The problem of iterative detection/decoding of data symbols transmitted over an additive white Gaussian noise (AWGN) channel in the presence of phase uncertainty is addressed in this paper. By modelling the phase uncertainty either as an unknown deterministic variable/process or random variable/ process with a known a priori probability density function, a number of non-Bayesian and Bayesian detection algorithms with various amount of suboptimality have been proposed in the literature to solve the problem. In this paper, a new set of suboptimal iterative detection algorithms is obtained by utilizing the variational bounding technique. Especially, applying the generic variational Bayesian (VB) framework, efficient iterative joint estimation and detection/decoding schemes are derived for the constant phase model as well as for the dynamic phase model. In addition, the relation of the VB-based approach to the optimal noncoherent receiver as well as to the classical approach via the expectation-maximization (EM) algorithm is provided. Performance of the proposed detectors in the presence of a strong dynamic phase noise is compared to the performance of the existing detectors. Furthermore, an incremental scheduling of the VB (or EM) algorithm is shown to reduce the overall complexity of the receiver.

Many algorithms in signal processing and digital communications must deal with the problem of computing the probabilities of the hidden state variables given the observations, i.e., the inference problem, as well as with the problem of estimating the model parameters. Such an inference and estimation problem is encountered, for example, in adaptive turbo equalization/demodulation where soft information about the transmitted data symbols has to be inferred in the presence of the channel uncertainty, given the received signal samples anda prioriinformation provided by the decoder. An exact inference algorithm computes thea posterioriprobability (APP) values for all transmitted symbols, but the computation of App-s is known to be an NP-hard problem, thus rendering this approach computationally prohibitive in most cases. We show in this paper how may of the well-known low complexity soft-input soft-output (SISO) equalizers, including the channel matched filter based linear SISO equalizers and minimum mean square error (MMSE) SISO equalizers as well as the expectation-maximization (EM) algorithm based SISO demodulators in the presence of the Rayleigh fading channel, can be formulated as solutons to a variational optimization problem. The variational optimization is a well-established methodology for low-complexity inference and estimation, originating from statistical physics. Importantly, the imposed variational optimization framework provides an interesting link between the APP demodulators and the linear SISO equalizers. Moreover, it provides a new set of insights into the structure and performance of these widely celebrated linear SISO equalizers while suggesting some fine tuning of them as well.

Many algorithms in signal processing and digital communications must deal with the problem of computing the probabilities of the hidden state variables given the observations, i.e., the inference problem, as well as with the problem of estimating the model parameters. Such an inference and estimation problem is encountered, for e.g., in adaptive turbo equalization/demodulation where soft information about the transmitted data symbols has to be inferred in the presence of the channel uncertainty, given the received signal samples and aprioriinformation provided by the decoder. An exact inference algorithm computes the aposterioriprobability (APP) values for all transmitted symbols, but the computation of APPs is known to be an NP-hard problem, thus, rendering this approach computationally prohibitive in most cases. In this paper, we show how many of the well-known low-complexity soft-input soft-output (SISO) equalizers, including the channel-matched filter-based linear SISO equalizers and minimum mean square error (MMSE) SISO equalizers, as well as the expectation-maximization (EM) algorithm-based SISO demodulators in the presence of the Rayleigh fading channel, can be formulated as solutions to a variational optimization problem. The variational optimization is a well-established methodology for low-complexity inference and estimation, originating from statistical physics. Importantly, the imposedvariationaloptimization framework provides an interesting link between the APP demodulators and the linear SISO equalizers. Moreover, it provides a new set of insights into the structure and performance of these widely celebrated linear SISO equalizers while suggesting their fine tuning as well.

In this paper, we show how many of the well-known low complexity linear turbo equalizers, including the zero-forcing (ZF) and minimum mean square error (MMSE) soft-input soft-output (SISO) equalizers, can be obtained as solutions to the variational optimization problem, originating from statistical physics. The imposed variational optimization framework provides an interesting link between the a posteriori probability (APP) based demodulators and the linear SISO equalizers, enabling us to gain new insight into the optimality of these equalizers in the context of turbo processing. Moreover, it suggests improved designs which either tune the known ones or combine the linear filtering and the nonlinear message-passing algorithms. Finally, simulation results are provided to confirm the advantages of the proposed new designs for the MIMO systems

In this paper, we focus on the joint estimation of the carrier frequency offset and statistical parameters of the multipath mobile channel. By modeling the multipath fading channel with a complex bandpass autoregressive (AR) model, we show how the estimates for the frequency offset, Doppler spread, and power profile of the multipath channel can be extracted from the estimated complex AR coefficients obtained via the expectation-maximization algorithm. A main advantage of the proposed joint estimator is that while it has a capability of performing equally well in all scattering environments, it can provide accurate estimates even in high-mobility channel conditions. We also demonstrate how the complexity of the estimator can be significantly reduced, while only slightly trading off performance, by applying the mean field approximation technique. Moreover, we derive a fully adaptive joint synchronization and channel-estimation scheme, as well as a novel Kalman-smoother-based frequency-error detector that can be used in feedback frequency-recovery schemes and is particularly well suited for fast-fading channel conditions. Finally, we revisit the Cramer-Rao lower bound analysis, and show how the Fisher information matrix can be conveniently computed in the presence of a frequency-selective Rayleigh fading channel

Many algorithms in signal processing and digital communications must deal with the problem of computing the probabilities of the hidden state variables given the observations, i.e., the inference problem, as well as with the problem of estimating the unknown model parameters. In this paper, we present an unified framework for approximate joint inference and estimation in the cases where an exact inference becomes computationally intractable. Specifically, approximate inference via variational minimization technique is obtained by operating a general message-passing algorithm in the distributed factor graph where the coupling between the multiple Markov chains is removed by minimizing the Kullback-Leibler distance between the original and the variational objective functions. Importantly, we demonstrate how this framework can be exploited in deriving reduced-complexity turbo receiver structures for coded single transmit antenna and space-time coded multiple transmit antenna systems over the multipath fading channels. Despite the significant reduction in complexity, the performance simulations showed that the derived turbo receivers are able to provide close to optimal performance.

The problems of adaptive maximum a posteriori (MAP) symbol detection for uncoded transmission and of adaptive soft-input soft-output (SISO) demodulation for coded transmission of data symbols over time-varying frequency-selective channels are explored within the framework of the expectation-maximization (EM) algorithm. In particular, several recursive forms of the classical Baum-Welch (BW) algorithm and its Bayesian counterpart (often referred to a Bayesian EM algorithm) are derived in an unified way. In contrast to earlier developments of the BW and BEM algorithms, these formulations lead to computationally attractive algorithms which avoid matrix inversions while using sequential processing over the time and trellis branch indices. Moreover, it is shown how these recursive versions of the BW and BEM algorithms can be integrated with the well-known forward-backward processing SISO algorithms resulting in adaptive SISOs with embedded soft decision directed (SDD) channel estimators. An application of the proposed algorithms to iterative "turbo-processing" receivers illustrates how these SDD channel estimators can efficiently exploit the extrinsic information obtained as feedback from the SISO decoder in order to enhance their estimation accuracy.

In this paper, the problem of adaptive MAP symbol detection in the uncoded transmission as well as the problem of adaptive APP demodulation in the coded transmission of data symbols over the frequency-selective Rayleigh fading channel are explored within the framework of the Bayesian expectation-maximization (BEM) algorithm. In particular, two novel versions of the BEM-based detection and demodulation algorithms are derived. In contrast to the earlier developments of BEM algorithms, the formulations derived in this paper lead to the computationally efficient algorithms which avoid the matrix inversions while using sequential processing over the time and trellis branch indexes. In addition, it is shown how the recursive versions of the BEM algorithms can be combined with the well-known forward-backward processing soft-input soft-output (SISO) algorithms resulting in adaptive SISOs with soft decision directed (SDD) channel estimators. An application of the proposed algorithms to the iterative "turbo-processing" receivers illustrates how these SDD channel estimators can efficiently exploit the extrinsic information obtained from the SISO decoder in order to enhance their estimation accuracy.

The problem of adaptive equalization of the time-varying frequency-selective channel is studied by using the adaptive Baum-Welch algorithm. In particular, blind LMS and blind RLS channel estimators are derived and they are combined with the well-known forward-backward processing SISO algorithm for known channels. An application of the proposed algorithms to the iterative "turbo-processing" receivers illustrated how the blind LMS and blind RLS estimators can efficiently utilize the extrinsic information obtained from the decoder SISO in order to enhance their estimation accuracy.

In this paper, the problem of carrier phase recovery of linearly modulated signals in the presence of unknown data symbols is studied within the expectation-maximization (EM) framework using two different approaches. First, an iterative joint maximum likelihood (ML) phase estimator and data detector is derived using conditional maximization steps. Secondly, an iterative joint ML nondata-aided (NDA) phase and power spectral density estimator is derived for uncoded and coded transmitted signals. The main difference between the approaches is that the first estimator uses hard decisions whereas the second estimator uses soft decisions about the data symbols at each iteration. Furthermore, the mean-squared performance of the second estimator is approximated by the inverse of the empirical observed Fisher information. Finally, it is shown that the recursive approximations of both estimators lead to the Costas type of loops introduced earlier in the literature.