Extrinsic Log-likelihood Ratio Research Articles

Density Evolution Analysis of the Iterative Joint Ordered-Statistics Decoding for NOMA

In this paper, we develop a density evolution (DE) framework for analyzing the iterative joint decoding (JD) for non-orthogonal multiple access (NOMA) systems, where the ordered-statistics decoding (OSD) is applied to decode short block codes. We first investigate the density-transform feature of the soft-output OSD (SOSD), by deriving the density of the extrinsic log-likelihood ratio (LLR) with known densities of the priori LLR. Then, we represent the OSD-based JD by bipartite graphs (BGs), and develop the DE framework by characterizing the density-transform features of nodes over the BG under the binary phase shift keying (BPSK) transmission. Numerical examples show that the proposed DE framework accurately tracks the evolution of LLRs during the iterative decoding, especially at moderate-to-high SNRs. Based on the DE framework, we further analyze the BER performance of the OSD-based JD, and the convergence points of the two-user and equal-power systems.

Low-complexity soft ML detection for generalized spatial modulation

Generalized Spatial Modulation (GSM) is a recent Multiple-Input Multiple-Output (MIMO) scheme, which achieves high spectral and energy efficiencies. Specifically, soft-output detectors have a key role in achieving the highest coding gain when an error-correcting code (ECC) is used. Nowadays, soft-output Maximum Likelihood (ML) detection in MIMO-GSM systems leads to a computational complexity that is unfeasible for real applications; however, it is important to develop low-complexity decoding algorithms that provide a reasonable computational simulation time in order to make a performance benchmark available in MIMO-GSM systems. This paper presents three algorithms that achieve ML performance. In the first algorithm, different strategies are implemented, such as a preprocessing sorting step in order to avoid an exhaustive search. In addition, clipping of the extrinsic log-likelihood ratios (LLRs) can be incorporating to this algorithm to give a lower cost version. The other two proposed algorithms can only be used with clipping and the results show a significant saving in computational cost. Furthermore clipping allows a wide-trade-off between performance and complexity by only adjusting the clipping parameter.

Reliability Ratio-Based Serial Algorithm of LDPC Decoder for Turbo Equalization Schemes

Serial decoding algorithms of low-density parity-check (LDPC) code converge efficiently with low errors. Previously, a serial decoding algorithm, named a shuffled belief-propagation (SBP), was applied in turbo equalization of bit-patterned magnetic recording (BPMR) systems. With the SBP algorithm, an LDPC decoder converged twice as fast as one using conventional BP algorithms. We further improved the convergence speed of SBP by updating the messages in an adaptive order, which played a flexible role throughout decoding. We proposed two adaptive-serial algorithms for LDPC codes in turbo equalization. One updated the messages using the extrinsic loglikelihood ratio (LLR) and the result of the parity-check equation checking. The second contained an additional rule that tracked the LLR sign changes in each iteration. Both algorithms converged faster and with lower bit error rates (BERs) than the SBP and previous adaptive-serial algorithms in a BPMR system with media noise.

Joint Channel Estimation and Generalized Approximate Messaging Passing-Based Equalization for Underwater Acoustic Communications

Acquiring channel state information and mitigating multi-path interference are challenging for underwater acoustic communications under time-varying channels. We address the issues using a superimposed training (ST) scheme with a least squares (LS) based channel estimation algorithm. The training sequences with a small power are linearly superimposed with the symbol sequences, and the training signals are transmitted over all time, resulting in enhanced tracking capability to deal with time-varying underwater acoustic channels at the cost of only a small power loss. To realize the full potentials of the ST scheme, we develop a LS based channel estimation algorithm with superimposed training, where the Toeplitz matrix is used, which is formed by the training sequences, enabling channel estimation with superimposed training. In particular, a low-complexity channel equalization algorithm based on generalized approximate messaging passing (GAMP) is proposed, where the a priori, a posteriori, extrinsic means and variances of interleaved coded bits are computed, and then convert them into extrinsic log likelihood ratios for BCJR decoding. Its computational complexity is only in a logarithmic order per symbol. Moreover, the channel estimation, GAMP equalization and decoding are performed jointly in an iterative manner, so that the estimated symbol sequences can also be used as virtual training sequences to improve the channel estimation and tracking performance, thereby remarkably enhance the overall system performance. Moving communication experiments in Jiaozhou Bay (communication frequency 12 kHz, bandwidth 6 kHz, sampling frequency 96 kHz, symbol transmission rate 4 ksym/s) were carried out, and the experimental results verify the effectiveness of the proposed technique.

Efficient Adaptive Turbo Equalization for Multiple-Input–Multiple-Output Underwater Acoustic Communications

An efficient adaptive turbo equalization (ATEQ) scheme is proposed for multiple-input–multiple-output underwater acoustic (UWA) communications. The proposed ATEQ scheme utilizes two layers of iterative processing: The inner-layer iteration is the soft-decision-based equalizer parameters adaptation and filtering of received signals in the equalizer, and the outer-layer iteration is the Turbo exchange of extrinsic log-likelihood ratio between the equalizer and decoder. In contrast, the existing ATEQ schemes use hard-decision symbols for filter adaptation but soft symbols for filtering and Turbo iteration. When the adaptive filters are designed and updated via the normalized least mean squares (NLMS) or the improved proportionate NLMS algorithms for low computational complexity and good channel tracking, the soft symbols utilized in both the pilot-assisted and the decision-directed modes of the proposed ATEQ scheme achieve fast convergence with short training sequences, thus achieving high spectrum efficiency. The proposed scheme is evaluated by the field trail data collected in the 2008 Surface Processes and Acoustic Communications Experiment. The results demonstrate that the proposed ATEQ scheme is robust against the severe triply-selective UWA channels and mitigate slow-convergence problem commonly suffered by direct-adaptation equalizers.

Distributed Joint Source-Channel Decoding Using Systematic Polar Codes

This letter proposes a novel joint decoding scheme for correlated sources using systematic polar codes. In the proposed scheme, each source independently encodes its message into a systematic polar code word and sends the codeword via a binary-input additive white Gaussian noise channel to the common destination, where adaptive cyclic redundancy check-aided successive cancellation list (CA-SCL) decoders are employed. During the iterative decoding process, the log likelihood ratios (LLRs) fed to each adaptive CA-SCL decoder are iteratively combined with the extrinsic LLRs from other decoders. The correlation among the sources is exploited to improve the decoding performance. Simulation results show that, in comparison with independent decoding, the proposed scheme can achieve significantly improved decoding performance without increased complexity.

Does Vector Gaussian Approximation After LMMSE Filtering Improve the LLR Quality?

In this letter, we investigate the extrinsic log-likelihood ratio (LLR) computation of a soft-input soft-output equalizer used in a turbo equalization system. The optimum LLRs are obtained by a maximum a posteriori -based equalizer, which may be computationally expensive. Thus, several reduced-complexity equalizers have been proposed. The most promising approach first applies linear minimum mean square error filtering to the channel output and then computes the LLRs based on a scalar Gaussian approximation of the filter output. The resulting LLRs can be viewed as an approximation of the optimum LLRs. In order to improve the approximation, we investigate the computation of the LLRs based on a vector Gaussian approximation of the filter output, which incorporates the correlation between the estimated symbols after filtering. Surprisingly, it turns out that both approaches, although their derivation is different, give the same LLRs. We verify this remarkable result through an analytical proof and bit error ratio simulations.

Analysis of Turbo Code Behavior with Extrinsic Information Transfer Charts in High-Speed Wireless Data Services

This paper examines turbo codes that are currently introduced in many international standards and implemented in numerous advanced communication systems, and evaluates the process of extrinsic information transfer (EXIT). The convergence properties of the iterative decoding process, associated with a given turbo-coding scheme, are estimated using the analysis technique based on so-called extrinsic information transfer (EXIT) charts. This approach provides a possibility to predict the bit-error rate (BER) of a turbo code system with only the extrinsic information transfer chart. It is shown that extrinsic information transfer charts are powerful tools to analyze and optimize the convergence behavior of iterative systems utilizing the turbo principle, i.e., systems exchanging and refining extrinsic information. The idea is to consider the associated soft-input soft-output (SISO) stages as information processors, which map input a priori log likelihood ratios (LLRs) onto output extrinsic LLRs, the information content being obviously assumed to increase from input to output, and introduce them to the design of turbo systems without the reliance on extensive simulation. Compared with the other methods for generating extrinsic information transfer functions, the suggested approach provides insight into the iterative behavior of linear turbo systems with substantial reduction in numerical complexity.

An Efficient Joint Channel-Network Decoder for Multiuser Diversity Network Codes With LDPC Coding

Diversity network codes (DNCs) were recently proposed for multiuser cooperative networks, where nonbinary network codes are superimposed over channel codes to yield high diversity gains. In this letter, we present a joint iterative channel-network decoder for use at the base station in a system employing a DNC. The scheme is based on hybrid soft/hard message passing between constituent low-complexity low-density parity-check (LDPC) decoders. It is shown that the formation of the extrinsic log-likelihood ratios to be passed between the LDPC decoders has a low-complexity implementation involving a set of permutation operations together with the well-known Jacobian logarithm. The proposed joint channel-network decoder has a significantly reduced complexity compared with the decoding structure originally presented for maximum-diversity-achieving DNCs, and achieves significant coding gain as well as full diversity gain.

Turbo Equalization for Two Dimensional Magnetic Recording Using Voronoi Model Averaged Statistics

This paper considers turbo equalization for 2-D magnetic recording. Magnetic grains are modeled as Voronoi regions of randomly distributed nuclei. Bits read from the magnetic grain model flow into a 2-D intersymbol interference (2D-ISI) model including additive white Gaussian noise. At high bit densities, some bits are not written on any grain, and hence are effectively “overwritten” by surrounding bits. The proposed system iteratively exchanges log-likelihood ratios (LLRs) between a 2D-ISI equalizer based on the forward-backward algorithm and an irregular repeat-accumulate (IRA) decoder. To combat bit overwrites, the system employs a non-linear function to map 2D-ISI extrinsic output LLRs to IRA decoder input LLRs. To pass back LLRs from the IRA decoder to the 2D-ISI equalizer, we design a simple likelihood-ratio-based LLR estimator. Simulations of the proposed system that employ the perturbed-bit-centers grain model proposed in a 2010 IEEE Transactions on Magnetics paper show a 6.5% increase in user bits per grain (U/G) and a 16.4 dB signal-to-noise ratio (SNR) gain compared with the previous paper, without iterative turbo equalization. Utilizing the LLR estimator to do iterative detection results in SNR gains of up to 1.7 dB compared with non-iterative detection. The random Voronoi model employed in this paper appears to be more difficult to equalize than the grain model in the 2010 paper. The proposed system with random Voronoi model achieves 0.4422 U/G at $\mathrm {SNR}=11.6$ dB, i.e., about 8.8 Tb/in2 at (typically assumed future grain density) 20 Tgr/in2; this is almost ten times the density of current systems at 10 Tgr/in2.

Multi-stage turbo equalization for MIMO systems with hybrid ARQ

A multi-stage turbo equalization scheme based on the bit-level combining (BLC) is proposed for multiple-input multiple-output (MIMO) systems with hybrid automatic repeat request (HARQ). In the proposed multi-stage turbo equalization scheme, the minimum mean-square-error equalizer at each iteration calculates the extrinsic log-likelihood ratios for the transmitted bits in a subpacket and the subpackets are sequentially replaced at each iteration according to the HARQ rounds of received subpackets. Therefore, a number of iterations are executed for different subpackets received at several HARQ rounds, and the transmitted bits received at the previous HARQ rounds as well as the current HARQ round can be estimated from the combined information up to the current HARQ round. In addition, the proposed multi-stage turbo equalization scheme has the same computational complexity as the conventional bit-level combining based turbo equalization scheme. Simulation results show that the proposed multi-stage turbo equalization scheme outperforms the conventional BLC based turbo equalization scheme for MIMO systems with HARQ.

Frequency-Domain Iterative SDFE for MIMO-OFDM Systems

This paper proposes a frequency-domain iterative soft decision feedback equalization (SDFE) scheme for multiple-input---multiple-output orthogonal frequency division multiplexing communication systems in multi-path fading channels. Error propagation can seriously affect the performance of the adaptive decision feedback equalization. We propose an iterative SDFE to not only combat error propagation but also be suitable for multilevel modulation systems. Under a Gaussian approximation of the a priori information and the SDFE output, the output of an equivalent AWGN channel and expressions for the a posteriori log likelihood ratio (LLR) and the extrinsic LLR are derived for multilevel modulation. In addition, we also investigate the iterative receivers by employing irregular low-density parity check codes (LDPC) and turbo codes. The performance of the LDPC-coded iterative receiver is better than that of the turbo-coded iterative receiver. Some simulation examples are given to show the effectiveness and comparisons of the proposed receiver.

EXIT-Chart-Based Design Method for Noncoherent LDPC-Coded Modulation in Correlated Rayleigh Fading Channels

In the paper, we investigate LDPC-coded differential amplitude/phase modulation (APM) schemes for correlated Rayleigh fading channels. Through the observation for the mutual information of the extrinsic log-likelihood ratio (LLR) values of the variable nodes in the LDPC decoder, it is found that the code design methods investigated in the previous literature loses their accuracy. Consequently, a modified code design method for the outer LDPC code is proposed in this paper. It is shown that the searched LDPC code effectively improves the error performance, when compared to the convolutional-coded scheme investigated in the previous literature.

From Nominal to True A Posteriori Probabilities: An Exact Bayesian Theorem Based Probabilistic Data Association Approach for Iterative MIMO Detection and Decoding

It was conventionally regarded that the approximate Bayesian theorem based existing probabilistic data association (PDA) algorithms output the estimated symbol-wise a posteriori probabilities (APPs) as soft information. In our recent work, however, we demonstrated that these probabilities are not the true APPs in the rigorous mathematical sense, but a type of nominal APPs, which are unsuitable for the classic architecture of iterative detection and decoding (IDD) aided receivers. To circumvent this predicament, in this paper we propose an exact Bayesian theorem based logarithmic domain PDA (EB-Log-PDA) method, whose output has similar characteristics to the true APPs, and hence it is readily applicable to the classic IDD architecture of multiple-input-multiple-output (MIMO) systems using the general M-ary modulation. Furthermore, we investigate the impact of the EB-Log-PDA algorithm's inner iteration on the design of EB-Log-PDA aided IDD receiver. We demonstrate that introducing inner iterations into EB-Log-PDA, which is common practice in conventional-PDA aided uncoded MIMO systems, would actually degrade the IDD receiver's performance, despite significantly increasing the overall computational complexity of the IDD receiver. Finally, we investigate the relationship between the extrinsic log-likelihood ratios (LLRs) of the proposed EB-Log-PDA and of the approximate Bayesian theorem based logarithmic domain PDA (AB-Log-PDA) reported in our previous work. Despite their difference in extrinsic LLRs, we also show that the IDD schemes employing the EB-Log-PDA and the AB-Log-PDA without incorporating any inner PDA iterations have a similar achievable performance close to that of the optimal maximum a posteriori (MAP) detector based IDD receiver, while imposing a significantly lower computational complexity in the scenarios considered.

Combining-after-Decoding Turbo Hybrid ARQ by Utilizing Doped-Accumulator

This letter proposes a doped-accumulator assisted packet-combining-after-decoding (ACC-CAD) technique with different doping rates per transmission phases for hybrid automatic repeat request (ARQ). Horizontal iteration (HI) performed for decoding the serially concatenated codes (SCC) is followed by Vertical iteration (VI) to exchange extrinsic log-likelihood ratio (LLR) of the uncoded (systematic) bits, and the HI/VI chain is repeated. The doped-accumulator enables the two extrinsic information transfer (EXIT) curves of the SCC to match very well and the convergence tunnel to open until a point very close to the (1.0,1.0) mutual information point. Excellent performance of our technique is verified through EXIT analysis as well as frame-error-rate (FER) and throughput simulations.

Approximate Bayesian Probabilistic-Data-Association-Aided Iterative Detection for MIMO Systems Using Arbitrary $M$-ary Modulation

In this paper, the issue of designing an iterative-detection-and-decoding (IDD)-aided receiver, relying on the low-complexity probabilistic data association (PDA) method, is addressed for turbo-coded multiple-input-multiple-output (MIMO) systems using general <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> -ary modulations. We demonstrate that the classic candidate-search-aided bit-based extrinsic log-likelihood ratio (LLR) calculation method is not applicable to the family of PDA-based detectors. Additionally, we reveal that, in contrast to the interpretation in the existing literature, the output symbol probabilities of existing PDA algorithms are not the true a posteriori probabilities (APPs) but, rather, the normalized symbol likelihoods. Therefore, the classic relationship, where the extrinsic LLRs are given by subtracting the a priori LLRs from the a posteriori LLRs, does not hold for the existing PDA-based detectors. Motivated by these revelations, we conceive a new approximate Bayesian-theorem-based logarithmic-domain PDA (AB-Log-PDA) method and unveil the technique of calculating bit-based extrinsic LLRs for the AB-Log-PDA, which facilitates the employment of the AB-Log-PDA in a simplified IDD receiver structure. Additionally, we demonstrate that we may dispense with inner iterations within the AB-Log-PDA in the context of IDD receivers. Our complexity analysis and numerical results recorded for Nakagami- <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i> fading channels demonstrate that the proposed AB-Log-PDA-based IDD scheme is capable of achieving a performance comparable with that of the optimal maximum a posteriori (MAP)-detector-based IDD receiver, while imposing significantly lower computational complexity in the scenarios considered.

Asymptotic Coded BER Analysis for MIMO BICM-ID with Quantized Extrinsic LLR

In this paper, we derive a closed-form expression for the probability density/mass function (PDF/PMF) and the moment generating function (MGF) of the quantized detector soft output, i.e., extrinsic log-likelihood ratio (LLR), for multiple-input multiple-output (MIMO) bit-interleaved coded modulation with iterative decoding (BICM-ID) systems. The effect of either LLR clipping or LLR clipping with rounding, often applied in practical implementations, are considered. Using the derived expression, we analyze the asymptotic coded bit error rate (BER) for MIMO BICM-ID systems with the two quantization operations under a flat Rayleigh fading channel. The error rate degradation caused by quantizing the extrinsic LLR is then interpreted as an additional signal-to-noise ratio (SNR) loss. Rather than Monte Carlo simulations, this theoretical treatment provides a more convenient alternative to determining the clipping level and optimal word-length (the number of bits needed to represent signals) for LLR in a BICM-ID implementation. Finally, several other applications of this proposed theoretical analysis are also demonstrated.

Spatially Concatenated Codes With Turbo Equalization for Correlated Sources

This paper proposes for single carrier signaling a simple structure that combines turbo equalization and decoding of correlated sources in multipath-rich multiuser Rayleigh fading multiple access channels (MAC), where the correlation between the sources is exploited by vertical iterations (VI) between the decoders. The bit-flipping model with a flipping probability <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">e</sub> is used to express the correlation ρ between the sources as ρ = 1-2 <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">e</sub> . The proposed simple structure, spatially concatenated codes (SpCC), can achieve turbo-like performance over MAC channels suffering from severe inter-symbol interference (ISI), even though it uses short memory convolutional codes. First of all, to achieve turbo-like performance we introduce <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">VI</i> s to exploit the knowledge about the source correlation by exchanging extrinsic log-likelihood ratio (LLR) between the two decoders. We then add a rate-1 doped accumulator (D-ACC) to flexibly adapt for the variation of correlations between the sources. The results of computer simulations and extrinsic information transfer (EXIT) analysis confirm that in multipath-rich environments the proposed structure can achieve excellent performances, 1.02-1.28 dB away from the Slepian-Wolf-Shannon limit, at 1% outage probability for 0 <; ρ ≤ 1.

Soft–Input Soft–Output Single Tree-Search Sphere Decoding

Soft-input soft-output (SISO) detection algorithms form the basis for iterative decoding. The computational complexity of SISO detection often poses significant challenges for practical receiver implementations, in particular in the context of multiple-input multiple-output (MIMO) wireless communication systems. In this paper, we present a low-complexity SISO sphere-decoding algorithm, based on the single tree-search paradigm proposed originally for soft-output MIMO detection in Studer (“Soft-output sphere decoding: Algorithms and VLSI implementation,” IEEE J. Sel. Areas Commun., vol. 26, no. 2, pp. 290-300, Feb. 2008). The new algorithm incorporates clipping of the extrinsic log-likelihood ratios (LLRs) into the tree-search, which results in significant complexity savings and allows to cover a large performance/complexity tradeoff region by adjusting a single parameter. Furthermore, we propose a new method for correcting approximate LLRs - resulting from sub-optimal detectors - which (often significantly) improves detection performance at low additional computational complexity.

Decoding of High Rate Convolutional Codes Using the Dual Trellis

This paper deals with a posteriori probability (APP) decoding of high-rate convolutional codes, using the dual code's trellis. After deriving the dual APP (DAPP) algorithm from the APP relation, its trellis-based implementation is addressed. The challenge involved in practical implementation of a DAPP decoder is then highlighted. Metric representation schemes similar to the log domain used for log-APP decoding are shown to be unattractive for DAPP decoding due to quantization requirements. After explaining the nature of the DAPP metrics, an arc hyperbolic tangent (AHT) scheme is proposed and its equivalent arithmetic operations derived. By using an efficient approximation, an addition is translated to an addition in the AHT domain. Efficient techniques for normalization and extrinsic log-likelihood ratio (LLR ) calculation are presented which reduce implementation complexity significantly. Simulation results with different high-rate codes are given to show that the AHT-DAPP decoder performs similarly to a log-APP decoder and at the same time performs better than a decoder for a punctured code. A fully fixed-point model of an AHT-DAPP decoder is shown to perform close to an optimum decoder. The decoding complexity of the log-APP and AHT-DAPP decoders are listed and compared for several rate-k/(k+1) codes. It is shown that an AHT-DAPP decoder starts to be less complex from a code rate of 7/8 . When compared against a max-log-APP decoder, the AHT-DAPP decoder is less complex at a code rate of 9/10 and above.