Exact Error Probability Research Articles

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.

Random Linear Streaming Codes in the Finite Memory Length and Decoding Deadline Regime—Part I: Exact Analysis

<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Streaming codes</i> take a string of source symbols as input and output a string of coded symbols in real time, which eliminate the queueing delay of traditional <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">block codes</i> and are thus especially appealing for delay sensitive applications. Existing works on streaming code performance either focused on the asymptotic error-exponent analyses, or on the optimal code construction under <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">deterministic adversarial channel models</i> . In contrast, this work analyzes the exact error probability of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">random linear streaming codes</i> (RLSCs) in the large field size regime over the stochastic i.i.d. symbol erasure channel model. A closed-form expression of the error probability of large-field-size RLSCs is derived under, simultaneously, the finite memory length and decoding deadline constraints. The result is then used to examine the intricate tradeoff between memory length (complexity), decoding deadline (delay), code rate (throughput), and error probability (reliability). Numerical evaluation shows that under the same code rate and error probability requirements, the end-to-end delay of RLSCs is 40–48% of that of the optimal block codes (i.e., MDS codes). This implies that switching from block codes to streaming codes not only eliminates the queueing delay completely (which accounts for the initial 50% of the delay reduction) but also improves the reliability (which accounts for the additional 2–10% delay reduction).

Error analysis of L-PPM modulated MIMO based multi-user NOMA-VLC system with perfect and imperfect SIC.

By offering a wide license-free spectrum, visible light communication (VLC) has emerged as a game-changing technology for networks beyond 5G. In this study, the multiple-input multiple-output (MIMO) technology is combined with a non-orthogonal multiple access VLC (NOMA-VLC) system to provide a reliable error-free high data rate network for multi-user scenarios. A generalized L-PPM modulated MIMO based multi-user NOMA-VLC system with M×N line-of-sight connections for each user is presented. Considering a practical scenario of imperfect successive interference cancellation, a novel closed form formulation of error probability for the proposed system is obtained. This study also includes a comprehensive analysis of two-user and three-user L-PPM modulated 2×2 MIMO-NOMA-VLC systems, and exact error probability expressions are derived. With the identical set of parameters, the L-PPM modulated MIMO-NOMA-VLC system totally outperforms the on-off keying modulated MIMO-NOMA-VLC system. As the number of photodetectors per user increases, the error probability of the considered system decreases. An L-PPM modulated 2×2 MIMO based three-user NOMA-VLC system offers the best performance at a power allocation co-efficient of 0.3. The simulation results validate the derived error probability expressions.

An Alternative Statistical Characterization of TWDP Fading Model.

Two-wave with diffuse power (TWDP) is one of the most promising models for the description of small-scale fading effects in 5G networks, which employs mmWave band, and in wireless sensor networks deployed in different cavity environments. However, its current statistical characterization has several fundamental issues. Primarily, conventional TWDP parameterization is not in accordance with the model’s underlying physical mechanisms. In addition, available TWDP expressions for PDF, CDF, and MGF are given either in integral or approximate forms, or as mathematically untractable closed-form expressions. Consequently, the existing TWDP statistical characterization does not allow accurate evaluation of system performance in all fading conditions for most modulation and diversity techniques. In this regard, physically justified TWDP parameterization is proposed and used for further calculations. Additionally, exact infinite-series PDF and CDF are introduced. Based on these expressions, the exact MGF of the SNR is derived in a form suitable for mathematical manipulations. The applicability of the proposed MGF for derivation of the exact average symbol error probability (ASEP) is demonstrated with the example of M-ary PSK modulation. The derived M-ary PSK ASEP expression is further simplified for large SNR values in order to obtain a closed-form asymptotic ASEP, which is shown to be applicable for SNR > 20 dB. All proposed expressions are verified by Monte Carlo simulation in a variety of TWDP fading conditions.

Error Rate Analysis of Non-Orthogonal Multiple Access With Residual Hardware Impairments

In this letter, the impact of residual hardware impairment (RHI) on the performance of non-orthogonal multiple access (NOMA) is investigated by taking into account the effect of imperfect successive interference cancellation (ipSIC). The new exact and asymptotic pairwise error probability (PEP) expressions with RHI under ipSIC are derived over Rayleigh fading channels. The asymptotic PEP expression is subsequently utilized to quantify the diversity orders achieved by NOMA users in the presence of RHI. In particular, we show that the effective diversity order of each NOMA user equals to user’s order for both ideal and non-ideal (RHI) scenarios. Leveraging the derived PEP, the union bound on the bit error rate of NOMA users are further evaluated. We demonstrate that the impact of RHI with ipSIC on each user’s error rate performance is negligible. In contrast, when perfect SIC is considered, such effects are significant. Numerical and Monte Carlo simulation results are presented to corroborate the analysis.

Soft Interference Cancellation for Random Coding in Massive Gaussian Multiple-Access.

In 2017, Polyanskiy showed that the trade-off between power and bandwidth efficiency for massive Gaussian random access is governed by two fundamentally different regimes: low power and high power. For both regimes, tight performance bounds were found by Zadik et al., in 2019. This work utilizes recent results on the exact block error probability of Gaussian random codes in additive white Gaussian noise to propose practical methods based on iterative soft decoding to closely approach these bounds. In the low power regime, this work finds that orthogonal random codes can be applied directly. In the high power regime, a more sophisticated effort is needed. This work shows that power-profile optimization by means of linear programming, as pioneered by Caire et al. in 2001, is a promising strategy to apply. The proposed combination of orthogonal random coding and iterative soft decoding even outperforms the existence bounds of Zadik et al. in the low power regime and is very close to the non-existence bounds for message lengths around 100 and above. Finally, the approach of power optimization by linear programming proposed for the high power regime is found to benefit from power imbalances due to fading which makes it even more attractive for typical mobile radio channels.

On Approximation, Bounding &amp; Exact Calculation of Average Block Error Probability for Random Code Ensembles

This paper presents a method to calculate the exact average block error probability of some random code ensembles under maximum-likelihood decoding. The proposed method is applicable to various channels and ensembles. The focus is on both spherical and Gaussian random codes on the additive white Gaussian noise channel as well as binary random codes on both the binary symmetric channel and the binary erasure channel. While for the uniform spherical ensemble Shannon, in 1959, argued with solid angles in N-dimensional space, the presented approach projects the problem into two dimensions and applies standard trigonometry. This simplifies the derivation and also allows for the analysis of the independent identically distributed (i.i.d.) Gaussian ensemble which turns out to perform better for short blocklengths and high rates. Moreover, a new lower bound on the average block error probability of the uniform spherical ensemble is found. For codes with more than three codewords, it is tighter than the sphere packing bound, but requires exactly the same computing effort. Furthermore, tight approximations are proposed to simplify the computation of both the exact average error probability and the two bounds. For the binary symmetric channel and the binary erasure channel, bounds on the average block error probability for i.i.d. random coding are derived and compared to the exact calculations.

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.

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.

GR and BP neural network-based performance prediction of dual-antenna mobile communication networks

The performance of a dual-antenna mobile communication network in 2-Rayleigh fading is investigated in this paper. Exact average symbol error probability (SEP) expressions with selection combining (SC) are derived for q-ary phase-shift keying (PSK) and pulse-amplitude modulation (PAM). Exact expressions are also given for the channel capacity. It is important to predict the performance of mobile communication networks in complex wireless environments. Thus, we propose generalized regression (GR) and back-propagation (BP) neural network-based SEP prediction methods. The theoretical results are used to generate training data. The proposed prediction methods are compared to the extreme learning machine (ELM), locally weighted linear regression (LWLR), support vector machine (SVM), and radial basis function (RBF) neural network methods. The results obtained verify that the proposed methods provide better SEP predictions.

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.

Performance Analysis of Full-Duplex Amplify-and-Forward Relay System with Hardware Impairments and Imperfect Self-Interference Cancellation

In this paper, we analyze the performance of a full-duplex (FD) amplify-and-forward (AF) relay system with imperfect hardware. Besides the aggregate hardware impairments of the imperfect transceiver, we also consider the impact of residual self-interference (RSI) due to imperfect cancellation at the FD relay node. An analytical framework for analyzing the system performance including exact outage probability (OP), asymptotic OP, and approximate symbol error probability (SEP) is developed. In order to tackle these impacts, we propose an optimal power allocation scheme which can improve the outage performance of the FD relay node, especially at the high signal-to-noise ratio (SNR) regime. Numerical results are presented for various evaluation scenarios and verified using the Monte Carlo simulations.

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.

Signal Detection for Superposition Transmission Protocols for Optical Wireless Scattering Broadcast Channel

Due to the scattering events in the ultra-violet (UV) spectrum, UV optical wireless communication enables transmission from a single transmitter to multiple receivers, which forms a broadcast channel. In this paper, we address the transmission protocols for such a broadcast channel. Based on the degradeness of such a broadcast channel, we propose time-dilation and time-shrinking superposition transmission. For the former, the duration of the base-layer symbols is shorter than that of the enhancement-layer symbols; while for the latter, the duration of the base-layer symbols is longer. We propose the signal detection of the two types of superposition transmission protocols, analyze the detection error probability and the associated upper bounds, and formulate the problems of power allocation among the signal layers. The numerical results on the sum power obtained from the exact detection error probability and the upper bounds are provided. It is shown that for the superposition with two signal layers, the sum power based on the upper bounds can be close to that based on the exact detection error probability.

Enhancing the sufficient condition of sparsity pattern recovery

The task of recovering a sparse signal from its noisy observations can be divided into two stages; recovering the positions of the nonzero entries or support and estimating their values. In this paper, fundamental limits of sparsity pattern recovery is investigated. Although this problem has been addressed in the recent literature, there is still a gap between those results and the exact limits. In this work, first a tight upper bound is derived on the error probability of the exact sparsity pattern recovery when the Maximum Likelihood (ML) decoder is exploited. Next, a sharp sufficient condition for perfect recovery of the support from noisy projections is derived. In their analyses, the authors treat random sampling matrix with standard normal entries and Gaussian noise. It is shown that the sufficient number of observations for perfect support recovery depends on the noise variance, the minimum nonzero entry of the sparse vector, and the sparsity parameter (k) which is in agreement with previous findings. The proposed upper bound is highly close to the exact probability of error, leading to tight sufficient condition which also matches the growth rate of the necessary conditions of the previous established works. More precisely, the proposed sufficient condition and the previously-established necessary condition on the number of samples are both O(k) (linear in k). The results are proved and supported both theoretically and by simulations.

Performance Analysis of Media-Based Modulation With Imperfect Channel State Information

Media-based modulation (MBM) is an attractive modulation scheme where information bits are conveyed by digitally controlling the on/off status of radio frequency mirrors (which are parasitic elements) placed near the transmit antenna. The MBM alphabet (which is the set of channel fade coefficients corresponding to all possible mirror on/off status vectors) is estimated at the receiver through pilot transmissions. In this paper, we analyze the effect of imperfect channel estimation on the bit error performance of MBM. We present the analysis for generalized spatial modulation MBM (GSM-MBM). We analyze the performance for two types of detectors, namely the commonly studied mismatched detector, and the true maximum-likelihood (ML) detector that maximizes the likelihood by taking the statistics of the channel estimate into account. First, we derive an exact average pairwise error probability (PEP) expression for the mismatched detector using characteristic function approach, and obtain a union bound based upper bound on the average bit error probability (BEP). Next, given the estimate of the MBM alphabet, we derive the true ML detector for GSM-MBM, and derive an exact average PEP expression (by averaging the conditional PEP over the statistics of the channel estimate) and an upper bound on the average BEP. The exactness of the average PEP and the tightness of the average BEP upper bounds of the detectors are validated through simulations.

Exact and Approximate Symbol Error Probability of cooperative systems with best relay selection and all participating relaying using Amplify and Forward or Decode and Forward Relaying over Nakagami-m fading channels

In this paper, we derive the theoretical Symbol Error Probability (SEP) of cooperative systems with best relay selection for Nakagami-m fading channels. For Amplify and Forward (AF) relaying, the selected relay offers the best instantaneous Signal to Noise Ratio (SNR) of the relaying link (source-relay-destination). In cooperative networks using Decode and Forward (DF), the selected relay offers the best instantaneous SNR of the link between the relay and the destination among the relays that have correctly decoded the transmitted information by the source. In the second part of the paper, we derive the SEP when all participating AF and DF relaying is performed. In the last part of the paper, we extend our results to cognitive radio networks where there is interference constraints : only relays that generate interference to primary receiver lower than a predefined threshold T can transmit. Both AF and DF relaying with and without relay selection are considered.

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.

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.