Study of a least-squares-based algorithm for autoregressive signals subject to white noise

Mathematical Problems in Engineering, Sep 2018

A simple algorithm is developed for unbiased parameter identification of autoregressive (AR) signals subject to white measurement noise. It is shown that the corrupting noise variance, which determines the bias in the standard least-squares (LS) parameter estimator, can be estimated by simply using the expected LS errors when the ratio between the driving noise variance and the corrupting noise variance is known or obtainable in some way. Then an LS-based algorithm is established via the principle of bias compensation. Compared with the other LS-based algorithms recently developed, the introduced algorithm requires fewer computations and has a simpler algorithmic structure. Moreover, it can produce better AR parameter estimates whenever a reasonable guess of the noise variance ratio is available.

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Study of a least-squares-based algorithm for autoregressive signals subject to white noise

Hindawi Publishing Corporation Mathematical Problems in Engineering STUDY OF A LEAST-SQUARES-BASED ALGORITHM FOR AUTOREGRESSIVE SIGNALS SUBJECT TO WHITE NOISE WEI XING ZHENG A simple algorithm is developed for unbiased parameter identification of autoregressive (AR) signals subject to white measurement noise. It is shown that the corrupting noise variance, which determines the bias in the standard least-squares (LS) parameter estimator, can be estimated by simply using the expected LS errors when the ratio between the driving noise variance and the corrupting noise variance is known or obtainable in some way. Then an LS-based algorithm is established via the principle of bias compensation. Compared with the other LS-based algorithms recently developed, the introduced algorithm requires fewer computations and has a simpler algorithmic structure. Moreover, it can produce better AR parameter estimates whenever a reasonable guess of the noise variance ratio is available. Estimation of the parameters of autoregressive (AR) signals from noisy measurements has been an important topic of research in the field of signal processing [2, 4, 6]. Since the standard least-squares (LS) method is unable to produce unbiased estimates of the AR parameters in the presence of noise, many identification algorithms have been developed with a view to achieving unbiasedness in AR signal estimation; for instance, the modified Yule-Walker (MYW) equations method [1], the maximum likelihood (ML) method [7], the recursive prediction error (RPE) method [3], the modified least-squares (MLS) method [5], and the improved least-squares (ILS) methods [8, 9]. It is of interest to note that the ILS-type algorithms are built on the simple idea of estimating the variance of the corrupting noise in an efficient way and then removing the noise-induced bias from the standard LS estimator in a straightforward way so as to attain unbiased AR parameter estimates. The good performances of the ILS-type algorithms are as follows. Firstly, as a linear regression-based method, the ILS-types methods require much less numerical efforts than the ML method, the RPE method, and the MLS method. Secondly, the ILS-type algorithms not only are well suited for online estimation, but also have much better numerical robustness than the MYW method. Thirdly, unlike the ML method and 1. Introduction the MYW method, the ILS-type algorithms can simultaneously estimate the corrupting noise variance and the signal power that may be required in certain signal processing applications. The objective of the present paper is to develop a simple algorithm for unbiased parameter identification of AR signals subject to white measurement noise. Note that the assumption on the measurement noise is a restriction, but it is not unrealistic. Like the other ILS-type algorithms, central to this new algorithm is the estimation of the corrupting noise variance, which determines the bias in the LS parameter estimator. However, it is observed that the other ILS algorithms need to compute some extra autocovariance estimates for the purpose of getting an estimate of the corrupting noise variance. This apparently requires added computations. In this paper, it is assumed that the ratio between the AR driving noise variance and the corrupting noise variance is given or obtainable in some way. Note that, on the one hand, this assumption may be considered as restrictive in some practical situations since it may be difficult to have information on both the driving noise and the corrupting one simultaneously. On the other hand, however, it may still conform to a number of signal processing application cases. For example, in speech processing, the level of background noise relative to a speech signal is sometimes predictable beforehand according to the experience so that a reasonable description of the noisy scenario (or the noise variance ratio) is admissible [4]. Under the imposed assumption, the corrupting noise variance can be estimated by simply using the expected LS errors. Then a new LS-based algorithm is established via the principle of bias compensation. Compared with the other ILS-type algorithms, the developed algorithm requires fewer computations and has a simpler algorithmic structure. Moreover, it can produce better AR parameter estimates once a sensible conjecture of the noise variance ratio is given. The sensitivity of the developed algorithm with respect to the noise variance ratio is also studied via computer simulations. 2. Signal model Assume that the AR signal x(t) is generated by a model of the form (2.1) (2.2) (2.3) where p is the order of the model, v(t) is the driving (white) noise with zero mean and finite variance σv2, and {ai, i = 1, . . . , p} are the AR parameters. Let be a noisy measurement of the AR signal, where w(t) is the corrupting (white) noise with zero mean and finite variance σw2 . The noisy AR model, which consists of (2.1) and (2.2), can be expressed in (...truncated)


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Wei Xing Zheng. Study of a least-squares-based algorithm for autoregressive signals subject to white noise, Mathematical Problems in Engineering, 2003, DOI: 10.1155/S1024123X03210012