Asymptotic Pairwise Error Probability Research Articles

Union Bound Minimization Approach for Designing Grassmannian Constellations

In this paper, we propose an algorithm for designing unstructured Grassmannian constellations for noncoherent multiple-input multiple-output (MIMO) communications over Rayleigh block-fading channels. Unlike the majority of existing unitary space-time or Grassmannian constellations, which are typically designed to maximize the minimum distance between codewords, in this work we employ the asymptotic pairwise error probability (PEP) union bound (UB) of the constellation as the design criterion. In addition, the proposed criterion allows the design of MIMO Grassmannian constellations specifically optimized for a given number of receiving antennas. A rigorous derivation of the gradient of the asymptotic UB on a Cartesian product of Grassmann manifolds, is the main technical ingredient of the proposed gradient descent algorithm. A simple modification of the proposed cost function, which weighs each pairwise error term in the UB according to the Hamming distance between the binary labels assigned to the respective codewords, allows us to jointly solve the constellation design and the bit labeling problem. Our simulation results show that the constellations designed with the proposed method outperform other structured and unstructured Grassmannian designs in terms of symbol error rate (SER) and bit error rate (BER), for a wide range of scenarios.

Generalized Space Shift Keying for Ambient Backscatter Communication

We consider a generalized space shift keying (GSSK)-enabled multiple-input multiple-output (MIMO) ambient backscatter communication (ABC) system. We propose a scheme to exploit the multiple antenna structure of the system to achieve a lower error-rate performance than conventional ABC systems. Furthermore, we present a novel low complexity energy-based maximum likelihood (EML) GSSK detector, which does not require the perfect knowledge of the ambient source’s signal, unlike the conventional ABC receivers. To gain insights into the performance of the proposed scheme, we derive the exact pairwise error probability (PEP) of the EML detector and further obtain an upper bound on the probability of error. We also derive a simple asymptotic PEP expression as the number of antennas of the reader becomes large. Finally, we derive a simple, asymptotic PEP at the reader when the noise variance approaches zero, i.e., under the large signal-to-interference ratio regime. We validate our analysis through Monte Carlo simulations and show a small performance loss due to the approximations.

An Investigation of the MIMO Space Time Block Code Based Selective Decode and Forward Relaying Network over n-u Fading Channel Conditions

In this paper, we examine the end-to-end average pairwise error probability (PEP) and output probability (OP) performance of the maximum ratio combining (MRC) based selective decode and forward (S-DF) system over an η–µ scattering environment considering additive white Gaussian noise (AWGN). The probability distribution function (PDF) and cumulative distribution function (CDF) expressions have been derived for the received signal-to-noise (SNR) ratio and the moment generating function (MGF) technique is used to derive the novel closed-form (CF) average PEP and OP expressions. The analytical results have been further simplified and are presented in terms of the Lauricella function for coherent complex modulation schemes. The asymptotic PEP expressions are also derived in terms of the Lauricella function, and a convex optimization (CO) framework has been developed for obtaining optimal power allocation (OPA) factors. Through simulations, it is also proven that, depending on the number of multi-path clusters and the modulation scheme used, the optimized power allocation system was essentially independent of the power relation scattered waves from the source node (SN) to the destination node (DN). The graphs show that asymptotic and accurate formulations are closely matched for moderate and high SNR regimes. PEP performance significantly improves with an increase in the value of η for a fixed value of µ. The analytical and simulation curves are in close agreement for medium-to-high SNR values.

Error rate performance of NOMA system with full-duplex cooperative relaying

This paper studies a FD cooperative non-orthogonal multiple access (NOMA) system in terms of pairwise error probability (PEP) while considering imperfect successive interference cancellation (ipSIC). Exact closed-form expressions of the PEP performance for the system users are derived. The PEP expressions are derived under both ideal condition and in the presence of self-interference due to the FD relaying. Moreover, we derive asymptotic PEP expressions to evaluate an effective diversity order for the considered system. Through the derived (asymptotic) expressions and simulation results, we demonstrate that self-interference due to FD relaying has a great influence on the performance of the full-duplex relay.

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.

Performance Analysis and Constellation Design for the Parallel Quadrature Spatial Modulation.

Spatial modulation (SM) is a multiple-input multiple-output (MIMO) technique that achieves a MIMO capacity by conveying information through antenna indices, while keeping the transmitter as simple as that of a single-input system. Quadrature SM (QSM) expands the spatial dimension of the SM into in-phase and quadrature dimensions, which are used to transmit the real and imaginary parts of a signal symbol, respectively. A parallel QSM (PQSM) was recently proposed to achieve more gain in the spectral efficiency. In PQSM, transmit antennas are split into parallel groups, where QSM is performed independently in each group using the same signal symbol. In this paper, we analytically model the asymptotic pairwise error probability of the PQSM. Accordingly, the constellation design for the PQSM is formulated as an optimization problem of the sum of multivariate functions. We provide the proposed constellations for several values of constellation size, number of transmit antennas, and number of receive antennas. The simulation results show that the proposed constellation outperforms the phase-shift keying (PSK) constellation by more than 10 dB and outperforms the quadrature-amplitude modulation (QAM) schemes by approximately 5 dB for large constellations and number of transmit antennas.

SWIPT-Enabled Cooperative NOMA With mth Best Relay Selection

Non-orthogonal multiple access (NOMA) was recently regarded as a potential technique for next generation wireless networks. Recent works on relay selection for cooperative NOMA systems have mainly addressed the best relay selection to forward its received signals to terminal nodes. Nonetheless, in practical scenarios, the best relay may be unavailable due to non-ideal conditions such as scheduling and overload constraints or possibly due to channel feedback delay. Therefore, there is compelling need to consider a more practical solution, in which the best available relay is selected. In this article, we examine the error rate performance for simultaneous wireless information and power transfer (SWIPT)-enabled NOMA, while considering the selection of the m th best available relay. In particular, we present an exact pairwise error probability (PEP) expression to obtain a bit error rate (BER) upper bound. The asymptotic PEP is investigated to evaluate the achievable diversity order for NOMA users. Finally, simulation results are provided to verify the accuracy of the derived PEP expressions and to give more insights into the system performance.

Pairwise Error Probability Performance of SM-MIMO and Spatially Modulated Cooperative Communication Employing SDF Protocol

In this paper, we analyze the performance of spatially modulated (SM) multiple input multiple output (MIMO) system and spatially modulated MIMO cooperative system with multiple relays employing selective decode and forward (SDF) protocol. Here the analysis is done for pre-determined orthogonal channel for SM MIMO and SM MIMO cooperative system using M-PSK modulation and ML detection. Firstly Pairwise error probability (PEP) expression is derived and optimal, sub-optimal detection is done in a spatially modulated MIMO system. Further, the analysis of SDF spatially modulated MIMO cooperative system is extended to single relay and multiple relay configurations. PEP expression is also derived for SDF spatially modulated MIMO cooperative system. Simulation results for the SM MIMO system show optimal detection performs better than suboptimal detection and system performance improves as the number of antenna increases. The simulation results of PEP analysis in the SM MIMO cooperative system include the theoretical PEP, exact PEP and asymptotic PEP and Optimal power allocation (OPA) is also derived for the same. Optimal power allocation for theoretical PEP, exact PEP and asymptotic PEP outperforms the same at equal power allocation.

Double Full Diversity Massive Unitary Space–Time Codes for MIMO Channels

In this paper, we consider a flat fading noncoherent wireless communication system with double transmitter antennas and massive multiple receiver antennas, in which the channel coherence time is divided into four orthogonal time slots, and these are used within a complete transmission cycle. For such a system, we systematically design a family of noncoherent unitary space–time codes. Then, within this family and with the noncoherent maximum likelihood (ML) detector, we propose the design of an optimal unitary space-time code that minimizes the worst-case pairwise error probability (PEP) subject to a constraint on total transmission bits. A closed-form optimal solution is attained by first characterizing the optimal structure for any fixed bits on each parameter space and then finding an optimal bit assignment that further minimizes the worst-case PEP. Also, asymptotic PEP performance analysis on such an optimal code further shows that it enables full receiver-diversity gain when the number of the receiver antennas goes to infinity, with the increasing rate of the coding gain in terms of signal-to-noise ratio (SNR) being the number of the transmitter antennas. In other words, it also provides full diversity gain, i.e., the product of the number of the receiver antennas and the number of the transmitter antennas, when SNR goes to infinity. Therefore, we call such a code double full diversity code. One of the significant advantages of the proposed optimal design for our considered massive multiple-input multiple-output system is that it has no error floor with SNR increasing. Another significant advantage is that it enables a fast ML detector.

Error Probability Analysis of Non-Orthogonal Multiple Access Over Nakagami-<inline-formula> <tex-math notation="LaTeX">$m$ </tex-math> </inline-formula> Fading Channels

Non-orthogonal multiple access (NOMA) is currently considered as a promising technology for the next-generation wireless networks. In this paper, the error rate performance of NOMA systems is investigated over Nakagami-m fading channels, while considering imperfect successive interference cancellation. In particular, this paper focuses on the pairwise error probability (PEP) analysis, where exact PEP expressions are derived to characterize the performance of all users under different fading conditions. The obtained PEP expressions are then used to derive an exact union bound on the bit error rate (BER). Through the derived PEP expressions, the asymptotic PEP analysis is presented to investigate the maximum achievable diversity gain of NOMA users. Moreover, using the derived BER bound, the power allocation problem for all users in NOMA systems is considered under average power and users BER constraints, which allows realizing the full potential of NOMA. Monte Carlo simulation and numerical results are presented to corroborate the derived analytical expressions and give valuable insights into the error rate performance of each user and the achievable diversity gain.

On the diversity analysis of symbol remapping in cooperative BICM–OFDM scheme for frequency selective Rayleigh fading channels

In this paper, diversity analysis of a symbol remapping scheme in cooperative systems is considered. We employ Bit Interleaved Coded Modulation and Orthogonal Frequency Division Multiplexing, which are widely used at wireless networks. In this paper, at the cooperative scheme, relays operate in DF mode and the effect of their erroneous detection will be considered. We present a closed form of the asymptotic worst-case pairwise error probability at the destination for our proposed scheme. It will be shown that this scheme exploits full spatial and frequency diversities. After that, we work on designing a new precoder matrix. Moreover, we propose two new subcarrier allocations for the proposed scheme. All new methods have been suggested to achieve the available diversities with a better performance than current state of the art. Finally, the simulation results reveal the analytical derivations and confirm the obtained diversity gain of the proposed systems.

Distributed Concatenated Recursive Alamouti-Circulant STBC for Two-Way Multi-Relay Networks

A general distributed multi-antenna two-way relaying network with multiple relays is considered in this paper. We first obtain a tight lower bound of pairwise error probability (PEP) of a maximum likelihood detector for the general distributed linear dispersion code for a half-duplex amplify-and-forward two-way relaying network (TWRN) consisting of two sources with each having single antenna and $N$ relays with each having two antennas. Furthermore, by jointly considering signal precoding at the sources and signal processing at the relays, a general distributed concatenated recursive Alamouti-circulant space-time block code is proposed for the considered TWRNs. Our design ensures that the equivalent channel matrices at both source nodes are the so-called the recursive Alamouti-circulant matrices, with each block being a product of the two Alamouti channel matrices. Based on a lower–upper bound strategy and an induction method, asymptotic PEP formula is attained to show that given the optimal angle rotation matrix and the precoding matrix, the code can meet the lower bound of the diversity gain, as well as the maximum coding gain. In addition, the proposed rate one code turns to be effectively decodable.

Exact and asymptotic PEP derivations for various quasi‐orthogonal space–time block codes

Insightful closed-form formulas for the exact pairwise error probability (PEP) of quasi-orthogonal space-time block codes in Rayleigh fading channels are presented. In addition, simple asymptotic expressions are proposed, from which the achievable asymptotic diversity orders are easily determined in the high signal-to-noise ratio regime. Finally, comparing analytical and simulation results, the accuracy of the derived analytical formulas is validated.

Performance analysis of modulation diversity with OSTBC transmission over Nakagami-m fading channels

In this paper performance of modulation diversity with multiple-input multiple-output transmission is studied over flat Nakagami-m fading channels with arbitrary fading parameter m. In the system, orthogonal space-time block coding and maximal ratio combining like combiner are used for transmission and reception, respectively. Exact pairwise error probability expression is derived to observe performance of the system. Moreover, in order to obtain the diversity order of the system, asymptotic pairwise error probability expression is also derived. Optimum rotation angles are analytically obtained for binary and quadrature phase shift keying modulations. Theoretical results are validated by Monte Carlo simulations.

Design of distributed collaborative space‐time block codes for two‐way relaying networks

Utilising the recently-developed unique factorisation of signals and distributed Alamouti coding, a distributed collaborative space-time block code design is presented for a two-way amplify-and-forward (AF) relaying network, where all nodes are equipped with a single antenna. Two asymptotic pairwise error probability (PEP) formulae are firstly derived for both fixed-gain AF and variable-gain AF over Rayleigh channels with the maximum-likelihood detector. Then, subject to the constraints on a fixed transmission bit rate and unity average transmission energy, the optimal constellation combinations with the optimal coefficients are attained by minimising the dominant term of PEP. It is shown that the PEP and the average block error rate of the newly-designed code are superior to those of the conventional distributed Alamouti code while the prior still has linear-decoding complexity.

Novel Distributed Quasi-Orthogonal Space-Time Block Codes for Two-Way Two-Antenna Relay Networks

A half-duplex amplify-and-forward two-way relay network consisting of two single-antenna sources and N relays with each having two antennas is considered in this paper. For such a system, a tight lower bound of pairwise error probability (PEP) of a maximum likelihood (ML) detector for any distributed linear dispersion code with a fixed average amplifier is derived, revealing that diversity gain function cannot decay faster than ln <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</sup> SNR/SNR <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2N</sup> , where rm SNR is signal to noise ratio. Particularly for the network having two relays, a novel distributed quasi-orthogonal space-time block code (QO-STBC) is proposed with two significant advantages. One is that the equivalent channel matrices at both source nodes turn out to be the block-circulant matrices, with each block being a product of the two Alamouti channel matrices. The other is that the equivalent noises at the both source nodes are Gaussian and white. Therefore, like QO-STBC for a multi-antennas system, the proposed code for the relay system still maintains low-decoding complexity for the ML detector. Asymptotic PEP formula is attained to show that the proposed code enables the optimal diversity gain function, i.e., meeting the lower bound of diversity gain, and the optimal coding gain. In addition, a near optimal power allocation that maximizes the received SNR of the worse link is presented and examined through comprehensive computer simulations for various kinds of asymmetric relay channels.

A Peak Power Efficient Cooperative Diversity using Star-QAM with Coherent/Noncoherent Detection

In this paper, we propose a new simple relaying strategy based on bit-interleaved convolutionally coded star-quadrature amplitude modulation (QAM) along with coherent/noncoherent detection. Star-QAM is composed of multiple concentric circles of phase-shift keying (PSK). Exploiting this property, a hard limiter is used to enhance power amplifier (PA) efficiency at the relay. Moreover, we show that the proposed approach retains differential detectability, which results in a significant reduction of receiver complexity with robustness against phase ambiguity. By analyzing our proposed cooperation with coherent/noncoherent detection in terms of asymptotic pairwise error probability (PEP), we show that the full diversity order can be achieved on the condition that the minimum free distance of the convolutional codes is larger than the predetermined value specified by the number of available relays. Furthermore, the effectiveness of the proposed scheme in terms of PA efficiency is confirmed by comparing the statistical distributions of the corresponding instantaneous signal powers. All the theoretical results agree with those obtained by computer simulations.

Performance Analysis of Differential Space–Time Modulation

In this paper, we analyze the performance of the differential space-time modulation (DSTM) in Rayleigh fading channels. By using the moment-generating function (MGF)-based approach, we derive a number of closed-form expressions for the pairwise error probability (PEP) of the DSTM in various fading environments. For spatially independent channels, we derive a closed-form expression for the exact PEP of the DSTM for both slow- and fast-fading cases. In the case of fast-fading channels, we show that an irreducible error floor exists in the PEP and in the block error rate of the DSTM, and derive a closed-form expression for the PEP error floor in terms of the fading rate. For spatially correlated channels, we derive a new closed-form expression for the PEP at asymptotically high signal-to-noise ratios and show that an existing asymptotic PEP expression derived by the residue method can also be obtained by using the MGF-based approach. Finally, we use computer simulations to verify the theoretical results.

Asymptotic Analysis of Space-Time Codes in Generalized Keyhole Fading Channels

The keyhole phenomenon, i.e., non-fully correlated but rank-deficient fading, can lead to significant performance degradations in multiple-input multiple-output (MIMO) systems and has to be carefully taken into account for system design. In this paper, we consider generalized keyhole fading including Rayleigh, Ricean, Nakagami-q, Nakagami-m, and Weibull fading. We derive simple and elegant closed-form expressions for the asymptotic pairwise error probability (PEP) of both single-antenna transmission and space-time coded transmission with two transmit antennas. The obtained PEP expressions provide significant insight into the impact of various system and channel parameters such as the types of fading at the transmitter and receiver side, the numbers of transmit and receive antennas, and the space-time code design on performance. This insight can be exploited for system design and to explain the differences in the performance of space-time codes in non-keyhole and keyhole fading channels. Furthermore, the presented asymptotic PEP expressions can be utilized to obtain tight asymptotic approximations for the bit, symbol, and frame error probabilities of arbitrary, not necessarily orthogonal, space-time codes. Simulation results confirm the accuracy and usefulness of the derived results.

On space-time coding for free-space optical systems

Atmospheric turbulence-induced fading is one of the main impairments affecting free-space optics (FSO) communications. In this paper, we consider FSO systems with intensity modulation and direct detection (IM/DD) and derive a closed- form expression for the asymptotic pairwise error probability of general FSO space-time codes (STCs) for two lasers and an arbitrary number of photodetectors for channels suffering from Gamma-Gamma fading. Furthermore, we provide a simple design criterion for FSO STCs which is used to establish the quasi-optimality of previously proposed FSO repetition codes. We also show that STCs optimized for RF systems achieve full diversity in FSO systems but are suboptimal as far as the coding gain is concerned. Simulation results confirm the analytical findings of this paper.