Reduced-Complexity BCJR Algorithm for Turbo Equalization

This paper presents an optimized turbo equalizer based on Laot's turbo equalization scheme by means of soft intersymbol interference (ISI) cancellation. Due to Berrou's turbo-principle, common turbo equalization schemes use the extrinsic information as a-priori input for the equalizer. In contrast to this, we propose the use of the full soft output channel decoder's information for the turbo interference canceller. The performance of this improved turbo equalizer by means of soft interference cancellation is evaluated in the environment of two channels, which introduce severe ISI. Simulation results show that the proposed turbo equalizer achieves a performance gain of up to 4 dB compared to Laot's original approach. The performance gain is achieved without any increase of computational complexity. Moreover, we apply a modified extrinsic information transfer (EXIT) chart to the proposed turbo equalizer and illustrate the convergence advantages in comparison to Laot's approach.

The turbo equalization principle, based on Berrou's (1993) turbo principle, is a high performance receiver scheme to mitigate intersymbol interference (ISI) caused by a frequency selective channel. Several turbo equalization (TEQ) schemes, like Douillard's (1995) MAP/MLSE TEQ, Laot's (1997) TEQ scheme by means of soft ISI cancellation and turbo equalization based on MMSE linear equalization have been presented different in performance and computational complexity. One of these schemes, turbo equalization based on MMSE linear equalization achieves nearly MAP/MLSE performance, while reducing the computational complexity tremendously. We extend this receiver scheme to 16-QAM and present simulation results for different ISI channels.

Turbo equalization schemes based on minimum mean square error (MMSE) criteria available in the literature for multiple-input multiple-output (MIMO) systems are computationally expensive, as they require a relatively large matrix inversion. In this paper, we propose a low complexity turbo equalization (TE) scheme with successive interference cancellation for orthogonal frequency division multiplexing (OFDM) based MIMO receiver systems (TE-SIC-MIMO). TE-SIC-MIMO leverages on the soft feedback symbol estimate to selectively remove the intercarrier interference (ICI) and co-antenna interference (CAI) from the received data, thus turning the system matrix into an easily invertible form. Extrinsic Information Transfer (EXIT) chart analysis supplemented with numerical simulation results shows that TE-SIC-MIMO performance under static and rapidly time varying multipath conditions is mostly on par with MMSE turbo equalization scheme available in the literature, that is based on a sub-matrix of the system matrix at signal-to-noise ratios (SNR) of practical interest. However, the computation complexity of TE-SIC-MIMO turns out to be much lesser than the TE scheme considered.

Communication over HF-ionospheric channel requires the use of a performant receiver based on turbo equalization scheme. In this receiver, channel estimation can be improved for each iteration using soft information fed back from the decoder. We propose a turbo equalizer based on a soft-input Kalman estimator that can operate without channel impulse response knowledge. Performance analysis of the channel estimator is tested using four paths channel for two Doppler spread values and records of realistic channel. Simulation results show the convergence of the algorithm versus the frequency dispersion of the paths and compare the BER of the proposed Kalman based turbo equalizer with that of the conventional turbo equalizer. Results show that the proposed equalizer provide a significant gains in BER performance and convergence without the need for training data.

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.

In this paper, an enhanced linear minimum mean square error (LMMSE) turbo equalization scheme is proposed for multiple-input multiple-output (MIMO) communication systems with bit-interleaved coded modulation (BICM) in the time domain and multiplexing in the space domain. The proposed turbo equalization scheme outperforms the conventional LMMSE turbo equalization by adopting two new signal processing techniques. First, it performs hybrid soft interference cancellation (HSOIC) by subtracting the soft decisions of the interfering symbols, and the soft decisions are calculated by using a hybrid of the a priori information at the equalizer input and the a posteriori information at the equalizer output. Second, it employs a novel block-wise reliability-based ordering scheme such that more “reliable” symbols are detected before the less “reliable” ones to reduce error propagation in HSOIC. The symbol reliability information is based on the symbol a priori probability, as a unique byproduct of turbo detection, thus can be obtained with very small overhead. A low-complexity approximation of the enhanced MIMO LMMSE turbo equalization is also proposed to balance the tradeoff between complexity and performance. The performance of the enhanced MIMO LMMSE turbo equalization has been verified through both numerical simulations and the undersea experimental data collected in the SPACE08 experiment launched near Martha's Vineyard, Edgartown, MA, in 2008.

Underwater acoustic channels usually suffer from the extended multipath propagation which causes inter symbol interference (ISI). At the receiving end, serious ISI would result in the decrease of the reliability of underwater acoustic communication systems. In this paper, Turbo equalization is applied to increase the reliability of communication systems. The realization of the channel equalizer in Turbo equalization is the linear minimum mean-square error (MMSE) estimate. To avoid performance loss, Turbo equalization and factor graph are combined together which provides a full-window method. Moreover, a modified extrinsic information feedback mechanism of turbo equalization is proposed. The reliability of communication systems is further enhanced by using this mechanism. Simulation results show that the proposed algorithm performs well with low complexity. This demonstrates the potential of the suggested algorithm in practice.

In this paper we consider a soft-input soft-output equalizer, for MIMO intersymbol interference channels, used in a turbo equalization scheme. We derive the equalizer on the basis of a cyclic factor graph representation and the sum-product algorithm, with two different message schedules (serial and parallel). In the sum-product algorithm we use Gaussian messages, which results in a low-complexity LMMSE equalizer implementation. Computer simulations show that, for the turbo equalization application, the proposed equalizer has a good bit error rate performance compared with the optimal MAP equalizer, while achieving a remarkable reduction in complexity.

In this paper, we analyze the sensitivity of MAP algorithm and its reduced ones, as max-log-MAP and scaled max-log-map, to SNR mismatch in turbo detection systems. And then, we propose a new reliability-based SNR estimation method using the soft extrinsic information output from SMLM decoder. Next, a new turbo equalization scheme is also provided with the new SNR estimation algorithm. Finally, the performance of turbo equalization systems are evaluated using our new scheme. The simulation results show that our equalization scheme performs very close to the LM algorithm with known channel SNR, although it does not need pilot symbols.

In this paper, a minimum mean squared error (MMSE) based turbo equalization scheme in delay-Doppler domain for a coded orthogonal time frequency space (OTFS) system is provided as a low complexity version requiring only a single matrix inversion. Also, we derived extrinsic log likelihood ratios (LLRs) for the turbo equalization, exploiting a priori information fed from a channel decoder, under the assumption that the equalized data symbol follows Gaussian distribution. We rigorously conducted link level evaluation to compare coded orthogonal frequency division multiplexing (OFDM) and OTFS over time-varying multipath channels (based on the channel model in 3GPP 5G New Radio) in 30 GHz band, which may cause severe inter-symbol interference (ISI) and/or inter-carrier interference (ICI) depending on numerology setup. The link level simulation verifies that OTFS is superior to OFDM in block error rate (BLER) performance, each being configured with the best numerology, over all the channel scenarios considered. OTFS shows better BLER performance with smaller subcarrier spacing (SCS) while OFDM has an intermediate SCS for the best BLER due to trade-off between ISI and ICI. Also, we measured normalized mean squared error (NMSE) to see how ISI and ICI affect the performance with respect to numerology, which is used to conjecture why the BLER performance has such peculiarity.

The purpose of this paper is to compare bit and symbol interleaving in turbo-equalization (TE) schemes. Considering a single binary encoder and memoryless mapper, the corresponding transmission schemes are respectively bit-interleaved coded modulation (BICM) and trellis-coded modulation (TCM) over a static frequency selective channel. Appropriate turbo processing at the receiver is ensured in both cases. For the sake of tractability, efficient low-complexity MMSE-based implementations are proposed. The asymptotic performance of the resulting bit- and symbol-interleaved TE schemes is then linked with that of the underlying coded modulation schemes. As an extension to [X. Li, J. Ritcey, 1999], this enables to show that bit-interleaved TE with carefully chosen constellation mapping and proper implementation can outperform its symbol-interleaved counterpart.

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.

Turbo codes are applied to magnetic recoding channels by treating the channel as a rate-one convolutional code that requires a soft a posteriori probability (APP) detector for channel inputs. The complexity of conventional APP detectors, such as the BCJR algorithm or the soft-output Viterbi algorithm (SOVA), grows exponentially with the channel memory length. This paper derives a new APP module for binary intersymbol interference (ISI) channels based on minimum mean squared error (MMSE) decision-aided equalization (DAE), whose complexity grows linearly with the channel memory length, and it shows that the MMSE DAE is also optimal by the maximum a posteriori probability (MAP) criterion. The performance of the DAE is analyzed, and an implementable turbo-DAE structure is proposed. The reduction of channel APP detection complexity reaches 95% for a five-tap ISI channel when the DAE is applied. Simulations performed on partial response channels show close to optimum performance for this turbo-DAE structure. Error propagation of the DAE is also studied, and two fixed-delay solutions are proposed based on combining the DAE with the BCJR algorithm.

The performance of turbo-codes relies on two major concepts: code concatenation so as to build a powerful global code and iterative decoding in order to efficiently approximate the optimal decoding process. A turbo equalization (TE) scheme based on convolutional code and normalized least mean square equalizer for coherent optical communication links is proposed and experimentally demonstrated. For counteracting PDM 16 QAM impairments in dispersion uncompensated coherent transmission links, the iterative TE technique is proved effective.

This paper presents a turbo equalization (TEQ) scheme, which employs a radial basis function (RBF)-based equalizer instead of the conventional trellis-based equalizer of Douillard et al. (1995). Structural, computational complexity, and performance comparisons of the RBF-based and trellis-based TEQs are provided. The decision feedback-assisted RBF TEQ is capable of attaining a similar performance to the logarithmic maximum a posteriori scheme in the context of both binary phase-shift keying (BPSK) and quaternary phase-shift keying (QPSK) modulation, while achieving a factor 2.5 and 3 lower computational complexity, respectively. However, there is a 2.5-dB performance loss in the context of 16 quadrature amplitude modulation (QAM), which suffers more dramatically from the phenomenon of erroneous decision-feedback effects. A novel element of our design, in order to further reduce the computational complexity of the RBF TEQ, is that symbol equalizations are invoked at current iterations only if the decoded symbol has a high error probability. This techniques provides 37% and 54% computational complexity reduction compared to the full-complexity RBF TEQ for the BPSK RBF TEQ and 16QAM RBF TEQ, respectively, with little performance degradation, when communicating over dispersive Rayleigh fading channels.