Browsing by Author "Li, Ta-Hsin"
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Item Asymptotic Normality of the Contraction Mapping Estimator for Frequency Estimation(1992) Li, Ta-Hsin; Kedem, Benjamin; Yakowitz, S.; ISRThis paper investigates the asymptotic distribution of the recently-proposed contraction mapping (CM) method for frequency estimation. Given a finite sample composed of a sinusoidal signal in additive noise, the CM method applies to the data a parametric filter that matches its parameter with the first-order autocorrelation of the filtered noise. The CM estimator is defined as the fixed-point of the parametrized first-order sample autocorrelation of the filtered data. In this paper, it is proved that under appropriate conditions, the CM estimator is asymptotically normal with a variance inversely related to the signal-to-noise ratio. A useful example of the AR(2) filter is discussed in detail to illustrate the performance of the CM method.Item Estimation and Blind Deconvolution of AR Systems with Nonstationary Binary Inputs(1992) Li, Ta-Hsin; ISRThe problem of parameter estimation and blind deconvolution of AR systems with independent nonstationary binary inputs is considered. The estimation procedure consists of applying an MA filter (equalizer) to the observed data and adjusting the parameters of the filter so as to minimize a criterion that measures the binariness of its output. The output sequence itself serves as an estimate of the unobservable binary input of the AR system. Without assuming stationarity of the inputs, it is shown that the proposed method produces a consistent estimator of the AR system, not only in the sense of converging to the true parameter as the sample size increases, but also attaining the true parameter of the AR system for sufficiently large sample size. For noisy data, the estimation criterion is modified based upon an asymptotic analysis of the effect of the noise. It is shown that the modified criterion is also consistent (in the usual sense) and its variability depends upon the filtered noise. Some simulation results are also presented to demonstrate the performance of the proposed method for parameter estimation as well as for blind deconvolution.Item Estimation of Multiple Sinusoids by Parametric Filtering(1992) Li, Ta-Hsin; Kedem, Benjamin; ISRThe problem of estimating the frequencies of multiple sinusoids from noisy observations is addressed in this paper. A parametric filtering approach, called the PF method, is proposed that leads to a consistent estimator of the AR representation of the sinusoidal signal, given the number of sinusoids. It is accomplished by using an iterative procedure to a fixed-point of the parametrized least squares estimator (from the filtered data) that comprises a contraction mapping in the vicinity of the true AR parameter. Employing appropriate filters, this method is able to achieve the accuracy of the nonlinear least squares estimator, with much less computational complexity and initialization requirement. It can also be implemented adaptively (recursively) in order to track time-varying frequencies. In this way, the PF method provides a flexible and efficient procedure of frequency estimation. An example of the AR filter is investigated in detail to illustrate the performance of the PF method.Item Multiple Frequency Estimation in Mixed-Spectrum Time Series by Parametric Filtering(1992) Li, Ta-Hsin; Kedem, B.; ISRA general parametric filtering procedure (the PF method) is proposed for the problem of multiple frequency estimation in mixed-spectrum times series (i.e., superimposed sinusoids in additive noise). The method is based on the fact that a sum of sinusoids satisfies an homogeneous autoregressive (AR) equation. The gist of the method is to parametrize a linear filter so that it possesses a certain parametrization property as suggested by the particular form of the bias encountered by Prony's (least squares) estimator. For any parametric filter with this property, in addition to some mild regularity conditions, the least squares estimator from the filtered data, as a function of the filter parameter, constitutes a contractive mapping - whose multivariate fixed-point serves as a consistent AR estimator. The chronic bias of Prony's estimator is thus eliminated. Coupled with the all- pole (AR) filter endowed with an extra bandwidth parameter, the PF method can achieve the accuracy of nonlinear least squares by a simple iterative procedure consisting of linear least squares estimation followed by linear recursive filtering. Crude initial guesses such as those from Prony's estimator are sufficient to initiate the iteration. The method is also capable of resolving closely-spaced frequencies which are unresolvable by periodogram analysis or DFT.To analyze the statistical properties of the PF method, some classical asymptotic results concerning the sample autocovariances are extended to accommodate mixed-spectrum time series and parametric filtering. In particular, under regularity conditions, uniform strong consistency and asymptotic normality are proved for the sample autocovariances of a mixed- spectrum time series after parametric filtering. Equipped with these results, some statistical properties of the PF method itself are investigated. These include the existence of the PF estimator as a fixed-point of the parametric least squares mapping, the convergence of an iterative algorithm that calculates the PF estimator, as well as the strong consistency and asymptotic normality of the PF estimator.
Computer simulations are also presented to demonstrate the effectiveness of the PF method. Direction for future research are briefly discussed at the end of the dissertation.
Item Strong Consistency of the Contraction Mapping Method for Frequency Estimation(1992) Li, Ta-Hsin; Kedem, Benjamin; ISRConsider the super position of a sinusoid plus noise. By the application of certain parametric filter, the first-order autocorrelation becomes a contraction mapping. The sample estimator of the first-order autocorrelation is also a contraction whose fixed point converges almost surely to the cosine of the frequency to be detected. The theory is illustrated by two specific examples corresponding to two different parametric filters,