Rician Random Variables Research Articles

SCMA Codebook for Uplink Rician Fading Channels

In this letter, the codebook design criteria for uplink Rician fading channels are explored. The probability distribution of the weighted sums of the Rician random variables from each frequency resource is derived based on the independent non-central circular complex Gaussian random variable. Then, considering the pairwise error probability, the key performance indicators of multidimensional constellations for uplink Rician channels are explored. A sub-optimal optimization approach is deployed to generate an enhanced codebook for various Rician scenarios. Finally, numerical results are provided to verify the key performance indicators in the uplink Rician fading channels.

Product of Three Random Variables and its Application in Relay Telecommunication Systems in the Presence of Multipath Fading

In this paper, the product of three random variables (RVs) will be considered. Distribution of the product of independent random variables is very important in many applied problems, including wireless relay telecommunication systems. A few of such products of three random variables are observed in this work: the level crossing rate (LCR) of the product of a Nakagami-m random variable, a Rician random variable and a Rayleigh random variable, and of the products of two Rician RVs and one Nakagami-m RV is calculated in closed forms and presented graphically. The LCR formula may be later used for derivation of average fade duration (AFD) of a wireless relay communication radio system with three sections, working in the multipath fading channel. The impact of fading parameters and multipath fading power on the LCR is analyzed based on the graphs presented.

On the ratio of independent complex Gaussian random variables

In this paper, we derive a closed form equation for the joint probability distribution $${{f_{{R}_{z}}},{\varTheta _{z}}}({r_{z}},{\theta _{z}})$$ of the amplitude $${R_{z}}$$ and phase $${\varTheta _{z}}$$ of the ratio $${Z=\frac{X}{Y}}$$ of two independent non-zero mean Complex Gaussian random variables $$X\sim CN(\nu _{x} \mathrm {e}^{j\phi _{x}},{\sigma ^{2}_{x}})$$ and $$Y\sim CN(\nu _{y} \mathrm {e}^{j\phi _{y}},{\sigma ^{2}_{y}})$$ . The derived joint probability distribution only contains a confluent hypergeometric function of the first kind $${_1F_{1}}$$ without infinite summations resulting in computational efficiency. We further derive the probability distribution for the ratio of two non-zero mean independent real Rician random variables containing an infinite summation generated by the estimation of the Cauchy product of equivalent series of two modified Bessel functions.

Dual-Hop Communication Over a Satellite Relay and Shadowed Rician Channels

A dual-hop amplify-and-forward (AF) relaying scheme over shadowed Rician fading channels is investigated. Specifically, the source and destination nodes are equipped with $N$ and $M$ antennas, respectively, whereas the relay is equipped with a single antenna. Communication via satellite relaying represents a direct application of the considered infrastructure. To this end, we study the scenario when the source and the destination are terrestrial nodes, whereas the end-to-end communication is established through an intermediate AF relay node, which is a satellite. To fully exploit the spatial diversity provided by multiple antennas, maximum ratio transmission and maximum ratio combining are implemented at the source and the destination, respectively. First, a new closed-form expression for the probability density function (pdf) of the sum of independent and identically distributed (i.i.d.) squared shadowed Rician random variables is derived by assuming integer distribution parameters. Capitalizing on the latter pdf, new closed-form results for the cumulative distribution function (cdf) and the moment function of the end-to-end signal-to-noise ratio (SNR) are obtained. Particularly, the proposed unified analysis includes the channel-state-information (CSI)-assisted and the fixed-gain AF relaying protocols. New expressions for important performance measures, namely, the outage probability, the average symbol error probability (ASEP), and the ergodic capacity of the end-to-end SNR, are presented for both AF schemes. Moreover, some useful engineering insights are manifested, such as simplified asymptotic outage performance results, the diversity order, and the impact on the number of antennas at the source and the destination.

Asymptotic Analysis of Different Multibranch Diversity Receivers With Arbitrarily Correlated Rician Channels

The asymptotic performances of multibranch diversity reception systems employing the popular and important combining techniques over arbitrarily correlated Rician channels are studied. Based on an integral representation of the joint probability density function of multivariate Rician random variables with arbitrary correlations, closed-form expressions for asymptotic error rate and asymptotic outage probability are derived for multibranch receiver diversity receptions. The important diversity schemes studied are maximal ratio combining, equal gain combining, selection combining, and hybrid maximal ratio combining as well as hybrid equal gain combining. Numerical results indicate that these analytical solutions can provide accurate estimation of error rate and outage probability in large-signal-to-noise-ratio regimes. The analysis provides some important theoretical and practical results concerning the performance characteristics of multibranch diversity receivers operating on arbitrarily correlated Rician fading channels.

Novel Representations for the Bivariate Rician Distribution

Novel single-integral representations for the bivariate probability density function and cumulative distribution function of two correlated Rician random variables are derived. The solutions are expressed in terms of well known functions which are available in common mathematical software. The two random variables are not necessarily identically distributed as required by some past solutions. The new form of the bivariate Rician cumulative distribution function is applied to give a new expression for the outage probability of dual selection combining diversity operating in correlated Rician fading.

Use of MATLAB in teaching the fundamentals of random variables

The way engineering is being taught has changed in recent years with the introduction of commercial and educational software thus allowing a better understanding of the subject matter and increased teaching efficiency. This paper presents the educational software package designed to work in PCs and provides an attractive way to introduce the fundamentals of random variables. The software can be used as a complement to theoretical classes or alone as a self-study tool. The package was implemented in MATLAB and uses MATLAB™ tool "makeshow" which allows the students to create interactive slideshows without building their own graphic interface. The software is easy to use and is in accordance with the established standards, mouse operations, icon menus, etc. The demos include Probability Density Function (PDF), and Cumulative Distribution Function (CDF), normal, lognormal, exponential, Rayleigh and Rician random variables. The package also demonstrates how to generate the desired random sequence, how to examine its characteristics and how to estimate its PDF and CDF.

Closed form expression and improved bound on pairwise error probability for performance analysis of turbo codes over Rician fading channels

This letter provides derivations for an exact expression and a bound on pair wise error probability over fully interleaved Rician fading channels under the assumption of ideal channel state information. The derivation which is based on the probability distribution of the sum of squared Rician random variables leads to an improved upper bound in comparison with the only known bound in literature. Pairwise error probability plots together with average union upper bounds for turbo codes having (1,7/5,7/5) and (1,5/7,5/7) generator polynomials are presented to demonstrate the effectiveness of the new results

Infinite-Series Representations Associated With the Bivariate Rician Distribution and Their Applications

Analytical expressions for the evaluation of the bivariate Rician cumulative distribution function (CDF), the covariance, and the characteristic function (CHF) are not known, despite their usefulness in wireless communications systems analysis. In this letter, motivated by the ability of the Rician model to describe fading in wireless communications, we derive infinite-series representations for the probability density function, the CDF, the covariance, and the CHF of two correlated Rician random variables. It is shown that the presented infinite-series expressions converge rapidly, and can be efficiently used to study several performance criteria for dual-diversity receivers operating over correlated Rician fading channels.

Multitone jamming of FH/BFSK in Rician channels

A new bit error rate (BER) expression for a noncoherent frequency-hopped binary-frequency-shift-keying (FH/BFSK) receiver in Rician channels, subjected to independent multitone and wideband noise interference, is derived. Unlike previously published analyzes, we combine the signal and jammer multipath components, the additive white Gaussian noise (AWGN), and other wideband interference as a single Gaussian process. This method leads to a BER expression which consists of only a single rather than a triple integral requiring numerical computation. We also present a new derivation of the probability density function (PDF) of the ratio of two Rician random variables, used in a previously published approximate BER analysis.

Multiple-symbol trellis-coded modulation applied to noncoherent continuous-phase frequency shift keying

This paper considers a maximum-likelihood (ML) noncoherent detection scheme for multiple full response continuous-phase frequency shift keying (CPFSK) waveforms and introduces a trellis-coded modulation (TCM) scheme for this noncoherent modulation. By utilizing a Gaussian approximation for Rician random variables, we express the pairwise error probability as a function of the equivalent normalized squared distance (ENSD). ENSD plays the same role as normalized squared Euclidean distance when evaluating error probability performance for coherent detection. We derive an analytical approximation on the bit-error probability by employing ENSD for both the coded system and the uncoded system. For the uncoded system we show that the bit-error probability of noncoherent detection approaches that of coherent symbol-by-symbol detection in the limit as the multiplicity of the symbol goes to infinity for large signal-to-noise ratio (SNR), We determine specific optimal trellis encoders for binary and 4-ary CPFSK with modulation index 1/2 and 1/4, respectively, by application of Ungerboeck's (1982) set partitioning approach.

Optimal Gabor filters for texture segmentation

Texture segmentation involves subdividing an image into differently textured regions. Many texture segmentation schemes are based on a filter-bank model, where the filters, called Gabor filters, are derived from Gabor elementary functions. The goal is to transform texture differences into detectable filter-output discontinuities at texture boundaries. By locating these discontinuities, one can segment the image into differently textured regions. Distinct discontinuities occur, however, only if the Gabor filter parameters are suitably chosen. Some previous analysis has shown how to design filters for discriminating simple textures. Designing filters for more general natural textures, though, has largely been done ad hoc. We have devised a more rigorously based method for designing Gabor filters. It assumes that an image contains two different textures and that prototype samples of the textures are given a priori. We argue that Gabor filter outputs can be modeled as Rician random variables (often approximated well as Gaussian rv's) and develop a decision-theoretic algorithm for selecting optimal filter parameters. To improve segmentations for difficult texture pairs, we also propose a multiple-filter segmentation scheme, motivated by the Rician model. Experimental results indicate that our method is superior to previous methods in providing useful Gabor filters for a wide range of texture pairs.