Journal ArticleDOI

# Bit Error Probability for MMSE Receiver in GFDM Systems

23 Feb 2018--Vol. 22, Iss: 5, pp 942-945

TL;DR: This letter considers the minimum mean-square error receiver for the generalized frequency division multiplexing system (GFDM) over frequency selective fading channels and derives an approximate probability density function for the signal-to-interference-plus-noise ratio.

AbstractIn this letter, we consider the minimum mean-square error receiver for the generalized frequency division multiplexing system (GFDM) over frequency selective fading channels. We derive an approximate probability density function for the signal-to-interference-plus-noise ratio. This expression allows us to obtain a new approximate, but rather accurate formulation for the bit error probability for a $\mathcal {M}$ -quadrature amplitude modulation scheme. Our results resort on the pivotal properties exhibited by eigenvalues of a circulant matrix. Since the entries of the channel matrix $\text {H}_{\text {ch}}$ are complex Gaussian distributed, and the eigenvalues are given as a weighted sum of its entries, the joint eigenvalue distribution is also Gaussian. Comparisons of the simulated and analytical results validate our formulation and allow a quick and efficient tool to compute the bit error rate for the GFDM system.

##### Citations
More filters
Journal ArticleDOI

25 May 2020
TL;DR: This work presents an accurate approximation and upper bounds for the bit error rate of 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.
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.

22 citations

### Cites methods from "Bit Error Probability for MMSE Rece..."

• ...BIT ERROR PROBABILITY The bit error probability for the M-QAM and BPSK modulations can be found according to [32], [33] respectively as...

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Journal ArticleDOI
, Lin Mei1
TL;DR: Considering relationships between zero forcing and minimum mean square error 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.
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.

7 citations

### Cites background from "Bit Error Probability for MMSE Rece..."

• ...From another point of view, although error probability with MMSE receiver is derived in [10], theoretical derivations are given according to an approximated distribution of signal-to-interference-plus-noise ratio (SINR) and the generality is not as clearly illustrated as that in [1], [2], and [7] through calculating NEF....

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Proceedings ArticleDOI

21 Oct 2019
TL;DR: 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 and a GFDM waveform–an emerging candidate waveform for the 5G mobile networks and beyond–is considered.
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.

5 citations

• ...where with the help of (14) and the constraint of (15), the probability density function (PDF) pγ pγnq can be approximated by [21] pγpγnq « 1 Γpκq θκ p1` γnq ́1 ́κ exp ˆ...

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Journal ArticleDOI
TL;DR: The closed form expressions of symbol error and outage probabilities with and without pointing errors are derived for GFDM over gamma–gamma channel model and verified with the Monte-Carlo simulations.
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.

5 citations

Proceedings ArticleDOI

01 Apr 2019
TL;DR: A design of the pulse shaping filter using computationally efficient optimisation method to reduce the peak-to-average power ratio (PAPR) of the GFDM system is proposed and the numerical results illustrate the effectiveness of the proposed method in reducing the PAPR of theGFDM transmitted signal.
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.

4 citations

### Cites background from "Bit Error Probability for MMSE Rece..."

• ...Generalized Frequency Division Multiplexing (GFDM) is a promising modulation candidate for the 5G waveform due to its capability in addressing various major 5G requirements, especially low latency in a tactile internet scenario [1], [3], [4]....

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##### References
More filters
Book
01 Jan 1977
TL;DR: The fundamental theorems on the asymptotic behavior of eigenvalues, inverses, and products of banded Toeplitz matrices and Toepler matrices with absolutely summable elements are derived in a tutorial manner in the hope of making these results available to engineers lacking either the background or endurance to attack the mathematical literature on the subject.
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.

2,231 citations

### "Bit Error Probability for MMSE Rece..." refers background or methods in this paper

• ...In order to address this constraint, we consider a matrix H given as [7]...

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• ...It, being a circulant matrix, possesses the normalized eigenvectors given by [7] φ j = 1 √ N 1, ω j , ω(2)j , ....

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• ...The corresponding eigenvalues are given by [7]...

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Journal ArticleDOI
Kyongkuk Cho
TL;DR: This work provides 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.
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.

942 citations

### "Bit Error Probability for MMSE Rece..." refers background in this paper

• ...As it was stated in [9], the BER for the lth bit error probability of M-QAM constellation can be expressed as...

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Journal ArticleDOI
Nicola Michailow
TL;DR: The flexible nature of GFDM makes this waveform a suitable candidate for future 5G networks, and its main characteristics are analyzed.
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.

711 citations

### "Bit Error Probability for MMSE Rece..." refers background in this paper

• ...In this letter, we have employed an analytical approach to calculate the SINR for GFDM waveform using the MMSE receiver, considering the influence of frequency selective fading channels....

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• ...So, x = Ad, where A is the modulation matrix GFDM and the vector d represents the complex data symbols d = [d0 d1 · · · dN−1]T with variance σ 2d ....

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• ...Index Terms— GFDM, Gamma approximation, MMSE, SINR....

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• ...One of these technologies that is being proposed for low latency and high throughput is called Generalized Frequency Division Multiplexing (GFDM) [1]....

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• ...After the CP removal, the received vector can be written as, y = HchAd + ν, (1) where ν is the AWGN vector of length N with variance σ 2ν and Hch is a circulant Toeplitz matrix based on vector h given as [5] Hch = ⎡ ⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ h1 0 · · · 0 hL · · · h2 h2 h1 · · · 0 0 · · · h3 ... . . . · · · ... hL hL−1 · · · · · · · · · · · · 0 0 hL · · · · · · · · · · · · 0 ... . . . · · · ... 0 0 hL · · · · · · h1 ⎤ ⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (2) The received vector y is distorted due to (i) self-interference coming from GFDM inherent non-orthogonality, and (i i) frequency selectivity introduced by the channel impulse response....

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Journal ArticleDOI
TL;DR: A Gamma distribution and a generalized Gamma distribution are proposed as approximations to the finite sample distribution of T and simulations suggest that these approximate distributions can be used to estimate accurately the probability of errors even for very small dimensions.
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).

277 citations

### "Bit Error Probability for MMSE Rece..." refers methods in this paper

• ...[6] have presented an expression for the SINR of the nth data symbol given as, γn = 1 MMSEn − 1 = 1 (IN + p N (HchA)†HchA)−1 nn − 1,...

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Journal ArticleDOI
, T. Parks1
TL;DR: The eigenvalues of a suitably normalized version of the discrete Fourier transform (DFT) are {1, -1,j, -j} and an eigenvector basis is constructed for the DFT.
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.

223 citations

### "Bit Error Probability for MMSE Rece..." refers background in this paper

• ...and is nothing but the normalized DFT matrix [8]....

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