Asymptotic Bit Error Rate Expression Research Articles

Performance Analysis and Resource Allocation for a Relaying LoRa System Considering Random Nodal Distances

In conventional star-topology LoRa networks, the gateways are expected to collect the data from all the nodes nearby. However, a major challenge for the conventional LoRa system is the performance degradation due to the long-range communication over fading channels. To resolve the challenging issue, this paper investigates a two-hop amplify-and-forward relaying LoRa network in a two-dimension plane, where random nodal distances are considered. Moreover, a relay-selection mechanism is developed for the proposed system. Based on the best relay-selection protocol, the analytical bit-error-rate (BER) and asymptotic BER expressions, achievable diversity order, coverage probability, and throughput of the proposed system are derived over the Nakagami-<inline-formula> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula> fading channel. Furthermore, to maximize the throughput of the proposed system, a two-dimensional resource allocation optimization problem (i.e., the spread factor selection and power allocation optimization) is formulated and investigated. The proposed optimal spread factor selection and power allocation scheme is verified to outperform the two baseline schemes. Simulation and numerical results show that although the proposed system reduces the throughput compared to the conventional LoRa system, it significantly improves the BER and coverage probability. Hence, the proposed system can be considered as a promising technique for low-power, long-range and highly reliable Internet-of-Things applications.

Accurate Variable-Order Approximations to the Sum of α – μ Variates With Application to MIMO Systems

Sum of fading signals arises in several communication applications such as maximal-ratio combining (MRC) used in multiple antenna systems and eliminating the inter-symbol interference in high speed wireless networks. The exact formulation of the statistics of the sum is known to be very complicated. This article presents two simple closed-form approximations to the probability density function (PDF) and cumulative distribution function (CDF) of the sum of $\alpha -\mu $ variates, resembling an urban mobile communication environment. By direct integration, it is demonstrated that the PDF of the sum of $\alpha -\mu $ random variables can be approximated by a weighted sum of few identical $\alpha -\mu $ distributions including order one coefficients. The proposed approximations are of variable-order to achieve the desired accuracy and also have specific parameters using which, multiple-input multiple-output (MIMO) diversity systems can be explicitly analyzed as simply as a single-input single-output (SISO) system. In addition, a MIMO wireless system utilizing orthogonal space-time block coding technique (OSTBC-MIMO) in an $\alpha -\mu $ frequency nonselective fading environment is analyzed, and expressions for outage probability (OP) and bit error rate (BER) of digital binary modulations as well as the capacity are obtained. Moreover, the asymptotic OP and BER expressions are derived from the declared approximate expressions. The asymptotic OP and BER performances clearly indicate a diversity order equal to the product of the $\frac {\alpha }{2}$ parameter, the $\mu $ parameter, the number of transmit antennas and the number of receive antennas. Analytical results are verified by simulation.

Performance Analysis of Uplink Massive Multiuser SM-MIMO System With Imperfect Channel State Information

In this paper, the error performance and spectrum efficiency (SE) of uplink massive multiuser spatial modulation (SM) MIMO systems with zero-forcing detection for perfect and imperfect channel state information (CSI) are investigated over composite Rayleigh fading channels. We firstly derive the error probabilities of transmit antenna index detection and transmitted symbol detection of each user. Using these results, the bit error rate (BER) performance of the system under perfect CSI is analyzed. Furthermore, a closed-form average BER of the system is derived. We generalize the analytical results of perfect CSI to those of imperfect CSI for practicality, and derive a closed-form BER for the system with imperfect CSI. Based on this, two asymptotic BER expressions are developed to characterize the error performance of the system for large-scale receive antennas and high SNR. By means of the asymptotic expressions, the diversity order of the system is attained. Besides, we derive a lower bound on the achievable SE of the system with imperfect CSI. With this lower bound, an approximate expression of the achievable SE is obtained in closed form. Simulation results indicate that our theoretical analysis is valid, and agrees with simulation results well.

Bit Error Rate for NOMA Network

This letter examines the bit error rate (BER) performance of downlink non-orthogonal multiple access networks for binary phase-shift keying modulation. Exact BER expression is derived for each user in closed-form under additive white Gaussian noise and Rayleigh fading channels in perfect and imperfect successive interference cancellation (SIC) cases. Next, in perfect SIC case, the asymptotic BER expression in a high signal-to-noise ratio (SNR) region is obtained to express the behavior of the network with diversity and array gains. On the other hand, in imperfect SIC case, the upper bound for BER is attained, and at high SNR values, the BER reveals an error floor, and hence a zero diversity gain is achieved. Then, a feasible range of power allocation coefficients is found such that a good BER performance can be provided for each user. Finally, through simulations and software-defined radio-based real-time tests, analytical results are validated.

Cooperative Time Successive SSK-M-ary Modulation for Decode-and-Forward Relay Systems

Space shift keying (SSK) is a single radio frequency (RF) chain multiple-input multiple-output (MIMO) scheme that uses the index of transmit antennas for information transmission. Time successive SSK- $ M$ -ary modulation (TSSM) and multi RF chain TSSM (MRF-TSSM) are recent MIMO techniques that achieve second order transmit diversity. We propose two cooperative relaying systems to attain distributed diversity in addition to transmit diversity of TSSM and MRF-TSSM. Cooperative-TSSM-1 (Coop-TSSM-1) uses multibranch relaying where TSSM is used in orthogonal time slots during the cooperative phase. Coop-TSSM-2 maps symbols to lower order constellations at multiple relays for simultaneous transmission of information using MRF-TSSM. We consider Decode-and-forward (DF) relaying where relays that correctly detect symbols forward it. We analyze the average number of relays that participate in cooperation, the end-to-end performance using a tight upper bound on the bit error rate (BER) and the asymptotic BER expressions to analyze diversity orders of both cooperative systems. We validate all analyzes using simulation results. From the analysis and results we show that Coop-TSSM-1 achieves better diversity order compared to Coop-TSSM-2 at the cost of higher number of time slots for completing the cooperative phase. Further in Coop-TSSM-2, results indicate that different constellation mappings used in MRF-TSSM during cooperative phase impact the BER performance. By applying spatial modulation based transmit diversity and $M$ -ary Modulation systems in the cooperative setups of Coop-TSSM schemes, performance comparison is given which highlights the significance of Coop-TSSM schemes.

Performance analysis of FSO coherent BPSK systems over Rician turbulence channel with pointing errors.

We investigate the performance of coherent free-space optical (FSO) system in terms of bit error rate (BER) evaluation by adopting the modified Rician distribution based coherent channel model which allows taking into consideration the composite effects of both Rician turbulence, including amplitude fluctuation and optical phase distortion, and pointing errors (PEs). By expanding the Rician distribution, a mathematically traceable expression of the probability density function (PDF) for the composite channel is derived in the form of the Meijer-G function. Based on the composite channel PDF, the exact BER expression is obtained, allowing the analysis of BER performance for single-input single-output (SISO) links. This analysis is extended to single-input multi-output (SIMO) links with maximal ratio combining (MRC). With the help of the moment generating function (MGF), the exact BER expression can be simplified into a single integral, facilitating the analysis with high accuracy and reducing calculation complexity. Engineering insights including high-SNR approximated channel PDF, asymptotic BER expression, coding and diversity gains, are investigated and cross-validated for both SISO and SIMO links. Through both analytical and numerical verifications, the impairment due to PEs as well as the effect of modal compensation on the BER performance are discussed in detail, unveiling the fact that their inner relations should be taken into account for optimization. These verify the effectiveness of our models for both SISO and SIMO links with a wide range of different conditions and can be feasibly applied for different types of coherent FSO links.

A Novel Relay Selection Strategy of Cooperative Network Impaired by Bursty Impulsive Noise

Best relay selection (BRS) is crucial in enhancing the performance of cooperative networks. In contrast to most previous works, where the guidelines for BRS are limited to Gaussian noise, in this paper, we propose a novel relay selection protocol for a decode-and-forward cooperative network taking into account the bursty impulsive noise (IN). The proposed protocol chooses the $N$ th best relay considering both the channel gains and the states of the IN of the source-relay and relay-destination links. For this scheme, to obtain the state of IN, we propose a state detection algorithm using maximum a posteriori (MAP) detection. To analyze the performance of the proposed protocol, we first derive closed-form expressions for the probability density function (PDF) of the received signal-to-noise ratio assuming all the relays know the state of IN perfectly (genie-condition). Then, these PDFs are used to derive closed-form expressions for the bit error rate (BER) and the outage probability. Finally, we also derive the asymptotic BER and outage expressions to quantify the diversity benefits. We show that the proposed MAP-based $N$ th BRS protocol attains the derived genie-aided analytical results and outperforms the conventional relay selection protocol, optimized for the Gaussian case, and which does not take into account the IN memory.

Dynamic SSK-BPSK System Under Transmitter Correlated Nonidentical Rayleigh Fading

Space shift keying (SSK) is a recent multiple-input multiple-output (MIMO) modulation scheme where information is conveyed by the index of the activated antenna. An advantage of SSK transmitter is that it requires only one radio frequency (RF) chain. To achieve transmit diversity in such a single RF chain two antenna transmitter, modulation is dynamically changed between SSK and a modified form of binary phase shift keying (BPSK) based on channel condition in dynamic SSK-BPSK (DSB). DSB achieves second-order transmit diversity which is attributed to the statistical independence of sum and difference of independent and identically distributed complex Gaussian channel vectors. In order to study the impact of correlated and nonidentically distributed channel vectors on the transmit diversity of DSB, we consider transmitter correlated nonidentical Rayleigh fading in this work. We derive the bit error rate (BER) expression of DSB and the probability of using BPSK in DSB for the considered fading channel. Using asymptotic BER expressions, we study the diversity order of DSB. Results show that the transmit diversity of DSB is unaffected by the transmitter correlation when channel fading is identical, but nonidentical fading affects transmit diversity. Finally, we corroborate our analyses with simulation results.

Performance Analysis for Spatial Modulation With AF Relaying Over Spatially Correlated Rayleigh Channels

As a low-complexity and energy-efficient multiple input multiple output (MIMO) technique, spatial modulation (SM) has received much interest recently. Combining SM with cooperative relaying, the performance of SM systems can be significantly improved. Considering joint transmit-receive-correlated Rayleigh channels, the error performance and power allocation (PA) of the amplify-and-forward (AF) relaying aided cooperative SM system are presented in this paper. Based on the moment generating function (MGF), the probability density function (PDF) of effective signal-to-noise ratio (SNR) is derived. With this PDF, a closed-form expression of the pairwise error probability (PEP) is obtained. Utilizing the obtained PEP and the union bound of bit error rate (BER), the approximate average BER expression is further derived, which will benefit the performance analysis of the system in spatially correlated channels. According to the asymptotical performance analysis at high SNR, an asymptotic BER expression is also derived. By minimizing this asymptotic BER, a suboptimal power allocation scheme is developed, and the closed-form PA coefficients are attained. Besides, the diversity order of system is also analyzed and derived for the performance evaluation. Numerical results show that our theoretical analysis is valid, and can match the corresponding simulation well. Moreover, the proposed PA scheme can achieve superior BER performance over the conventional equal PA scheme since the powers are well allocated in terms of channel information.

BER Performance of an Optically Pre-amplified FSO System Under Turbulence and Pointing Errors With ASE Noise

The performance of a free space optical (FSO) communication system is significantly affected by various atmospheric turbulence conditions and pointing errors (PEs) apart from the additive noise, which is assumed to be Gaussian. Optical pre-amplifiers are an essential component of FSO systems for improving the receiver sensitivity. However, optical pre-amplification results in amplified spontaneous emission (ASE), which dominates the receiver thermal and shot noises. The square law photodetection process at the receiver in a FSO system necessitates the consideration of chi-square statistics for the decision variable contrary to the Gaussian approximation that is widely used in the literature. In this paper, we evaluate the bit error rate (BER) performance of a FSO system assuming non-return-to-zero on–off keying modulation in the presence of ASE noise under weak, moderate to strong, and very strong atmospheric turbulence regimes and PEs. We also derive asymptotic BER expressions for the considered FSO system for large values of the signal-to-noise ratio in terms of simple elementary functions. Further insight into the system is provided by performing a diversity analysis.

Quantized Feedback-Based Differential Signaling for Free-Space Optical Communication System

We propose a quantized feedback-based differential signaling scheme for $2 \times 1$ free space optical (FSO) multiple-input single-output communication systems. The analytical bit error rate (BER) for the proposed differential signaling scheme is derived and compared with the existing differential signaling scheme for FSO systems. The asymptotic BER of the proposed scheme is also determined and it is shown by using the asymptotic BER expression that the proposed scheme achieves full spatial diversity of two. Moreover, the effect of the feedback error on the performance of the proposed scheme is studied and it is demonstrated by simulation-based and analytical results that the proposed scheme significantly outperforms a previously proposed differential signaling scheme in terms of turbulence fades reduction for tolerable feedback errors.

Power imbalance induced BER performance loss under limited-feedback CoMP techniques

Coordinated multipoint (CoMP) technology utilizes simultaneous transmission/reception from/to different access points, and it is considered as an important feature to exploit and/or mitigate intercell interference in fourth-generation mobile networks. Yet, channel power imbalance at the receiver is experienced in CoMP systems due to, e.g., spatially distributed transmissions. Traditional co-located multi-antenna systems may also experience power imbalance among antenna branches due to inaccurate antenna calibration. This paper presents a bit error rate (BER) analysis and derives asymptotic and approximate BER expressions for some practical CoMP transmission techniques under channel power imbalance. Besides the analytical results, numerical analysis is made to thoroughly capture the performance impact of channel power imbalance on the performance gain of the CoMP methods. The results demonstrate that power imbalance considerably affects BER performance and applying long-term amplitude information with fast phase feedback has insignificant benefit to effectively compensate the detrimental effect of large channel power imbalance when base stations use a single antenna. In this case, exploiting both short-term amplitude and phase information is a very good choice. On the contrary, for a large number of diversity antennas in base stations, using long-term amplitude information with a sparsely quantized phase shows BER performance close to the case where full channel state information is applied.

Relay Selection Based Double-Differential Transmission for Cooperative Networks with Multiple Carrier Frequency Offsets: Model, Analysis, and Optimization

AbstractDue to the distributed nature, cooperative networks are generally subject to multiple carrier frequency offsets (MCFOs), which make the channels time-varying and drastically degrade the system performance. In this paper, to address the MCFOs problem in detect-andforward (DetF) multi-relay cooperative networks, a robust relay selection (RS) based double-differential (DD) transmission scheme, termed RSDDT, is proposed, where the best relay is selected to forward the source's double-differentially modulated signals to the destination with the DetF protocol. The proposed RSDDT scheme can achieve excellent performance over fading channels in the presence of unknown MCFOs. Considering double-differential multiple phase-shift keying (DDMPSK) is applied, we first derive exact expressions for the outage probability and average bit error rate (BER) of the RSDDT scheme. Then, we look into the high signal-to-noise ratio (SNR) regime and present simple and informative asymptotic outage probability and average BER expressions, which reveal that the proposed scheme can achieve full diversity. Moreover, to further improve the BER performance of the RSDDT scheme, we investigate the optimum power allocation strategy among the source and the relay nodes, and simple analytical solutions are obtained. Numerical results are provided to corroborate the derived analytical expressions and it is demonstrated that the proposed optimum power allocation strategy offers substantial BER performance improvement over the equal power allocation strategy.

Cooperative Transmission in Mobile Wireless Sensor Networks with Multiple Carrier Frequency Offsets: A Double-Differential Approach

As a result of the rapidly increasing mobility of sensor nodes, mobile wireless sensor networks (MWSNs) would be subject to multiple carrier frequency offsets (MCFOs), which result in time-varying channels and drastically degrade the network performance. To enhance the performance of such MWSNs, we propose a relay selection (RS) based double-differential (DD) cooperative transmission scheme, termed RSDDCT, in which the best relay sensor node is selected to forward the source sensor node’s signals to the destination sensor node with the detect-and-forward (DetF) protocol. Assuming a Rayleigh fading environment, first, exact closed-form expressions for the outage probability and average bit error rate (BER) of the RSDDCT scheme are derived. Then, simple and informative asymptotic outage probability and average BER expressions at the large signal-to-noise ratio (SNR) regime are presented, which reveal that the RSDDCT scheme can achieve full diversity. Furthermore, the optimum power allocation strategy in terms of minimizing the average BER is investigated, and simple analytical solutions are obtained. Simulation results demonstrate that the proposed RSDDCT scheme can achieve excellent performance over fading channels in the presence of unknown random MCFOs. It is also shown that the proposed optimum power allocation strategy offers substantial average BER performance improvement over the equal power allocation strategy.

Error performance analysis for FSO systems with diversity reception and optical preamplification over gamma–gamma atmospheric turbulence channels

The error performance was investigated of free-space optical (FSO) systems that use optical preamplifiers and diversity reception in turbulent atmosphere with gamma–gamma distribution. More specifically, assuming that the dominant sources of the receiver noises are the background light and amplifier spontaneous emission noises, we develop an exact bit-error-rate (BER) expression for binary pulse position modulation (BPPM) by a moment-generating function (MGF) derivative-based method. To gain more insight, we also derive a closed-form asymptotic BER expression for BPPM at high received optical power, by which we obtain the diversity and coding gains of the considered systems. Our analytical results lead to the observations that the attained diversity gain only changes with the gamma-gamma distribution parameters and the number of spatial modes collected by the receiver, while independent on the number of temporal modes passed by the optical preamplifiers. We verify the analytical results by Monte Carlo simulations.

An analytical approach for performance evaluation of BICM transmission over Nakagami-m fading channels

Bit-interleaved coded modulation (BICM) has established itself as a quasi-standard for bandwidth- and power-efficient wireless communication. In this paper, we present an analytical approach to evaluate the performance of BICM transmission over frequency-flat fading additive white Gaussian noise channels. The statistic of the fading envelope is modeled as Nakagami-m distributed, which spans a wide range of practical multipath fading scenarios through adjustment of the m-parameter. For this setup, we derive approximations for the bit-error rate (BER) and cutoff rate of BICM. Different from previously proposed methods, our analysis is valid for general quadrature amplitude modulation and phase shift keying signal constellations and arbitrary bit-to-symbol mapping rules, and it results in simple closed-form expressions. The key idea is to use well-chosen subsets of signal points to approximate the probability density function of reliability metrics used for decoding. This approximation is accurate for signal-to-noise ratio regions of interest for BICM systems with moderate coding complexity such as, e.g., convolutional coded BICM systems. Based on this approximation we also derive an asymptotic BER expression, which reveals the diversity order and coding gain of BICM. The usefulness of the proposed analytical approach is validated through numerical and simulation results for a number of BICM transmission examples.

Adaptive Lp-norm diversity combining in non-gaussian noise and interference

In this paper, we introduce an adaptive Lp-norm metric for robust coherent, differential, and noncoherent diversity combining in non-Gaussian noise and interference. We consider the general case where all diversity branches may use different combining weights and different Lp-norms. We derive a general closed-form expression for the asymptotic bit error rate (BER) for Lp-norm combining in independent non-identically distributed Ricean fading and non-Gaussian noise and interference with finite moments. The asymptotic BER expression reveals that the diversity gain of Lp-norm combining is independent of the type of noise and the metric parameters. In contrast, the combining gain depends on both the type of noise and the metric parameters. Thus, the asymptotic BER can be minimized by optimizing the Lp-norm metric parameters for the underlying type of noise. For this purpose finite difference stochastic approximation (FDSA) and localized random search (LRS) algorithms are developed. Both adaptive algorithms do not require any a priori knowledge about the underlying noise and are able to track changes in the noise statistics. Simulation results confirm the validity of the derived asymptotic BER expressions, the effectiveness of the proposed adaptive algorithms, and the excellent performance of the proposed adaptive Lp-norm metric compared to other popular metrics.

Asymptotic Performance Analysis of Multihop Relayed Transmissions over Nakagami-m Fading Channels

This letter studies the asymptotic bit error rate (BER) performance of multihop communication systems with amplify-and-forward relaying over Nakagami-m fading channels. Since it is difficult to find the exact probability density function (PDF) of the output signal-to-noise ratio (SNR) at the destination, we resort to the series expansion of this PDF in the neighborhood of zero. Building upon this result, a closed-form expression for the average BER in the high SNR region is derived. Numeric results show that the derived asymptotic BER expression is accurate at medium and high SNR for both independent identically distributed (i.i.d.) and independent non-identically distributed (i.n.i.d.) channels.