Estimation of Autoregressive Parameters from Noisy Observations Using Iterated Covariance Updates

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

Errors-In-Variables-Based Approach for the Identification of AR Time-Varying Fading Channels

This letter deals with the identification of time-varying Rayleigh fading channels using a training sequence-based approach. When the fading channel is approximated by an autoregressive (AR) process, it can be estimated by means of Kalman filtering, for instance. However, this method requires the estimations of both the AR parameters and the noise variances in the state–space representation of the system. For this purpose, the existing noise compensated approaches could be considered, but they usually require a long observation window and do not necessarily provide reliable estimates when the signal-to-noise ratio is low. Therefore, we propose to view the channel identification as an errors-in-variables (EIV) issue. The method consists in searching the noise variances that enable specific noise compensated autocorrelation matrices of observations to be positive semidefinite. In addition, the AR parameters can be estimated from the null spaces of these matrices. Simulation results confirm the effectiveness of this approach, especially in presence of a high amount of noise

Dual optimal filters for parameter estimation of a multivariate autoregressive process from noisy observations

This study deals with the estimation of a vector process disturbed by an additive white noise. When this process is modelled by a multivariate autoregressive (M-AR) process, optimal filters such as Kalman or H1 filter can be used for prediction or estimation from noisy observations. However, the estimation of the M-AR parameters from noisy observations is a key issue to be addressed. Off-line or iterative approaches have been proposed recently, but their computational costs can be a drawback. Using on-line methods such as extended Kalman filter and sigma-point Kalman filter are of interest, but the size of the state vector to be estimated is quite high. In order to reduce this size and the resulting computational cost, the authors suggest using dual optimal filters. In this study, the authors propose to extend to the multi-channel case the so-called dual Kalman or H1 filters-based scheme initially proposed for single-channel applications. The proposed methods are first tested with a synthetic M-AR process and then with an M-AR process corresponding to a mobile fading channel. The comparative simulation study the authors carried out with existing techniques confirms the effectiveness of the proposed methods

Robust parametric modeling of speech in additive white Gaussian noise

ABSTRACT: In estimating the linear prediction coefficients for an autoregressive spectral model, the concept of using the Yule-Walker equations is often invoked. In case of additive white Gaussian noise (AWGN), a typical parameter compensation method involves using a minimal set of Yule-Walker equation evaluations and removing a noise variance estimate from the principal diagonal of the autocorrelation matrix. Due to a potential over-subtraction of the noise variance, however, this method may not retain the symmetric Toeplitz structure of the autocorrelation matrix and there- by may not guarantee a positive-definite matrix estimate. As a result, a significant decrease in es- timation performance may occur. To counteract this problem, a parametric modelling of speech contaminated by AWGN, assuming that the noise variance can be estimated, is herein presented. It is shown that by combining a suitable noise variance estimator with an efficient iterative scheme, a significant improvement in modelling performance can be achieved. The noise variance is esti- mated from the least squares analysis of an overdetermined set of p lower-order Yule-Walker eq- uations. Simulation results indicate that the proposed method provides better parameter estimates in comparison to the standard Least Mean Squares (LMS) technique which uses a minimal set of evaluations for determining the spectral parameters

Estimation of Autoregressive Fading Channels Based on Two Cross-Coupled H∞ Filters

This paper deals with the on-line estimation of time-varying frequency-flat Rayleigh fading channels based on training sequences and using H∞ filtering. When the fading channel is approximated by an autoregressive (AR) process, the AR model parameters must be estimated. As their direct estimations from the available noisy observations at the receiver may yield biased values, the joint estimation of both the channel and its AR parameters must be addressed. Among the existing solutions to this joint estimation issue, Expectation Maximization (EM) algorithm or crosscoupled filter based approaches can be considered. They usually require Kalman filtering which is optimal in the H2 sense provided that the initial state, the driving process and measurement noise are independent, white and Gaussian. However, in real cases, these assumptions may not be satisfied. In addition, the state-space matrices and the noise variances are not necessarily accurately estimated. To take into account the above problem,we propose to use two crosscoupled H∞ filters. This method makes it possible to provide robust estimation of the fading channel and its AR parameters

An Overdetermined System for Improved Autocorrelation Based Spectral Moment Estimator Performance

Autocorrelation based spectral moment estimators are typically derived using the Fourier transform relationship between the power spectrum and the autocorrelation function along with using either an assumed form of the autocorrelation function, e.g., Gaussian, or a generic complex form and applying properties of the characteristic function. Passarelli has used a series expansion of the general complex autocorrelation function and has expressed the coefficients in terms of central moments of the power spectrum. A truncation of this series will produce a closed system of equations which can be solved for the central moments of interest. The autocorrelation function at various lags is estimated from samples of the random process under observation. These estimates themselves are random variables and exhibit a bias and variance that is a function of the number of samples used in the estimates and the operational signal-to-noise ratio. This contributes to a degradation in performance of the moment estimators. This dissertation investigates the use autocorrelation function estimates at higher order lags to reduce the bias and standard deviation in spectral moment estimates. In particular, Passarelli's series expansion is cast in terms of an overdetermined system to form a framework under which the application of additional autocorrelation function estimates at higher order lags can be defined and assessed. The solution of the overdetermined system is the least squares solution. Furthermore, an overdetermined system can be solved for any moment or moments of interest and is not tied to a particular form of the power spectrum or corresponding autocorrelation function. As an application of this approach, autocorrelation based variance estimators are defined by a truncation of Passarelli's series expansion and applied to simulated Doppler weather radar returns which are characterized by a Gaussian shaped power spectrum. The performance of the variance estimators determined from a closed system is shown to improve through the application of additional autocorrelation lags in an overdetermined system. This improvement is greater in the narrowband spectrum region where the information is spread over more lags of the autocorrelation function. The number of lags needed in the overdetermined system is a function of the spectral width, the number of terms in the series expansion, the number of samples used in estimating the autocorrelation function, and the signal-to-noise ratio. The overdetermined system provides a robustness to the chosen variance estimator by expanding the region of spectral widths and signal-to-noise ratios over which the estimator can perform as compared to the closed system

Matlab-Based Algorithm for Real Time Analysis of Multiexponential Transient Signals

Multiexponential transient signals are particularly important due to their occurrences in many natural phenomena and human applications. For instance, it is important in the study of nuclear magnetic resonance (NMR) in medical diagnosis (Cohn-Sfetcu et al., 1975)), relaxation kinetics of cooperative conformational changes in biopolymers (Provencher, 1976), solving system identification problems in control and communication engineering (Prost and Guotte, 1982), fluorescence decay of proteins (Karrakchou et al., 1992), fluorescence decay analysis (Lakowicz, 1999). Several research work have been reported on the analysis of multicomponent transient signals following the pioneer work of Prony in 1795 (Prony, 1975) and Gardner et al. in 1959 (Gardner, 1979). Detailed review of several techniques for multicomponent transient signals’ analysis was recently reported in (Jibia, 2010)