Space-time turbo equalization in frequency-selective MIMO channels

A computationally efficient space-time turbo equalization algorithm is derived for frequency-selective multiple-input-multiple-output (MIMO) channels. The algorithm is an extension of the iterative equalization algorithm by Reynolds and Wang (see Signal Processing, vol.81, no.5, p.989-995, 2001) for frequency-selective fading channels and of iterative multiuser detection for code-division multiple-access (CDMA) systems by Wang and Poor (see IEEE Trans. Commun., vol.47, p.1046-1061, 1999). The proposed algorithm is implemented as a MIMO detector consisting of a soft-input-soft-output (SISO) linear MMSE detector followed by SISO channel decoders for the multiple users. The detector first forms a soft replica of each composite interfering signal using the log likelihood ratio (LLR), fed back from the SISO channel decoders, of the transmitted coded symbols and subtracts it from the received signal vector. Linear adaptive filtering then takes place to suppress the interference residuals: filter taps are adjusted based on the minimum mean square error (MMSE) criterion. The LLR is then calculated for adaptive filter output. This process is repeated in an iterative fashion to enhance signal-detection performance. This paper also discusses the performance sensitivity of the proposed algorithm to channel-estimation error. A channel-estimation scheme is introduced that works with the iterative MIMO equalization process to reduce estimation errors.     

Linear turbo equalization for parallel ISI channels

We propose a method for exploiting transmit diversity using parallel independent intersymbol interference channels together with an iterative equalizing receiver. Linear iterative turbo equalization (LITE) employs an interleaver in the transmitter and passes a priori information on the transmitted symbols between multiple soft-input/soft-output minimum mean-square error linear equalizers in the receiver. We describe the LITE algorithm, present simulations for both stationary and fading channels, and develop a framework for analyzing the evolution of the a priori information as the algorithm iterates.     

Adaptive turbo reduced-rank equalization for MIMO channels

An adaptive iterative (turbo) decision-feedback equalizer (DFE) for channels with intersymbol interference (ISI) is presented. The filters are computed directly from the soft decisions and received data to minimize a least-squares (LS) cost function. Numerical results show that this method gives a substantial improvement in performance relative to a turbo DFE computed from an exact channel estimate, assuming perfect feedback. Adaptive reduced-rank estimation methods are also presented, based on the multistage Wiener filter (MSWF). The adaptive reduced-rank turbo DFE for single-input/single-output channels is extended to multiple-input/multiple-output (MIMO) channels with ISI and multiple receive antennas. Numerical results show that for MIMO channels with limited training, the reduced-rank turbo DFE can perform significantly better than the full-rank turbo DFE.     

On the performance of turbo equalization for precoded ISI channels

In this paper, we present a semi‐analytical approach for analyzing the serial concatenation (SCC) of a high rate convolutional code and a precoded intersymbol interference (ISI) channel. The significance of the proposed approach is that it can be used to identify, for a given outer high rate code and fixed ISI channel, the precoder that results in the lowest error rate floor. In estimating the bit error performance in the floor region, we use analytical techniques developed for trellis codes to compute the overall system minimum squared Euclidean distance, and then use a semi‐analytical approach to find the corresponding multiplicity. We also demonstrate via simulations that the proposed technique may be extended to fading ISI channels with favorable results. We give examples that support our analysis. Copyright © 2006 John Wiley & Sons, Ltd.     

CRC-aided turbo equalization for MIMO frequency selective fading channels

This paper presents a CRC （Cyclic Redundancy Check）-aided turbo equalization approach to reduce the computational complexity. In this approach, CRC code bits are padded to the end of each transmit block, and a cyclic redundancy check is performed after decoding each block at the receiver en.d. If the check sum is zero, which means the receive block is correct, the corresponding LLRs （Log Likelihood Ratios） of this block are set high reliable values, and all the computations corresponding to this block can be cancelled for the subsequent outer iterations. With a lower computational complexity the proposed approach can achieve the same as or even better performance than the conventional non-CRC method.     

The soft-feedback equalizer for turbo equalization of highly dispersive channels

The complexity of a turbo equalizer based on the Bahl-Cocke-Jelinek-Raviv (BCJR) algorithm is manageable only for mildly dispersive channels having a small amount of memory. To enable turbo equalization of highly dispersive channels, we propose the soft-feedback equalizer(SFE). The SFE combines linear equalization and soft intersymbol-interference cancellation. Its coefficients are chosen to minimize the mean-squared error(MSE) between the equalizer output and the transmitted sequence, under a Gaussian approximation to the a priori information and the SFE output. The resulting complexity grows only linearly with the number of coefficients, as opposed to the quadratic complexity of previously reported minimum-MSE structures. We will see that an SFE-based turbo equalizer consistently outperforms another structure of similar complexity, and can outperform a BCJR-based scheme when complexity is taken into account.     

Structured set partitions and multilevel concatenated coding forpartial response channels

We present a systematic way to construct multilevel concatenated codes for partial response (PR) channels using: (1) a structured set partition (SSP) of multiple channel output sets and (2) a set of conventional block codes with different error correcting capabilities. A lower bound on the minimum squared Euclidean distance of the constructed codes is given. This bound is based on the interset minimal Euclidean distances of the SSP and the minimum Hamming distances of the used block codes. An example of SSP for the extended class 4 partial response channel (EPR4) is presented. Iterative suboptimal decoding, which combines Viterbi detection on the trellis of the PR channel with algebraic error detection/correction, can be applied to the constructed concatenated codes. Truncated versions of the iterative decoding scheme are simulated and compared with each other     

Iterative channel estimation for turbo equalization of time-varying frequency-selective channels

We investigate turbo equalization, or iterative equalization and decoding, as a receiver technology for systems where data is protected by an error-correcting code, shuffled by an interleaver, and mapped onto a signal constellation for transmission over a frequency-selective channel with unknown time-varying channel impulse response. The focus is the concept of soft iterative channel estimation, which is to improve the channel estimate over the iterations by using soft information fed back from the decoder from the previous iteration to generate "extended training sequences" between the actual transmitted training sequences.     

Bit-interleaved turbo equalization over static frequency-selective channels: constellation mapping impact

The purpose is to assess the performance of bit-interleaved turbo equalization (TE) over static frequency-selective channels. The asymptotic performance is therefore first pointed out, emphasizing the fundamental role played by the constellation mapping. This specific feature is then further analyzed using the extrinsic information-transfer chart technique, leading to an efficient optimization tool. This finally enables showing that bit-interleaved TE can outperform its symbol-interleaved counterpart.     

Cyclic delay diversity with frequency domain turbo equalization for uplink fast fading channels

Cyclic delay diversity (CDD) is an attractive diversity technique due to its low complexity and compatibility to existing wireless communication systems. This letter proposes a CDD with frequency domain turbo equalization (FDTE) for single carrier (SC) transmission, in order to achieve the full spatial diversity of frequency-selective multi-antenna channels. The frequency diversity inherent in SC is picked up from the increased channel selectivity of CDD. The noise or intersymbol interference enhanced by equalization for highly selective channels is then mitigated through applying FDTE at the receiver. Simulation results show that the performance of proposed system approaches the corresponding orthogonal spacetime block coding (STBC) system in slowly fading channels without any data rate loss, and considerably outperforms the STBC system in fast fading channels.     

LMMSE turbo equalization based on factor graphs

In this paper, a vector-form factor graph representation is derived for intersymbol interference (ISI) channels. The resultant graphs have a tree-structure that avoids the short cycle problem in existing graph approaches. Based on a joint Gaussian approximation, we establish a connection between the LLR (log-likelihood ratio) estimator for a linear system driven by binary inputs and the LMMSE (linear minimum mean-square error) estimator for a linear system driven by Gaussian inputs. This connection facilitates the application of the recently proposed Gaussian message passing technique to the cycle-free graphs for ISI channels. We also show the equivalence between the proposed approach and the Wang-Poor approach based on the LMMSE principle. An attractive advantage of the proposed approach is its intrinsic parallel structure. Simulation results are provided to demonstrate this property.     

Blind equalization of turbo trellis-coded partial-response continuous-phase modulation signaling over narrow-band Rician fading channels

In this paper, a blind maximum-likelihood channel estimation algorithm is developed for turbo trellis-coded/continuous-phase modulation (TTC/CPM) signals propagating through additive white Gaussian noise (AWGN) and Rician fading environments. We present CPM for TTC signals, since it provides low spectral occupancy and is suitable for power- and bandwidth-limited channels. Here, the Baum-Welch (BW) algorithm is modified to estimate the channel parameters. We investigate the performance of TTC/CPM for 16-CPFSK over AWGN and Rician channels for different frame sizes, in the case of ideal channel state information (CSI), no CSI, and BW estimated CSI.     

Joint MF combining and turbo equalization for wireless communications over ISI channels with diversity reception

A new method called joint Matched Filter (MF) combining and turbo equalization is proposed for wireless communications over Inter-Symbol Interference (ISI) channels with diversity reception. This method takes diversity combining and equalization as integrity and need just one turbo equalizer for all diversity branches. Computer simulations prove that our method can take advantage of turbo equalization and diversity reception to combat fading of wireless channels.     

Blind equalization of multilevel turbo coded‐continuous phase frequency shift keying over MIMO channels

In this paper, multilevel turbo coded‐continuous phase frequency shift keying (MLTC‐CPFSK) is introduced and its bit error performance in multiple input multiple output (MIMO) fading channels are investigated considering a blind maximum likelihood channel estimation. Multilevel turbo coded signals with continuous phase modulation (CPM) provides low spectral occupancy and is suitable for power and bandwidth‐limited channels. Besides, antenna diversity is one of the best method to combat with multipath fading effects. The performance of 2LTC for 4‐ary CPFSK over AWGN, Rician (for Rician channel parameter K=10 dB) and Rayleigh channels are given for 1Tx–1Rx, 2Tx–1Rx and 2Tx–2Rx antenna configurations. Channel capacities of 2LTC‐4CPFSK signals are obtained as ?5.26, ?7.65 and ?7.14 dB for 1Tx–1Rx, 2Tx–1Rx and 2Tx–2Rx antenna configurations, respectively. Baum–Welch (BW) algorithm is used to estimate the channel parameters. Bit error probabilities of 2 level turbo coded 4 CPFSK (2LTC‐4CPFSK) are drawn in the cases of no channel state information (CSI), BW estimation, and perfect CSI. Approximately 0.1 and 0.75 dB gains in E_s/N₀ are obtained using BW channel estimator for Rician and Rayleigh channels, respectively. Therefore, MLTC‐CPFSK with BW channel estimator has excellent performance in MIMO fading channels. Copyright © 2006 John Wiley & Sons, Ltd.     

Low complexity turbo equalization based on soft feedback interference cancellation

This letter presents a new soft feedback interference cancellation (SFIC) based equalizer suitable for iterative receivers applying turbo equalization. SFIC offers a very low computational complexity depending only linearly on the channel memory length. Despite its low complexity, SFIC shows a very good BER performance. Simulation results for the severely intersymbol interference distorted Proakis C channel show, that our approach performs within 0.5 dB to the powerful turbo equalization scheme based on MMSE linear filtering with time-varying coefficients and fails the mathematical optimum maximum a-posteriori (MAP) equalizer only by 1.2 dB.     

Faster-than-Nyquist signaling: on linear and non-linear reduced-complexity turbo equalization

In the framework of digital video broadcasting by satellite-second generation (DVB-S2), we analyze a faster-than-Nyquist (FTN) system based on turbo equalization and low-density parity-check (LDPC) codes. Truncated maximum a posteriori and minimum mean square error equalizers provide a reduced-complexity implementation of the FTN system. On the other hand, LDPC codes allow us to demonstrate attractive performance results over an additive white Gaussian noise channel while increasing spectral efficiency beyond the Nyquist rate and keeping a complexity comparable to that of a current DVB-S2 modem.     

Comparative study of turbo equalization schemes usingconvolutional, convolutional turbo, and block-turbo codes

Turbo equalizers have been shown to be successful in mitigating the effects of inter-symbol interference introduced by partial response modems and by dispersive channels for code rates of R⩽ 1/2. We comparatively studied the performance of a range of binary phase-shift keying turbo equalizers employing block-turbo codes, namely Bose-Chaudhuri-Hocquenghen (1960, 1959) turbo codes, convolutional codes, and convolutional turbo codes having high code rates, such as R=3/4 and R=5/6, over a dispersive five-path Gaussian channel and an equally weighted symbol-spaced five-path Rayleigh fading channel. These turbo equalization schemes were combined with an iterative channel estimation scheme in order to characterize a realistic scenario. The simulation results demonstrated that the turbo-equalized system using convolutional turbo codes was the most robust system for all code rates investigated     

Method to analyse convergence of turbo codes

The use of cross-entropy to analyse the convergence behaviour of a turbo decoder is proposed. Based on the new method, E/sub b//N/sub 0/ thresholds are predicted and compared with those predicted by existing techniques. Simulation results show that the new technique is effective and advantageous in practical applications.     

Performance of multilevel‐turbo codes with blind/non‐blind equalization over WSSUS multipath channels

In this paper, in order to improve error performance, we introduce a new type of turbo codes, called ‘multilevel‐turbo codes (ML‐TC)’ and we evaluate their performance over wide‐sense stationary uncorrelated scattering (WSSUS) multipath channels. The basic idea of ML‐TC scheme is to partition a signal set into several levels and to encode each level separately by a proper component of the turbo encoder. In the considered structure, the parallel input data sequences are encoded by our multilevel scheme and mapped to any modulation type such as MPSK, MQAM, etc. Since WSSUS channels are very severe fading environments, it is needed to pass the received noisy signals through non‐blind or blind equalizers before turbo decoders. In ML‐TC schemes, noisy WSSUS corrupted signal sequence is first processed in equalizer block, then fed into the first level of turbo decoder and the first sequence is estimated from this first Turbo decoder. Subsequently, the other following input sequences of the frame are computed by using the estimated input bit streams of previous levels. Here, as a ML‐TC example, 4PSK 2 level‐turbo codes (2L‐TC) is chosen and its error performance is evaluated in WSSUS channel modelled by COST 207 (Cooperation in the field of Science & Technology, Project #207). It is shown that 2L‐TC signals with equalizer blocks exhibit considerable performance gains even at lower SNR values compared to 8PSK‐turbo trellis coded modulation (TTCM). The simulation results of the proposed scheme have up to 5.5 dB coding gain compared to 8PSK‐TTCM for all cases. It is interesting that after a constant SNR value, 2L‐TC with blind equalizer has better error performance than non‐blind filtered schemes. We conclude that our proposed scheme has promising results compared to classical schemes for all SNR values in WSSUS channels. Copyright © 2005 John Wiley & Sons, Ltd.     

Turbo equalization of doubly selective channels

In this paper, turbo equalization for transmission over doubly selective channels is proposed. The maximum a posteriori probability (MAP) algorithm is used for channel detection as well as for channel decoding. The detection/decoding constituents can exchange soft information in an iterative manner resulting in the so‐called turbo equalization. The time‐varying multi‐path fading channel is modeled using the basis expansion model (BEM). In this BEM, the time‐varying channel is viewed as a bank of time‐invariant finite impulse response filters, and the time variation is captured by means of time‐varying complex exponential basis functions. Therefore, the time‐varying transition tables that characterize the time‐varying channel can also follow a similar BEM. The complexity of the MAP channel detector is rather prohibitive for practical applications. This motivates the search for lower‐complexity soft‐output channel detectors. For this purpose, soft‐output linear minimum‐mean square error (LMMSE)‐based channel detectors are proposed for single carrier as well as for multi‐carrier systems. With the use of Gaussian approximation, expressions for the a posteriori and extrinsic log‐likelihood ratios have been derived. The performance of the proposed turbo equalization schemes are evaluated using numerical simulations. Copyright © 2012 John Wiley & Sons, Ltd.