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Recursive maximum likelihood estimation of complex autoregressive processes
Abstract
The recursive maximum likelihood estimation (RMLE) algorithm
conceived by Kay (see ibid., vol.ASSP-31, p.56, 1983) is extended to
complex data sets. The complex version requires the same level of
computation as that for real data. The original development was
restricted to the case of realm data. The purpose is to extend RMLE to
the more universal realm of complex data. The derivation is discussed.
It is argued, without direct proof, that the algorithm is stable in the
sense that the magnitude of the reflection coefficient at each step is
less than unity
A maximum-likelihood (ML) algorithm is developed for the recursive estimation of reflection coefficients uniquely determining a Toeplitz correlation matrix. The algorithm is tested for efficiency with reference to signal extraction in noise, both as part of iterative methods to obtain the trial solution and as an estimation technique in its own right.
The authors apply the method of projections onto convex sets (POCS) to reconstruct an image in computer tomography. Although POCS has been used before to recover missing portions of the data set, this is the first time that it is directly used to invert raysum data. Since POCS is geometry-free, it can be directly applied to cases of incomplete data. The strength of POCS compared to other geometry-free methods is its systematic use of a priori information in a natural and consistent way along with the raysum data
An algorithm is proposed for the reconstruction of a sparse spike train from an incomplete set of its Fourier components. It is shown that as little as 20-25% of the Fourier spectrum is sufficient in practice for a high-quality reconstruction. The method employs linear programming to minimize the L1-norm of the output, because minimization of this norm favors solutions with isolated spikes. Given a wavelet, this technique can be used to perform deconvolution of noisy seismograms when the desired output is a sparse spike series. Relative reliability of the data is assessed in the frequency domain, and only the reliable spectral data are included in the calculation of the spike series. Equations for the unknown spike amplitudes are solved to an accuracy compatible with the uncertainties in the reliable data. In examples with 10% random noise, the output is superior to that obtained using conventional least-squares techniques. -Authors
A new method of autoregressive parameter estimation is presented. The technique is a closer approximation to the true maximum likelihood estimator than that obtained using linear prediction techniques. The advantage of the new algorithm is mainly for short data records and/or sharply peaked spectra. Simulation results indicate that the parameter bias as well as the variance is reduced over the Yule-Walker and the forward-backward approaches of linear prediction. Also, spectral estimates exhibit more resolution and less spurious peaks. A stable all-pole filter estimate is guaranteed. The algorithm operates in a recursive model order fashion, which allows one to successively fit higher order models to the data.
A new method of signal restoration is presented for applications where the signal is a sparse, positive series of delta functions. The method is defined by solving the constrained maximization problem maximize f^{T}f subject to f geq 0 and parallelg - Hfparallel^{2} = parallelnparallel^{2} . An algorithm to obtain the solution to this problem is given. Results are shown that demonstrate the successful application of this method.
This paper presents a number of results concerning the eigenvectors of a symmetric Toeplitz matrix and the location of the zeros of the filters (eigenfilters) whose coefficients are the elements of the eigenvectors. One of the results is that the eigenfilters corresponding to the maximum and minimum eigenvalues, if distinct, have their zeros on the unit circle, while the zeros of the other eigenfilters may or may not have their zeros on the unit circle. Even if the zeros of the eigenfilters of a matrix are all on the unit circle, the matrix need not be Toeplitz. Examples are given to illustrate the different properties.
Non-Gaussian reflectivity, entropy, and deconvolu-tion Iterative image restoration with ring-ing suppression using the method of POCS
- A Papoulis
- C Chamzasio
- A T Walden
A. Papoulis and C. Chamzas, " Detection of hidden periodicities by adaptive extrapolation, " IEEE Trans. Acoust., Speech, Signal Pro-cessing, vol. ASSP-27, no. 5, pp. 492-500, Oct. 1979. [IO] A. T. Walden, " Non-Gaussian reflectivity, entropy, and deconvolu-tion, " Geophysics, vol. 50, no. 12, pp. 2862-2888, Dec. 1985. [ I I] M. I. Sezan and A. M. Tekalp, " Iterative image restoration with ring-ing suppression using the method of POCS, " in Proc. IEEE Int. Conf. Acoust., Speech, Signal Processing (New York, NY), Apr. 1988, pp. 1300-1303. 871-874.
Deconvolution with the I , norm
- H L Taylor
- S C Banks
- J F Mccoy
I] H. L. Taylor, S. C. Banks, and J. F. McCoy, " Deconvolution with the I, norm, " Geophysics, vol. 44, no. 1, pp. 39-52, Jan. 1979.
Fast algorithms for I,-deconvolution An iterative threshold method for spiky deconvolution
- R Yargaladda
- J B Bedmar
- T L Watt
R. Yargaladda, J. B. Bedmar, and T. L. Watt, " Fast algorithms for I,,-deconvolution, " IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-33, no. 1, pp. 174-182, Feb. 1985. 141 A. R. Figueiras-Vidal, D. Docampo-Amoedo. and J. M. PBez-Bor-rallo, " An iterative threshold method for spiky deconvolution, " in Proc. Euro. Signal Processing Conf. (Grenoble, France), Sept. 1988.