Worst-case Pairwise Error Probability Research Articles

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.

Average Worst-Case PEP Optimality of Repetition Coding Among Rate-1 DC-Offset STBCs for MIMO Optical Intensity Channels

An optical wireless intensity-modulation direct-detection multiple-input multiple-output communication system is considered. The performance of $M$ -PAM rate-1 direct current offset space-time block codes is studied in terms of average worst-case pairwise error probability (WC-PEP) in quasi-static channels. It is shown that within this code class, the average WC-PEP is minimized by repetition coding (RC) under both electrical and optical individual power constraints, irrespective of channel statistics. This agrees with previously published results related to ON–OFF keying RC. This is further extended to sum power constraints, where it is shown that spatial beamforming minimizes the average WC-PEP within this code class, which simplifies to RC if the channel matrix has independent and indentically distributed columns and a sum electrical power constraint. Under a sum optical power constraint, this also holds true at high signal-to-noise ratio (SNR), but not at low SNR. Generally, the time dimension of this code class is redundant from an average WC-PEP perspective. Numerical results are provided to support the theoretical findings and to show that the average WC-PEP leads to a good approximation of the actual error probability at high SNR.

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.

Optimal Power Control and Error Performance for Full-Duplex Dual-Hop AF Relaying Under Residual Self-Interference

This paper investigates optimal power allocation schemes at the relay and their impact on the error performance for a full-duplex, dual-hop, amplify-and-forward system using finite constellations and under residual self-interference. The variance of the interference is assumed to be proportional to the λ-th power of the transmitted power. The worst-case pairwise error probability (PEP) is considered as the performance criterion. To find the optimal power allocation, we first derive a closed-form expression for the derivative of the PEP. A simple bisection method is then applied to obtain the optimal scheme. An asymptotically optimal allocation in closed form is also obtained assuming high power regions. From this analysis, it is shown that power allocation leads to a better diversity performance than full power transmission. In addition, the error floor behavior observed in previous works can be eliminated through power allocation. Numerical results are finally presented to illustrate the advantage of the proposed scheme over full power usage.

SLNC for Multi-Source Multi-Relay BICM-OFDM Systems

In this paper, we study the application of bit-interleaved coded modulation (BICM) and orthogonal frequency division multiplexing (OFDM) to reap the benefits of wireless multiuser network coding in practical frequency-selective fading channels. We propose a mapping based symbol level network coding (SLNC) scheme for a cooperative diversity system comprising multiple sources, multiple relays, and one common destination. A simple cooperative maximum-ratio combining scheme is used at the destination and is shown to successfully exploit both the full spatial and the full frequency diversity offered by the channel for arbitrary numbers of sources, arbitrary numbers of relays, and arbitrary linear modulation schemes. To gain analytical insight for system design, we derive a closed-form upper bound for the asymptotic worst-case pairwise error probability (PEP) and obtain the diversity gain of the considered SLNC scheme for BICM-OFDM systems. These analytical results reveal the influence of the various system parameters, such as the number of sources, the free distance of the code, and the frequency diversity of the involved links, on performance. Furthermore, we propose two different relay selection schemes for the considered system: a) bulk selection, i.e., a single best relay is selected to transmit on all sub-carriers, and b) per-subcarrier selection, where a best relay is selected on each sub-carrier. Last but not least, we exploit the derived PEP expression for selecting a subset of sources from the set of active sources when the number of active sources is larger than the number of available orthogonal relay channels. We study the achievable diversity gain for the proposed relay and source subset selection schemes. Numerical results corroborate the derived diversity gain expressions and confirm the performance gains.

Diversity and Delay Analysis of Buffer-Aided BICM-OFDM Relaying

In this paper, we study a cooperative diversity scheme for wireless systems where the relay is equipped with a buffer. We consider practical frequency—selective channels and adopt the combination of bit interleaved coded modulation and orthogonal frequency division multiplexing (BICM—OFDM). We propose a novel link selection protocol for BICM—OFDM systems where the relay either transmits or receives in a given time slot depending on the quality of the links. We derive a closed—form upper bound for the asymptotic worst—case pairwise error probability (PEP) and the diversity gain of the considered buffer—aided relaying scheme for both infinite and finite buffer size. We show that significant diversity gains can be achieved with buffer—aided relaying compared to conventional relaying at the expense of larger packet delays. In fact, for buffers of infinite (or very large) size, the diversity gain is doubled for links with identical frequency diversity, and an even higher diversity gain advantage is possible for links with non—identical frequency diversities. Furthermore, we perform an exact closed—form average delay analysis for buffers of both finite and infinite size which provides important insight into the achieved delay—performance tradeoff. The derived analytical results and performance gains are corroborated by extensive simulation results.

Network Coded Multi-Source Cooperative Communication in BICM-OFDM Networks

In this paper, we study a cooperative diversity scheme for wireless systems employing network coding and the combination of bit-interleaved coded modulation (BICM) and orthogonal frequency division multiplexing (OFDM). The considered system comprises multiple sources, one relay, and one destination. The relay decodes the signal received from all sources and performs network coding before forwarding the signal to the destination. We propose a simple cooperative maximum-ratio combining scheme for the destination which can successfully exploit the full spatial and frequency diversity offered by the channel for arbitrary numbers of sources and arbitrary linear modulation schemes. Furthermore, we propose techniques to reduce the signaling overhead and the decoding complexity at the destination. To gain insight for system design, we derive a closed-form upper bound for the asymptotic worst-case pairwise error probability and the diversity gain of the considered network coded cooperative BICM-OFDM system. These analytical results reveal the influence of various system parameters, including the number of sources, the free distance of the code, and the frequency diversity of the involved links, on performance. Based on the derived analytical results, we develop schemes for optimal relay placement and power allocation. Simulation results corroborate the derived analytical results and confirm the effectiveness of the developed optimization framework.

Analysis and Design of Cooperative BICM-OFDM Systems

In this paper, we propose a novel cooperative diversity scheme for wireless systems employing the combination of bit-interleaved coded modulation (BICM) and orthogonal frequency division multiplexing (OFDM). The proposed scheme utilizes an amplify-and-forward protocol where relays are assigned to multiple groups. Relays in the same group transmit concurrently over disjoint sets of sub-carriers and relays in different groups transmit in different time slots. We derive closed-form expressions for the asymptotic worst-case pairwise error probability and the diversity gain of the proposed cooperative BICM-OFDM scheme. Based on the derived analytical results we develop design guidelines for sub-carrier allocation, relay grouping, and relay selection. Simulation results corroborate the derived analytical results and confirm the effectiveness of the developed optimization framework.

Robust Transmit Processing for BICM-OFDM Systems

In this paper, we introduce robust cyclic space-frequency (CSF) filtering for systems combining bit-interleaved coded modulation (BICM) and orthogonal frequency division multiplexing (OFDM). The proposed robust CSF filtering scheme exploits imperfect channel state information at the transmitter (CSIT) and takes into account the reliability of the CSIT via a Bayesian model. To further improve robustness, we combine CSF filtering with orthogonal space-time block coding in the frequency domain and refer to the resulting scheme as SFC-CSF filtering. We also propose a linear prediction method for improving the quality of the CSIT via post-processing. Based on an upper bound on the worst-case pairwise error probability we formulate an optimization problem for the robust CSF filters which can be solved exactly for certain special cases. For the general case, we obtain an approximate solution by solving a related problem with additional constraints. This approximate solution can be further improved with a gradient algorithm for the original problem. Simulation results confirm the excellent performance of BICM-OFDM with robust CSF and SFC-CSF filtering and the effectiveness of the proposed CSIT post-processing method.

Full-Diversity Codes for MISO Systems Equipped With Linear or ML Detectors

In this paper, a general criterion for space time block codes (STBC) to achieve full-diversity with a linear receiver is proposed for a wireless communication system having multiple transmitter and single receiver antennas (MISO). Particularly, the STBC with Toeplitz structure satisfies this criterion and therefore, enables full-diversity. Further examination of this Toeplitz STBC reveals the following important properties: a) The symbol transmission rate can be made to approach unity. b) Applying the Toeplitz code to any signalling scheme having nonzero distance between the nearest constellation points results in a non-vanishing determinant. In addition, if QAM is used as the signalling scheme, then for independent MISO flat fading channels, the Toeplitz codes is proved to approach the optimal diversity-vs-multiplexing tradeoff with a ZF receiver when the number of channel uses is large. This is, so far, the first non-orthogonal STBC shown to achieve the optimal tradeoff for such a receiver. On the other hand, when ML detection is employed in a MISO system, the Toeplitz STBC achieves the maximum coding gain for independent channels. When the channel fading coefficients are correlated, the inherent transmission matrix in the Toeplitz STBC can be designed to minimize the average worst case pair-wise error probability.

Trace-Orthonormal Full-Diversity Cyclotomic Space–Time Codes

In this paper, we consider the design of full-diversity space-time codes for a coherent multiple-input multiple-output (MIMO) communication system. Starting from both the information theoretic and detection error viewpoints, we first establish that a desirable property for general linear dispersion (LD) codes is to have an interunitary as well as an intraunitary structure-a structure we call trace-orthonormality. By imposing the trace-orthonormal structure on an LD code and applying cyclotomic number theory, we establish, for an arbitrary number of transmitter and receiver antennas, a systematic and simple method to jointly design a unitary cyclotomic matrix, the Diophantine number, and the corresponding constellation for an LD code. As a result, this enables us to construct full-diversity rectangular cyclotomic LD codes with any symbol transmission rate less than or equal to the number of transmitter antennas. In addition, for the case when the number of transmitter antennas is greater than the number of receiver antennas, by taking advantage of the delay, we also arrive at the design of a special trace-orthonormal full-diversity cyclotomic space-time block code which, for the number of transmitter antenna being equal to 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> , can be proved to minimize the worst case pairwise error probability of a maximum-likelihood (ML) detector for a q-ary quadrature amplitude modulation (QAM) signal constellation and, therefore, achieves optimal coding gain. Computer simulations show that these codes have bit-error performance advantages over currently available codes

Tools for Performance Analysis and Design of Space–Time Block Codes

Space-time block codes (STBCs) have attracted recent interest due to their ability to take advantage of both space and time diversity to reliably transmit data over a wireless fading channel. In many cases, their design is based on asymptotically tight performance criteria, such as the worst-case pairwise error probability (PEP) or the union bound. However, these quantities fail to give an accurate performance picture, especially at low signal-to-noise ratio, because the classical union bound is known to be loose in this case. This paper develops tighter performance criteria for STBCs which yield considerably better bounds. First, the union bound is developed as the average of the exact PEPs. By noting that some of the terms in the bound are redundant, a second bound is obtained by expurgation. Since this still yields a loose bound, a tighter bound, denoted as the progressive union bound (PUB), is obtained. Because the PUB cannot be computed in closed form, in its most general case, and to avoid computing a high-dimensional numerical integration, its saddlepoint approximation is developed. In addition to the significant improvement of the PUB analysis over other bounding methods, it is also shown that codes designed to optimize the PUB can perform better than those obtained by the looser criteria

Distance Spectra and Performance Bounds of Space-Time Trellis Codes Over Quasi-Static Fading Channels

This correspondence presents a general approach to upper bounding coded system performance over quasistatic fading channels (QSFC). This approach has the advantage of yielding a closed-form upper bound that converge for all signal-to-noise ratios (SNRs). The proposed approach is used to upper-bound the performance of space-time trellis codes (STTC) over QSFCs. The resulting upper bounds for STTCs are better adapted to the QSFC and present an improvement over worst case pairwise error probability (PEP) analysis used so far. In its second part, this correspondence investigates several ways to reduce the complexity of computing the distance spectrum of STTCs. The combined result obtained from using the new upper bounds and the computed distance spectra are shown to be close to simulated performance for all SNRs.