High-order Yule-Walker Equations Research Articles

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

This paper presents four new methods for estimating the parameters of an autoregressive (AR) process based on observations corrupted by white noise. The first three methods are iterative, while the last one is non-iterative. One method is designed to achieve an unbiased estimation of the AR parameters by undermining the destructive impact of observation noise in terms of utilizing the null space of the AR parameter vector. Another one uses both low- and high-order Yule-Walker equations to construct a constrained least squares optimization problem, in which the variance of observation noise is estimated by alternating between two equations. One more method exploits an approximation which leads to reducing the problem of estimating the AR parameters with arbitrary order p to estimating just two parameters, while the last one estimates the variance of the observation noise using the minimum eigenvalue of the enlarged autocorrelation matrix. The performance of the proposed methods is evaluated in terms of various numerical examples, which demonstrate their superiority in terms of accuracy and robustness against the observation noise compared to state-of-the-art existing methods in most simulation examples. It makes the proposed methods a good fit for practical analysis of data contaminated by observation noise, when AR modeling is applicable, and gives a range of choices of methods for different data analysis situations.

Identification of noisy input-output FIR models with colored output noise

This paper deals with the identification of FIR models corrupted by white input noise and colored output noise. An identification algorithm that exploits the properties of both the dynamic Frisch scheme and the high-order Yule-Walker (HOYW) equations is proposed. It is shown how the HOYW equations allow to define a selection criterion for identifying the input noise variance (and then the FIR coefficients) within the Frisch locus of solutions. The proposed approach does not require any a priori knowledge about the input and output noise variances. The algorithm performance is assessed by means of Monte Carlo simulations.

Two-dimensional noisy autoregressive estimation with application to joint frequency and direction of arrival estimation

The present study addresses the problem of two-dimensional autoregressive estimation in the presence of additive white noise. The estimation method is based on combining the low-order and high-order Yule-Walker equations. The noise-compensated YW equations are solved using an iterative algorithm. The proposed method is also applied to joint frequency and direction of arrival estimation in uniform linear arrays. Using simulation study, the performance of the proposed algorithm is evaluated and compared with other methods.

Identifying an autoregressive process disturbed by a moving-average noise using inner–outer factorization

This paper deals with the identification of an autoregressive (AR) process disturbed by an additive moving-average (MA) noise. Our approach operates as follows: Firstly, the AR parameters are estimated by using the overdetermined high-order Yule–Walker equations. The variance of the AR process driving process can be deduced by means of an orthogonal projection between two types of estimates of AR process correlation vectors. Then, the correlation sequence of the MA noise is estimated. Secondly, the MA parameters are obtained by using inner–outer factorization. To study the relevance of the resulting method, we compare it with existing algorithms, and we analyze the identifiability limits. The identification approach is then combined with Kalman filtering for channel estimation in mobile communication systems.

Adaptive Algorithm for Multichannel Autoregressive Estimation in Spatially Correlated Noise

This paper addresses the problem of multichannel autoregressive (MAR) parameter estimation in the presence of spatially correlated noise by steepest descent (SD) method which combines low-order and high-order Yule-Walker (YW) equations. In addition, to yield an unbiased estimate of the MAR model parameters, we apply inverse filtering for noise covariance matrix estimation. In a simulation study, the performance of the proposed unbiased estimation algorithm is evaluated and compared with existing parameter estimation methods.

Parameter estimation of autoregressive signals from observations corrupted with colored noise

This paper presents a new method for estimation of the parameters of a noisy autoregressive (AR) signal using observations corrupted with colored noise. This method is an improved least-squares (ILS) based method that combines low-order and high-order Yule–Walker equations. The performance of the proposed method is illustrated using computer simulations.

Gradient-based approaches for recursive Frisch scheme identification

An algorithm for recursive Frisch scheme system identification of linear single-input single-output errors-in-variables systems is developed. For the update of the estimated model parameters, a recursive bias-compensating least squares algorithm, which is based on the well-known recursive least squares technique, is considered. The estimate of the output measurement noise variance is determined using a conjugate gradient method, which tracks the smallest eigenvalue of a slowly varying matrix. For the update of the input measurement noise estimate, a steepest gradient search is applied. It tracks the minimum of a model selection cost function, which is based on a set of high order Yule-Walker equations.

Iterative Estimation Algorithm of Autoregressive Parameters

This paper presents an iterative autoregressive system parameter estimation algorithm in the presence of white observation noise. The algorithm is based on the parameter estimation bias correction approach. We use high order Yule-Walker equations, sequentially estimate the noise variance, and exploit these estimated variances for the bias correction. The improved performance of the proposed algorithm in the presence of white noise is demonstrated via Monte Carlo experiments.

PARAMETER ESTIMATION OF MULTICOMPONENT TRANSIENT SIGNALS USING DECONVOLUTION AND ARMA MODELLING TECHNIQUES

Parameter estimation of transient signals, having real decaying exponential constants, is a difficult but important problem that often arises in many areas of scientific disciplines. The frequency domain method of analysis that involves Gardner transformation and conventional inverse filtering often degrades the quality of the deconvolved data, leading to inaccurate results, especially for noisy data. An improved method that is based on the combination of Gardner transformation, optimal compensation deconvolution, and signal modelling techniques is suggested in this paper. In this method of analysis the exponential signal is converted to a convolution model whose input is a train of weighted delta function that contains the signal parameters to be determined. The resolution of the estimated decay rates is poor if the conventional fast Fourier transform (FFT) algorithm is used to analyse the resulting deconvolved data. Using an autoregressive moving (ARMA) model whose AR parameters are determined by solving high-order Yule–Walker equations (HOYWE) via the singular value decomposition (SVD) algorithm can alleviate this shortcoming. The effect of sampling conditions, noise level, number of components and relative sizes of the signal parameters on the performance of this modified method of analysis is examined in this paper. Simulation results show that high-resolution estimates of decay constants can be obtained when the above signal processing techniques are used to analyse multiexponential signals with varied signal-to-noise ratio (SNR). This approach also provides a graphical procedure for detecting and validating the number of exponential signals present in the data. Some computer simulation results are presented to justify the need for this modified method of analysis.

Model parameter estimation for reciprocal Gaussian random processes

The problem of estimating the model parameters of a discrete-index reciprocal Gaussian random process from a limited number of noisy observations is addressed. The general case of a first-order multivariate process is analyzed, stating its basic properties and deriving a linear equation set that relates the model parameters (including the unknown variance of the observation noise) to the (generally nonstationary) autocorrelation function of the observed process. It generalizes to the reciprocal processes the so-called 'high-order Yule-Walker equations' for AR processes. Based on these results, a practical estimation algorithm is proposed. >

Optimal high-order Yule-Walker estimation of sinusoidal frequencies

The asymptotic properties of sinusoidal frequency estimators based on the high-order Yule-Walker (HOYW) equations were analyzed recently. The results of that analysis are used to propose two classes of frequency estimators; one class uses singular value decomposition, and the other uses a sparse matrix solution. Both classes entail two estimation steps: the first step generates initial estimates which are used to obtain an optimal weighting matrix, and the second step generates an optimally weighted estimate. Each two-step method produces asymptotically minimum variance estimates over all estimators of their class. The implementation of the proposed estimators is described in detail, and numerical examples are presented to evaluate their performance. >

High-order Yule-Walker equations for estimating sinusoidal frequencies: The complete set of solutions

The user of the high-order Yule-Walker (HOYW) methodology for sinusoidal frequency estimation faces the problems of choosing (a) the dimensions of the HOYW system of equations; (b) the relative weighting of the HOYW equations and (c) one specific solution out of an infinite number of possible HOYW solutions. To aid all these choices, the present work devices the asymptotic variances and covariances of the general HOYW frequency estimates. This derivation represents a significant extension of the previously available results which considered only particular cases of HOYW estimates, such as the minimum-norm estimate.

New insights into the high-order Yule-Walker equations

Some new analytical results on the high-order Yule-Walker (HOYW) equations are presented. A simple singularity condition for the high-order autocorrelation matrix is found which makes it possible to determine situations where application of the HOYW equations could be especially unsuitable. It is also shown that using the HOYW equations is equivalent to performing conventional noise compensation in the main diagonal of the low-order autocorrelation matrix. This is in connection with the possibility of matching a given set of (the first) 2M+1 autocorrelation estimates to a noisy autoregressive process of order M.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A noise-compensated long correlation matching method for AR spectral estimation of noisy signals

A noise-compensated long correlation matching (NCLCM) method is proposed for autoregressive (AR) spectral estimation of the noisy AR signals. This method first computes the AR parameters from the high-order Yule-Walker equations. Next, it employs these AR parameters and uses the low-order Yule-Walker equations to compensate the zeroth autocorrelation coefficient for the additive white noise. Finally, it solves the low- as well as high-order Yule-Walker equations in a least-squares sense to determine the AR parameters. It is shown that for the noisy AR signals the NCLCM method performs better than the conventional Burg method and the high-order Yule-Walker method.

Autoregressive spectral estimation in additive noise

The estimation of the spectral density of a discrete-time stationary Gaussian autoregressive process AR (p) from a finite set of noise observations is considered. A modified spectral estimator based on the high-order Yule-Walker equations is considered. Joint asymptotic normality of this spectral estimator is established; a precise asymptotic expression for the covariance matrix of the limiting distribution is obtained. The special case of AR(1) plus noise is considered in some detail. >

Estimation of noise variance from the noisy AR signal and its application in speech enhancement

In a number of applications involving the processing of noisy signals, it is desirable to know a priori the noise variance. The author proposes a method of estimating the noise variance from the autoregressive (AR) signal corrupted by the additive white noise. This method first estimates the AR parameters from the high-order Yule-Walker equations, and then uses these AR parameters to estimate the noise variance from the low-order Yule-Walker equations. The method is used in a speech enhancement application where its performance is studied for stationary as well as nonstationary noise conditions. The results are found to be encouraging. >

Asymptotic properties of high-order Yule-Walker estimates of the AR parameters of an ARMA time series

The high-order Yule-Walker equations are used to estimate the autoregressive parameters of an autoregressive moving-average time series. The asymptotic statistical properties of these estimates are derived. It is shown that they are asymptotically unbiased and normal, the covariance matrix of the limit distribution is derived. The special case of estimating the autoregressive parameters of a noise corrupted autoregressive series is also examined.

Spectral estimation via the high-order Yule-Walker equations

A new spectral estimator is developed from the high-order Yule-Walker equations and pseudoinverse solutions to the equations. The approach circumvents the usual difficulties associated with inverting ill-conditioned matrices and allows the choice of high model order. It also provides high resolution spectral estimates as substantiated by simulation results. A byproduct of this spectral estimator is a constant false alarm rate detector for a single sinusoid in noise. The detector's receiver operating characteristics curves are constructed from histograms, based on 1000 runs, of the detection statistics.