Exact Symbol Error Probability Research Articles

Channel Norm-Based Energy Detection With Diversity Reception for Amplitude-Shift Keying in Rayleigh Fading

Optimum and suboptimum energy detection receivers with diversity reception in Rayleigh fading, that only require limited channel state information based on the channel norm, are introduced. The importance of these channel norm based energy detection (CNBED) receivers is that they achieve the same diversity order as that of a coherent receiver. Furthermore, their performance degradation when compared to a coherent receiver can be made sufficiently low by appropriate choice of parameters. Due to their performance and the low complexity of energy detection, the CNBED techniques have potential applications to internet of things. Analysis of the CNBED receivers is performed for multi-level amplitude-shift keying (ASK). For the optimum receiver, an analytical approximation of the symbol error probability (SEP) as a sum of univariate integrals involving averaging of Gaussian Q-functions is derived. For the suboptimum receiver, closed form expressions for the exact SEP and for the approximate SEP under the condition of high signal-to-noise ratio (SNR) are obtained; in addition, using asymptotic approximations, expressions for the optimal ASK transmit levels with high SNR and large number of receive diversity branches, that minimize the SEP subject to an energy constraint, are found. Numerical results are provided to demonstrate the performance of the receivers.

Union bounds on the symbol error probability of LoRa modulation for flat Rician block fading channels

In this paper the symbol error performance of LoRa modulation is addressed for flat Rician block fading channels. First the exact symbol error probability of the LoRa modulation on Rician fading is derived. Then the upper and lower union bounds are employed on the derived symbol error probability. The proposed bounds are compared against the exact symbol error probability, the numerical evaluation of the symbol error probability and the state-of-art approximation of the LoRa symbol error probability. Numerical results show that while the proposed upper bound is very tight to the exact symbol error probability, there is approximately a 2.5 dB gap for the lower bound.

On Minimizing Symbol Error Rate Over Fading Channels With Low-Resolution Quantization

We analyze the symbol error probability (SEP) of $M$-ary pulse amplitude modulation ($M$-PAM) receivers equipped with optimal low-resolution quantizers. We first show that the optimum detector can be reduced to a simple decision rule. Using this simplification, an exact SEP expression for quantized $M$-PAM receivers is obtained when Nakagami-$m$ fading channel is considered. The derived expression enables the optimization of the quantizer and/or constellation under the minimum SEP criterion. Our analysis of optimal quantization for equidistant $M$-PAM receiver reveals the existence of error floor which decays at a double exponential rate with increasing quantization bits, $b$. Moreover, by also allowing the transmitter to optimize the constellation based on the statistics of the fading channel, we prove that the error floor can be eliminated but at a lower decay exponent than the unquantized case. Characterization of this decay exponent is provided in this paper. We also expose the outage performance limitations of SEP-optimal uniform quantizers. To be more precise, its decay exponent does not improve with $b$. Lastly, we demonstrate that the decay exponent of a quantized receiver can be complemented by receive antenna diversity techniques.

Symbol error probability optimisation in three‐phase bidirectional amplify‐and‐forward relaying systems

This study deals with the error probability optimisation problem in three-phase bidirectional amplify and forward relaying systems. Different from the common two-phase strategy, in the three-phase counterpart, the relay power is split into two different terminals in terms of data error probability. With such a three-phase division, the authors present a unified optimisation algorithm based on the exact symbol error probability (SEP) and its upper bound expression. In addition, by virtue of the dual decomposition algorithm, the authors derive the optimal power allocation and demonstrate its uniqueness at both terminals and the relay node. Moreover, the unified optimisation algorithm can be reduced to some special cases, such as the maximum minimisation problem and unbalanced link problem. The proposed results reveal that the proposed algorithm has a significant impact on SEP reduction.

SEP Bounds for MPSK With Low SNR

We derive new bounds on the symbol error probability (SEP) of $M$ -ary phase-shift keying (MPSK) with $M>4$ , in additive white Gaussian noise. The exact SEP expressions require numerical integration or the computation of a two-dimensional function. Differently, the new SEP bounds only depend on simple one-dimensional functions and hence are easier to compute than the exact SEP. We show that the new bounds are tighter than the existing SEP bounds for MPSK at low signal-to-noise ratio (SNR), while preserving accuracy at large SNR. In addition, simple and accurate SEP approximations for MPSK with $M>4$ are derived.

Symbol error probability evaluation for PSK modulation schemes in generalized non‐Gaussian noise channels

SummaryThe average symbol error probability (ASEP) performance over the faded radio frequency (RF) link when the noise is additive in nature and has generalized Laplacian distribution is evaluated in this paper. The additive white generalized Laplacian noise (AWGLN) distribution can model different impulsive and non‐Gaussian noise environments often encountered in practice and provides a robust alternative to Gaussian distribution. A new expression for evaluating the exact symbol error probability over a multilevel M‐ary PSK‐modulated AWGLN channel is derived. Based on the obtained expression, the ASEP for the single‐hop RF link that models the shadowing and fading conditions over the RF channel by a generalized –K (GK) distribution is derived. Further, the error performance of a decode‐and‐forward relayed and GK‐distributed two‐hop RF link is discussed, and the results are validated through numerical plots.

Performance Analysis of Coherent PLC With MPSK Signaling in Nakagami-$m$ Noise Environment

An $N$ -branch receive diversity power line communication (PLC) system subject to Rayleigh fading and corrupted by additive Nakagami- $m$ background noise is considered. A suboptimal maximal-ratio-combining (MRC), an optimal diversity combining, and a Gauss-optimal receiver are proposed for the PLC system. A closed form expression for the exact symbol error probability (SEP) and a union bound SEP expression are obtained for the proposed MRC and Gauss-optimal receivers, respectively, using a characteristic function (c.f.) approach for the system employing $M$ -ary phase-shift keying (MPSK) at the transmitter, conditioned to $mN$ being an integer. The asymptotic expression for the SEP obtained for the MRC receiver at high signal-to-noise (SNR) demonstrates that the diversity order of the system is independent of the shape parameter $m$ of the additive noise which is identical to the diversity order of the optimal and the Gauss-optimal receivers. Further, numerical studies justify the advantages of employing receive diversity to achieve superior SEP performance. The effect of increasing the diversity branches on the PLC system performance with varying $m$ and the power efficiency of the PLC system are also presented.

Optical Spatial Modulation for FSO IM/DD Communications With Photon-Counting Receivers: Performance Analysis, Transmit Diversity Order and Aperture Selection

This paper investigates two pulse-based Optical Spatial Modulation (OSM) schemes as cost-efficient solutions for multi-aperture Free-Space Optical (FSO) communications with Intensity-Modulation and Direct-Detection (IM/DD). Namely, we consider Optical Space Shift Keying (OSSK) where information is encoded in the index of the pulsed optical source and Spatial Pulse Position Modulation (SPPM) where additional bits determine the position of the transmitted optical pulse resulting in higher transmission rates. A performance analysis is carried out over gamma-gamma channels with the exact Poisson photon-counting detection model. Exact Symbol Error Probability (SEP) expressions, simple upper bounds and the achievable transmit diversity orders are derived for both the open-loop and closed-loop scenarios. Based on the presented performance analysis, a transmit aperture selection scheme capable of maximizing the transmit diversity order is proposed for OSSK and SPPM in the closed-loop case. Results show that for open-loop OSSK, open-loop SPPM and closed-loop OSSK, the transmit diversity order does not depend on the severity of scintillation unlike the closed-loop SPPM case.

Energy-Efficient and Low-Latency Massive SIMO Using Noncoherent ML Detection for Industrial IoT Communications

To enable ultrareliable low-latency wireless communications required in the Industrial Internet of Things, in this paper we develop an energy-based modulation [i.e., non-negative pulse amplitude modulation (PAM)] constellation design framework for noncoherent detection in massive single-input multiple-output (SIMO) systems. We consider that one single-antenna transmitter communicates to a receiver with a large number of antennas over a Rayleigh fading channel, and the receiver decodes the transmitted information at the end of every symbol. For such an SIMO system with non-negative PAM modulation, we first propose a fast noncoherent maximum-likelihood decoding algorithm and derive a closed-form expression of its symbol error probability (SEP). We then enhance the system energy efficiency by finding the optimal PAM constellation that minimizes the exact SEP subject to a total signal power constraint for such a system with an arbitrary number of receiver antennas, signal-to-noise ratio (SNR), and constellation size. Furthermore, the closed-form upper and lower bounds on the optimal SEP are derived. Based on these bounds, the exact expression for coding gain of the dominant term of the SEP is presented for such an optimal massive SIMO system. We also present an asymptotic SEP expression at a high SNR regime and the approximate diversity gain of the system. Simulation results for the proposed optimal PAM constellation validate the theoretical analysis, and show that our presented optimal constellation attains significant performance gains over the currently available minimum-distance-based constellation systems.

Symbol Error Probability Analysis of DFrFT-Based OFDM Systems With CFO and STO in Frequency Selective Rayleigh Fading Channels

In this paper, we derive exact symbol error probability (SEP) expressions for orthogonal frequency-division multiplexing (OFDM) based on discrete fractional Fourier transform (DFrFT) when symbol timing offset (STO) and carrier frequency offset (CFO) are present together. Closed form expressions are derived for binary phase shift keying, quadrature phase shift keying, and 16-ary quadrature amplitude modulation in the case of transmission over a frequency selective Rayleigh fading channel. As it is well known, in OFDM the presence of CFO destroys the orthogonality between the sub-carriers, generating inter-carrier interference (ICI), while the presence of STO introduces intersymbol interference, between successive symbols, and phase rotations in frequency domain, within the same symbol. It is well recognized that an OFDM system based on DFrFT is more robust to ICI introduced by CFO than that based on discrete Fourier transform (DFT). Motivated by this, the combined effect of CFO and STO on the SEP of a DFrFT-based OFDM system is analyzed. Our results show that the DFrFT-based OFDM system outperforms that based on the DFT for different values of the DFrFT “angle parameter.” Monte Carlo simulations are used to demonstrate the correctness of the derived analytical expressions.

Average SEP of AF Relaying in Nakagami-m Fading Environments

This paper is devoted to an investigation of an exact average symbol error probability (SEP) for amplify and forward (AF) relaying in independent Nakagami-m fading environments with a nonnegative integer plus one-half m, which covers many actual scenarios, such as one-side Gaussian distribution (m=0.5). Using moment generating function approach, the closed-form SEP is expressed in the form of Lauricella multivariate hypergeometric function. Four modulation modes are considered: rectangular quadrature amplitude modulation (QAM), M-ary phase shift keying (MPSK), M-ary differential phase shift keying (MDPSK), and π/4 differential quaternary phase shift keying (DQPSK). The result is very simple and general for a nonnegative integer plus one-half m, which covers the same range as integer m. The tightness of theoretical analysis is confirmed by computer simulation results.

A simplified exact expression of SEP for cross QAM in AWGN channel from M × N rectangular QAM and its usefulness in Nakagami-m fading channel

This paper derives a simple exact expression of symbol error probability (SEP) for general order cross QAM constellation in the presence of additive white Gaussian noise (AWGN) channel. The key idea is to obtain this novel expression from a simple analysis of the corresponding rectangular QAM. The analysis of the expression involves simple one dimensional Gaussian Q functions unlike other complex SEP expressions. A simple tight bound approximation of the proposed exact SEP is also given, which provides performance improvement over the existing SEP approximations, particularly for low signal to noise ratios (SNRs). Also, with the help of simulation results, we show that the proposed approximation is in excellent agreement with the exact SEP curve. Moreover, the proposed expressions prove to be useful for accurate estimation of the SEP in Nakagami-m fading channel, including the special case of Rayleigh fading (m=1).

Performance Comparison of Selection Combining With Full CSI and Switch-and-Examine Combining With and Without Post-Selection

Selection combining (SC) with full channel state information (CSI) achieves full diversity order in a decode-and-forward (DF) relaying system. However, the estimation of CSI of all the links is a cumbersome process. As an alternative, switched-diversity techniques have been proposed in the literature, in which CSI of all the links are not simultaneously required. In this paper, we compare the variants of the switched-diversity and full-CSI SC schemes through a paired error approach for a multi-DF-relaying-based cooperative diversity system. We derive the exact symbol error probability (SEP) expressions for the full-CSI SC, switch-and-examine combining (SEC), and SEC with post-selection (SECps) schemes, considering $M$ -ary phase-shift keying (MPSK) signaling over slow, flat, independent, and nonidentical Nakagami- $m$ fading channels. It is to be noted that the derived SEP expressions are valid for integer values of fading severity parameter $m$ . In addition, we also present the closed-form asymptotic SEP expressions for MPSK signaling to evaluate the diversity order. Concisely, from the numerical results, it is observed that the SECps scheme is able to offer the same performance as that of the full-CSI SC scheme. Furthermore, the SECps scheme outperforms the SEC scheme with more CSI.

Tight bounds and invertible average error probability expressions over composite fading channels

The focus in this paper is on obtaining tight, simple algebraic-form bounds and invertible expressions for the average symbol error probability (ASEP) of M-ary phase shift keying (MPSK) in a class of composite fading channels. We employ the mixture gamma (MG) distribution to approximate the signal-to-noise ratio (SNR) distributions of fading models, which include Nakagami-m, Generalized-K (K G ), and Nakagami-lognormal fading as specific examples. Our approach involves using the tight upper and lower bounds that we recently derived on the Gaussian Q-function, which can easily be averaged over the general MG distribution. First, algebraic-form upper bounds are derived on the ASEP of MPSK for M > 2, based on the union upper bound on the symbol error probability (SEP) of MPSK in additive white Gaussian noise (AWGN) given by a single Gaussian Q-function. By comparison with the exact ASEP results obtained by numerical integration, we show that these upper bounds are extremely tight for all SNR values of practical interest. These bounds can be employed as accurate approximations that are invertible for high SNR. For the special case of binary phase shift keying (BPSK) (M = 2), where the exact SEP in the AWGN channel is given as one Gaussian O-fu notion, upper and lower bounds on the exact ASEP are obtained. The bounds can be made arbitrarily tight by adjusting the parameters in our Gaussian bounds. The average of the upper and lower bounds gives a very accurate approximation of the exact ASEP. Moreover, the arbitrarily accurate approximations for all three of the fading models we consider become invertible for reasonably high SNR.

Performance Analysis of the M2M Network over Keyhole Nakagami Fading Channels

The performance of the mobile-to-mobile (M2M) network over keyhole Nakagami fading channels is investigated in this paper. Using the method of the probability density function (PDF) of the signal-to-noise ratio (SNR), the exact and approximate closed-form average symbol error probability (ASEP) expressions are derived for pulse-amplitude modulation (PAM), phase shift keying (PSK),and frequency shift keying modulation (FSK). The exact closed-form expressions for channel capacity are also presented. Then the ASEP and channel capacity performance under different conditions is evaluated through numerical simulations to verify the analysis. The simulation results showed that the performance of the M2M network is improved with the transmitting/receiving antennas and the fading coefficient increased.

Performance analysis of two‐way decode‐and‐forward relaying in the presence of co‐channel interferences

This study investigates the performance of two-way decode-and-forward (DF) relaying networks, considering transmissions over independent but not necessarily identically distributed (i.n.i.d.) Rayleigh fading channels, in the presence of multiple co-channel interferers at both the relay and end-source nodes. Both asymmetrical and symmetrical cases, of whether the channels from source terminals to the relay are identically distributed or not, are considered. Specifically, closed-form expressions for the cumulative distribution function of the equivalent signal-to-interference-plus-noise ratio (SINR) in different cases are derived, based on which the exact symbol error probability (SEP) and the systems’ achievable rate are derived and analysed. Based on the analytic results, the authors study the impacts of system parameters, such as interference power and number of interferers on the performance of the system. Furthermore, the system behavior at high signal-to-noise ratio (SNR) values is studied via deriving the asymptotic SEP. The results of this study are attested through Monte Carlo simulations.

Performance of Hybrid Selection and Switch-and-Stay Combining With Decode-and-Forward Relaying

In this letter, an uncoded cooperative diversity system with a source, two decode-and-forward (DF) relays, and a destination (D) is considered. We propose a hybrid selection and switch-and-stay combining (HSSSC) scheme at D. Furthermore, we also derive the exact symbol error probability (SEP) of this scheme for DF relaying over slow and flat Rayleigh fading channels considering <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="TeX">$M$</tex-math></inline-formula> -ary phase-shift keying (MPSK) signaling using paired error approach. Numerical results reveal that full diversity order is achieved in the case of HSSSC scheme and improvement in the SEP performance is observed when compared to the distributed selection combining (SC), distributed scaled selection combining, and distributed switch-and-stay combining schemes proposed in the literature. In addition, the HSSSC scheme requires less channel state information (CSI) at D compared to the distributed full CSI SC scheme proposed in the literature.

Exact Symbol Error Probability of Hybrid/Integrated Satellite-Terrestrial Cooperative Network

In this paper, we study the Symbol Error Probability (SEP) performance of a hybrid/integrated satellite-terrestrial cooperative network. In particular, we focus on the case of mobile relays that forward the satellite signal to a masked mobile destination node. The Selective Decode-and-Forward (SDF) transmission scheme is implemented and only the relay nodes which can successfully decode the satellite message are selected to retransmit the signal. The destination node exploits the spatial diversity advantages by implementing a typical Maximum Ratio Combining (MRC) technique. The closed-form expressions for the exact average SEP of the arbitrary M-ary phase shift keying and M-ary quadrature amplitude modulation signaling with MRC diversity reception over independent but not necessarily identically distributed fading channels are derived using a Moment Generating Function (MGF) approach. These closed-form expressions are represented in terms of a finite sum of Lauricella hypergeometric functions. The analytical expressions show excellent agreement with the simulation results. Numerical results show that for a system using QPSK under the frequent heavy shadowed fading condition, the diversity gain of approximately 7 dB can be obtained at the SEP of 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1</sup> with respect to the direct transmission, when only one relay is used. It increases to around 12 dB in the case of 3 relays.

Error Performance of Multidimensional Lattice Constellations—Part I: A Parallelotope Geometry Based Approach for the AWGN Channel

Multidimensional lattice constellations which present signal space diversity (SSD) have been extensively studied for single-antenna transmission over fading channels, with focus on their optimal design for achieving high diversity gain. In this two-part series of papers we present a novel combinatorial geometrical approach based on parallelotope geometry, for the performance evaluation of multidimensional finite lattice constellations with arbitrary structure, dimension and rank. In Part I, we present an analytical expression for the exact symbol error probability (SEP) of multidimensional signal sets, and two novel closed-form bounds, named Multiple Sphere Lower Bound (MLSB) and Multiple Sphere Upper Bound (MSUB). Part II extends the analysis to the transmission over fading channels, where multidimensional signal sets are commonly used to combat fading degradation. Numerical and simulation results show that the proposed geometrical approach leads to bounds that can be used for the performance evaluation and the design of multidimensional lattice constellations, both in Additive White Gaussian Noise (AWGN) and fading channels.

Cooperative mobile-to-mobile communications over double Nakagami-m fading channels

Double Nakagami-m fading model can be used to provide a realistic description of the mobile-to-mobile (M2M) channel. The moment generating function (MGF)-based approach is used to derive the symbol error probability (SEP) expressions for the multiple-mobile-relay-based M2M system. Under the consideration of better performance and lower complexity, both decode-and-forward (DF) and amplify-and-forward (AF) protocols are addressed in this study. For adaptive DF (ADF) relaying, the asymptotic SEP at high signal-to-noise ratio regime in addition to the exact SEP formulations are derived. Furthermore, the impacts of channel estimation error for the insight of non-ideal channel estimation are analysed. For fixed-gain AF (FAF) relaying, the authors present approximate SEP and asymptotic SEP to indicate the cooperative gain. For L mobile relays and under the assumption of independent and identically distributed (i.i.d) source- relay channels and i.i.d relay-destination channels, it is shown in this paper that asymptotic diversity orders of L × min(mRD,1, mRD,2) + min(mSD,1, mSD,2) and L × min(mSR,1, mSR,2, mRD,1, mRD,2) + min(mSD,1, mSD,2) can be achieved for ADF-M2M and FAF-M2M schemes, respectively. Simulations are performed to verify the analytical results.