The use of non-orthogonal signals has several benefits over orthogonal signals in multi-coded communications. We provide a novel, theoretical study of non-orthogonal signaling to expand the applicability of these schemes. Motivated by a class of multi-carrier spread spectrum systems, this paper presents a thorough symbol error rate analysis of the broad class of multi-code signaling methods when they make use of codes which are not necessarily orthogonal. Our analysis is also extended to the case where the code set includes the negative of each code vector, i.e., an extension to biorthogonal signaling. Moreover, it is shown that the symbol error rate results derived in this paper reduce to those available in the literature when the multi-codes are orthogonal or have equal correlation between vectors. Additionally, we show how Monte Carlo integration can be used to evaluate the integrals in the error probability calculation and derive low complexity upper bounds on the error probabilities. We show that by combining these techniques, the error probability can be efficiently computed across the full SNR regime. Finally, we use the upper bound of the error probability to develop some analytical insights about the impacts of non-orthogonality among the code vectors on the symbol error probability.

ABSTRACTMillimeter‐wave (mmWave) communication has wide applications in 5G broadband cellular communication, wireless backhaul connections, wireless personal area networks, vehicular area networks, and mobile ad hoc networks. Major issues in using mmWave communication for multicasting are blockage and penetration losses in signal propagation. To overcome these losses, the proposed system model includes decode‐and‐forward (DF) relay nodes between source and destination nodes in the multicast network. The performance of the relay network is further improved by adopting physical layer network coding (PLNC) at the relay node. In addition, spatial modulation (SM) is adopted as a modulation technique due to its inherent advantage of improved spectral efficiency. In this paper, a multicast relay network using SM and PLNC in mmWave communications is proposed for indoor line‐of‐sight (LoS) environments. The error performance analysis of the proposed system is investigated in terms of deriving analytical expressions by considering orthogonal channel conditions among the nodes. The overall end‐to‐end pairwise error probability (PEP) and symbol error rate (SER) performances of the proposed system are derived from the performances at the links between the source nodes to relay and destination nodes during the first time slot and relay nodes to destination nodes during the second time slot. The advantage of the PLNC‐based system is identified by comparing it with the non‐PLNC‐based multicast system. Analytical results are compared with Monte Carlo simulations.

This paper investigates the error rate performance of simultaneous wireless information and power transfer (SWIPT) systems employing opportunistic amplify-and-forward (OAF) relaying under Rayleigh fading conditions. To support both data forwarding and energy harvesting at relays, a power splitting (PS) mechanism is applied. We derive exact and asymptotic symbol error rate (SER) expressions using moment-generating function (MGF) methods, providing analytical insights into how the power splitting ratio ρ and the quality of source-relay (SR) and relay-destination (RD) links jointly affect system behavior. Additionally, we propose a novel approximation that interprets the SWIPT-OAF configuration as an equivalent non-SWIPT OAF model. This enables tractable performance analysis while preserving key diversity characteristics. The framework is extended to include scenarios with partial channel state information (CSI) and Nth best relay selection, addressing practical concerns such as limited relay availability and imperfect decision-making. Extensive simulations validate the theoretical analysis and demonstrate the robustness of the proposed approach under a wide range of signal-to-noise ratio (SNR) and channel conditions. These findings contribute to a flexible and scalable design strategy for SWIPT-OAF relay systems, making them suitable for deployment in emerging wireless sensor and internet of things (IoT) networks.

This paper studies the symbol error rate (SER) performance of a wireless-powered device-to-device (D2D) communication system operating under a time-switching (TS) protocol in the presence of multiple co-channel interferers (CCI). The considered model involves a battery-less source harvesting energy from multiple dedicated power beacons and transmitting to a multi-antenna destination over quasi-static Rayleigh fading channels. Both selection combining (SC) and maximal ratio combining (MRC) schemes are examined at the destination. The analysis focuses on the impact of key system parameters, including the interference power level, interferer-to-destination distance, energy harvesting efficiency, and modulation type, on the SER performance. The obtained results offer valuable insights into the design of energy-constrained D2D systems operating in spectrum-sharing environments, serving as a reference for future enhancements and practical deployments.

Reconfigurable Intelligent Surface (RIS)-assisted communication has recently attracted significant attention for enhancing wireless performance in challenging environments, making accurate error analysis under practical hardware constraints and imperfect channel state information (CSI) conditions crucial for future multi-antenna systems. This paper presents a unified theoretical framework for the symbol error rate (SER) analysis of RIS-assisted multiple-antenna systems employing orthogonal space–time block codes (OSTBC), considering practical reflection models with amplitude-dependent and quantized phase responses under channel estimation errors (CEEs). By exploiting the Gramian structure of the cascaded channel f, we derive exact moment-generating function (MGF) expressions of the nonzero eigenvalue of ff for small RIS sizes. For large-scale RIS deployments, where closed-form analysis becomes intractable, we employ Saddle Point Approximation (SPA) to approximate the eigenvalue distribution. Using these results, we derive unified SER expressions using exact and SPA-based MGF formulations, applicable to arbitrary RIS sizes, phase configuration, and both identical and non-identical amplitude responses. Extensive Monte Carlo simulations confirm the accuracy of the proposed SER expressions, demonstrating very close agreement for all configurations and under imperfect channel state information (CSI) scenarios. In addition, by applying asymptotic SNR analysis on the SPA-based SER formulation, we mathematically establish that the coding gain is inversely proportional to the Nt-th negative moment of the SPA-approximated probability density function (PDF) corresponding to the nonzero eigenvalue of the cascaded RIS–receiver Gram matrix. This insight motivates a negative moment minimization algorithm that efficiently identifies hardware-constrained RIS phase configurations, achieving near-optimal SER performance with low complexity.

Optical wireless communication (OWC) enables wireless connectivity using ultraviolet bands, infrared or visible. With its advantages features as high bandwidth, low cost, and operation in an unregulated spectrum. Free-space optical (FSO) communication systems are near terrestrial as a communication link between transceivers, the link is line-of-sight and successfully transmitted optical signals. Nevertheless, the optical signals transmissions over the FSO channels bring challenges to the system. To overcome the challenges posed by the FSO channels, the most common technique is to use relay stations, the most recent is the reconfigurable intelligent surfaces (RISs) technique. This study introduces a Weibull distribution model for a free-space optical communication link with RISs assisted, the parameter used to evaluate the performance of the system is the average symbol error rate (ASER). The RISs effect is examined by considering the influence of the transmitter beam waist radius, shape parameter, aperture radius, scale parameter, and signal-to-noise ratio on the ASER.

This paper studies an uplink cell-free massive multiple-input multiple-output (MIMO) system with imperfect channel state information and maximum-ratio combining detectors over Rayleigh fading channel. For such a system, this paper analyzes the performance of symbol error rate (SER), which plays a key role in system reliability. Firstly, the probability density function of the desired signal and the equivalent noise are analytically calculated for <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i>-ary pulse amplitude modulation, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i>-ary quadrature amplitude modulation, and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i>-ary phase-shift keying, respectively. Next, based on these analyses, the novel analytical expressions for the SER of different modulations are derived, which is verified by numerical results. Then, in order to enhance the SER performance of all users in different geographical locations, a geometric programming based power control (GPPC) algorithm is proposed. Finally, numerical results show that compared with four benchmark schemes of the full power algorithm, the fractional power control algorithm, the traditional max-min fairness power control algorithm, and the genetic algorithm, the proposed GPPC algorithm can effectively minimize the maximum SER among all users through collaborative control of the power control coefficient while ensuring lower computational complexity.

A single-phase binary/quadrature phase-shift keying (BPSK/QPSK) demodulator basing on a phase-locked loop (PLL) is described. The demodulator relies on a linear characteristic a rising-edge RESET/SET flip-flop (RSFF) employed as a phase detector. The phase controller takes the average output from the RSFF and performs a sub-ranging/re-scaling operation to provide an input signal to a voltage-controlled oscillator (VCO). The demodulator is truly modular which theoretically can be extended for a multiple-PSK (m-PSK) signal. Symbol-error rate analysis has also been extensively carried out. The proposed BPSK and QPSK demodulators have been fabricated in a 0.18&lt;em&gt;-&lt;/em&gt;mm digital complementary metal–oxide–semiconductor (CMOS) process where they operate from a single supply of 1.8 V. At a carrier frequency of 60 MHz, the BPSK and QPSK demodulators achieved maximum symbol rates of 25 and 12.5 Msymb/s while consuming 0.68 and 0.79 mW, respectively. At these maximum symbol rates, the BPSK and QPSK demodulators deliver symbol-error rates less than 7.9×10&lt;sup&gt;-10&lt;/sup&gt; and 9.8×10&lt;sup&gt;-10&lt;/sup&gt;, respectively where their corresponding energy per bit figures were at 27.2 and 31.7 pJ.

Orthogonal frequency division multiplexing (OFDM) is widely used and works efficiently for the communication, but emerging applications requires OFDM to be flexible to meet sensing requirements. The time-frequency waveform design of OFDM for dual-functional radar-communications (DFRC) is critical to achieve the future communication and sensing requirements. Therefore, we propose a novel method to minimize Cramér-Rao bounds (CRBs) of the delay and Doppler estimation to improve radar performance of an OFDM DFRC system. Although some methods are proposed in the literature to improve the CRBs, these methods either require feedforward signaling or subcarrier reservation. However, it is possible to exploit the constellation extension of quadrature amplitude modulation (QAM) to achieve lower CRBs without these requirements. Therefore, the proposed method provides a transparent communication along with the CRB minimization for conventional OFDM systems. For the evaluation of the proposed method, CRB and symbol error rate (SER) are considered in the simulation results. Furthermore, the theoretical SER analysis of the proposed method is derived to understand the effects of CRB minimization on the communication performance.

Theoretical analysis of orthogonal frequency division multiplexing (OFDM) systems, equipped at the receiver by a non-linear impulsive noise suppressor, is very challenging. Indeed, although an exact closed-form expression for the output signal-to-noise ratio (SNR) of such OFDM systems is available for widely used impulsive noise models, theoretical analysis of the associated symbol error rate (SER) is still open. The analytical SER available in the literature, exploit a Gaussian approximation of the non-linear distortion noise, which however holds true only under specific conditions. Conversely, this work develops an accurate analytical closed-form expression of the distortion noise distribution at the nonlinearity output, as well as its approximation by a Gaussian mixture model (GMM). By using GMMs we unify the SER analysis for communication systems equipped by non-linear impulsive noise suppressors, typically also used in vehicular communications. Closed form expressions for the SER are derived both for non-fading and frequency-selective Rayleigh/Rician fading channels affected by impulsive noise, which is also modeled by a GMM, including Bernoulli-Gaussian (BG), Middleton Class-A, and alpha-stable noises. Theoretical SER performance are compared with simulations, showing very good agreement for all the impulsive noise scenarios and the non-linear suppressors.

Intelligent reflecting surface (IRS)-assisted wireless communication has recently become a promising technology to relax the line-of-sight requirement and also to provide a communication link in uncovered and remote areas. In this work, an alternate implementation of unmanned aerial vehicle (UAV)-based IRS-assisted hybrid free space optics/radio frequency wireless communications is presented. Phase shift errors occur due to the presence of reflecting elements in the IRS and affect the reflected beam. We consider the effect of phase shift error along with atmospheric turbulence, which is modeled as gamma-gamma distribution. Furthermore, a statistical model is considered for pointing errors due to the position and orientation fluctuation of the mounted IRS on the UAV. We derive closed-form expressions of the average symbol error rate and spectral efficiency of the considered system, under the combined effects of atmospheric turbulence, pointing error, and phase shift error. In addition, we present asymptotic analysis of the average symbol error rate of the system in a high SNR region and evaluate the diversity order and coding gain. The derived results are further validated using Monte Carlo simulation.

In this work, we propose adaptable decision regions in the successive interference cancellation (SIC) decoder for the three-user uplink/downlink non-orthogonal multiple access (NOMA) scheme by exploiting the knowledge of the channel gains. We present scenarios in which the modified SIC decoder outperforms the traditional one and is able to replicate the performance achieved by a deep learning-based decoder. We present analytical symbol error rate (SER) expressions for the three users, compare them to simulation results, and discuss possible consequences of our analysis, such as the optimal power allocation. We also employ heatmaps to analyze the joint influence on the SER of some system parameters, such as signal-to-noise ratio and channel gains.

A camera with an image sensor is an important alternative device to a photodiode-based receiver of visible-light communication in outdoor scenarios. The intrinsic color separation capability of a camera qualifies color shift keying (CSK) modulation as an intuitive solution to enhance the achievable data rate. The symbol error rate (SER) of CSK modulation is considerably important to system design and performance evaluation, and has not been extensively investigated for outdoor optical camera communication systems from the viewpoint of camera-based channel and image processing-based demodulation. In this study, a two-level channel model is proposed to characterize CSK transmission in a single pixel and in the entire image. A general framework of SER analysis for arbitrary CSK constellations was proposed by directly calculating the upper bounds from the empirical distribution of the noise light in the CIE 1931 color space. Through numerical simulations, the influence of the image detector on CSK demodulation was evaluated. The results indicated that an accurate target region is important for maintaining the SER, and an enlarged target region is beneficial when the maximum ratio combination and selective combination algorithms are used in pixel combination.

The recently proposed intelligent reflecting surface (IRS) is considered as a promising technology to combat the propagation distance problem in the future communications. However, due to the lack of quantitative analysis and IRS selection scheme from the perspective of symbol error rate (SER) performance, the performance analysis method of asymptotically tight approximation SER is proposed. Firstly, a communication system model aided by an IRS is established. Then, through the reasonable reconstruction between the reflection coefficient and the link coefficient of IRS, the SNR on the forwarding link of IRS satisfies the harmonic mean form. Finally, based on the calculation method of moment generating function (MGF), the asymptotically tight approximation SER formula of the system with simple structure is derived. The simulation results show that, the asymptotically tight approximation SER formula can accurately describe the system SER performance under the low SNR condition. Before the number of IRS reflection unit increases to a certain value, the system SER performance can be effectively improved. Compared with the AF multi-relay system, IRS aided system has the slightly better SER performance only in the case of low SNR. Our work will provide the important theoretical basis for the IRS selection in practical communications.

A composite α-µ/Lognormal fading channel is proposed with several channel performance criteria. This model considers the most effective occurrences in a fading channel, mainly non-linearity, multi-cluster nature of propagation medium, and shadowing effects. The new generation of communication systems is moving towards the use of millimetre waves (mmW). In this type of propagation, large-scale effects of fading channel on the received signal are significant, so in the proposed composite model, the lognormal distribution is considered to model large-scale effects of fading, which is the most accurate distribution to model shadowing. The Gaussian-Hermite quadrature sum is used to approximate the probability distribution function (PDF) of the proposed model. After calculating the statistics, the symbol error rate (SER) and ergodic capacity are computed. The Mellin transform technique is used to calculate the SER expression of different modulation schemes; then, ergodic capacity is computed for a diverse frequency spectrum. Finally, the Monte Carlo method is used to evaluate the analyses.

Analysis on the exact error rate of jointly optimal (JO) maximum likelihood (ML) multiuser detection is limited in the existing literature owing to its high computational complexity. In this paper, the JO ML detection of two quadrature phase-shift keying (QPSK) signals with an arbitrary relative phase offset is investigated. Exact, analytical symbol error rate (SER) and bit error rate (BER) formulas are derived for two users with no less than 3 dB power difference between them. Closed-form union bounds on the user SER and BER with an arbitrary phase offset and an arbitrary user power ratio are also given as a simpler approach to acquire the asymptotic error rate performance. Furthermore, insights on the potentials and properties of the JO ML detector regarding its error rate behaviors for each user are revealed through comparisons with the successive interference cancellation receiver. The accuracies of the obtained formulas are demonstrated through simulation. The results show the error rates for the two-user QPSK case no longer follow the classic knowledge on the error rate behaviors in the single signal detection. In addition, the performance and dB penalty employing different modulation types are also compared and discussed.

The use of unmanned aerial vehicle, as Aerial Base Stations (ABSs) has received high attention in academia and industry for supporting the communication traffic growth. In this article, we focus on obtaining the optimal altitude of an ABS using two criteria - maximum cell coverage area and minimum Symbol Error Rate (SER), implemented on a probabilistic Air-to-Ground (A2G) channel model, developed for low altitude aerial platforms via simulations on a commercial ray tracing software, for various scenarios such as Urban High Rise, Urban and Suburban. We present a system model based on Generalized Frequency Division Multiplexing (GFDM) used for SER analysis in a time-frequency grid compatible with Long Term Evolution (LTE) by implementing parameters for low latency communication of Physical Layer (PHY). Also, we provide the probability distributions of the received power of the ground users and power delay profile at optimal ABS altitude. We demonstrate the variation of optimal altitude with cell area. We further analyze the impact of “Better than Nyquist” pulses on the GFDM system and evaluate SER performance. From the proposed results, significant improvement is demonstrated compared to Nyquist pulses.

The opportunistic relaying scheme design and symbol error rate analysis for PLC networks in smart homes

Symbol error rate analysis of non-orthogonal multiple access systems

In this article, we analyze the symbol error rate (SER) performance of the simultaneous wireless information and power transfer (SWIPT) enabled three-node differential decode-and-forward (DDF) relay networks, which adopt the power splitting (PS) protocol at the relay. The use of non-coherent differential modulation eliminates the need for sending training symbols to estimate the instantaneous channel state information (CSI) at all network nodes, and therefore improves the power efficiency, as compared with the coherent modulation. However, performance analysis results are not yet available for the state-of-the-art detectors such as the maximum-likelihood detector (MLD) and approximate MLD. Existing works rely on the Monte-Carlo simulation method to show the existence of an optimal PS ratio that minimizes the overall SER. In this work, we propose a near-optimal detector with linear complexity with respect to the modulation size. We derive an approximate SER expression and prove that the proposed detector achieves the full diversity order. Based on our expression, the optimal PS ratio can be accurately estimated without requiring any Monte-Carlo simulation. We also extend the proposed detector and its SER analysis for adopting the time switching (TS) protocol at the relay. Simulation results verify the effectiveness of our proposed detector and the accuracy of our SER results in various network scenarios for both PS and TS protocols.