Maximum A Posteriori Detection Research Articles

Temporal Convolutional Network-Based Signal Detection for Magnetotactic Bacteria Communication System.

Molecular communication (MC) aims to use signaling molecules as information carriers to achieve communication between biological entities. However, MC systems severely suffer from inter symbol interference (ISI) and external noise, making it virtually difficult to obtain accurate mathematical models. Specifically, the mathematically intractable channel state information (CSI) of MC motivates the deep learning (DL) based signal detection methods. In this paper, a modified temporal convolutional network (TCN) is proposed for signal detection for a special MC communication system which uses magnetotactic bacteria (MTB) as information carriers. Results show that the TCN-based detector demonstrates the best overall performance. In particular, it achieves better bit error rate (BER) performance than sub-optimal maximum a posteriori (MAP) and deep neural network (DNN) based detectors. However, it behaves similarly to the bidirectional long short term memory (BiLSTM) based detector that has been previously proposed and performs worse than the optimal MAP detector. When both BER performance and computational complexity are taken into account, the proposed TCN-based detector outperforms BiLSTM-based detectors. Furthermore, in terms of robustness evaluation, the proposed TCN-based detector outperforms all other DL-based detectors.

A Simple Variational Bayes Detector for Orthogonal Time Frequency Space (OTFS) Modulation

The emerging orthogonal time frequency space (OTFS) modulation technique has shown its superiority to the current orthogonal frequency division multiplexing (OFDM) scheme, in terms of its capabilities of exploiting full time-frequency diversity and coping with channel dynamics. The optimal maximum a posteriori (MAP) detection is capable of eliminating the negative impacts of the inter-symbol interference in the delay-Doppler (DD) domain at the expense of a prohibitively high complexity. To reduce the receiver complexity for OTFS scheme, this paper proposes a variational Bayes (VB) approach as an approximation of the optimal MAP detection. Compared to the widely used message passing algorithm, we prove that the proposed iterative algorithm is guaranteed to converge to the global optimum of the approximated MAP detector regardless the resulting factor graph is loopy or not. Simulation results validate the fast convergence of the proposed VB receiver and also show its promising performance gain compared to the conventional message passing algorithm.

Tail‐biting in optimal CPM detectors

The maximum a posteriori (MAP) detection of a continuous phase modulation (CPM) signal deploys the Bahl-Cocke-Jelinek-Raviv (BCJR) algorithm that utilises the trellis structure of the CPM signal. For optimal results the start as well as the end state must be known to the MAP detector. While the start state is usually the zero state by convention, the end of the trellis must be terminated properly to yield a defined end state. This leads to an overhead, which size is dependent on the particular CPM parameters. To overcome the decrease of spectral efficiency, a tail-biting technique is presented and compared to termination and truncation in a fair comparison regarding spectral efficiency.

Comment on “Bayesian MMSE Estimation of a Gaussian Source in the Presence of Bursty Impulsive Noise”

In the above-titled letter by Alam et al. (see ibid., vol.22, no.9, p.1846–49, September 2018), the authors claimed to derive the minimum-mean-squared-error (MMSE) estimator for a scalar Gaussian source in the presence of correlated noise, modeled by a two-state Gauss-Markov process. In the proposed approach, first, the state of the noise process at time epoch k is determined according to the maximum a posteriori (MAP) decision rule based on the whole sequence of observations. Then, given the hard decision on the noise state, the corresponding MMSE estimator (i.e., conditional mean) is employed to estimate the source signal at time epoch k . However, performing estimation based on the hard decision of the MAP detector is not optimal in terms of minimizing the mean-squared-error (MSE). In this comment, we show that the optimal MMSE estimator for the considered problem is obtained as a weighted average of the MMSE estimators corresponding to distinct states of the noise probable at time epoch k and the weights are equal to the a posteriori probabilities of the noise state computed based on the whole sequence of observations.

Asymptotic Comparison of ML and MAP Detectors for Multidimensional Constellations

A classical problem in digital communications is to evaluate the symbol error probability (SEP) and bit error probability (BEP) of a multidimensional constellation over an additive white Gaussian noise channel. In this paper, we revisit this problem for nonequally likely symbols and study the asymptotic behavior of the optimal maximum a posteriori (MAP) detector. Exact closed-form asymptotic expressions for SEP and BEP for arbitrary constellations and input distributions are presented. The well-known union bound is proven to be asymptotically tight under general conditions. The performance of the practically relevant maximum likelihood (ML) detector is also analyzed. Although the decision regions with MAP detection converge to the ML regions at high signal-to-noise ratios, the ratio between the MAP and ML detector in terms of both SEP and BEP approach a constant, which depends on the constellation and a priori probabilities. Necessary and sufficient conditions for asymptotic equivalence between the MAP and ML detectors are also presented.

Non-Coherent Detection and Denoise-and-Forward Two-Way Relay Networks

We investigate non-coherent detection of $M$ -ary orthogonal signaling in a denoise-and-forward two-way relay scheme in fast fading, as well as block fading channels. We first design the maximum likelihood, as well as maximum a-posteriori detectors, and then propose a simple suboptimal version for this digital network coding system in Rayleigh fast fading environment. We find exact expression of bit error rate for this suboptimal detector. Interestingly, this suboptimal detector performs only less than 0.25 dB away from optimum, i.e., this expression is a tight upper bound for the performance of the optimal one. Moreover, we derive the optimal detector for the case where the channel amplitudes are known at the relay. To make this detector practical, we propose efficient estimators for the channel amplitudes at the relay for block fading environments. Finally, our simulations reveal the effectiveness of the proposed methods compared with the state-of-the-art method.

Transmission Performance of 448 Gb/s Single-Carrier and 1.2 Tb/s Three-Carrier Superchannel Using Dual-Polarization 16-QAM With Fixed LUT Based MAP Detection

Transmission of single-carrier 448 Gb/s dual-polarization 16-ary quadrature-amplitude-modulation (DP 16-QAM) electrical time-division multiplexed (ETDM) signal and a 1.206 Tb/s three-carrier superchannel ETDM signal using DP 16-QAM is demonstrated by applying a fixed look-up table (LUT) based on maximum-a-posteriori (MAP) detection. The MAP detector is used to mitigate the effect of pattern dependent distortion in the transmitted signal arising from the high symbol rates. Different decision rules for the MAP detector are considered based on multiple observations of the same symbol as the detection window advances through the received symbol sequence. The back-to-back optical signal-to-noise ratios for a bit error ratio of $\mathbf {10^{-3}}$ are 26.9 and 26.4 dB for the 448 Gb/s signal (12% overhead with 7% for forward error correction (FEC) coding overhead) and 1.206 Tb/s superchannel signal (20% overhead with 15% for FEC coding overhead), respectively. Compared to conventional MAP detection, a seven-symbol minimum distance MAP detector reduces the required OSNR by about 0.8 and 1 dB at the FEC thresholds of $\mathbf {3.8\times 10^{-3}}$ and $\mathbf {1.9\times 10^{-2}}$ for the 448 Gb/s signal and 1.206 Tb/s superchannel signal, respectively. Transmission over 1200 and 1500 km of standard single-mode fiber with EDFA amplification is achieved for the 448 Gb/s signal and 1.206 Tb/s superchannel signal, respectively.

Comparison of Advanced Detection Techniques for QPSK Signals in Super-Nyquist WDM Systems

We compare the performance of detecting quadrature-phase-shift-keying (QPSK) in a super-Nyquist wavelength division multiplexing system as a QPSK and quadrature duobinary (QDB) signal using maximum likelihood sequence estimation (MLSE) and maximum a posteriori (MAP) criterion with both simulations and experiments. Three different detection schemes are studied: 1) QPSK detection with a multitap linear equalizer followed by a MLSE or MAP detector; 2) QPSK detection with only a MLSE or MAP detector; and 3) QDB detection with a MLSE or MAP detector, where a multitap linear equalizer and a delay-&-add operation are placed before the MLSE/MAP detector to convert a QPSK signal to a QDB signal. Our simulations and experiments show that the QDB detection with multitap MLSE does not provide better performance than that with two-tap MLSE, and the QDB detection with two-tap MLSE has better filtering tolerance than the QPSK detection with linear equalization followed by a multitap MLSE/MAP detector.

Stochastic Digital Backpropagation

In this paper, we propose a novel detector for single-channel long-haul coherent optical communications, termed stochastic digital backpropagation (SDBP), which takes into account noise from the optical amplifiers in addition to handling deterministic linear and nonlinear impairments. We discuss the design approach behind this detector, which is based on the maximum a posteriori (MAP) principle. As closed-form expressions of the MAP detector are not tractable for coherent optical transmission, we employ the framework of Bayesian graphical models, which allows a numerical evaluation of the proposed detector. Through simulations, we observe that by accounting for nonlinear signal-noise interactions, we achieve a significant improvement in system reach with SDBP over digital backpropagation (DBP) for systems with periodic inline optical dispersion compensation. In uncompensated links with high symbol rates, the performance difference in terms of system reach for SDBP over DBP is small. In the absence of noise, the proposed detector is equivalent to the well-known DBP detector.

Approximate maximum a posteriori detection for multiple‐input–multiple‐output systems with bit‐level lattice reduction‐aided detectors and successive interference cancellation

For iterative detection and decoding (IDD) in multiple-input–multiple-output (MIMO) systems, although the maximum a posteriori (MAP) detector can achieve an optimal performance, because of its prohibitively high computational complexity, various low-complexity approximate MAP detectors are studied. Among the existing MIMO detectors for the non-IDD receivers, lattice reduction (LR)-aided detectors can provide a near maximum-likelihood (ML) detector's performance with reasonably low complexity, and they could be modified to be used in the IDD receivers. In this study, the authors propose a bit-level LR-aided MIMO detector whose performance can approach that of the MAP detector, where a priori information is taken into account for soft-decisions. Furthermore, the proposed method can be extended to large dimensional MIMO systems by channel matrix decomposition and successive interference cancellation, by which a significant complexity reduction can be achieved. Through simulations and complexity analysis, it is shown that a near-optimal performance is obtained by the authors proposed low-complexity bit-level LR-aided detector for the IDD in the MIMO systems.

Successive Interference Cancellation via Rank-Reduced Maximum A Posteriori Detection

This paper proposes a codeword-based iterative detection and decoding (IDD) algorithm for multiple-input multiple-output (MIMO) systems. In the proposed algorithm, multiple streams in a codeword are jointly detected at the rank-reduced (RR) maximum a posteriori (MAP) receiver and inter-stream interference is mitigated with successive interference cancelation (SIC). Thus, the algorithm is abbreviated to RR-MAP-SIC. Recent wireless standards such as Long-Term Evolution require the system to encode data bits per codeword, not per stream. As a result, conventional SIC algorithms could lose joint information among streams in a codeword because different streams of the same codeword are treated as interference. Instead, the proposed RR-MAP-SIC minimizes the loss of joint information by using the rank-reduced MAP detector. In addition, this paper compares the detector complexity of RR-MAP-SIC and investigates how the probability of symbol error is changed in terms of the covariance of the residual interference. As the number of iterations increases, the covariance is reduced so that the error events also decrease. Lastly, the extrinsic information transfer (EXIT) chart is used to analyze the performance of RR-MAP-SIC. Simulation results demonstrate the superiority of RR-MAP-SIC over the conventional algorithm and numerically verify the EXIT chart analysis.

Reduced-Complexity Iterative-Detection-Aided Generalized Space-Time Shift Keying

A novel reduced-complexity soft decision (SoD)-aided detector is proposed for the recent concept of space-time shift keying (STSK), where the detector's achievable performance is capable of closely approaching that of the optimal maximum a posteriori (MAP) detector. More specifically, we exploit a hybrid combination of the modified matched filtering concept and of reduced-complexity exhaustive search for the sake of reducing the MAP detector's decoding complexity. Furthermore, we extended this detector to support the class of generalized STSK (GSTSK) scheme that subsumes diverse multiple-input–multiple-output (MIMO) arrangements. The proposed reduced-complexity SoD-aided GSTSK detector also attains significantly lower complexity than the MAP detector while imposing only marginal performance degradation, which is in the range of 1–2 dB. As an optional means of further reducing complexity, the Markov Chain Monte Carlo (MCMC) algorithm is invoked for the proposed GSTSK detector. Our EXtrinsic Information Transfer (EXIT) chart analysis reveals that the proposed STSK detector is capable of closely approaching the optimal performance, whereas the GSTSK detector advocated exhibits a modest performance gap with respect to the max-log MAP detector.

Sparsening filter design for iterative soft-input soft-output detectors

A large body of research exists around the idea of channel shortening, where a prefilter is designed to reduce the effective channel impulse response to some smaller number of contiguous taps. This idea was originally conceived to reduce the complexity of Viterbi-based maximum-likelihood equalizers. Here, we consider a generalization of channel shortening which we term "channel sparsening". In this case, a prefilter is designed to reduce the effective channel to a small number of nonzero taps which do not need to be contiguous. When used in combination with belief-propagation-based maximum a posteriori (MAP) detectors, an analogous complexity reduction can be realized. We address the design aspects of sparsening filters, including several approaches to minimize the bit error rate of MAP detectors. We devote attention to the interaction of the sparsening filter and detector, and demonstrate the performance gains through simulation.

Hardware oriented, quasi-optimal detectors for iterative and non-iterative MIMO receivers

Abstract In this article we study hardware-oriented versions of the recently appeared Layered ORthogonal lattice Detector (LORD) and Turbo LORD (T-LORD). LORD and T-LORD are attractive Multiple-Input Multiple-Output (MIMO) detection algorithms, that aim to approach the optimal Maximum-Likelihood (ML) and Maximum-A-Posteriori (MAP) performance, respectively, yet allowing a complexity quadratic in the number of transmitting antennas rather than exponential. LORD and T-LORD are also well suited to a hardware (e.g., ASIC or FPGA) implementation because of their regularity, parallelism, deterministic latency, and complexity. Nevertheless, their complexity is still high in case of high cardinality constellations, such as the 64-QAM foreseen by the 802.11n standard. We show that, when only global latency constraints exist, e.g., a fixed time to detect the whole OFDM symbol, the LORD and T-LORD complexity can be remarkably reduced, still approaching the ML and MAP performance, respectively. Notwithstanding the suboptimal low-complexity and hardware-oriented implementation, LORD and T-LORD approach the EXtrinsic Information Transfer of the ML and MAP detectors, respectively. To focus on a specific setting, we consider the indoor MIMO wireless LAN 802.11n standard, taking into account errors in channel estimation and a frequency selective, spatially correlated channel model.

Hardware oriented, quasi-optimal detectors for iterative and non-iterative MIMO receivers

In this article we study hardware-oriented versions of the recently appeared Layered ORthogonal lattice Detector (LORD) and Turbo LORD (T-LORD). LORD and T-LORD are attractive Multiple-Input Multiple-Output (MIMO) detection algorithms, that aim to approach the optimal Maximum-Likelihood (ML) and Maximum-A-Posteriori (MAP) performance, respectively, yet allowing a complexity quadratic in the number of transmitting antennas rather than exponential. LORD and T-LORD are also well suited to a hardware (e.g., ASIC or FPGA) implementation because of their regularity, parallelism, deterministic latency, and complexity. Nevertheless, their complexity is still high in case of high cardinality constellations, such as the 64-QAM foreseen by the 802.11n standard. We show that, when only global latency constraints exist, e.g., a fixed time to detect the whole OFDM symbol, the LORD and T-LORD complexity can be remarkably reduced, still approaching the ML and MAP performance, respectively. Notwithstanding the suboptimal low-complexity and hardware-oriented implementation, LORD and T-LORD approach the EXtrinsic Information Transfer of the ML and MAP detectors, respectively. To focus on a specific setting, we consider the indoor MIMO wireless LAN 802.11n standard, taking into account errors in channel estimation and a frequency selective, spatially correlated channel model.

Multiple description quantizer design for multiple-antenna systems with MAP detection

We study the design of index assignment based multiple description quantizers for multiple-antenna systems with slow Rayleigh fading. A time-interleaver is employed at the transmitter to provide independent channel instances for the multiple descriptions, and a maximum a posteriori (MAP) detector is employed at the receiver to jointly decode the multiple descriptions. The concatenation of index assignment, binary phase-shift keying modulators, space-time orthogonal block coder, multiple-antenna channel, and MAP detector is modeled as a discrete memoryless channel. We optimize multiple description vector quantizers using an upper bound of the channel transition probability achieved by the MAP detector. The proposed scheme of combining multiple description coding and space-time orthogonal block coding is shown to be superior to each individual coding scheme.

Row-Column Soft-Decision Feedback Algorithm for Two-Dimensional Intersymbol Interference

We present a novel iterative row-column soft decision feedback algorithm (IRCSDFA) for detection of binary images corrupted by 2-D intersymbol interference and additive white Gaussian noise. The algorithm exchanges weighted soft information between row and column maximum a posteriori (MAP) detectors. Each MAP detector exploits soft-decision feedback from previously processed rows or columns. The new algorithm gains about 0.3 dB over the previously best published results for the 2times2 averaging mask. For a non-separable 3times3 mask, the IRCSDFA gains 0.8 dB over a previous soft-input/soft-output iterative algorithm which decomposes the 2-D convolution into 1-D row and column operations.

Belief Propagation over SISO/MIMO Frequency Selective Fading Channels

In this letter, we propose an iterative belief propagation (BP) channel detector (equalizer) over single-input single- output (SISO) and multiple-input multiple-output (MIMO) frequency selective fading channels as an alternative to the typically used maximum a-posteriori (MAP) or maximum likelihood (ML) detectors. The proposed detector has a parallel structure, resulting in fast hardware implementations. Moreover, BP detector is less complex than the MAP detector and it has a short decoding delay. We analyze the bit error rate and the mutual information and show that, over frequency selective fading channels, the proposed BP detector achieves a near-optimal performance, even in the presence of the length 4 cycles in the corresponding channel factor graph.

Suboptimal soft-output MAP detector with lattice reduction

The use of lattice reduction (LR) has been shown to be a very effective method for signal detection in multiple antenna systems. In this letter, the performance of the suboptimal soft-output maximum a posteriori (MAP) detector with lattice reduction is studied. We propose two polynomial-time methods for flat fading channels to generate the desired size candidate list in the reduced lattice. The suboptimal soft-output MAP detector with these list generation methods provides a near-optimal performance in uncorrelated multiple-input multiple-output systems but with significantly reduced computational complexity.

Reduced-dimension MAP turbo-BLAST detection

The Bell Labs layered space-time (BLAST) architecture is a simple and efficient multiantenna coding structure that can achieve high spectral efficiency. Many BLAST detectors require more receiver antennas than transmitter antennas. We propose two novel turbo-processing BLAST detectors that can operate in systems with fewer receiver antennas than transmitter antennas. Both detectors are based on the group-detection strategy. The first proposed detector, the reduced-dimension maximum a posteriori (RDMAP) detector uses a dynamically formed group for each bit decision, while the second proposed detector, the group maximum a posteriori (GMAP) uses a static grouping. For both detectors, a maximum a posteriori (MAP) decision is made using a group of transmitted symbols, and the remaining signal contribution is treated as interference. The interference is characterized as nonzero mean colored-noise source that is whitened before a decision is made. Both proposed detectors are generalizations of the MAP detector and the turbo-processing minimum mean-squared error (MMSE) detector in Sellathurai and Haykin, and Abe and Matsumoto. An uncoded bit-error rate analysis for an independent Rayleigh fading environment is also presented. Simulated results are presented which show that both the RDMAP and GMAP detectors have a performance improvement over the MMSE detector, especially in systems having an excess number of transmitter antennas.