Log-likelihood Ratio Calculation Research Articles

Low-complexity soft-output signal detector based on AI-SSOR preconditioned conjugate gradient method over massive MIMO correlated channel

Massive multiple-input multiple-output (MIMO) techniques with a large number of antenna elements at base station (BS) have been proved as an alternative to provide potential opportunity to increase the spectrum and energy efficiency. However, in the system, there generally exists a spatial correlation effect due to insufficient antenna elements spacing and/or the lack of rich scattering at BS. The minimum mean square error (MMSE) method performs signal detection at the expense of large-scale matrix inversion operation. Thus, the conjugate gradient (CG) method has received a lot of attentions to realize the MMSE detection efficiently. Unfortunately, this efficiency can be compromised due to the ill-conditioned equalization matrix of MMSE method over the correlated channel environments. Moreover, the hard output signal detection exhibits a sharply degradation in performance for higher-order quadrature amplitude modulation (QAM). Therefore, the modern communication systems use the soft-output information, i.e., log-likelihood ratio (LLR) along with the forward error-correcting code (FEC) to achieve satisfactory performance. The LLR computation along with a higher-order QAM remains challenging due to the exhaustive search of symbol in the modulation constellation. In this paper, a low-complexity soft-output signal detector based on approximate inverse symmetric successive over-relaxation preconditioned conjugate gradient (AI-SSOR-CG-SOD) method is proposed to realize MMSE method detection for uplink multiuser massive MIMO correlated channel. In the proposed method, a new preconditioner, an AI-SSOR, which is based on the Neumann series approximation of the inverse of the conventional SSOR preconditioner is firstly developed to handle ill-conditioned matrix, and then incorporated with CG method to improve the convergence rate and performance. According to the characteristic of the Gray-coding that adjacent symbols in the constellation set have only one different bit, the constellation set is divided multiple times based on the bits of the inphase and the quadrature components of the symbol, which reduces the complexity of the LLR computation of the transmitted bits by avoiding the exhaustive search process. Simulation results show that the AI-SSOR preconditioner is robust against spatial correlation effect, and the proposed detector converges at 3 iterations. Simulation results also show that the proposed detector achieves a better trade-off between the complexity and the performance compared to other existing detectors.

Simple and Robust Log-Likelihood Ratio Calculation of Coded MPSK Signals in Wireless Sensor Networks for Healthcare

The simple and robust log-likelihood ratio (LLR) computation of coded Multiple Phase Shift Keying (MPSK) signals in Wireless Sensor Networks (WSNs) is considered under both phase noncoherent and Rayleigh fading channels for healthcare applications. We first simplify the optimal LLR for phase noncoherent channel, the estimation of the instantaneous channel state information (CSI) for both the fading amplitude and the additive white Gaussian noise (AWGN) is successfully avoided, and the complexity-intensive process for zero-order Bessel function of the first kind is also perfectly eliminated. Furthermore, we also develop the simplified LLR under Rayleigh fading channel. Correspondingly, the variance estimation for both AWGN and the statistical characteristic of the fading amplitude is no longer required, and the complicated process for implementation of the exponential function is also successfully avoided. Compared to the calculation of optimal LLR with full complexity, the proposed method is implementation-friendly, which is practically desired for energy-limited WSNs. The simulations are developed in the context of low-density parity-check (LDPC) codes, and the corresponding results show that the detection performance is extremely close to that of the full-complexity LLR metrics. That is, the performance degradation is efficiently prevented, whereas complexity reduction is also successfully achieved.

Improving Log-Likelihood Ratio Estimation with Bi-Gaussian Approximation under Multiuser Interference Scenarios

Accurate estimation of channel log-likelihood ratio (LLR) is crucial to the decoding of modern channel codes like turbo, low-density parity-check (LDPC), and polar codes. Under an additive white Gaussian noise (AWGN) channel, the calculation of LLR is relatively straightforward since the closed-form expression for the channel likelihood function can be perfectly known to the receiver. However, it would be much more complicated for heterogeneous networks where the global noise (i.e., noise plus interference) may be dominated by non-Gaussian interference with an unknown distribution. Although the LLR can still be calculated by approximating the distribution of global noise as Gaussian, it will cause performance loss due to the non-Gaussian nature of global noise. To address this problem, we propose to use bi-Gaussian (BG) distribution to approximate the unknown distribution of global noise, for which the two parameters of BG distribution can easily be estimated from the second and fourth moments of the overall received signals without any knowledge of interfering channel state information (CSI) or signaling format information. Simulation results indicate that the proposed BG approximation can effectively improve the word error rate (WER) performance. The gain of BG approximation over Gaussian approximation depends heavily on the interference structure. For the scenario of a single BSPK interferer with a 5 dB interference-to-noise ratio (INR), we observed a gain of about 0.6 dB. The improved LLR estimation can also accelerate the convergence of iterative decoding, thus involving a lower overall decoding complexity. In general, the overall decoding complexity can be reduced by 25 to 50%.

Accurate Log-Likelihood Ratio Calculation for Vector Perturbation Precoding

Vector perturbation (VP) precoding is a promising technique to achieve near optimal capacity by exploring the channel state information (CSI) at the multiple-input multiple-output (MIMO) transmitter. To further enhance the bit error rate (BER) performance associated with forward error correction (FEC) coding, soft decision should be invoked in VP. However, the modulo operation in VP may influence the accuracy of soft decision. For alleviating this problem, the calculation of log-likelihood ratio (LLR) for VP is discussed and we further propose a sub-constellation shifted (SCS) scheme for VP precoding toward more accurate LLR. Simulation results demonstrate that the proposed scheme is capable of improving the BER performance of VP systems by accurate LLR calculation with moderate computational complexity.

A robust soft demapper algorithm for OFDM communication systems

In this article, we propose a noise-robust joint channel estimation and soft demapper algorithm for OFDM systems which is based on log-likelihood ratio calculation that taking into account multiplicative and additive error variance. The simulation results show that the proposed algorithm outperforms conventional techniques, such as LMMSE channel estimation about 1.1-1.8 dB in threshold signal-to-noise ratio in different wireless mobile channels.

Cube-Split: A Structured Grassmannian Constellation for Non-Coherent SIMO Communications

In this paper, we propose a practical structured constellation for non-coherent communication with a single transmit antenna over Rayleigh flat and block fading channel without instantaneous channel state information. The constellation symbols belong to the Grassmannian of lines and are defined up to a complex scaling. The constellation is generated by partitioning the Grassmannian of lines into a collection of bent hypercubes and defining a mapping onto each of these bent hypercubes such that the resulting symbols are approximately uniformly distributed on the Grassmannian. With a reasonable choice of parameters, this so-called cube-split constellation has higher packing efficiency, represented by the minimum distance, than the existing structured constellations. Furthermore, exploiting the constellation structure, we propose low-complexity greedy symbol decoder and log-likelihood ratio computation, as well as an efficient way to associate it to a multilevel code with multistage decoding. Numerical results show that the performance of the cube-split constellation is close to that of a numerically optimized constellation and better than other structured constellations. It also outperforms a coherent pilot-based scheme in terms of error probability and achievable data rate in the regime of short coherence time and large constellation size.

Error Rate-Based Log-Likelihood Ratio Processing for Low-Density Parity-Check Codes in DNA Storage

Due to the advantages of high storage densities and longevity, DNA storage has become one of the attractive technologies for future data storage systems. However, the writing/reading cost is still high and more efficient techniques for DNA storage are required. In this paper, we propose improved log-likelihood ratio (LLR) processing schemes based on observed statistics for low-density parity-check (LDPC) code decoding to reduce reading cost while encoding schemes are kept unchanged. Due to the mismatch between the real channel and the observed statistics and also the limit of maximum decoder input value, scaling the magnitude of LLR can lead to a better error correcting performance. Therefore, we propose two strategies: 1) directly scaling LLRs and 2) scaling pairwise substitution error rates, which changes the magnitude of LLRs. We also suggest the relation between substitution error rate and scaling values in the strategies by using curve fitting methods. Simulation results show that the error correcting performance from the proposed LLR calculation is better than that from the conventional scheme. Finally, we verify that the proposed LLR methods can be generally applied in DNA storage systems, and present practical methods to calculate error rates.

Coarsely Quantized Decoding and Construction of Polar Codes Using the Information Bottleneck Method

The information bottleneck method is a generic clustering framework from the field of machine learning which allows compressing an observed quantity while retaining as much of the mutual information it shares with the quantity of primary relevance as possible. The framework was recently used to design message-passing decoders for low-density parity-check codes in which all the arithmetic operations on log-likelihood ratios are replaced by table lookups of unsigned integers. This paper presents, in detail, the application of the information bottleneck method to polar codes, where the framework is used to compress the virtual bit channels defined in the code structure and show that the benefits are twofold. On the one hand, the compression restricts the output alphabet of the bit channels to a manageable size. This facilitates computing the capacities of the bit channels in order to identify the ones with larger capacities. On the other hand, the intermediate steps of the compression process can be used to replace the log-likelihood ratio computations in the decoder with table lookups of unsigned integers. Hence, a single procedure produces a polar encoder as well as its tailored, quantized decoder. Moreover, we also use a technique called message alignment to reduce the space complexity of the quantized decoder obtained using the information bottleneck framework.

Log-Likelihood Ratio Calculation for Pilot Symbol Assisted Coded Modulation Schemes With Residual Phase Noise

This paper presents a novel log-likelihood ratio (LLR) calculation for high order coded modulation schemes over an additive white Gaussian noise channel at the presence of residual phase noise (RPN). Residual phase noise is known to significantly degrade the error rate performance of such systems, particularly at lower error rates, resulting in an early error floor. To model RPN, we consider the commonly used pilot symbol assisted modulation schemes for carrier recovery. We derive the exact formula for the calculation of LLR for such systems. To simplify the implementation, we also derive an approximation of LLR which reduces the complexity significantly with almost no loss in performance. The simulation results are presented for coded modulation schemes based on quadrature amplitude modulations and low-density parity-check codes. The simulations demonstrate significant performance improvement in the error rate as a result of using the new LLR calculation instead of the conventional calculation of LLR which ignores the RPN.

Novel non‐linear demapper for soft decision decoder of LDPC codes in impulsive noise

Impulsive noises severely degrade the performance of a communication system. This study deals with the performance of the soft decision decoder for low-density parity-check codes over impulsive noise channels. To simplify the calculation of log likelihood ratio (LLR) and cooperate with the soft decision decoder, a new non-linear approximation named inverse demapper of LLR over the impulsive noise is proposed. Without carrying the noise statistics, the inverse demapper performs close to the optimal demapper. In addition, the density evolution is employed to obtain optimal parameters of the inverse demapper. Then, the extrinsic information transfer chart analysis and simulation results are presented to verify the effectiveness of the authors' proposed demapper.

High-Rate Regular APSK Constellations

The majority of modern communication systems adopt quadrature amplitude modulation (QAM) constellations as transmission schemes. Due to their square structure, however, QAM do not provide satisfying protection to phase noise effects, as the number of constellation points grows, increasing at the same time their peak-to-average-power ratio. This requires an expensive power amplifier and an oscillator at the transmitter to guarantee low distortion, complicating the adoption of dense transmission schemes in practical high-data rate systems. In this paper, we construct a coded modulation scheme based on regular amplitude and phase shift keying modulations. We propose a novel multilevel coding labeling for the constellation points separating the amplitude and phase domains. We provide a novel multistage decoding scheme allowing for a low-complexity log-likelihood ratio calculation for soft-input decoding of component codes, along with a suitable rate design. Finally, we compare the proposed scheme with the state-of-the-art QAM constellations and optimize the constellations in the presence of phase noise.

Efficient Embedded Soft Nodes Identification Using Block Coding in Relay Systems with High Order Modulations

Embedded signaling is an efficient method to blindly estimate different parameters in wireless communication systems. In this paper, we propose a new nodes identification method based on embedded signaling for wireless relay systems. Data packets, modulated with high order modulations, are transmitted by a source node and forwarded by a relay node to a destination node. In the proposed identification scheme, source and relay nodes incorporate precoding sequences, based on codewords generated from block codes, into the data sequence. A novel soft treatment based on the computation of log-likelihood ratios is performed at the destination node to identify source and relay nodes involved in packets transmissions. This soft treatment avoids noise amplification encountered in previous works. Two types of identification schemes are proposed: the hierarchical scheme and the non-hierarchical scheme. The hierarchical scheme allows the destination to recover hierarchically the identities of transmitting source and relay nodes of received packets in two shots. The non-hierarchical scheme allows the destination to efficiently recover in one shot the identities of transmitting nodes. We study the complexities of both identification schemes and compare their performance in terms of identification error rate. Since the knowledge of the channel is necessary in the identification process, we propose a blind estimation technique allowing the destination to estimate the channel. Simulations results show that our proposed methods significantly outperform previous identification methods based on embedded signaling and hard decoding. Indeed, gains of approximately 1.5 dB and 2 dB are obtained at high signal to noise ratio when the non-hierarchical scheme is used respectively for 8 Phase Shift Keying and 16PSK modulations. Simulations show also that the non-hierarchical scheme always outperforms the hierarchical scheme.

On the Concatenated Transmission Scheme With the Low-Complexity Symbol-Level Watermark Decoder for Recovering the Synchronization

The Davey–MacKay (DM) concatenated code employing the symbol-level watermark decoding algorithm is able to correct a large number of binary insertions, deletions, and substitutions, while it has the high computational complexity for recovering the synchronization. In the DM concatenated scheme, at the large insertion/deletion probability, in order to achieve the reliable output, the watermark decoder needs to perform the forward and backward passes on a very large trellis. In this paper, a threshold is selected to prevent the paths having very low forward and backward quantities from participating in the calculation of log-likelihood ratios from the watermark decoder. Simulation results show that great reduction in the complexity of the decoding algorithm is achieved at a very slight expense of accuracy.

FTN Signaling-Aided Space-Time Multi-Mode Index Modulation Systems With a GMP-Based Receiver

In this paper, we propose a faster-than-Nyquist (FTN) signaling-aided space-time multi-mode index modulation (IM) scheme with a low-complexity Gaussian message passing (GMP) based receiver. The advantages of spatial domain IM, time-domain IM relying on multiple modes and FTN signaling are explored. Specifically, multi-mode IM is employed in time domain by activating all time slots, hence the spectral efficiency is improved substantially compared to its single-mode counterpart. Meanwhile, the transmission rate is further increased by utilizing FTN signaling. At the receiver, based on the state-space model of IM symbols, a two-layer Forney style factor graph (FG) representation for the system is constructed to exploit the truncated inter-symbol interference structure imposed by FTN. Belief propagation is invoked to update the messages over the FG. The discrete messages from the channel decoder are properly approximated to be Gaussian for reducing the complexity of joint detection of multidimensional index modulated symbols. Moreover, the extrinsic information from the equalizer is derived with a concise representation. A low-complexity log-likelihood ratio calculation method is further proposed, where the search space is significantly reduced using a Gaussian mixture model. Simulation results demonstrate that the proposed scheme outperforms the existing schemes in terms of bit error rate given the same spectral efficiency. The outstanding performance and low complexity of the proposed GMP-based receiver are also verified by comparisons with existing methods.

Exact and approximate log-likelihood ratio of M-ary QAM with two-time dimensions

Abstract A ( 3 × 2 m )-ary QAM scheme maps a binary symbol into two-consecutive signal points (signal vector). It provides very lower PAPR than the conventional QAM because it removes corner signal points. For the scheme, this paper discusses an exact analysis of log-likelihood ratio calculation of the signal vector and signal point, and an approximated method to reduce hardware complexity for the channel coding. Numerical results show that the approximated method provides lower hardware complexity and small performance gap compared to exact one. Also, the PAPR and error performances of ( 3 × 2 m )-ary QAM are compared with the one of conventional QAM.

Low-Complexity LLR Calculation for OFDM With Index Modulation

Orthogonal frequency division multiplexing with index modulation (OFDM-IM) is a promising multicarrier transmission technique. In OFDM-IM, only a subset of subcarriers are activated to transmit modulated symbols and the indices of the activated subcarriers are used to convey information bits implicitly. In the coded OFDM-IM system, we need to calculate the log likelihood ratio (LLR) value of each bit, whose computational complexity is extremely high due to the dependence between the subcarriers caused by the subcarrier-index modulation. In this letter, we propose a novel simplified LLR calculation algorithm for OFDM-IM. Simulation results show that the proposed algorithm achieves near-optimal coded bit error rate performance with considerably low computational complexity.

Mixed Numerologies Interference Analysis and Inter-Numerology Interference Cancellation for Windowed OFDM Systems

Extremely diverse service requirements are one of the critical challenges for the upcoming fifth-generation (5G) radio access technologies. As a solution, mixed numerologies transmission is proposed as a new radio air interface by assigning different numerologies to different subbands. However, coexistence of multiple numerologies induces the inter-numerology interference (INI), which deteriorates the system performance. In this paper, a theoretical model for INI is established for windowed orthogonal frequency division multiplexing (W-OFDM) systems. The analytical expression of the INI power is derived as a function of the channel frequency response of interfering subcarrier, the spectral distance separating the aggressor and the victim subcarrier, and the overlapping windows generated by the interferer's transmitter windows and the victim's receiver window. Based on the derived INI power expression, a novel INI cancelation scheme is proposed by dividing the INI into a dominant deterministic part and an equivalent noise part. A soft-output ordered successive interference cancelation (OSIC) algorithm is proposed to cancel the dominant interference, and the residual interference power is utilized as effective noise variance for the calculation of log-likelihood ratios for bits. Numerical analysis shows that the INI theoretical model matches the simulated results, and the proposed interference cancelation algorithm effectively mitigates the INI and outperforms the state-of-the-art W-OFDM receiver algorithms.

Aggregation-Assisted Combining for MIMO Multiple ARQ Systems

This letter proposes an aggregation-assisted combining (AAC) for multiple-input multiple-output multiple automatic repeat request systems. The proposed AAC can be considered as a hybrid scheme of the symbol-level combining (SLC) and bit-level combining (BLC), which consists of the aggregation-assisted log-likelihood ratio (LLR) calculation and subsequent LLR combining stages. The aggregation-assisted LLR calculation stage is performed in a similar manner on the SLC except that the interference-to-noise reformulation is performed instead of the terminated-packet elimination; thus, the remaining interference from the terminated packets is regarded as additional noise. This additional noise can degrade the LLR quality from the aggregation-assisted LLR calculation stage. Therefore, the proposed AAC combines the currently calculated LLRs with the previously calculated LLRs similar to the BLC when the previously calculated LLRs have less LLR contaminations, i.e., they experience fewer numbers of terminated packets. The simulation results verify that the proposed AAC outperforms the BLC and even achieves a performance comparable with the SLC with simplified reception procedures.

Iterative Demodulation and Decoding Algorithm for 3GPP/LTE-A MIMO-OFDM Using Distribution Approximation

Soft iterative detection/decoding algorithms are fundamentally necessary for multiple-input multiple-output orthogonal frequency-division multiplexing (MIMO-OFDM) adopted in the Third Generation Long Term Evolution (LTE)-Advanced in order to increase the capacity and achieve high data rates. However, their high performance critically requires log likelihood ratio computations with prohibitive complexity. This challenge will be addressed in this paper. We first use the assumption of Gaussian transmit symbols to show the equivalence among several existing algorithms. We next develop a non-Gaussian approximation for high-order constellations, which paves the way for interference cancellation-based detectors. Based on both Gaussian and non-Gaussian approximations, we thus develop several capacity-achieving iterative MIMO-OFDM demodulation and decoding algorithms. To this end, we adopt $K$ -best algorithms to take advantage of both the types of approximations and the list decoder. Unlike existing algorithms, our proposed $K$ -best algorithms make use of the a priori probabilities to generate the list. Simulations of standard-compliant LTE systems demonstrate that the proposed algorithms outperform the existing ones.

Low Complexity Log-Likelihood Ratio Calculation Scheme with Bit Shifts and Summations

Low Complexity Log-Likelihood Ratio Calculation Scheme with Bit Shifts and Summations