Iterative decoding and turbo equalization: The Z-crease phenomenon

We track the density of extrinsic information in iterative turbo decoders by actual density evolution, and also approximate it by symmetric Gaussian density functions. The approximate model is verified by experimental measurements. We view the evolution of these density functions through an iterative decoder as a nonlinear dynamical system with feedback. Iterative decoding of turbo codes and of serially concatenated codes is analyzed by examining whether a signal-to-noise ratio (SNR) for the extrinsic information keeps growing with iterations. We define a for the iterative decoder, such that the turbo decoder will converge to the correct codeword if the noise figure is bounded by a number below zero dB. By decomposing the code's noise figure into individual curves of output SNR versus input SNR corresponding to the individual constituent codes, we gain many new insights into the performance of the iterative decoder for different constituents. Many mysteries of turbo codes are explained based on this analysis. For example, we show why certain codes converge better with iterative decoding than more powerful codes which are only suitable for maximum likelihood decoding. The roles of systematic bits and of recursive convolutional codes as constituents of turbo codes are crystallized. The analysis is generalized to serial concatenations of mixtures of complementary outer and inner constituent codes. Design examples are given to optimize mixture codes to achieve low iterative decoding thresholds on the signal-to-noise ratio of the channel.

Iterative turbo equalization is capable of achieving impressive performance gains over the conventional non-iterative equalization having the same complexity, when communicating over channels that suffer from intersymbol interference (ISI). The state-of-the-art turbo equalizers employ the logarithmic Bahl-Cocke-Jelinek-Raviv (Log-BCJR) algorithm. However, due to the specific nature of serial data processing, the Log-BCJR algorithm introduces significant processing delays at the receiver. Therefore, in low-latency applications having a high throughput, the turbo equalizer might be deemed less attractive than its conventional counterparts. In order to circumvent this problem, in this paper, we conceived a novel fully parallel turbo equalization algorithm, which is capable of significantly reducing the data processing delay and, hence, improving both the processing latency and the attainable throughput at the receiver. The fully parallel equalizer is then combined with the fully parallel turbo decoder for improving the system performance achieved in terms of the bit error ratio. Furthermore, we propose a novel odd-even interleaver design for employment between the fully parallel equalizer and the fully parallel turbo decoder in order to reduce complexity by 50% in fully parallel turbo equalization arrangements, while retaining a comparable performance. Finally, we compare the computational complexity, latency, throughput, hardware resource requirements, and the bit error ratio of the proposed fully parallel scheme to those of a Log-BCJR-based turbo equalizer benchmarker.

A number of authors have sought to combine equalization and decoding in an iterative system in order to reduce the effects of frequency and temporal dispersiveness of time-varying frequency-selective channels. In the recent literature, several architectures based on these iterative systems have been proposed. One can note architectures which combine a maximum a posteriori (MAP) equalizer and decoder, architectures formed by an interference canceller and MAP decoder, architectures implementing a decision feedback equalizer and a MAP decoder, etc. Most of these architectures require accurate channel estimation to adapt the equalizer filters or to execute the MAP equalizers. This article presents a turbo equalizer architecture for time-varying frequency-selective channels without channel estimators. The proposed turbo equalizer consists of an interference canceller in direct-adaptation mode and a turbo decoder. In order to reduce noise correlation, the addition of a transverse filter to the interference-canceller architecture is proposed. The reliability factor for variable time-varying frequency-selective channels is also redefined in order to improve the performance of the turbo decoder. The architecture of the proposed turbo equalizer reduces considerably the effects of frequency and temporal dispersiveness of time-varying frequency-selective channels depending on the normalized Doppler frequency range.

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 proposes an improved Max-Log-maximum a posteriori (MAP) algorithm for turbo decoding and turbo equalization. The proposed algorithm utilizes the MacLaurin Series to expand the logarithmic term in the Jacobian logarithmic function of the Log-MAP algorithm. In terms of complexity, the proposed algorithm can easily be implemented by means of adders and comparators as this is the case for the Max-Log-MAP algorithm. In addition, simulation results show that the proposed algorithm performs very closely to the Log-MAP algorithm for both turbo decoding over additive-white-Gaussian-noise channels and turbo equalization over frequency-selective channels. Further, it is shown than even in a high-loss intersymbol-interference channel, the proposed algorithm preserves its performance close to that of the Log-Map algorithm, while there is a wide gap between the performance of the Log-MAP and Max-Log-MAP turbo equalizers

A more robust version of the nonlinear M-decorrelating detector through iterative (turbo) decoding is proposed. The turbo code performance in the linear decorrelating detector is also studied. Simulation results indicate that combined nonlinear signal processing and turbo decoding can achieve highly robust performance in impulsive noise.

In this paper, we extend the iterative demapping technique to observation separation (OS), which is an iterative channel estimation and symbol detection procedure. We consider a bit interleaved coded modulation with iterative decoding (BICM-ID) on a multipath channel with coded packet oriented transmission system with 16QAM. The channels considered are the time-invariant multipath and the channel estimation corresponds to the pseudo-inverse type. We propose here to study the influence of mapping over the performance of turbo equalization. Turbo equalization is a receiving process performing iteratively equalization and channel decoding; soft information generated by each receiving function is improved through the iterations until the optimum behavior of the system reached. In the studied system the turbo equalization and the channel estimation part is based on our previous work. In the context of bit interleaved coded modulation (BICM), the iterative demapping is an iterative process between the soft demapping device and the channel decoding. We analyze here the improvement of mapping optimization for turbo equalization and we proposed an optimized binary mapping of 16-QAM constellation in order to increase coding gain, which can be used for some practically important cases. It is demonstrated that owning to a carefully designed mapping, different to the classical Gray mapping, bit error rate performance is improved.

We consider differential MIMO architecture based on Cayley transform (Hassibi and Hochwald, 2002) when the channel state information is available neither at the transmitter nor the receiver. Within this framework, a low complexity iterative receiver is proposed based on a linearly complex space matched filtering signal detection, iterative probabilistic data association (PDA) and turbo equalization. Simulation results on Rayleigh faded MIMO channels illustrate good performances of the designed receiver.

A novel low-complexity scheme of Turbo Trellis-coded modulation (TCM) iterative equalisation and decoding is presented, which serially concatenates a soft-input-soft-output equaliser and symbol-by-symbol Turbo TCM (TTCM) decoder through an iterative process in order to realise iterative equalisation and decoding process. This scheme can not only improve the bandwidth efficiency, but also reduce the intersymbol interference. The equaliser with maximum a posteriori is replaced with linear one based on minimum mean-square error at the receiver so that the complexity of the system can be reduced greatly. Computer simulation results show that the novel low-complexity receiver on fading channels can achieve comparable performance with TTCM on Gaussian channels.

Turbo codes are proposed in most of the advanced digital communication standards, such as 3GPP-LTE. However, due to its computational complexity, the turbo decoder is one of the most power hungry blocks in digital baseband. To alleviate this issue, one way is to avoid surplus computing phases thanks to the early termination of the iterative decoding process. The use of stopping criteria is one of the most common algorithm level power reduction methods in literature. These methods always come with some hardware overhead. In this paper, a new trellis based stopping criterion is proposed. The novelty of this approach is the lower hardware overhead thanks to the use of trellis states as key parameter to stop the iterative process. Results are showing the importance of this added hardware in terms of method efficiency. Compared to state-of-the-art Log Likelihood Ratio (LLR) based techniques, proposed Low Complexity Trellis Based (LCTB) is demonstrating 23% less power consumption on average, for comparable performance level in terms of Bit Error Rate (BER) and Frame Error Rate (FER).

The author proposed soft decision feedback equalisation for orthogonal frequency division multiplexing (OFDM) transmission over doubly selective channels which is used for turbo equalisation. The doubly selective channel is approximated using the basis expansion model (BEM). The decision feedback equaliser (DFE) is obtained and updated for every sub-carrier which results in a per-tone DFE. On the basis of the Gaussian approximation, expressions for the a priori, the a posteriori and the extrinsic information have been derived for quadrature phase shift keying (QPSK) modulation. For turbo equalisation, where iterative joint channel equalisation and channel decoding is executed, in the first iteration, estimates are based on the feed-forward filter which are fed to the channel decoder. These estimates are not reliable and may incur error propagation and poor turbo equalisation performance. For more reliable estimates in the first iteration, the DFE is initialised as a linear minimum mean square error (LMMSE) equaliser. The per-tone LMMSE soft-output equaliser is also derived, and expressions for the a priori, a posteriori and extrinsic information are obtained viewing the LMMSE equaliser as a special case of the DFE. The extrinsic information in both cases was derived based on the Gaussian approximation. Numerical results for bit-error rate performance assuming the BEM channel model and Jakes' channel model are shown to confirm our findings.

In this paper, we propose a low complexity turbo coded BICM-ID (Bit-Interleaved Coded Modulation with Iterative Decoding) scheme. To increase the spectral efficiency, capacity approaching turbo codes can be combined with high order modulations and form a turbo coded modulation (CM) or a turbo-BICM (Bit-Interleaved Coded Modulation). On the other hand, it is also well known that the performance of turbo-BICM can be further improved by performing iterative demodulation and decoding between demodulator and turbo decoder which known as turbo-BICM-ID. However, compared to turbo-BICM, the decoding complexity of turbo-BICM-ID is highly increased. To reduce the required complexity of turbo-BICM-ID, we propose a low complexity turbo-BICM-ID scheme which offers virtually identical performance compared to turbo-BICM-ID.

Iterative or Turbo decoding of parallel concatenated binary error control codes in Binary Phase Shift Keyed (BPSK) Communication System corrupted by both Additive White Gaussian Noise (AWGN) and oscillator phase noise is discussed. As an example of application of the discussed theory, iterative decoding technique of Parallel Concatenated block Codes (PCBC) is applied to a coherent Optical Code Division Multiple Access (OCDMA) network in the presence of Multiple Access Interference (MAI), detector shot noise and laser phase noise. Remarkable improvement in performance resulting from the use of turbo decoding in the OCDMA network is also discussed.

A novel receiver is proposed for CDMA links with recursive turbo coding, based on a soft-input soft-output multistage interference canceler (IC) and an iterative turbo decoder (TD). After performing IC and TD a first time, the coder state estimates available from the two soft-output Viterbi algorithms in the last TD iteration are exploited to calculate the data bits and the redundancy bits. These are employed for preliminary hard cancellation in the first iteration of a supplementary IC run, and then the TD is run again. An alternative solution with soft feedback is obtained by approximating the reliabilities of the redundancy bits with those of the coder state estimates, and using them for preliminary soft cancellation. Simulation results show the effectiveness of the proposed solutions.

Several researches investigate the performance of the conventional adaptive equalizers for the medium data rates HF waveforms (up to 8 PSK) and show that they are suitable for mitigating the HF channel. However, the performance of these types of equalizers is very bad for the high data rates HF waveforms (up to 64QAM). Most of the later researches in the high data rates HF waveforms are based on the soft turbo equalizers which are very complicated. In this paper, we try to design the lowest complex equalizer can mitigate the HF channel for the high data rates waveforms. We investigate the performance of the minimum mean square error decision feedback equalizer (MMSE-DFE) indirectly adaptive through channel estimation based on least square estimation (LS). We show that the basic performance of the MMSE-DFE cannot satisfy the standard requirements, and then we introduce two additional enhancements. First, an effective channel estimation technique (double estimation) is proposed. The new estimator shows a clear performance enhancement over simple least square estimation with linear interpolation (about 3 dB gain at 1e-5 BER for the 64QAM). The second enhancement is to use the iterative structure through decoder using the same conventional equalizer and decoder and this gives another 3 dB gain at 1e-5 BER for the 64QAM. Iterative structure is less complicated than turbo equalizers for sure, as turbo equalizers require a lot of changes in the equalizer, decoder and demapper blocks to be SISO modules. Final equalizer model satisfies all the strict requirements of both (STANAG 4539 annex B) and (MIL-STD-188-110B app. C) standards under the ITU-R Poor channel conditions. In addition, we compare our performance with two different vendors of HF modems and other different research and show that our performance is very competitive.