Exact Pairwise Error Probability Research Articles

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.

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.

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.

Layered spatial modulation

In this paper, we propose a layered spatial modulation (L-SM) scheme that combines a multi-layered coding scheme with the spatial modulation. At each antenna selection, the number of active transmit antennas equals the number of layers. The proposed L-SM scheme maximizes the diversity order and coding gain at each antenna selection. The exact pairwise and upper bound bit error probabilities are derived. Several examples of different spatial modulation schemes and the proposed L-SM scheme are simulated with the same number of transmit antennas and same spectral efficiency. As expected, the proposed L-SM scheme satisfies significant increase in the performance compared to all existing spatial modulation schemes.

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

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

An improved scheme based on log-likelihood-ratio for lattice reduction-aided MIMO detection

Lattice reduction (LR)-aided detectors have been shown great potentials in wireless communications for their low complexity and low bit-error-rate (BER) performance. The LR algorithms use the unimodular transformation to improve the orthogonality of the channel matrix. However, the LR algorithms only utilize the channel state information (CSI) and do not take account for the received signal, which is also important information in enhancing the performance of the detectors. In this paper, we make a readjustment of the received signal in the LR domain and propose a new scheme which is based on the log-likelihood-ratio (LLR) criterion to improve the LR-aided detectors. The motivation of using the LLR criterion is that it utilizes both the received signal and the CSI, so that it can provide exact pairwise error probabilities (PEPs) of the symbols. Then, in the proposed scheme, we design the LLR-based transformation algorithm (TA) which uses the unimodular transformation to minimize the PEPs of the symbols by the LLR criterion. Note that the PEPs of the symbols affect the error propagation in the vertical Bell Laboratories Layered Space-Time (VBLAST) detector, and decreasing the PEPs can reduce the error propagation in the VBLAST detectors; thus, our LLR-based TA-aided VBLAST detectors will exhibit better BER performance than the previous LR-aided VBLAST detectors. Both the BER performance and the computational complexity are demonstrated through the simulation results.

Error rate analysis of quasi‐orthogonal space–time block codes with power scaling

The exact pairwise error probability (PEP) of quasi-orthogonal space–time block codes (QSTBCs) employing the power scaling technique in quasi-static Rayleigh fading channels is derived. By utilising the derived PEP formula, the tight union upper bound on the bit error rate (BER) for the QSTBC with power scaling (PS-QSTBC) is evaluated and compared with the Monte Carlo simulation results to verify the analytical results that full diversity rate-1 transmission can be achieved. Furthermore, considering the closed-loop system, the optimal value of the power scaling factor is also investigated in terms of minimising the union bound on the BER.

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.

A Distributed Differential Space-Time Coding Scheme With Analog Network Coding in Two-Way Relay Networks

In this correspondence, we consider general two-way relay networks (TWRNs) with two source and <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> relay nodes. A distributed differential space time coding with analog network coding (DDSTC-ANC) scheme is proposed. A simple blind estimation and a differential signal detector are developed to recover the desired signal at each source. The pairwise error probability (PEP) and block error rate (BLER) of the DDSTC-ANC scheme are analyzed. Exact and simplified PEP expressions are derived. To improve the system performance, the optimum power allocation (OPA) between the source and relay nodes is determined based on the simplified PEP expression. The analytical results are verified through simulations.

Multi-cell cooperative transmission based on unitary space-time modulation

As a non-coherent transmission scheme that does not require channel state information at both transmitters and receivers, unitary space-time modulation is a promising technique that can be applied in high mobility scenario where the fading coefficients are changing too fast to be tracked and estimated. This article proposes a multi-cell cooperative transmission scheme based on unitary space-time modulation. Each cooperative base-station transmits an individual unitary signal from the common constellation set to the mobile unit which is located at the cell edge and suffers from severe inter-cell interference. Compared with traditional unitary space-time modulation, cooperation among multi-cells not only eliminates the inter-cell co-channel interference but also increases the transmission rate by expanding the constellation size. Performance of error probability is analyzed for the proposed scenario with maximum-likelihood decoding, in which the exact pairwise error probability is derived. Additionally, constellation optimization for cooperative transmission is also discussed to achieve the balance between transmission efficiency and reliability. Simulation results are provided to confirm the effectiveness of the proposed scheme in both block-fading channels and fast-fading channels.

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.

An Exact Close-form PEP and a new PEP for Space-Time Codes in Rayleigh Fading Channels

A close-form expression for the exact Pair-wise Error Probability (PEP) of Space-Time (S-T) codes in Rayleigh fading channel is derived using the general and close-form solution for the probability-density function (PDF) of a sum of independent exponential distributed random variables. The expression requires evaluating the coefficients for partial fraction expansion, so an easy analytical way is proposed for doing this. The exact PEP is subsequently used to develop a simple PEP using the upper bound. Both PEPs are used in the Union bound for error rate evaluation. Numerical calculations and Monte Carlo computer simulation are used to study the accuracies of these Union bounds for error rate evaluation of a rotation-based diagonal S-T code (D code) in Rayleigh fading channels. Four other PEPs based on different bounds, i.e., the Chernoff bound, the asymptotic bound, the tight asymptotic bound, and the Eigen-Geometric-Mean (EGM) bound, are also studied for comparison. Results show that our derived close-form PEP is an exact PEP and our proposed PEP is a very tight bound to the exact PEP.

Space-time trellis codes over rapid rayleigh fading channels with channel estimation-part ii: performance analysis and code design for non-identical channels

We consider the maximum likelihood (ML) receiver design, performance analysis and code design for space-time trellis codes (STTC) over non-identical, rapid fading channels with imperfect channel state information (CSI). The exact pairwise error probability (PEP) and PEP bounds for the ML receiver are obtained. A new code design criterion exploiting the statistical information of the channel estimates is proposed, which can minimize the performance loss caused by channel estimation error. New codes are obtained via an iterative search algorithm with reduced complexity. Under actual channel estimation conditions, our codes perform better than the existing codes in the literature which are designed on the assumption of identical channels, and perfect CSI at the receiver. More performance gain can be achieved by our codes when the degree of imbalance among the links is higher.

MAP decoding for multi-antenna systems with non-uniform sources: exact pairwise error probability and applications

We study the maximum a posteriori (MAP) decoding of memoryless non-uniform sources over multiple-antenna channels. Our model is general enough to include space-time coding, BLAST architectures, and single-transmit multi-receive antenna systems which employ any type of channel coding. We derive a closed-form expression for the codeword pairwise error probability (PEP) of general multi-antenna codes using moment generating function and Laplace transform arguments. We then consider space-time orthogonal block (STOB) coding and prove that, similar to the maximum likelihood (ML) decoding case, detection of symbols is decoupled in MAP decoding. We also derive the symbol PEP in closed-form for STOB codes. We apply these results in several scenarios. First, we design a binary antipodal signaling scheme which minimizes the system bit error rate (BER) under STOB coding. At a BER of 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-6</sup> , this constellation has a channel signal-to-noise ratio (CSNR) gain of 4.7 dB over conventional BPSK signaling for a binary nonuniform source with p <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> Delta <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">=</sub> P(0) = 0.9. We next design space-time linear dispersion (LD) codes which are optimized for the source distribution under the criterion of minimizing the union upper bound on the frame error rate (FER). Two codes are given here: one outperforms V-BLAST by 3.5 dB and Alamouti's code by 12.3 dB at an FER of 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-2</sup> for a binary source with p <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> = 0.9, and the other outperforms V-BLAST by 4.2 dB at an FER of 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-3</sup> for a uniform source. These codes also outperform the LD codes of constructed under a different criteria. Finally, the problem of bit-to-signal mapping is studied. It is shown that for a binary source with p <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> = 0.9, 64-QAM signaling, and SER = 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-3</sup> , a gain of 3.7 dB can be achieved using a better-than-Gray mapping. For a system with one transmit and two receive antennas that uses trellis coding with 16-QAM signaling, a 1.8 dB gain over quasi-Gray mapping and ML decoding is observed when MAP decoding is used for binary sources with p <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> = 0.9.

Exact Pairwise Error Probability for Block-Fading MIMO OFDM Systems

In this letter, we derive a closed-form solution of the exact pairwise error probability (PEP) for Rayleigh fading orthogonal frequency-division multiplexing (OFDM) channels with multiple transmit and receive antennas. We begin with the case of frequency-selective quasi-static fading OFDM channel model. The analysis is then generalized to the frequency-selective block fading OFDM channel model. This is a suitable model for cooperative space-time coded OFDM systems, where, due to the spatial separation between the cooperating users, the signal-to-noise ratio (SNR), the number of antennas, and the number of nonzero channel taps during each user's transmission (hence, in each block) may be different. We provide numerical examples to illustrate the usefulness of the derived exact PEP expressions.

On the error probability of quasi‐orthogonal space–time block codes

AbstractIn this paper, we derive the exact pairwise error probabilities (PEPs) of various quasi‐orthogonal space–time block codes (QO‐STBCs) using the moment generating function. By classifying the exact PEPs of QO‐STBCs into three types, we derive the closed‐form expression for each type of PEP. Based on these closed‐form expressions, we obtain the union bounds on the symbol error probability and bit error probability for QPSK modulation. Through simulation, it is shown that these union bounds are quite tight. Copyright © 2008 John Wiley &amp; Sons, Ltd.

Performance evaluation of high rate space–time trellis-coded modulation using Gauss–Chebyshev quadrature technique

The performance analysis of high rate space-time trellis-coded modulation (HR-STTCM) using the Gauss-Chebyshev quadrature technique is presented. HR-STTCM is an example of space-time codes that combine the idea used in trellis coded modulation (TCM) design that is signal set expansion and set partitioning into its construction. HR-STTCM construction is based on the concatenation of an outer TCM encoder and inner space-time block code. This paper evaluates the exact pairwise error probability of HR- STTCM based on the Gauss-Chebyshev quadrature formula. Comparison of numerical and simulation results shows that the proposed method is accurate. The method used is shown to be computationally simpler than those in the literature.

Error Probability Bounds for Bit-Interleaved Space–Time Trellis Coding Over Block-Fading Channels

In this correspondence, we investigate the error probability bounds of a bit-interleaved space-time trellis coding (BISTTC) scheme, which concatenates bit-interleaved coded modulation (BICM) with a space-time trellis code (STTC). We focus on general block-fading channels, wherein each data packet or frame spans a number of independent fading blocks. BICM applied to such channel environments can effectively exploit both time and frequency selectivity, while a STTC maximizes the spacial diversity order. The exact pairwise error probability (PEP) and weight enumeration function (WEF) of BISTTC are used to evaluate the error bound. Due to the concatenation of an outer error correction code (ECC) and a STTC, the overall WEF of BISTTC is obtained by combining the WEF of the outer code with that of the STTC through an uniform interleaver. The main challenge here is to compute the WEF of a STTC for block-fading channels with reasonable complexity. We rely on constructing a composite state transition matrix based on a number of single-step virtual trellises, each corresponding to an independent fading block within a frame. We discuss how this approach reduces storage and computational requirements in the bound analysis, compared to the existing method of obtaining the state transition matrix through accumulation of single-step transition matrices. The derived bound is applicable to both spatially uncorrelated and correlated channels as well as to both flat and frequency-selective block-fading channels. The bound is shown to provide a reasonably close estimate of the simulated performance based on the turbo equalizer-like iterative processing of soft information between the STTC decoder and the ECC decoder.