Inverse filtering based method for estimation of noisy autoregressive signals

New estimation methods for autoregressive process in the presence of white observation noise

This paper is concerned with identification of autoregressive (AR) model parameters using observations corrupted with colored noise. A novel formulation of an auxiliary least-squares estimator is introduced so that the autocovariance functions of the colored observation noise can be estimated in a straightforward manner. With this, the colored-noise-induced estimation bias can be removed to yield the unbiased estimate of the AR parameters. The performance of the proposed unbiased estimation algorithm is illustrated by simulation results. The presented work greatly extends the author's previous methods that were developed for identification of AR signals observed in white noise.

In this article, the least squares (LS) estimates of the parameters of periodic autoregressive (PAR) models are investigated for various distributions of error terms via Monte-Carlo simulation. Beside the Gaussian distribution, this study covers the exponential, gamma, student-t, and Cauchy distributions. The estimates are compared for various distributions via bias and MSE criterion. The effect of other factors are also examined as the non-constancy of model orders, the non-constancy of the variances of seasonal white noise, the period length, and the length of the time series. The simulation results indicate that this method is in general robust for the estimation of AR parameters with respect to the distribution of error terms and other factors. However, the estimates of those parameters were, in some cases, noticeably poor for Cauchy distribution. It is also noticed that the variances of estimates of white noise variances are highly affected by the degree of skewness of the distribution of error terms.

Fusion identification and estimation of multisensor multichannel AR signals with missing measurements and sensor biases

Bilinear equation method for unbiased identification of linear FIR systems in the presence of input and output noises

In a regression model, prediction is typically performed after model selection. The large variability in the model selection makes the prediction unstable. Thus, it is essential to reduce the variability in model selection and improve prediction accuracy. To achieve this goal, a parametric bootstrap smoothing can be applied. In this method, model selection is performed for each resampling from a parametric distribution, and these models are then averaged such that the distribution of the selected models is considered. Here, the prediction accuracy is highly dependent on the choice of a distribution for resampling. In particular, an experimental study shows that the choice of error variance significantly changes the distribution of the selected model and thus plays a key role in improving the prediction accuracy. We also observe that the true error variance does not always provide optimal prediction accuracy. Therefore, it is not always appropriate to use unbiased estimators of the true parameters or standard estimators of the parameters for the resampling distribution. In this study, we propose employing cross-validation to choose a suitable resampling distribution rather than unbiased estimators of parameters. Our proposed method was applied to several data sets. The results indicate that the proposed method provides a better prediction accuracy than the existing method.

This paper presents a novel nonlinear filter and parameter estimator for narrow band interference suppression in code division multiple access spread-spectrum systems. As in the article by Rusch and Poor (1994), the received sampled signal is modeled as the sum of the spread-spectrum signal (modeled as a finite state independently identically distributed (i.i.d.) process-here we generalize to a finite state Markov chain), narrow-band interference (modeled as a Gaussian autoregressive process), and observation noise (modeled as a zero-mean white Gaussian process). The proposed algorithm combines a recursive hidden Markov model (HMM) estimator, Kalman filter (KF), and the recursive expectation maximization algorithm. The nonlinear filtering techniques for narrow-band interference suppression presented in Rusch and Poor and our proposed HMM-KF algorithm have the same computational cost. Detailed simulation studies show that the HMM-KF algorithm outperforms the filtering techniques in Rusch and Poor. In particular, significant improvements in the bit error rate and signal-to-noise ratio (SNR) enhancement are obtained in low to medium SNR. Furthermore, in simulation studies we investigate the effect on the performance of the HMM-KF and the approximate conditional mean (ACM) filter in the paper by Rusch and Poor, when the observation noise variance is increased. As expected, the performance of the HMM-KF and ACM algorithms worsen with increasing observation noise and number of users. However, HMM-KF significantly outperforms ACM in medium to high observation noise.

In this paper we investigate the block- type pilot channel estimation for orthogonal frequency division multiplexing (OFDM) systems. The estimation is based on the minimum mean square error (MMSE) estimator and the least square (LS) estimator. We derive the MMSE and LS estimators’ architecture and investigate their performances. We prove that the MMSE estimator performance is better but computational complexity is high, contrary the LS estimator has low complexity but poor performance. For reducing complexity we proposed two different solutions which are the Simplified Least Square (SLS) estimator and the modified MMSE estimator. In the SLS estimator, we apply an auto-correlation function with the LS estimator to remove the noise. In the modified MMSE estimator, we consider only the significant energy samples and ignore the remaining noisy samples. Based on this idea we introduce the modified MMSE estimator. We evaluate estimator’s performance on basis of mean square error and symbol error rate for 16 QAM systems using MATLAB.

Weibull distribution is widely used in various areas such as life tables, failure rates, and definition of wind speed distribution. Therefore, parameter estimation for the Weibull distribution is an important problem in many real data applications. The least square (LS), the weighted least square (WLS) and the maximum likelihood (ML) are the most popular methods for the parameter estimation in the Weibull distribution. In this study, based on the LS, WLS and ML estimation methods, a multi-objective programming approach is proposed for the parameter estimation of two-parameter Weibull distribution. This new approach evaluates together LS, WLS and ML methods in the estimation process. NSGA-II method, which is a multi-objective heuristic optimization method, is used to solve the proposed multi-objective estimation model. To evaluate the applicability and performance of the proposed approach, a detailed Monte Carlo simulation study based on deficiency criteria and a real data application are designed. The results illustrated that the proposed multi-objective programming approach provides quite accurate parameter estimates for the two parameter Weibull distribution with respect to deficiency criteria.

Estimating ultrasonic time of flight using envelope and quasi maximum likelihood method for damage detection and assessment

A two-stage autocorrelation method is proposed for frequency estimation of a complex sinusoidal signal in complex additive white Gaussian noise. In the first stage, a function about signal frequency is constructed via an autocorrelation procedure. By the Least-Square (LS) principle, the signal frequency is estimated from the function. In the second stage, the estimated error of signal frequency in the first stage is defined. Applying the Taylor series, a function about the estimated error of signal frequency is conducted from the autocorrelation function. Applying the LS estimation once more, the estimate of estimated error of frequency in the first stage is obtained. Then the fine estimate of signal frequency is obtained. Simulation results show that the performance of proposed method can approach the Cramer–Rao lower bound in the whole frequency range.

Estimating the parameters of the autoregressive (AR) random process is a problem that has been well-studied. In many applications, only noisy measurements of AR process are available. The effect of the additive noise is that the system can be modeled as an AR model with colored noise, even when the measurement noise is white, where the correlation matrix depends on the AR parameters. Because of the correlation, it is expedient to compute using multiple stacked observations. Performing a weighted least-squares estimation of the AR parameters using an inverse covariance weighting can provide significantly better parameter estimates, with improvement increasing with the stack depth. The estimation algorithm is essentially a vector RLS adaptive filter, with time-varying covariance matrix. Different ways of estimating the unknown covariance are presented, as well as a method to estimate the variances of the AR and observation noise. The notation is extended to vector autoregressive (VAR) processes. Simulation results demonstrate performance improvements in coefficient error and in spectrum estimation.

In this paper, we present two Data-Aided channel estimators for Continuous Phase Modulation (CPM) in the case of transmissions over doubly -selective channels. They both capitalize on the Basis Expansion Model (BEM), widely used for OFDM systems and for Single Carrier transmission with linear modulation. However, in the case of CPM signals, we need to work on a over-sampled received signal (fractionally-spaced representation) as the equalization techniques are also working on the over-sampled received signal. The first one is a classical Least Squares (LS) estimation of the BEM parameters whereas the second channel estimator introduces first a parametric dependence on the paths delays. Indeed, in the case where those delays are known (by estimation or by geometrical consideration as for the aeronautical channel by satellite), the second LS estimation on the BEM parameters is improved and less computationally demanding. Simulations results are provided and show good performance of our parametric LS estimation.

Covariance shaping least squares estimator is a new kind of linear estimator based on covariance shaping in quantum signal processing (QSP) framework, it is a biased estimator directed at improving the performance of traditional leastsquares (LS) estimator. In this paper, we discuss the covariance shaping least squares (CSLS) estimator, and apply it in multi-user detection that leads to a new detection algorithm, and then we employ it in the detection of MIMO systems. The numerical results in synchronous DS-CDMA communications system show that the performance of covariance shaping multi-user detection which depends only on the users' signatures is better than that of traditional MF and decorrelation detection, and is an approach to the MMSE detection method. The results in MIMO systems also show that the performance of system is better than that of detection algorithm with Zero-forcing and MMSE solutions.

The best accuracy for estimated spectra is obtained with parsimonious time series models, which have the smallest number of parameters to guarantee unbiased models. Durbin 's method for the estimation of moving average (MA) parameters uses the estimated parameters of a long autoregressive (AR) model to calculate MA parameters. Probably all pejorative remarks on the quality of Durbin's method in the literature are based on suboptimal or wrong choices for the method of AR estimation or for the order of the intermediate AR model. That AR order should be considerably higher than the order of the best predicting AR model and it should grow with the sample size. Furthermore, the Burg estimates for the AR parameters give the best results because they have the smallest variance of all AR methods with a small bias. The triangular autocorrelation bias of the popular Yule-Walker method of AR estimation can cause large bias errors in finite samples, which makes it unsuited. Durbin's method applied to the proper number of AR parameters estimated with Burg's method outperforms all other known MA estimation methods, asymptotically as well as in finite samples. The accuracy is generally close to the Cramer-Rao bound.