Bit Error Probability for MMSE Receiver in GFDM Systems

Abstract: In this work, we investigate the probability distribution function of the channel fading between a base station, an array of intelligent reflecting elements, known as large intelligent surfaces (LIS), and a single-antenna user. We assume that both fading channels, i.e., the channel between the base station and the LIS, and the channel between the LIS and the single user are Nakagami- $m$ distributed. Additionally, we derive the exact bit error probability considering quadrature amplitude ( $M$ -QAM) and binary phase-shift keying (BPSK) modulations when the number of LIS elements, $n$ , is equal to 2 and 3. We assume that the LIS can perform phase adjustment, but there is a residual phase error modeled by a Von Mises distribution. Based on the central limit theorem, and considering a large number of reflecting elements, we also present an accurate approximation and upper bounds for the bit error rate. Through several Monte Carlo simulations, we demonstrate that all derived expressions perfectly match the simulated results.

Abstract: In this letter, we investigate theoretical BER performances of generalized frequency division multiplexing (GFDM) systems with receivers based on Gabor window. Considering relationships between zero forcing and minimum mean square error (MMSE) receivers, theoretical BER expressions of GFDM systems with an MMSE receiver are derived over AWGN and fading channels through calculating noise enhancement factor at the receiver. To demonstrate the generality of proposed methods, analytical BER expressions of DFT and weighted-type fractional Fourier transform precoded GFDM systems are also given and verified by simulation results.

Abstract: In this paper, the concept of wireless power transfer and wireless information transmission, which have recently received special attention for improving energy efficiency in wireless communication systems, are investigated. In particular, we focus on a cooperative communication network consisting of a source node, an energy-constrained relay node and a destination node. In contrast to conventional wireless-powered cooperative networks, a GFDM waveform–which is an emerging candidate waveform for the 5G mobile networks and beyond–is considered. The performance of the proposed GFDM-based relaying network with energy harvesting is studied in terms of the average BER for the general M-QAM constellation set. Numerical and simulation results are provided to give useful insights and to assess the accuracy of our mathematical derivations.

Abstract: Generalized frequency division multiplexing (GFDM) with the free space optical channel can be a desirable option for free space optical communication systems as it has numerous benefits such as high spectral efficiency, flexibility, low out of band radiation and low peak-to-average power ratio. In this paper, the closed form expressions of symbol error and outage probabilities with and without pointing errors are derived for GFDM over gamma–gamma channel model. The theoretical results are verified with the Monte-Carlo simulations. Moreover, the impact of turbulence levels and comparison with orthogonal frequency division multiplexing are also explored.

Abstract: Generalized Frequency Division Multiplexing (GFDM) has been proposed as a potential modulation candidate for 5G communication systems. Unlike the current Orthogonal Frequency Division Multiplexing (OFDM), GFDM has the flexibility to cover various types of waveforms. However, GFDM inherits the peak-to-average power ratio (PAPR) problem from the OFDM system. The high PAPR of the GFDM transmitted signal introduces non-linearity which may lead to power inefficiency in the high power amplifier (HPA). This is a crucial issue for low power devices. This paper proposes a design of the pulse shaping filter using computationally efficient optimisation method to reduce the PAPR of the GFDM system. The numerical results illustrate the effectiveness of the proposed method in reducing the PAPR of the GFDM transmitted signal.

Abstract: The fundamental theorems on the asymptotic behavior of eigenvalues, inverses, and products of banded Toeplitz matrices and Toeplitz matrices with absolutely summable elements are derived in a tutorial manner. Mathematical elegance and generality are sacrificed for conceptual simplicity and insight in the hope of making these results available to engineers lacking either the background or endurance to attack the mathematical literature on the subject. By limiting the generality of the matrices considered, the essential ideas and results can be conveyed in a more intuitive manner without the mathematical machinery required for the most general cases. As an application the results are applied to the study of the covariance matrices and their factors of linear models of discrete time random processes.

Abstract: Quadrature amplitude modulation (QAM) is an attractive technique to achieve high rate transmission without increasing the bandwidth. A great deal of attention has been devoted to the study of bit error rate (BER) performance of QAM, and approximate expressions for the bit error probability of QAM have been developed in many places in the literature. However, the exact and general BER expression of QAM with an arbitrary constellation size has not been derived yet. We provide an exact and general closed-form expression of the BER for one-dimensional and two-dimensional amplitude modulations, i.e., PAM and QAM, under an additive white Gaussian noise (AWGN) channel when Gray code bit mapping is employed. The provided BER expressions offer a convenient way to evaluate the performance of PAM and QAM systems for various cases of practical interest. Moreover, simple approximations can be found from our expressions, which are the same as the well-known approximations, if only the dominant terms are considered.

Abstract: Cellular systems of the fourth generation (4G) have been optimized to provide high data rates and reliable coverage to mobile users. Cellular systems of the next generation will face more diverse application requirements: the demand for higher data rates exceeds 4G capabilities; battery-driven communication sensors need ultra-low power consumption; and control applications require very short response times. We envision a unified physical layer waveform, referred to as generalized frequency division multiplexing (GFDM), to address these requirements. In this paper, we analyze the main characteristics of the proposed waveform and highlight relevant features. After introducing the principles of GFDM, this paper contributes to the following areas: 1) the means for engineering the waveform's spectral properties; 2) analytical analysis of symbol error performance over different channel models; 3) concepts for MIMO-GFDM to achieve diversity; 4) preamble-based synchronization that preserves the excellent spectral properties of the waveform; 5) bit error rate performance for channel coded GFDM transmission using iterative receivers; 6) relevant application scenarios and suitable GFDM parameterizations; and 7) GFDM proof-of-concept and implementation aspects of the prototype using hardware platforms available today. In summary, the flexible nature of GFDM makes this waveform a suitable candidate for future 5G networks.

Abstract: This correspondence studies the statistical distribution of the signal-to-interference-plus-noise ratio (SINR) for the minimum mean-square error (MMSE) receiver in multiple-input multiple-output (MIMO) wireless communications. The channel model is assumed to be (transmit) correlated Rayleigh flat-fading with unequal powers. The SINR can be decomposed into two independent random variables: SINR=SINR/sup ZF/+T, where SINR/sup ZF/ corresponds to the SINR for a zero-forcing (ZF) receiver and has an exact Gamma distribution. This correspondence focuses on characterizing the statistical properties of T using the results from random matrix theory. First three asymptotic moments of T are derived for uncorrelated channels and channels with equicorrelations. For general correlated channels, some limiting upper bounds for the first three moments are also provided. For uncorrelated channels and correlated channels satisfying certain conditions, it is proved that T converges to a Normal random variable. A Gamma distribution and a generalized Gamma distribution are proposed as approximations to the finite sample distribution of T. Simulations suggest that these approximate distributions can be used to estimate accurately the probability of errors even for very small dimensions (e.g., two transmit antennas).

Abstract: The principal results of this paper are listed as follows. 1) The eigenvalues of a suitably normalized version of the discrete Fourier transform (DFT) are {1, -1,j, -j} . 2) An eigenvector basis is constructed for the DFT. 3) The multiplicities of the eigenvalues are summarized for an N×N transform as follows.