The simulation of independent Rayleigh faders

Multiple independent Rayleigh fading waveforms are often required for the simulation of wireless communications channels. Jakes (1974) Rayleigh fading model and its derivatives based on the sum-of-sinusoids provide simple simulators, but they have major shortcomings in their simulated correlation functions. A novel sum-of-sinusoids fading model is proposed and verified, which generates Rayleigh fading processes satisfying the theoretical independence requirements and providing desired power spectral densities with ideal second-order moment. The effects of replacing sinusoids in the proposed model by their approximate waveforms are also analyzed and tested. Performance evaluation and comparison are provided, using the quality measures of the mean-square-error of autocorrelation function and the second-order moment of the power spectral density.     

An Improved Deterministic SoS Channel Simulator for Multiple Uncorrelated Rayleigh Fading Channels

The generation of multiple uncorrelated Rayleigh fading waveforms is often demanded for simulating wideband fading channels, multiple-input multiple-output (MIMO) channels, and diversity-combined fading channels. In this letter, an improved deterministic sum-of-sinusoids (SoS) channel simulator with a new parameter computation method is proposed to simulate a large number of uncorrelated Rayleigh fading processes. Compared with the existing SoS channel simulators, the proposed deterministic SoS model yields a much better simulation efficiency while still preserving satisfactory approximations to the desired statistical properties of the reference model.     

Accurate Simulation of multiple cross-correlated Rician fading channels

The computer generation of multiple cross-correlated Rician fading channels is investigated. We prove that the output sequences of existing multichannel fading simulators are restricted to have cross-correlation statistics that have the same functional form as the component autocorrelation functions. To overcome this limitation, vector autoregressive stochastic models are proposed for the generation of multiple Rician fading processes with specified realizable autocorrelation and cross-correlation statistics. This capability is desirable, for example, to permit realistic performance assessments of space-time modem designs by enabling the simulation of space-time-selective wireless channel models. The utility of the simulation approach is demonstrated by the accurate synthesis of some bandlimited multichannel Rayleigh and Rician processes.     

On the Correlation and Ergodic Properties of the Squared Envelope of SOC Rayleigh Fading Channel Simulators

In this paper, we investigate the correlation and ergodic properties of the squared envelope of a class of autocorrelation-ergodic (AE) sum-of-cisoids (SOC) simulation models for mobile Rayleigh fading channels. Novel closed-form expressions are presented for both the ensemble and the time autocorrelation functions (ACFs) of the SOC simulation model’s squared envelope. These expressions have been derived by assuming that the SOC model’s inphase and quadrature (IQ) components have arbitrary autocorrelation and cross-correlation properties. This consideration makes the results herein presented more general than those given previously in other papers, where it is assumed that the IQ components of the simulation model are strictly uncorrelated. We show that under certain conditions, the squared envelope of the SOC model is an AE random process. In addition, we evaluate the performance of three fundamental methods for the computation of the model parameters—namely the generalized method of equal areas, the L _p -norm method, and the Riemann sum method—regarding their accuracy for emulating the squared envelope ACF of a reference narrowband Rayleigh fading channel model. The obtained results are important to design efficient simulators for the performance analysis of systems and algorithms sensitive to the correlation properties of the channel’s squared envelope, such as speed estimators and handover mechanisms.     

The generalized method of equal areas for the design of sum‐of‐cisoids simulators for mobile Rayleigh fading channels with arbitrary Doppler spectra

We present in this paper a generalized version of the method of equal areas (MEA) that is well suited for the design of sum‐of‐cisoids (SOC) simulators for narrowband mobile Rayleigh fading channels characterized by any type of Doppler power spectral densities (DPSDs). Unlike the original MEA, the generalized MEA (GMEA) can be applied to the simulation of fading channels with asymmetrical DPSDs. This is an important feature because the simulation of such channels is of great interest for the laboratory analysis of mobile communication systems under non‐isotropic scattering conditions. We show that irrespective of the underlying DPSD, the GMEA results in a very good approximation to the autocorrelation function, average Doppler shift, Doppler spread, and envelope distribution of the channel. We compare the performance of the GMEA with that of the L_p‐norm method (LPNM), which is the method that has been most widely used for designing SOC simulators for fading channels with arbitrary DPSDs. The obtained results demonstrate that the performance of the GMEA measures up to that of the LPNM. In addition, the results show that the determination of the model parameters is easier and less time‐consuming when applying the GMEA. Copyright © 2011 John Wiley & Sons, Ltd.     

STBC-Based (Turbo) STTC Codes Built by Set Partitioning for Three Transmit Antennas: Construction and Performances

Simulating wideband fading channels, multiple-input multiple-output (MIMO) channels, and diversity-combined fading channels often demands the generation of multiple uncorrelated Rayleigh fading waveforms. In this letter, two appropriate parameter computation methods, namely the method of exact Doppler spread (MEDS) and L_p -norm method (LPNM), for deterministic sum-of-sinusoids (SoS) channel simulators are investigated to guarantee the uncorrelatedness between different simulated Rayleigh fading processes. Numerical and simulation results show that the resulting deterministic SoS channel simulator can accurately and efficiently reproduce all the desired statistical properties of the reference model     

Stochastic Modeling and Simulation of Frequency-Correlated Wideband Fading Channels

For the simulation of practical frequency-diversity wireless communication systems, such as frequency-hopping systems, multicarrier code-division multiple-access systems, and orthogonal frequency-division multiplexing systems, it is often desirable to produce multiple Rayleigh fading processes with given frequency correlation properties. In this paper, a novel stochastic wide-sense stationary sum-of-sinusoids channel simulator is proposed to emulate frequency-correlated wideband fading channels, where the frequency correlation properties are controlled by only adjusting the constant phases. Closed-form expressions are provided for all the parameters of the simulation model. This enables us to investigate analytically the overall correlation properties (not only the correlation coefficients) of the simulated processes with respect to both time separation and frequency separation. It is shown that the wideband channel simulator will be reduced to a narrowband Rayleigh fading-channel simulator by removing the frequency selectivity. Furthermore, the COST 207 typical-urban and rural-area channels are applied to evaluate the performance of the resulting wideband and narrowband channel simulators, respectively. The correlation properties of the simulation models approach the desired ones of the underlying reference models as the number of exponential functions tends to infinity, while very good approximations are achieved with the chosen limited number of exponential functions     

Limitations of sum-of-sinusoids fading channel simulators

Rayleigh signal fading due to multipath propagation in wireless channels is widely modeled using sum-of-sinusoids simulators. In particular, Jakes' (1994) simulator and derivatives of Jakes' simulator have gained widespread acceptance. Despite this, few in-depth studies of the simulators' statistical behavior have been reported in the literature. Here, the extent to which Jakes' simulator adequately models the multipath Rayleigh fading propagation environment is examined. The results show that Jakes' simulator does not reproduce some important properties of the physical fading channel. Some possible improvements to Jakes' simulator are examined. The significances of the number and the symmetries of the Doppler frequency shifts on the validity of the simulator's reproduction of the physical fading channel are elucidated     

Improved models for the generation of multiple uncorrelatedRayleigh fading waveforms

An improved sum-of-sinusoids simulation model is proposed for Rayleigh fading channels. The new model employs random initial phase, and conditional random Doppler frequency for all individual sinusoids. The second-order statistics of the new simulator match the desired ones exactly even if the number of sinusoids is a single-digit integer. Other key statistics of the new simulator approach the desired ones of Clarke's (1968) reference model as the number of sinusoids approaches infinity, while good convergence is achieved when the number of sinusoids is small. Moreover, the new simulator can be directly used to generate multiple uncorrelated fading waveforms; it is also pointed out that a class of 16 different simulators, which have identical statistical properties, can be developed for Rayleigh fading channels     

A New Design Concept for High-Performance Fading Channel Simulators Using Set Partitioning

In this paper, we introduce a new technique for the design of high-performance Rayleigh fading channel simulators. The proposed design method uses set partitioning – a technique, which plays a key role in the design of trellis-coded modulation schemes. We show how set partitioning can be used to design multiple uncorrelated fading waveforms enabling the simulation of Rayleigh fading channels. For the important case of isotropic scattering, we show that the sample average of the generated waveforms results in a deterministic process, the autocorrelation function (ACF) of which tends to the zeroth-order Bessel function of the first kind as the number of sample functions increases. The proposed procedure is completely deterministic. The comparison with a stochastic procedure using Monte Carlo techniques will be made. A study of the performance shows clearly that the new technique using set partitioning outperforms by far existing Monte Carlo methods.     

Asymptotic SEP for M-PSK Signals overα-μFading Channels

The α-μfading is a general physical model recently rewritten from the Stacy (or generalized gamma) distribution, encompassing a variety of fading environments such as Rayleigh, Nakagami-m, and Weibull fading. We characterize the asymptotic average symbol error probability (SEP) at a high signalto- noise ratio (SNR)?specifically, the high-SNR slope and power gain?in terms of two physical fading parametersαandμ. We further extend the asymptotic SEP analysis to a multiple-input multiple-output diversity system employing orthogonal space? time block codes inα-μfading channels.     

Simulation models with correct statistical properties for Rayleigh fading channels

In this paper, new sum-of-sinusoids statistical simulation models are proposed for Rayleigh fading channels. These new models employ random path gain, random initial phase, and conditional random Doppler frequency for all individual sinusoids. It is shown that the autocorrelations and cross correlations of the quadrature components, and the autocorrelation of the complex envelope of the new simulators match the desired ones exactly, even if the number of sinusoids is as small as a single-digit integer. Moreover, the probability density functions of the envelope and phase, the level crossing rate, the average fade duration, and the autocorrelation of the squared fading envelope which contains fourth-order statistics of the new simulators, asymptotically approach the correct ones as the number of sinusoids approaches infinity, while good convergence is achieved even when the number of sinusoids is as small as eight. The new simulators can be directly used to generate multiple uncorrelated fading waveforms for frequency selective fading channels, multiple-input multiple-output channels, and diversity combining scenarios. Statistical properties of one of the new simulators are evaluated by numerical results, finding good agreements.     

A general approach to generate Rayleigh fading simulators with correct statistical properties

Rayleigh fading channel is widely modeled using sum‐of‐sinusoids simulators. In the paper, the construction method of an accuracy simulator is of concern. The intrinsic relationships between path gain, angle of incoming wave (angle of arrival), and initial phase are analyzed. At the same time, rules of simulator construction were proposed. According to these rules, the second‐order statistical properties of designed simulators match to the desired ones perfectly. Furthermore, our method can be an elicitation to generate simulators with correct high‐order statistical properties. In addition, a realization based on the construction method is designed to example how to use these construction rules. The new realization can be used to generate multiple uncorrelated fading waveforms for frequency‐selective fading channels, multiple‐input multiple‐output channels, and diversity combining scenarios. Copyright © 2011 John Wiley & Sons, Ltd.     

Design of high-speed simulation models for mobile fading channelsby using table look-up techniques

This paper describes a procedure for the design of fast simulation models for Rayleigh fading channels. The presented method is based on an efficient implementation of Rice's sum of sinusoids by using table look-up techniques. The proposed channel simulator is composed of a few numbers of adders, storage elements, and simple modulo operators, whereas time-consuming operations like multiplications and trigonometric operations are not required. Such a multiplier-free simulation model is introduced as a high-speed channel simulator. It is shown that the high-speed channel simulator can be interpreted as a finite state machine which generates deterministic output envelope sequences with approximately Rayleigh distribution. The statistical properties of the designed high-speed channel simulator are investigated analytically and compared with the statistics of the underlying Rayleigh reference model. Results of experiments for measuring the speed of the presented and other types of channel simulators are also presented     

Performance analysis of lognormally shadowed generalized Gamma fading channels

A new composite channel model is proposed for the performance analysis of shadowed fading channels. This model is represented as a mixture of generalized Gamma (GG) multipath fading and lognormal shadowing. GG distribution includes the Rayleigh, Nakagami, and Weibull as special cases; hence the presented model, which is referred to as GG‐L, is a generic model that covers many well‐known composite fading models, including the Rayleigh–lognormal (R‐L), Nakagami–lognormal (N‐L), and Weibull–lognormal (W‐L). The main drawback of the lognormal‐based composite models is that the composite probability density function (PDF) is not in closed form, thereby making the performance evaluation of communication links in these channels cumbersome. To bypass this problem, an approximation method is developed which makes it possible to derive a closed‐form, analytical expression for GG‐L composite distribution. The proposed method only needs the mean and the variance of the underlying lognormal distribution, and hence, bypasses the required complicated integration needed to calculate the PDF of the received signal envelope in GG‐L channel. Based on this method, the most statistical characteristics, such as cumulative density function (CDF) and moments of the GG‐L composite distribution, are derived and used for the performance analysis of a single receiver operating over GG‐L fading channel. Copyright © 2010 John Wiley & Sons, Ltd.     

Performance of coherent OFDM-CDMA for broadband mobile communications

Uncoded orthogonal frequency division multiplexing (OFDM) transmission technique applied in a multipath environment has a bit error rate (BER) comparable with a narrowband radio channel because the fading of each subcarrier is frequency-nonselective. To overcome this behaviour and to reduce the BER, a combination of OFDM and CDMA has been proposed recently. In an OFDM-CDMA system the energy of each information symbol is spread over several subcarriers. Therefore a diversity gain can be obtained in a broadband fading channel.In this paper we discuss the performance of OFDM-CDMA with coherent QPSK signalling over a frequency-selective Rayleigh fading channel. Channel estimation and demodulation are integral parts that determine the performance of the system. The method for channel estimation presented in this paper is based on a two-dimensional array of pilot symbols with second-order regression in the time domain and interpolation in the frequency domain. Quantitative comparison of four different detection algorithms in frequency-selective Rayleigh fading with noisy channel state information (CSI) will be presented in this paper: conventional correlation (equal gain correlation, EGC), orthogonality restoring correlation (ORC), ORC with a threshold in order to suppress subcarriers with low signal strength (TORC), and an iterative improvement based on a maximum likelihood approach. With TORC and iterative improvement a gain of approximately 9 dB over conventional OFDM can be obtained at a BER of 10^–3 in Rayleigh fading.     

一种新的Jakes模型参数计算方法

本文分析了小尺度无线衰落信道的理论参考模型—Clark模型,比较了两种较为重要的确定性仿真模型的参数计算方法—精确多普勒扩展法（MEDS）和随机精确多普勒扩展法（R_MEDS）,提出了一种新的用于仿真平坦衰落信道Jakes模型的参数计算方法,即在MEDS的基础上对离散多普勒频率引入随机抖动。仿真数据和图形表明,在模型复杂度较低的情况下,新的信道模型的自相关和互相关特性较R_MEDS有很大改善,而且与原始MEDS相比,新模型多普勒频率为随机变量,可以有效仿真瑞利衰落、多个不相关的瑞利衰落和频率选择性衰落信道等,并且可以结合工程需要设定随机抖动幅度,保证了衰落信道的不相关性和遍历性的有机统一。      

A Comparison of Energy Detectability Models for Cognitive Radios in Fading Environments

In this work, we first analyze the accuracy of different energy detector models in approximating the exact solution in AWGN. These models motivate us to develop approximation analysis to address energy detection for fading channels. Our analysis develops approximation that has almost the same performance as the exact solution in Rayleigh channels. Our new model is simple enough to derive the relationship between the required number of samples (N) and the signal-to-noise ratio for a single Rayleigh channel similar to the one obtained for AWGN channels. We also define a fading margin for link budget calculations that relates N in fading channels to AWGN channels. Furthermore, we analyze the impact of multiple antennas for cognitive radios considering two receiver diversity schemes and quantify the improvement in performance regarding this margin. All the analytical results derived in this paper are verified by simulations. Finally, we have implemented and verified energy detection models in our multiple antenna wireless testbed.     

Non-unitary precoder design for reducing feedback overhead in MIMO-OFDM systems

The precoder feedback overhead in MIMO-OFDM systems is heavy as the number of subcarriers is usually large. Most of the current limited feedback works focus on unitary precoding schemes. Based on the fact that the channel matrices of neighboring subcarriers in MIMO-OFDM systems are correlated, an alternative transceiver is proposed here. In this paper, non-unitary precoders shared by several adjacent subcarriers, and the corresponding individual equalizers are jointly designed under partial CSI. As the number of precoding matrices is smaller, the feedback overhead can be reduced. The transceivers are designed to minimize the detection mean-squared error (MSE) under the total transmit power constraint. A convergent iterative algorithm based on the Lagrange multipliers method is proposed. The necessary conditions for an optimal transceiver, Karush–Kuhn–Tucker (KKT) conditions, are satisfied in each iteration. Numerical results under the frequency-selective MIMO-OFDM channel, which is generated by the classical Zheng and Xiao's model, prove that the proposed transceiver can significantly reduce the feedback overhead without severe performance degradation.     

On the performance analysis of wireless receiver using generalized-gamma fading model

In this paper, we provide a unified analysis for wireless system over generalized fading channels that is modeled by the two parameter generalized gamma model. This model is versatile enough to represent short-term fading such as Weibull, Nakagami-m, or Rayleigh as well as shadowing. The performance measures such as the amount of fading, average bit error rate, and signal outage are considered for analysis. With the aid of moment generating function (MGF) approach and Padé approximation (PA) technique, outage probability and average bit error rate have been evaluated for a variety of modulation formats. We first use the PA technique to find a simple way to evaluate compact rational expressions for the MGF of output signal-to-noise ratio, unlike previously derived intricate expressions in terms of Fox’s H and MeijerG functions. Using these rational expressions, we evaluate the performance of wireless receivers under a range of representative channel fading conditions. Our results are validated through computer simulations, which shows perfect match.