Turbo Equalization Scheme Research Articles

Iterative Equalization and Decoding over an Additive White Gaussian Noise Channel with ISI Using Low-Density Parity-Check Codes

In this article we present an iterative system of equalization and decoding to manage the intersymbol interference over an additive white Gaussian noise (AWGN) channel. Following the classic turbo equalization scheme, the proposed system consists of low-density parity-check (LDPC) coding at the transmitter side; we applied a Log maximum a posteriori probability (Log-MAP) equalizer and min-sum LDPC decoding at the receiver side. The equalizer and decoder, linked through interleaving and deinterleaving, iteratively update each other’s information. We performed the performance analysis of the proposed system, bit error rate (BER) vs. signal-to-noise ratio (SNR), considering three different impulse responses of the channel (h). Our experimental results indicated that increasing the number of iterations performed by the LDPC decoder from 10 to 20 during the iterative process of equalization and decoding leads to better outcomes. The proposed system was compared with turbo equalization and separate equalization, performed before the decoding process with minimum mean-square error (MMSE) and LDPC decoding, in terms of BER vs. SNR, considering the three different h. Based on the analyzed results, it can be concluded that the equalization performance depends on both the impulse responses of the channel and the chosen decoding and equalization method; therefore, the equalization method does not always offer good results for any h.

Expectation-maximization vector approximate message passing-based frequency-domain turbo equalization for underwater acoustic communications.

Channel equalization plays a crucial role in single-carrier underwater acoustic (UWA) communications. Recently, a frequency-domain turbo equalization (FDTE) scheme enabled by the vector approximate message passing (VAMP) algorithm, was proposed, and it outperformed classic linear minimum mean square error FDTE at acceptable complexity cost. The operation of the VAMP-FDTE requires knowledge of noise power, which is predetermined before the equalization starts. In practice, however, it is difficult to obtain prior knowledge of noise power due to factors of unknown channel estimation errors and dynamic underwater environments. Motivated by this fact, we propose an enhanced VAMP-FDTE scheme, which learns the noise power knowledge during the equalization process via the expectation-maximization (EM) algorithm. The EM-based noise power estimation makes use of intermediate results of the VAMP-FDTE and, thus, only incurs a small extra computational overhead. The improved VAMP-FDTE, named EM-VAMP-FDTE, was tested by experimental data collected in shallow-sea horizontal UWA communication trials with MIMO configuration. It showed better performance than the existing VAMP-FDTE scheme, attributed to the online noise power learning.

Turbo DFE Assisted Time-Frequency Packing for Probabilistically Shaped Terabit Superchannels

We present a probabilistically shaped (PS) time-frequency-packing (TFP) wavelength-division multiplexing superchannel system employing higher order modulation (HoM) formats, with an objective to improve the spectral efficiency (SE). However, TFP systems suffer from inter-symbol interference (ISI) and/or inter-carrier interference (ICI). Additionally, the bandwidth limitations of the wavelength selective switches in the fiber link may cause severe ISI for the edge sub-channels (SCs) in a superchannel. Mitigation of such interferences is particularly challenging for HoM systems, since the implementation of the well-known turbo equalization schemes, such as the Bahl-Cocke-Jelinek-Raviv equalizer, is computationally challenging for larger constellations. In this paper, we investigate an expectation propagation based, computationally efficient, turbo decision feedback equalizer for ISI cancellation, in tandem with a parallel interference cancellation based ICI mitigation. By optimizing the parameters in the shaping and the packing dimensions, we show through our numerical results that the proposed PS-TFP superchannels enabling 1.8 Tbps data rates offer up to 1.05 dB packing gain over an unpacked system using the same modulation format, and a 1.15 dB shaping gain over an unshaped system that uses a lower modulation order to achieve the same SE.

Adaptive Turbo Equalization for Nonlinearity Compensation in WDM Systems

In this paper, the performance of adaptive turbo equalization for nonlinearity compensation (NLC) is investigated. A turbo equalization scheme is proposed where a recursive least-squares (RLS) algorithm is used as an adaptive channel estimator to track the time-varying intersymbol interference (ISI) coefficients associated with inter-channel nonlinear interference (NLI) model. The estimated channel coefficients are used by a MIMO $2\times 2$ soft-input soft-output (SISO) linear minimum mean square error (LMMSE) equalizer to compensate for the time-varying ISI. The SISO LMMSE equalizer and the SISO forward error correction (FEC) decoder exchange extrinsic information in every turbo iteration, allowing the receiver to improve the performance of the channel estimation and the equalization, achieving lower bit-error-rate (BER) values. The proposed scheme is investigated for polarization multiplexed 64QAM and 256QAM, although it applies to any proper modulation format. Extensive numerical results are presented. It is shown that the scheme allows up to 0.7 dB extra gain in effectively received signal-to-noise ratio (SNR) and up to 0.2 bits/symbol/pol in generalized mutual information (GMI), on top of the gain provided by single-channel digital backpropagation.

Adaptive Turbo Equalization for Probabilistic Constellation Shaped Underwater Acoustic Communications

To increase the spectral efficiency of the underwater acoustic (UWA) communication system, the high order quadrature amplitude modulations (QAM) are deployed. Recently, the probabilistic constellation shaping (PCS) has been a novel technology to improve the spectral efficiency. The PCS with high-order QAM is introduced into the UWA communication system. A turbo equalization scheme with PCS was proposed to cancel the severe inter-symbol interference (ISI). The non-zero a priori information is available for the equalizer and decoder before turbo iteration. A priori hard decision approach is proposed to improve the detection performance and the equalizer convergence speed. At the initial turbo iteration, the relation between the a priori information and the probability of the amplitude of 16QAM symbols in one dimension is given. The simulation results verified the efficiency of the proposed method, and compared to the uniform distribution (UD), the PCS-16QAM had a significant improvement of the bit error rate (BER) performance with PCS-adaptive turbo equalization (PCS-ATEQ). The UWA communication experiments further verified the performance superiority of the proposed method.

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.

Channel Equalization With Expectation Propagation at Smoothing Level

In this paper we propose a novel turbo equalizer based on the expectation propagation (EP) algorithm. Optimal equalization is computationally unfeasible when high-order modulations and/or large memory channels are used. In these scenarios, low-cost and suboptimal equalizers, such as those based on the linear minimum mean square error (LMMSE), are commonly used. The LMMSE-based equalizer can be efficiently implemented with a Kalman smoother (KS), i.e., a forward and backward Kalman filtering whose predictions are merged in a posterior smoothing step. Recently, it was shown that applying EP at the forward and backward stages of a KS equalizer could significantly improve its performance. In this paper, we investigate applying EP at the smoothing level instead. Also, we propose some further modifications to better exploit the information coming from the channel decoder in turbo equalization schemes. Overall, we remarkably reduce the computational complexity while highly improving the performance in terms of bit error rate.

MMSE‐based iterative processing with imperfect channel and parity check in MIMO systems

It is known that the acquisition of the complete channel state information at receivers is difficult in multiple-input multiple-output (MIMO) systems. Channel estimation error is unavoidable in practical applications. Under imperfect channel conditions, the channel estimate is directly applied to the equalisation process in the conventional minimum mean-square error (MMSE)-based turbo equalisation scheme. A few studies treat the channel estimation error as an independent component from the channel estimate and slightly enhanced performance is achieved. Unlike the existing work, the authors derive the MMSE-based iterative processing conditioned on channel estimate. Moreover, they note that in low-density parity check coded systems, the parity-check procedure is also involved. The pass in parity check indicates that the message bitstream is successfully recovered. This information can be utilised to reduce the overall computational complexity by degrading the MIMO size since the unknown parameters are reduced. By extending the analysis in a small-scale MIMO system to a large-scale one, they propose to utilise the normalised transmission power in the development. Numerical results show the proposed schemes outperform the existing schemes in terms of system bit error rate and computational complexity performance.

Iterative Interference Cancellation for Coded Filter Bank Multicarrier Systems

Filter bank multicarrier is one of the candidates for future communication systems. Simple equalization methods cannot be directly applied due to the high interference from adjacent channels. In this article, the authors derive a soft-input/soft-output (SISO) equalizer based on the minimum mean square error (MMSE) criterion for the bit-interleaved coded system using a filter bank multi-carrier scheme with offset quadrature amplitude modulation (FBMC/OQAM). The authors use this SISO-MMSE equalizer in a turbo-equalization scheme for each sub-carrier. The difficulty in this implementation comes from the required processing delay per turbo-iteration due to the non-causal nature of the interference in this system. Therefore, the number of turbo-iterations is limited in order to limit the processing delay. The authors evaluate the performance of the proposed turbo-equalizer over the International Telecommunication Union (ITU) Vehicular B channel by mean of numerical simulations. The obtained results show the effectiveness of the proposed equalizer in term of signal-to-noise ratio (SNR) gain.

Nonlinear MMSE Equalizer for Impulsive Noise Mitigation in OFDM-Based Communications

Destructive effects of impulsive noise has been broadly observed not only in wireless communication systems but also in power-line communications. Impulsive noise is a common impediment in orthogonal frequency division multiplexing (OFDM) based communication systems for industry applications. This non-Gaussian noise degrades the performance of conventional equalizers and, hence, elicit a modified version of the equalizer that fits the non-Gaussian description of the noise. This letter proposes a nonlinear minimum mean square error equalizer for OFDM systems where the characteristics of the added noise to the system is known. The soft values were obtained based on the derivation of the equalizer for a memory-less channel impaired with impulsive noise. Obtaining such values are required for the implementation of a turbo-equalization scheme. The validity of such an equalizer is tested through simulations and the result of simulations shows that the nonlinear equalizer is successful in combating the effect of an impulsive noise. Thus, the turbo-coded OFDM system shows a significant boost at low signal-to-noise ratios.

Iterative Approaches for Massive MIMO Uplink Processing Under Imperfect Channel Conditions

Obtaining channel state information in massive multiple-input multiple-output (MIMO) systems is challenging because of significant multiple-access interference and very limited resources available for channel estimation. Most existing equalization schemes for massive MIMO systems assume perfect channel knowledge at the receivers. When channel estimation error is present, however, significant performance degradation will be experienced, which has been demonstrated in the literature. In this paper, we model the maximum-likelihood channel estimation as the sum of real channel and random channel estimation error, and develop a minimum mean-square error (MMSE) based turbo equalization method conditioned on the channel estimate. We demonstrate that our newly developed iterative processing approach is a general expression, whereas existing methods can be viewed as special cases. We further derive an MMSE-sorted QR decomposition (MMSE-SQRD) based turbo equalization to further improve the performance for massive MIMO systems under large user numbers and severe channel estimation error conditions. Numerical results and performance comparison show that the proposed MMSE-based turbo equalization scheme greatly outperforms the existing schemes. With a large number of users being simultaneously served, our proposed MMSE-SQRD-based scheme can further enhance system performance over the MMSE-based turbo equalization scheme.

Time-Frequency Domain Turbo Equalization for Single-Carrier Underwater Acoustic Communications

Single-carrier frequency-domain equalization (SC-FDE) is a promising technique for coherent high-rate underwater acoustic (UWA) communications. In this paper, a time-frequency domain turbo equalization (TFD-TE) scheme is proposed for single-carrier modulation. The TFD-TE scheme is to couple a multi-channel frequency domain equalizer with phase rotation compensation and a low-order single-channel time-domain turbo equalizer. To mitigate the error propagation and further enhance receiver performance, a bidirectional equalizer that combined the soft outputs of a conventional equalizer and a backward equalizer is introduced into the proposed TFD-TE scheme (Bi-TFD-TF). The sparse channel is estimated by the improved proportionate normalized least mean squares (IPNLMS) algorithm block by block. Both the simulated and experimental results show the proposed TFD-TE can effectively improve system performance with acceptable complexity. The proposed TFD-TE scheme is evaluated using communication data collected from the DJK15 lake experiment. For the $1\times 4$ single-input multiple-output (SIMO) communication system using quadrature phase shift keying (QPSK) and 8-phase shift keying (8PSK) modulations, the proposed scheme achieves error-free transmission after a few iterations. Our receiver achieves a maximum raw data rate of 12 kbit/s and a reliable net data rate of 4 kbit/s within 4-kHz bandwidth at a range of 10.8 km. Compared with conventional frequency domain turbo equalization (FD-TE), the proposed TFD-TE scheme can enhance 0.8–1.3 dB in terms of output signal-to-noise ratio (OSNR). The related bi-TFD-TE provides an additional 0.8–1.7-dB OSNR improvements.

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.

MMSE Turbo Equalization and Detection for Multicarrier Faster-Than-Nyquist Signaling

In this paper, two minimum mean-squared error (MMSE) turbo equalization and detection schemes for spectral efficient multicarrier faster-than-Nyquist (MFTN) signaling system are presented. First, the two-dimensional (2-D) soft input and soft output (SISO) MMSE equalization intended for magnetic recording channels, i.e., 2-D intersymbol interference (ISI), is extended to the detection of MFTN signals. Although the 2-D MMSE turbo equalization exhibits an attractive bit-error-rate (BER) performance under certain time-frequency spacing packing between adjacent symbols/subcarriers, the BER performance will be degraded when introducing serious intercarrier interference. Second, a 1-D SISO MMSE equalization combined with soft successive interference cancellation (SIC) is further proposed for the detection of MFTN signals, and it shows that the MMSE equalization combined with SIC could asymptotically approach the maximum a posterior equalization combined with SIC with negligible BER performance loss. Our computational complexity analysis and numerous simulation results demonstrate that the proposed MMSE turbo equalization schemes may be more preferred choice in the practical detection of MFTN signals.

Soft-Decision-Driven Sparse Channel Estimation and Turbo Equalization for MIMO Underwater Acoustic Communications

Multi-input multi-output (MIMO) detection based on turbo principle has been shown to provide a great enhancement in the throughput and reliability of underwater acoustic (UWA) communication systems. Benefits of the iterative detection in MIMO systems, however, can be obtained only when a high quality channel estimation is ensured. In this paper, we develop a new soft-decision-driven sparse channel estimation and turbo equalization scheme in the triply selective MIMO UWA. First, the Homotopy recursive least square dichotomous coordinate descent (Homotopy RLS-DCD) adaptive algorithm, recently proposed for sparse single-input single-output system identification, is extended to adaptively estimate rapid time-varying MIMO sparse channels. Next, the more reliable a posteriori soft-decision symbols, instead of the hard decision symbols or the a priori soft-decision symbols, at the equalizer output, are not only feedback to the Homotopy RLS-DCD-based channel estimator but also to the minimum mean-square-error (MMSE) equalizer. As the turbo iterations progress, the accuracy of channel estimation and the quality of the MMSE equalizer are improved gradually, leading to the enhancement in the turbo equalization performance. This also allows the reduction in pilot overhead. The proposed receiver has been tested by using the data collected from the SHLake2013 experiment. The performance of the receiver is evaluated for various modulation schemes, channel estimators, and MIMO sizes. Experimental results demonstrate that the proposed a posteriori soft-decision-driven sparse channel estimation based on the Homotopy RLS-DCD algorithm and turbo equalization offer considerable improvement in system performance over other turbo equalization schemes.

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.

Two‐dimensional iterative soft decision interference cancellation turbo equaliser

A novel iterative equaliser, the 2-dimensional iterative soft decision interference cancellation (ISDIC) (2D ISDIC) equaliser, is introduced. The 2D ISDIC is the first equaliser that bridges the gap between the low-complexity ISDIC equaliser which has poor performance, and the high-complexity inverse-channel equalising methods, such as the LMMSE and the MMSE ISDIC equalisers, which have better performances. The 2D ISDIC will be incorporated into the turbo equalisation scheme, resulting in an enhanced turbo equaliser, that performs better than the conventional, state of the art, linear turbo equaliser, with lower complexity and lower amount of iterations.

Expectation Propagation as Turbo Equalizer in ISI Channels

In probabilistic equalization of channels with inter-symbol interference, the BCJR algorithm and its approximations become intractable for high-order modulations, even for moderate channel dispersions. In this paper, we introduce a novel soft equalizer to approximate the symbol a posteriori probabilities (APP), where the expectation propagation (EP) algorithm is used to provide an accurate estimation. This new soft equalizer is presented as a block solution, denoted as block-EP (BEP), where the structure of the matrices involved is exploited to reduce the complexity order to $ {\mathcal {O}}( L N ^{2})$ , i.e., linear in the length of the channel, $ L$ , and quadratic in the frame length, $ N$ . The solution is presented in complex-valued formulation within a turbo equalization scheme. This algorithm can be cast as a linear minimum-mean-squared-error (LMMSE) turbo equalization with double feedback architecture, where constellations being discrete is a restriction exploited by the EP that provides a first refinement of the APP. In the experiments included, the BEP exhibits a robust performance, regardless of the channel response, with gains in the range 1.5–5 dB compared with the LMMSE equalization.

Equivalence of Frequency Domain Turbo Equalization Schemes for Single Carrier Multiple-Input Multiple-Output Systems

This paper investigates two typical frequency-domain turbo equalization (FDTE) schemes, i.e., frequency-domain equalization (FDE) with parallel interference cancelation (FDE-PIC) and FDE with frequency-domain decision feedback (FDE-FDDF) in single-carrier (SC) multiple-input–multiple-output (MIMO) systems. These two FDTE schemes are proven to be equivalent under the condition that we derived. We discover that the covariance matrix of the low-complexity FDE-PIC scheme and the correlation matrix of the FDE-FDDF scheme satisfy a unique relationship for constant-modulus modulations, and the original FDE-FDDF scheme suffers numerical instability for non-constant-modulus modulations because the estimation of the correlation matrices leads to the inversion of ill-conditioned matrices. By proposing a new estimation method of the correlation matrices, we guarantee that the non-constant-modulus FDE-FDDF scheme is numerically stable and that the relationship between these two FDTE schemes holds for both constant-modulus and non-constant-modulus modulations. Furthermore, we prove that the FDE-PIC and FDE-FDDF schemes are equivalent under this relationship. The equivalence of these two FDTE schemes is also verified through numerical simulations in terms of bit error rate (BER) at each turbo iteration and the estimation of the correlation matrices.

Joint ML Estimation of CFO and Channel, and a Low Complexity Turbo Equalization Technique for High Mobility OFDMA Uplinks

In this paper, we propose a pilot aided joint estimation technique for carrier frequency offsets (CFOs) and doubly selective channels (DSCs), and a low complexity turbo equalization (TE) scheme for orthogonal frequency division multiple access (OFDMA) uplink systems with high mobility users. We propose the use of Chebyshev polynomials of the first kind (CPF) for accurately representing the DSCs. As the exact solution to the estimation problem is computationally intensive, we make use of space alternating generalized expectation maximization (SAGE) algorithm, that replaces the complex multidimensional search with many one dimensional searches. We derive the exact Cramer-Rao bound (CRB) of the joint estimation problem and the mean squared error of the proposed estimation technique is shown to be closely comparable to it. The convergence characteristics are also studied and it is proved analytically that the proposed method converges. Additionally, we propose a low complexity turbo equalization method with iterative multiple access interference (MAI) cancellation, which uses the soft feedback symbol estimate for removing the MAI and self inter-carrier interference (ICI) in each iteration. Our proposed estimation and equalization technique can be applied to any type of carrier assignment scheme (CAS) in OFDMA and offers very good performance even at high mobile speeds.