In studying the performance of coded communications over memoryless channels (with or without fading), the results are given as upper bounds on the average bit error probability (BEP). In principle, there are three different approaches to arriving at these bounds, all of which employ obtaining the so-called pairwise error probability , or the probability of choosing one symbol sequence over another for a given pair of possible transmitted symbol sequences, followed by a weighted summation over all pairwise events. In this chapter, we will focus on the results obtained from the third approach since these provide the tightest upper bounds on the true performance. The first emphasis will be placed on evaluating the pairwise error probability with and without CSI, following which we shall discuss how the results of these evaluations can be used via the transfer bound approach to evaluate average BEP of coded modulation transmitted over the fading channel.

Coded Communication over Fading Channels

The principle of positioning with the Global Positioning System (GPS) is based on determining the distance of at least four GPS satellites to a receiver. For that purpose the so-called code measurements can be used, so that the position of the receiver can be determined up to several meters. In order to allow for navigation with centimeter accuracy, it is however required to use the very precise carrier phase measurements, since the phase of the carrier wave of the GPS signal can be measured up to several millimeters. Unfortunately, a receiver can only measure the phase of the carrier A¢â¬â that is the fraction of the wavelength at the time of arrival. The wavelength is approximately 20 centimeters, and it is not known how many complete waves have preceded the one at the time of arrival. Therefore the phase measurements are called ambiguous. And that brings the problem of 'integer ambiguity resolution'. The goal of this research has been to set up a procedure for ambiguity resolution, such that there is a sound theoretical foundation for the estimation as well as the validation procedure. Until now that was not the case. Different approaches have been considered, and it was shown that an optimal method can be set up, based on the new principle of Integer Aperture estimation. But there are also alternative methods, which are computationally more attractive, and for which it has been shown that their solution is close to optimal. Furthermore, it has been shown that the principle of Integer Aperture estimation gives a theoretical foundation for the traditional validation procedures, as used until now. The result of this research is, that there is now a complete theory available, which gives a solution to the problem of integer ambiguity resolution. This allows for the first time to make inferences on the statistical reliability of the estimated ambiguities. This is an important step, since for many applications not only precision but also reliability is a very important measure in order to decide whether or not one can safely use the navigation system.

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The GNSS integer ambiguities: Estimation and validation

This paper investigates a natural generalization of the κ - μ fading channel in which the line-of-sight (LOS) component is subject to shadowing. This fading distribution has a clear physical interpretation and good analytical properties and unifies the one-side Gaussian, Rayleigh, Nakagami- m, Rician, κ - μ, and Rician shadow fading distributions. The three basic statistical characterizations, i.e., probability density function (pdf), cumulative distribution function (cdf), and moment-generating function (mgf), of the κ - μ shadowed distribution are obtained in closed form. Then, it is also shown that the sum and maximum distributions of independent but arbitrarily distributed κ - μ shadowed variates can be expressed in closed form. This set of new statistical results is finally applied to modeling and analysis of several wireless communication systems, e.g., the proposed distribution has applications to land mobile satellite (LMS) communications and underwater acoustic communications (UAC).

Statistical Characterization of $\kappa{ - }\mu$ Shadowed Fading

We introduce the fluctuating two-ray (FTR) fading model, a new statistical channel model that consists of two fluctuating specular components with random phases plus a diffuse component. The FTR model arises as the natural generalization of the two-wave with diffuse power (TWDP) fading model; this generalization allows its two specular components to exhibit a random amplitude fluctuation. Unlike the TWDP model, all the chief probability functions of the FTR fading model (PDF, CDF, and MGF) are expressed in closed-form, having a functional form similar to other state-of-the-art fading models. We also provide approximate closed-form expressions for the PDF and CDF in terms of a finite number of elementary functions, which allow for a simple evaluation of these statistics to an arbitrary level of precision. We show that the FTR fading model provides a much better fit than Rician fading for recent small-scale fading measurements in 28 GHz outdoor mm-wave channels. Finally, the performance of wireless communication systems over FTR fading is evaluated in terms of the bit error rate and the outage capacity, and the interplay between the FTR fading model parameters and the system performance is discussed. Monte Carlo simulations have been carried out in order to validate the obtained theoretical expressions.

The Fluctuating Two-Ray Fading Model: Statistical Characterization and Performance Analysis

In this Letter we derive exact closed-form expressions for the outage probability (OP) in η-μ fading channels. First, a general expression in terms of the confluent Lauricella function is derived for arbitrary values of μ. Next, we restrict the analysis to physical η-μ channel models, i.e. to integer values of 2μ, and obtain exact closed-form expressions for the OP in terms of Marcum Q, Bessel and elementary functions. The results in this Letter are applicable to the OP analysis of maximal ratio combining (MRC) over i.i.d. η-μ or Hoyt fading channels.

Outage probability analysis for η-μ fading channels

A general framework for the bit error rate (BER) analysis of quadrature amplitude modulation (QAM) systems is presented. This analysis is valid for any QAM system with independent bit mapping for in-phase and quadrature components and includes previous analyses in the literature as particular cases. We use this methodology to analyze BER in maximal ratio combining (MRC) systems with channel estimation errors in Ricean fading channels with Ricean-faded interference. An exact closed-form expression for the BER is obtained in this scenario, as well as an accurate simplified expression when perfect channel estimation is assumed, for the common case of Rayleigh-faded interference.

Generalized BER Analysis of QAM and Its Application to MRC Under Imperfect CSI and Interference in Ricean Fading Channels

In this paper, we present an analytical study of the bit error rate (BER) for single-carrier frequency-division multiple access (SC-FDMA) transmission over frequency-selective fading channels when zero-forcing frequency-domain equalization is applied. SC-FDMA, which can be described as a precoded version of orthogonal frequency-division multiple access (OFDMA), is regarded as a promising candidate for next mobile communication systems due its favorable envelope characteristics and low peak-to-average-power ratio (PAPR), compared with that of OFDMA. We focus on Nakagami-m fading channels and provide a method to calculate BER values with a single numerical computation. We provide a closed-form expression for the BER with binary phase-shift keying (BPSK) and square M-ary quadrature amplitude modulation (M-QAM) under the assumption of independence among channel frequency responses for allocated subcarriers.

BER Analysis for Zero-Forcing SC-FDMA Over Nakagami-m Fading Channels

In this paper, we present a novel analytical framework for the calculation of the level crossing rate (LCR) and the average fade duration (AFD) of fading channels sampled at a certain sampling period TS. These expressions are valid for arbitrary fading distributions with arbitrary correlation and can be easily computed in terms of the cumulative distribution function (cdf) of the fading envelope and its bivariate cdf. This approach yields interesting insights into the effect of finite sampling in the higher order statistics of fading processes. We also demonstrate that the proposed expressions for sampled fading process converge with the existing expressions for continuous fading processes as the sampling period tends to zero. As a direct application, exact closed-form expressions are given for the LCR and AFD of sampled Rayleigh fading processes, which are suitable to characterize the higher order statistics of the equivalent frequency-domain fading process in multipath Rayleigh fading.

Higher Order Statistics of Sampled Fading Channels With Applications

Significant throughput improvements can be obtained in multiple-input multiple-output (MIMO) fading channels by merging beamforming at the transmitter and maximal ratio combining (MRC) at the receiver. In general, accurate channel state information (CSI) is required to achieve these performance gains. In this paper, we analyze the impact of channel prediction error on the bit error rate (BER) of combined beamforming and MRC in slow Rayleigh fading channels. Exact closed-form BER expressions are obtained in terms of elementary functions. Numerical results show that imperfect CSI causes little BER degradation using channel prediction of moderate complexity.

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Exact BER analysis for M-QAM modulation with transmit beamforming under channel prediction errors

This paper provides new results for a family of incomplete Lipschitz-Hankel integrals (ILHIs) which also lead to the evaluation of certain integrals involving the generalized Marcum Q function. These mathematical results are then applied to analyze the bit error rate (BER) of adaptive modulation over multiple-input-multiple-output (MIMO) fading channels under imperfect channel state information (CSI). A novel exact closed-form expression for the average BER of adaptive modulation under MIMO transmit beamforming with maximal ratio combining, assuming prediction errors at the receiver for the adaptation CSI required by the transmitter, is obtained. The benefit of this result with respect to previous analysis is threefold. First, the expression is an exact closed form. Second, it is applicable to any antenna configuration, and third, it allows a design improvement of the cutoff SNR thresholds, which leads to better system performance in terms of average spectral efficiency at no extra cost.

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Unai Fernandez-Plazaola

Papers

Generalized BER Analysis of QAM and Its Application to MRC Under Imperfect CSI and Interference in Ricean Fading Channels

BER Analysis for Zero-Forcing SC-FDMA Over Nakagami-m Fading Channels

Higher Order Statistics of Sampled Fading Channels With Applications

Exact BER analysis for M-QAM modulation with transmit beamforming under channel prediction errors

Analysis of Adaptive MIMO Transmit Beamforming Under Channel Prediction Errors Based on Incomplete Lipschitz–Hankel Integrals