Error Floor Research Articles

Spatially coupled serially concatenated codes: Analysis and extension

Spatially coupled serially concatenated codes (SC-SCCs) perform well in the error floor region over additive white Gaussian noise (AWGN) channel. In this paper, firstly, we investigate the impact of the component codes on the performances of SC-SCCs. Simulation results show that, by properly selecting the component codes, we can trade off the performances of the SC-SCCs over the error floor region and the waterfall region. Secondly, we introduce a class of spatially coupled multiple-concatenated codes, the recursive-superposition-based SC-SCCs (RS-SC-SCCs). We derive the exact density evolution (DE) equations of the RS-SC-SCCs over the binary erasure channels (BECs). These equations can be used to guide the selection of the component codes and the coupling ratios. Simulation results over the AWGN channels show that the proposed RS-SC-SCCs perform better than the SC-SCC in both the waterfall region and the error floor region.

Bit error rate evaluation of relay-aided cooperative NOMA with energy harvesting under imperfect SIC and CSI

In this paper, we investigate a decode-and-forward relay-assisted cooperative non-orthogonal multiple access (NOMA) scheme, where the relay implements a time-switching (TS) based energy harvesting (EH). The impacts of the imperfect channel state information (CSI), inter-cell interference (ICI) and imperfect successive interference cancellation (SIC) are taken into account. We derive the end-to-end bit error rate (BER) expressions under imperfect CSI and ICI for both users. The effect of the EH parameters under the imperfect CSI on users’ BER performance is also examined. Furthermore, we discuss the impact of ICI on BER performance. Computer simulations are used for numerical analysis validation. The results reveal that the CSI deterioration reduces the SIC performance in each node despite the increase in EH parameters and causes an error floor at the higher signal-to-noise ratio (SNR). Furthermore, the BER performance of the users increases by increasing the EH parameters. Also, the ICI affects the SIC and degrades the BER of users.

UP-GDBF: A 19.3 Gbps Error Floor Free 4KB LDPC Decoder for NAND Flash Applications

An error floor phenomenon, decoding performance, and throughput are three major concerns for LDPC decoders in NAND Flash applications. With a penalty method and an active iteration mechanism, we present a Unified Penalty Gradient Descent Bit Flipping (UP-GDBF) decoding algorithm, which not only possesses error-floor free property but also improves convergence speed in decoding performance. To fulfill the high-throughput requirement while maintaining reliable error correction capability, we propose an energy-based backtracking scheme to reduce 40% latency with a negligible 0.8% area overhead. Implemented in TSMC 16nm process, the proposed 4KB LDPC decoder can achieve a throughput of 19.3 Gbps with 0.120 mm2 area to satisfy ONFI 5.0 throughput requirement. Compared to existing approaches, our decoder architecture provides superior data rate and decoding performance in both 1KB and 4KB LDPC codes.

Joint Phase and Timing Estimation With 1-Bit Quantization and Oversampling

Digital receivers based on 1-bit quantization and oversampling w.r.t. the transmit signal bandwidth promise lower energy consumption. However, since 1-bit quantization is a highly non-linear operation, standard off the shelf receiver algorithms cannot be applied. In this paper we consider an unknown phase rotation and timing offset and a fully digital receiver. To reduce the non-linear behavior introduced by 1-bit quantization, we assume that the receiver applies uniform phase and sample dithering, which can be implemented by sampling at an irrational normalized intermediate frequency and with an irrational oversampling factor, respectively. Based on the least squares objective function we derive a typical digital matched filter receiver with a data- and timing-aided phase estimator and square time recovery based timing estimation. Our main contribution is to show that both estimators are consistent under very general assumptions, e.g., arbitrary colored noise and stationary ergodic transmit symbols. Performance evaluations are done via simulations and are compared against a numerically computable upper bound of the Cramér–Rao lower bound. For low signal-to-noise ratio the estimators perform well but for high signal-to-noise ratio they run into an error floor. The performance loss of the phase estimator due to decision-directed operation or estimated timing information is marginal.

Polarization Combining and Equalization in 5G Mobile-to-Mobile Systems

This article develops maximum likelihood (ML) detection for post-FFT processing of dual-polarized antenna outputs with a cyclic-prefix OFDM (CP-OFDM) waveform for a frequency selective multipath fading in a 5G mobile-to-mobile setting. The suggested maximum likelihood detector (MLD) comprises a combiner applied to the channel matched filter outputs (designated MLC) followed by a decision rule based on correlation and signal energy. When MLC is coupled with a frequency-domain equalizer, this structure is called MLC+FDE. The designed MLD and MLC+FDE are compared to the traditional combining techniques: maximum ratio combining, equal gain combining, and selection combining. Computer simulations performed over a stochastic channel model with polarization state information. The simulation results show that the difference between MLD and maximum ratio combining (the best performing method), MLC+FDE (almost the best performing method) and selection diversity with frequency-domain equalization (the worst performing method) is 2 dB. This diversity improvement is limited by the correlation between the two channels that characterize the two polarization states. This correlation makes it difficult to achieve reliable transmission through combining alone; some form of error control coding is also needed. When channel estimates are used in place of perfect channel knowledge, an error floor is observed.

Machine learning of irreducible error floor in the space-time trellis code

Machine learning of irreducible error floor in the space-time trellis code

Zipper Codes

Zipper codes are a framework for describing spatially-coupled product-like codes. Many well-known codes, such as staircase codes and braided block codes, are subsumed into this framework. New types of codes such as tiled diagonal and delayed diagonal zipper codes are introduced along with their software simulation results. Stall patterns that can arise in iterative decoding are analyzed, giving a means of error floor estimation.

Optimized Power Control Design for Over-the-Air Federated Edge Learning

<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Over-the-air federated edge learning</i> (Air-FEEL) has emerged as a communication-efficient solution to enable distributed machine learning over edge devices by using their data locally to preserve the privacy. By exploiting the waveform superposition property of wireless channels, Air-FEEL allows the “one-shot” over-the-air aggregation of gradient-updates to enhance the communication efficiency, but at the cost of a compromised learning performance due to the aggregation errors caused by channel fading and noise. This paper investigates the transmission power control to combat against such aggregation errors in Air-FEEL. Different from conventional power control designs (e.g., to minimize the individual <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">mean squared error</i> (MSE) of the over-the-air aggregation at each round), we consider a new power control design aiming at directly maximizing the convergence speed. Towards this end, we first analyze the convergence behavior of Air-FEEL (in terms of the optimality gap) subject to aggregation errors at different communication rounds. It is revealed that if the aggregation estimates are unbiased, then the training algorithm would converge exactly to the optimal point with mild conditions; while if they are biased, then the algorithm would converge with an error floor determined by the accumulated estimate bias over communication rounds. Next, building upon the convergence results, we optimize the power control to directly minimize the derived optimality gaps under the cases without and with unbiased aggregation constraints, subject to a set of average and maximum power constraints at individual edge devices. We transform both problems into convex forms, and obtain their structured optimal solutions, both appearing in a form of regularized channel inversion, by using the Lagrangian duality method. Finally, numerical results show that the proposed power control policies achieve significantly faster convergence for Air-FEEL, as compared with benchmark policies with fixed power transmission or conventional MSE minimization.

Bit Error Floor Reduction for MQAM Signals in the Presence of Phase Error

Bit Error Floor Reduction for MQAM Signals in the Presence of Phase Error

Performance of Non-Line of Sight Underwater Optical Wireless Communication Links with Spatial Diversity

Line-of-sight (LOS) underwater optical wireless communication (UOWC) transmission may suffer blocking and are not always possible due to obstructions from sea creatures, bubbles, large suspended particles and features of the seabed, especially in coastal and turbid water environments. Thus, we present the performance of a spatially diverse non-line-of-sight (NLOS) UOWC system employing continuous phase modulation (CPM), which is shown to offer sensitivity benefits of several dBs over on–off keying (OOK) without coherent reception. We obtain the channel impulse response (CIR) by using Monte Carlo simulation, including absorption and multiple scattering. Turbulence is included by conditioning the CIR on log-normal statistics. To mitigate the resultant fading, we exploit spatial diversity with equal gain combining at the receiver side. Photon counting at the receiver is employed to accommodate shot noise. We compare the saddlepoint and Gaussian approximations for bit error rate (BER) calculations, using the latter for later calculations as it delivers excellent results and is simpler. Our results show that spatial diversity offers performance improvements, for example an 8 dB sensitivity gain at 10^-9 BER using 1 Gbps 3×1 multiple-input single-output (MISO) transmission over a 20 m link with 0.16 log-amplitude variance. We determine using an upper bound that Intersymbol Interference (ISI) has a significant impact at high bit rates, producing error floors for multiple-output arrangements.

Machine learning of irreducible error floor in the space-time trellis code

The phenomenon of irreducible error floor in the space-time trellis code (STTC) is not fully understood. This comes from the fact that the connection between the trellis structure of the generator matrix G and the instigation of an irreducible error floor is uncertain. Given this difficulty, the present study attempts to gain a better insight into the ordeal via a data-driven approach. The classification and regression trees (CART) machine learning model is employed to predict the occurrence of the irreducible error floor from the trellis structure. Further analysis of the combinatorial characterisation of the trellis structure unveils a series of dominant patterns that consistently instigate the irreducible error floor. Furthermore, simulation also reveals that the codewords within the 'initial state' of the trellis structure are primal in the occurrence of the irreducible error floor. CART can achieve approximately 0.92 accuracy in predicting the irreducible error floor, with an average prediction time of 0.3833 μs.

Construction of Protograph-Based Partially Doped Generalized LDPC Codes

In this paper, we propose a new code design technique, called partial doping, for protograph-based generalized low-density parity-check (GLDPC) codes. While the conventional construction method of protograph-based GLDPC codes is to replace some single parity-check (SPC) nodes with generalized constraint (GC) nodes applying to multiple variable nodes (VNs) that are connected in the protograph, the proposed technique can select any VNs in the protograph to be protected by GC nodes. In other words, the partial doping technique facilitates finer tuning of doping, which in turn enables a sophisticated code optimization with higher degree of freedom. We construct the proposed partially doped GLDPC (PD-GLDPC) codes using the partial doping technique and optimize the PD-GLDPC codes by the protograph extrinsic information transfer (PEXIT) analysis. In addition, we propose a condition guaranteeing the linear minimum distance growth of the PD-GLDPC codes and use the condition for the optimization. Experimental results show that the optimized PD-GLDPC codes outperform the conventional GLDPC codes and have competitive performance compared to the state-of-the-art protograph-based LDPC codes without the error floor phenomenon over the binary erasure channel (BEC).

Joint Design of Improved Spectrum and Energy Efficiency With Backscatter NOMA for IoT

To develop emerging transmission techniques for potential applications of Internet of Things (IoT), the system performance analysis of a cognitive radio (CR)-enabled ambient backscatter (AmBC) system will be studied in this paper with functionality of non-orthogonal multiple access (NOMA). In the proposed scheme, a base station communicates with two destinations via a designated backscatter device. It is assumed that the relay node is fitted with two different interfaces and can simultaneously collect/decode and backscatter the received source signals. Such transmission mechanism benefits to design various applications in IoT as well as wireless systems with improved performance. To exhibit system performance metrics, the outage probability and the ergodic capacity of the recipient nodes are derived analytically. Furthermore, it is shown that employing AmBC NOMA together with CR for secondary communication can significantly improve overall network performance in terms of the achievable throughput in delay-limited and delay-tolerant modes and outage probability. Numerical results show that: <i>1) The proposed system can improve the spectrum efficiency by employing both CR and NOMA techniques; 2) Compared with the orthogonal multiple access (OMA)-aided AmBC systems, the considered CR AmBC system relying on NOMA can obtain better reliability in the whole range of SNR; 3) There are error floors for the outage probability in the high SNR regime due to required target rates; 4) There exists a trade-off between system performance of IoT devices and power allocation coefficients associated with NOMA; 5) We find energy efficiency factor as evident of further improvement in such system</i>.

Comparative Analytical Study of SCMA Detection Methods for PA Nonlinearity Mitigation.

Non-orthogonal multiple access (NOMA) has emerged as a promising technology that allows for multiplexing several users over limited time-frequency resources. Among existing NOMA methods, sparse code multiple access (SCMA) is especially attractive; not only for its coding gain using suitable codebook design methodologies, but also for the guarantee of optimal detection using message passing algorithm (MPA). Despite SCMA’s benefits, the bit error rate (BER) performance of SCMA systems is known to degrade due to nonlinear power amplifiers at the transmitter. To mitigate this degradation, two types of detectors have recently emerged, namely, the Bussgang-based approaches and the reproducing kernel Hilbert space (RKHS)-based approaches. This paper presents analytical results on the error-floor of the Bussgang-based MPA, and compares it with a universally optimal RKHS-based MPA using random Fourier features (RFF). Although the Bussgang-based MPA is computationally simpler, it attains a higher BER floor compared to its RKHS-based counterpart. This error floor and the BER’s performance gap are quantified analytically and validated via computer simulations.

Design of Code Pair for Protograph-LDPC Codes-Based JSCC System With Joint Shuffled Scheduling Decoding Algorithm

Although there are many studies on code optimization of the joint source-channel coding (JSCC) system based on double protograph low-density parity-check codes with the joint belief propagation (JBP) algorithm, but it is still unknown whether the source code and channel code (as a code pair) can perform well when the joint shuffled scheduling decoding (JSSD) algorithm is adopted. In this letter, two decoding threshold analysis algorithms, including joint shuffled protograph extrinsic information transfer (PEXIT) and source shuffled PEXIT algorithm, are proposed to calculate joint/source decoding thresholds for this system with the JSSD algorithm. With the proposed algorithms, it is found that the optimized code pairs for this system with the JBP algorithm may not perform well with the JSSD algorithm, implying that code pair with the JSSD algorithm needs to be redesigned. Then, a two-stage optimized framework is proposed to design the code pair for this system with the JSSD algorithm. Simulations and decoding threshold analysis both show that the proposed code pair for this system with the JSSD algorithm can obtain lower error floor and better waterfall performance than the existing code pairs.

Optical OFDM Error Floor Estimation by Means of OTDR Enhanced by Front-End Optical Preamplifier.

Optical time-domain reflectometer (OTDR) enables simple identification and localization of a plethora of refractive and reflective events on a fiber link, including splices, connectors and breaks, and measuring insertion/return loss. Specifically, large enough OTDR dynamic range (DR) and thus high signal-to-noise-ratio (SNR) enable clear far-end visibility of longer fibers. We point out here that, under such conditions, the optical bit-error-rate (BER) floor is dominantly determined by reflective events that introduce significant return loss. This complements the OTDR legacy tests by appropriate optical BER floor estimation in the field. As high SNR implies inter-symbol interference as dominating error generating mechanism, we could apply the classical time-dispersion channel model for the optical BER floor determined by the root-mean-square (rms) delay spread of the actual fiber channel power-delay profile. However, as the high-SNR condition is not always fulfilled mostly due to insufficient DR, we propose here inserting a low-noise optical preamplifier as the OTDR front-end to reduce noise floor and amplify the backscattered signal. In order to verify the model for the exemplar test situation, we measured BER on the same fiber link to find very good matching between the measured BER floor values and the ones predicted from the OTDR trace.

On Minimizing Symbol Error Rate Over Fading Channels With Low-Resolution Quantization

We analyze the symbol error probability (SEP) of $M$-ary pulse amplitude modulation ($M$-PAM) receivers equipped with optimal low-resolution quantizers. We first show that the optimum detector can be reduced to a simple decision rule. Using this simplification, an exact SEP expression for quantized $M$-PAM receivers is obtained when Nakagami-$m$ fading channel is considered. The derived expression enables the optimization of the quantizer and/or constellation under the minimum SEP criterion. Our analysis of optimal quantization for equidistant $M$-PAM receiver reveals the existence of error floor which decays at a double exponential rate with increasing quantization bits, $b$. Moreover, by also allowing the transmitter to optimize the constellation based on the statistics of the fading channel, we prove that the error floor can be eliminated but at a lower decay exponent than the unquantized case. Characterization of this decay exponent is provided in this paper. We also expose the outage performance limitations of SEP-optimal uniform quantizers. To be more precise, its decay exponent does not improve with $b$. Lastly, we demonstrate that the decay exponent of a quantized receiver can be complemented by receive antenna diversity techniques.

Multi-User Joint Maximum-Likelihood Detection in Uplink NOMA-IoT Networks: Removing the Error Floor

The Internet of Things (IoT) framework requires a massive number of connection thus demanding spectral efficient solutions such as Non-Orthogonal Multiple Access (NOMA). However, the main drawback of NOMA with successive interference canceler (SIC)-based detectors is the error floor in the uplink. In this letter, a reliable multi-user detection in uplink IoT NOMA is guaranteed by a Joint Maximum-Likelihood (JML) detector (i.e., optimum detection algorithm). We derive a closed-form upper bound of bit error rate (BER) of JML over Rayleigh fading channels for arbitrary number of IoT devices and an adaptive M-ary phase shift keying (M-PSK). Based on the extensive simulations, the derived expressions are validated and it is revealed that the JML improves the error performance in uplink NOMA and removes the error floor. Furthermore, regardless of the number of the IoT devices and modulation order, a full diversity order (i.e., number of receiving antennas) is guaranteed for each device.

A New QoS-Guarantee Strategy for NOMA Assisted Semi-Grant-Free Transmission

Semi-grant-free (SGF) transmission has recently received significant attention due to its capability to accommodate massive connectivity and reduce access delay by admitting grant-free users to channels which would otherwise be solely occupied by grant-based users. In this paper, a new SGF transmission scheme that exploits the flexibility in choosing the decoding order in non-orthogonal multiple access (NOMA) is proposed. Compared to existing SGF schemes, this new scheme can ensure that admitting the grant-free users is completely transparent to the grant-based users, i.e., the grant-based users’ quality-of-service experience is guaranteed to be the same as for orthogonal multiple access. In addition, compared to existing SGF schemes, the proposed SGF scheme can significantly improve the robustness of the grant-free users’ transmissions and effectively avoid outage probability error floors. To facilitate the performance evaluation of the proposed SGF transmission scheme, an exact expression for the outage probability is obtained and an asymptotic analysis is conducted to show that the achievable multi-user diversity gain is proportional to the number of participating grant-free users. Computer simulation results demonstrate the performance of the proposed SGF transmission scheme and verify the accuracy of the developed analytical results.

Performance Analysis of OTFS Under In-Phase and Quadrature Imbalance at Transmitter

The influence of in-phase and quadrature (IQ) imbalance on the performance of orthogonal time frequency space (OTFS) transmission is investigated in this work. OTFS is a recent two-dimensional transmission scheme with promising robustness against the time and frequency selectivity of the time-varying wireless channel, and IQ impairment arises from inevitable imperfections between in-phase and quadrature branches in the transceiver design. In this paper, the mathematical model is presented to illustrate the physical effect of the IQ mismatched transmitter in OTFS, which results in a new type of interference named mirror-Doppler interference (MDI), occurring only along the Doppler axis. Moreover, a closed-form expression for the average bit error rate (BER) of OTFS with IQ imbalanced transmitter is derived taking into account M-ary symbol with a constant envelope and maximum likelihood (ML) detection principle. Importantly, our findings prove that IQ imbalanced transmitter causes a significant reduction in the diversity gain of OTFS transmission, but does not lead to an error floor in the BER performance as in one-dimensional current communication systems. Monte-Carlo simulations are provided to validate the derived mathematical formulas for OTFS transmission with constant envelope M-ary symbols under different system configurations.