Pilot symbol assisted modulation in frequency selective fadingwireless channels

Frequency-selective fading channels are typically modeled either as a combination of Doppler components or as lowpass stochastic processes. In both cases, accurate parameter and/or Doppler frequency estimation is impeded by the fact that the Doppler frequencies are typically very low (compared with the data rate) and closely spaced. This problem is mitigated in pilot symbol assisted modulation (PSAM) systems that employ distributed training. Those systems can provide information about a time-undersampled version of the channel that may be easier to identify. We address the problem of estimating the fading channel's correlation matrices from the received data by exploiting the distributed training symbols. Multichannel autoregressive (AR) models are estimated to fit the channel's variations, and the Doppler frequencies are identified through the peaks of the AR spectrum. The performance of the proposed methods is studied through analytical and experimental results. Finally, Kalman filtering ideas are employed to track the time-varying channel taps based on the estimated AR model     

Error exponents for AR order testing

This paper is concerned with error exponents in testing problems raised by autoregressive (AR) modeling. The tests to be considered are variants of generalized likelihood ratio testing corresponding to traditional approaches to autoregressive moving-average (ARMA) modeling estimation. In several related problems, such as Markov order or hidden Markov model order estimation, optimal error exponents have been determined thanks to large deviations theory. AR order testing is specially challenging since the natural tests rely on quadratic forms of Gaussian processes. In sharp contrast with empirical measures of Markov chains, the large deviation principles (LDPs) satisfied by Gaussian quadratic forms do not always admit an information-theoretic representation. Despite this impediment, we prove the existence of nontrivial error exponents for Gaussian AR order testing. And furthermore, we exhibit situations where the exponents are optimal. These results are obtained by showing that the log-likelihood process indexed by AR models of a given order satisfy an LDP upper bound with a weakened information-theoretic representation.     

Autocorrelation model-based identification method for ARMA systems in noise

A novel method for parameter estimation of minimum-phase autoregressive moving average (ARMA) systems in noise is presented. The ARMA parameters are estimated using a damped sinusoidal model representation of the autocorrelation function of the noise-free ARMA signal. The AR parameters are obtained directly from the estimates of the damped sinusoidal model parameters with guaranteed stability. The MA parameters are estimated using a correlation matching technique. Simulation results show that the proposed method can estimate the ARMA parameters with better accuracy as compared to other reported methods, in particular for low SNRs.     

ARMA-cepstrum Recursion Algorithm for the Estimation of the MA Parameters of 2-D ARMA Models

This paper considers the problem of estimating the moving average (MA) parameters of a two-dimensional autoregressive moving average (2-D ARMA) model. To solve this problem, a new algorithm that is based on a recursion relating the ARMA parameters and cepstral coefficients of a 2-D ARMA process is proposed. On the basis of this recursion, a recursive equation is derived to estimate the MA parameters from the cepstral coefficients and the autoregressive (AR) parameters of a 2-D ARMA process. The cepstral coefficients are computed benefiting from the 2-D FFT technique. Estimation of the AR parameters is performed by the 2-D modified Yule–Walker (MYW) equation approach. The development presented here includes the formulation for real-valued homogeneous quarter-plane (QP) 2-D ARMA random fields, where data are propagated using only the past values. The proposed algorithm is computationally efficient especially for the higher-order 2-D ARMA models, and has the advantage that it does not require any matrix inversion for the calculation of the MA parameters. The performance of the new algorithm is illustrated by some numerical examples, and is compared with another existing 2-D MA parameter estimation procedure, according to three performance criteria. As a result of these comparisons, it is observed that the MA parameters and the 2-D ARMA power spectra estimated by using the proposed algorithm are converged to the original ones     

A Novel Technique for the Identification of ARMA Systems Under Very Low Levels of SNR

In this paper, a novel technique for the identification of minimum-phase autoregressive moving average (ARMA) systems from the output observations in the presence of heavy noise is presented. First, starting from the conventional correlation estimator, a simple and accurate ARMA correlation (ARMAC) model in terms of the poles of the ARMA system is presented in a unified manner for white noise and impulse-train excitations. The AR parameters of the ARMA system are then obtained from the noisy observations by developing and using a residue-based least-squares correlation-fitting optimization technique that employs the proposed ARMAC model. As for the estimation of the MA parameters, it is preceded by the application of a new technique intended to reduce the noise present in the residual signal that is obtained by filtering the noisy ARMA signal via the estimated AR parameters. A scheme is then devised whereby the task of MA parameter estimation is transformed into a problem of correlation-fitting of the inverse autocorrelation function corresponding to the noise-compensated residual signal. In order to demonstrate the effectiveness of the proposed method, extensive simulations are performed by considering synthetic ARMA systems of different orders in the presence of additive white noise and the results are compared with those of some of the existing methods. It is shown that the proposed method is capable of estimating the ARMA parameters accurately and consistently with guaranteed stability for signal-to-noise ratio (SNR) levels as low as $-{5}~{hbox {dB}}$ . Simulation results are also provided for the identification of a human vocal-tract system using natural speech signals showing a superior performance of the proposed technique in terms of the power spectral density of the synthesized speech signal.      

On the use of first-order autoregressive modeling for Rayleigh flat fading channel estimation with Kalman filter

This letter deals with the estimation of a flat fading Rayleigh channel with Jakes's spectrum. The channel is approximated by a first-order autoregressive (AR(1)) model and tracked by a Kalman filter (KF). The common method used in the literature to estimate the parameter of the AR(1) model is based on a correlation matching (CM) criterion. However, for slow fading variations, another criterion based on the minimization of the asymptotic variance (MAV) of the KF is more appropriate, as already observed in few works (Barbieri et al., 2009 [1]). This letter gives analytic justification by providing approximated closed-form expressions of the estimation variance for the CM and MAV criteria, and of the optimal AR(1) parameter.     

Multiscale causal connectivity analysis by canonical correlation: theory and application to epileptic brain

Multivariate Granger causality is a well-established approach for inferring information flow in complex systems, and it is being increasingly applied to map brain connectivity. Traditional Granger causality is based on vector autoregressive (AR) or mixed autoregressive moving average (ARMA) model, which are potentially affected by errors in parameter estimation and may be contaminated by zero-lag correlation, notably when modeling neuroimaging data. To overcome this issue, we present here an extended canonical correlation approach to measure multivariate Granger causal interactions among time series. The procedure includes a reduced rank step for calculating canonical correlation analysis (CCA), and extends the definition of causality including instantaneous effects, thus avoiding the potential estimation problems of AR (or ARMA) models. We tested this approach on simulated data and confirmed its practical utility by exploring local network connectivity at different scales in the epileptic brain analyzing scalp and depth-EEG data during an interictal period.     

LD2-ARMA identification algorithm

The authors present the LD²-ARMA identifier, a novel algorithm that solves the essentially nonlinear autoregressive moving-averaged (ARMA) identification problem with a linear procedure, in two steps: an order selection algorithm followed by an ARMA parameter estimator. The determination of the AR and MA coefficients involves the solution of two dual systems of linear equations. These systems decouple the estimation of the autoregressive component from the estimation of the moving average component. The selection of the number of poles and of the number of zeros is accomplished by a scheme that minimizes the mismatch of the data to each proposed model. Simulated experiments on the proposed order selection procedure are presented     

Time-Frequency ARMA Models and Parameter Estimators for Underspread Nonstationary Random Processes

Parsimonious parametric models for nonstationary random processes are useful in many applications. Here, we consider a nonstationary extension of the classical autoregressive moving-average (ARMA) model that we term the time-frequency autoregressive moving-average (TFARMA) model. This model uses frequency shifts in addition to time shifts (delays) for modeling nonstationary process dynamics. The TFARMA model and its special cases, the TFAR and TFMA models, are shown to be specific types of time-varying ARMA (AR, MA) models. They are attractive because of their parsimony for underspread processes, that is, nonstationary processes with a limited time-frequency correlation structure. We develop computationally efficient order-recursive estimators for the TFARMA, TFAR, and TFMA model parameters which are based on linear time-frequency Yule-Walker equations or on a new time-frequency cepstrum. Simulation results demonstrate that the proposed parameter estimators outperform existing estimators for time-varying ARMA (AR, MA) models with respect to accuracy and/or numerical efficiency. An application to the time-varying spectral analysis of a natural signal is also discussed.     

Tutorial on higher-order statistics (spectra) in signal processingand system theory: theoretical results and some applications

A compendium of recent theoretical results associated with using higher-order statistics in signal processing and system theory is provided, and the utility of applying higher-order statistics to practical problems is demonstrated. Most of the results are given for one-dimensional processes, but some extensions to vector processes and multichannel systems are discussed. The topics covered include cumulant-polyspectra formulas; impulse response formulas; autoregressive (AR) coefficients; relationships between second-order and higher-order statistics for linear systems; double C(q,k) formulas for extracting autoregressive moving average (ARMA) coefficients; bicepstral formulas; multichannel formulas; harmonic processes; estimates of cumulants; and applications to identification of various systems, including the identification of systems from just output measurements, identification of AR systems, identification of moving-average systems, and identification of ARMA systems     

Semiblind identification of nonminimum-phase ARMA models via orderrecursion with higher order cumulants

This paper develops a novel identification methodology for nonminimum-phase autoregressive moving average (ARMA) models of which the models' orders are not given. It is based on the third-order statistics of the given noisy output observations and assumed input random sequences. The semiblind identification approach is thereby named. By the order-recursive technique, the model orders and parameters can be determined simultaneously by minimizing well-defined cost functions. At each updated order, the AR and MA parameters are estimated without computing the residual time series (RTS), with the result of decreasing the computational complexity and memory consumption. Effects of the AR estimation error on the MA parameters estimation are also reduced. Theoretical statements and simulations results, together with practical application to the train vibration signals' modeling, illustrate that the method provides accurate estimates of unknown linear models, despite the output measurements being corrupted by arbitrary Gaussian noises of unknown pdf     

Multivariate ARMA modeling by scalar algorithms

An algorithm for multichannel autoregressive moving average (ARMA) modeling which uses scalar computations only and is well suited for parallel implementation is proposed. The given ARMA process is converted to an equivalent scalar, periodic ARMA process. The scalar autoregressive (AR) parameters are estimated by first deriving a set of modified Yule-Walker-type equations and then solving them by a parallel, order recursive algorithm. The moving average (MA) parameters are estimated by a least squares method from the estimates of the input samples obtained via a high-order, periodic AR approximation of the scalar process     

Fast adaptive algorithms for AR parameters estimation using higherorder statistics

Time-varying statistics in linear filtering and linear estimation problems necessitate the use of adaptive or time-varying filters in the solution. With the rapid availability of vast and inexpensive computation power, models which are non-Gaussian even nonstationary are being investigated at increasing intensity. Statistical tools used in such investigations usually involve higher order statistics (HOS). The classical instrumental variable (IV) principle has been widely used to develop adaptive algorithms for the estimation of ARMA processes. Despite, the great number of IV methods developed in the literature, the cumulant-based procedures for pure autoregressive (AR) processes are almost nonexistent, except lattice versions of IV algorithms. This paper deals with the derivation and the properties of fast transversal algorithms. Hence, by establishing a relationship between classical (IV) methods and cumulant-based AR estimation problems, new fast adaptive algorithms, (fast transversal recursive instrumental variable-FTRIV) and (generalized least mean squares-GLMS), are proposed for the estimation of AR processes. The algorithms are seen to have better performance in terms of convergence speed and misadjustment even in low SNR. The extra computational complexity is negligible. The performance of the algorithms, as well as some illustrative tracking comparisons with the existing adaptive ones in the literature, are verified via simulations. The conditions of convergence are investigated for the GLMS     

Spectral estimation of nonstationary EEG using particle filtering with application to event-related desynchronization (ERD)

This paper proposes non-Gaussian models for parametric spectral estimation with application to event-related desynchronization (ERD) estimation of nonstationary EEG. Existing approaches for time-varying spectral estimation use time-varying autoregressive (TVAR) state-space models with Gaussian state noise. The parameter estimation is solved by a conventional Kalman filtering. This study uses non-Gaussian state noise to model autoregressive (AR) parameter variation with estimation by a Monte Carlo particle filter (PF). Use of non-Gaussian noise such as heavy-tailed distribution is motivated by its ability to track abrupt and smooth AR parameter changes, which are inadequately modeled by Gaussian models. Thus, more accurate spectral estimates and better ERD tracking can be obtained. This study further proposes a non-Gaussian state space formulation of time-varying autoregressive moving average (TVARMA) models to improve the spectral estimation. Simulation on TVAR process with abrupt parameter variation shows superior tracking performance of non-Gaussian models. Evaluation on motor-imagery EEG data shows that the non-Gaussian models provide more accurate detection of abrupt changes in alpha rhythm ERD. Among the proposed non-Gaussian models, TVARMA shows better spectral representations while maintaining reasonable good ERD tracking performance.     

Maximum entropy spectral analysis and ARMA processes (Corresp.)

The maximum entropy spectral analysis for discrete-time stationary processes was first proposed by J. P. Burg. He showed that if a finite number of covariance lag values of a stationary process are known, then an autoregressive (AR) process with the given autocorrelation values best fits the given constraints in the sense of maximizing thc differential entropy rate of the model. A more general type of prior knowledge of the process is considered, and it is shown that the maximum entropy method, subject to our constraints, is equivalent to fitting a mixed autoregressive moving average (ARMA) model.     

基于零极点预测模型的码激励线性预测语音编码算法

针对传统码激励线性预测（Code Excited Linear Predictive，CELP）语音编码器在预测模型和参数估计方面的不足，提出了一种基于零极点预测模型的CELP语音编码新算法。该算法采用零极点预测模型来更准确地描述语音信号的短时相关性，并采用梯度法来同时对零极点模型的参数和激励码本增益进行联合优化求解。实验结果表明所提语音编码算法可显著降低CELP编码器合成语音的归一化均方误差，有效提高合成语音的质量。     

Evaluation of Simple Algorithms for Spectral Parameter Analysis of the Electroencephalogram

Simple autoregressive moving-average (ARMA) and autoregressive (AR) algorithms were tested for use in spectral parameter analysis (SPA) of the background electroencephalogram (EEG). In studies on simulated EEG, both algorithms successfully extracted estimates of the spectral component parameters, and their performance was relatively independent of assumed model order. The ARMA algorithm was unbiased. The AR algorithm, though biased, was simpler and more precise and, thus, may be the most suitable for on-line use. The test results on simulated data were supported by the successful application of the algorithms to human EEG recorded during surgery.     

A new algorithm for linear and nonlinear ARMA model parameter estimation using affine geometry

A linear and nonlinear autoregressive (AR) moving average (MA) (ARMA) identification algorithm is developed for modeling time series data. The new algorithm is based on the concepts of affine geometry in which the salient feature of the algorithm is to remove the linearly dependent ARMA vectors from the pool of candidate ARMA vectors. For noiseless time series data with a priori incorrect model-order selection, computer simulations show that accurate linear and nonlinear ARMA model parameters can be obtained with the new algorithm. Many algorithms, including the fast orthogonal search (FOS) algorithm, are not able to obtain correct parameter estimates in every case, even with noiseless time series data, because their model-order search criteria are suboptimal. For data contaminated with noise, computer simulations show that the new algorithm performs better than the FOS algorithm for MA processes, and similarly to the FOS algorithm for ARMA processes. However, the computational time to obtain the parameter estimates with the new algorithm is faster than with FOS. Application of the new algorithm to experimentally obtained renal blood flow and pressure data show that the new algorithm is reliable in obtaining physiologically understandable transfer function relations between blood pressure and flow signals.     

A new recursive pseudo least squares algorithm for ARMA filteringand modeling. II

For pt.I see ibid., vol.40, no.11, p.2766-74 (Nov. 1992). A recursive algorithm for ARMA (autoregressive moving average) filtering has been developed in a companion paper. These recursions are seen to have a lattice-like filter structure. The ARMA parameters, however, are not directly available from the coefficients of this filter. The problem of identification of the ARMA model from the coefficients of this filter is addressed here. Two new update relations for certain pseudoinverses are derived and used to obtain a recursive least squares algorithm for AR parameter estimation. Two methods for the estimation of the MA parameters are also presented. Numerical results demonstrate the usefulness of the proposed algorithms