ m ≤ 1) and Gaussian probability density distributions (PDFs) (m ≥ 1), which lacks efficiency (65.75 and 66.67%, respectively), with a set of constant coefficients determined empirically. Reference [7] gives a universal hat function denoted by a second-order polynomial, which can be used for arbitrary m but becomes inefficient as m increases (about 60-63%, for m ≥ 4). In this Letter, we propose a highly efficient rejection scheme based on a novel exponential decay hat function for generating independent Nakagami-m RVs, which yields an excellent accuracy and high efficiency.

Almost rejectionless sampling from Nakagami-m distributions (m≥1)

“…Results: To analyse the performance of the algorithm, we have compared the acceptance rate (AR), am of our approach and the Gaussian proposal used in [7] to draw samples from a Nakagami PDF without truncation. The AR of our technique can be obtained analytically:…”

“…with T(m) denoting the gamma function, whereas the AR for the proposal used in [7] can be approximated for m > 4 as…”

“…Fig. la shows an example of the target, p(x), our proposal, 7T(X), and the proposal used in [7] for an unbounded domain, which fits the true PDF in a much looser way than ours, thus leading to worse acceptance rates. (Fig.…”

“…m -n/2 with n G M), independent samples can be generated through the square root of a sum of squares of n zero-mean independent identically distributed (IID) Gaussian random variables (RVs). On the other hand, for m ^ n/2 several techniques have been proposed for drawing correlated samples from (1) [2][3][4], but all of them present limitations in terms of complexity, applicability or poor performance for some values of m. Alternatively, several simple and efficient acceptance-rejection methods, using different proposals and with increasing accuracy, have been recently introduced [5][6][7]. Currently, the best results are provided by [7] using a truncated Gaussian PDF as the proposal.…”

Efficient random variable generation: ratio of uniforms and polar rejection sampling

“…Uniform sampling inside the sector delimited by angles 0i and 0 2 and radius r 0 can be achieved by drawing two independent samples from: (3) is slightly more involved, requiring the generation of two independent RVs, v, w, ~ U([0, 1]). The desired radius is then: (4) In this Letter we propose to combine the RoU and polar RS (i.e. RS using a sector of a circle, S p , as the embedding region where we sample uniformly using polar coordinates) to improve the efficiency in the generation of some RVs, such as truncated Cauchy and Gaussian.…”