Conference Paper

Performance comparisons of the minimum free energy algorithms with the reduced rank modified covariance eigenanalysis algorithm

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Low SNR simulations comparing the minimum-free-energy (MFE) spectral estimation algorithms of Silverstein and Pimbley (1988) with the reduced rank modified covariance eigenanalysis algorithm of Tufts and Kumaresan (1982) have been performed. Two different MFE algorithms are discussed and simulated. The results of the statistical analyses demonstrate that both MFE algorithms are robust, low-variance spectral estimators capable of making reliable frequency estimations of closely spaced sources at very low SNRs, from single snapshot data. They are applicable to the general domain of spectral estimation, while the eigenanalysis algorithms are primarily restricted to line spectra frequency estimation. Nonetheless, for simulations of closely spaced line spectra at low SNRs, the MFE estimations are considerably more robust than the Tufts-Kumaresan estimations

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A new method of parametric spectral estimation, which is called minimum-free-energy (MFE) estimation, is introduced. The MFE method produces a generic theoretic estimation model that is particularly relevant to signal-analysis problems that suffer from incomplete and/or noisy data. In the general MFE formulation, the objective function is defined as a linear combination of a mean-square-error-energy expression and a signal entropy expression. This objective function form is analogous to a free-energy function in statistical thermodynamics. The negative coefficient of the entropy term is represented by an effective signal-processing temperature that drives noise-induced fluctuations in the statistical model. The model parameters that characterize the spectrum are determined commensurate with a minimum of the objective function. The mathematical details and solution methods are developed for a specific embodiment of the MFE method, called the MFE-ACS method, in which the error energy is defined as the window-weighted sum of the absolute square of the difference between the initial and final estimated values of the autocorrelation sequence. The order of the autocorrelation sequence used corresponds to the parametric model order for the spectral estimation procedure. Simulations for a variety of narrow-band and broadband test signals and combinations thereof are presented. These simulations are performed for a variety of signal-to-noise-ratio (SNR) scenarios. The MFE algorithms have a broad application domain because they are not restricted to narrow-band sources as are the signal-noise-subspace algorithms. The MFE-ACS algorithm is shown to compare quite favorably with the signal-subspace Tufts-Kumaresan noise-reduced modified-covariance algorithm for closely separated narrow-band sources in the low-SNR regime (~10 dB).
We introduce a 3-D discrete transform that depends on the choice of a certain pair of 3-D arrays called inverse pairs. Many choices of inverse pairs are possible, and each choice gives a new 3-D transform. A number of new 2-D and 1-D transforms are also derived
We present a 2-D extension of the minimum free energy (MFE) autoregressive parameter and spectral estimation technique. MFE can outperform the multidimensional Levinson algorithm producing superior spectral estimation of 2-D sinusoids in white Gaussian noise. MFE spectral estimates are found to be comparable to modified covariance method estimates. MFE also outperforms the conventional Fourier method for closely spaced sinusoids at low signal-to-noise ratio and low data set size
A separable quasi 2-D spectral estimation algorithm is presented. This hybrid algorithm combines the 1-D minimum free-energy method with the 1-D periodogram. The algorithm reduces to the corresponding maximum entropy hybrid spectral estimator in some cases. The minimum free-energy hybrid algorithm is preferable to the maximum entropy version when the estimated autocorrelation sequence is not extendable or when the signal-to-noise ratio is low. A simulation helps illustrate this point
Simplified linear versions of the nonlinear minimum free energy (MFE) noise suppression algorithms are introduced. The LMFE algorithms result from the addition of a smoothness penalty function to the linear prediction cost function. The autoregressive (AR) parameters are chosen commensurate with a global minimum of the modified cost function. It is shown that this constrained optimization procedure reduces to a form of matrix regularization. The LMFE algorithms are calculable in real time as they require a negligible increase in complexity over the conventional autoregressive algorithms, including fast computational versions thereof. Linear MFE extensions are applicable to all conventional AR algorithms, and should in each case substantially increase the useful SNR range of these algorithms. Simulation results illustrating the single snapshot performance of these algorithms are given for both narrowband sources and combinations of narrowband and broadband sources subjected to various levels of Gaussian white noise
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The frequency-estimation performance of the forward-backward linear prediction (FBLP) method of Nuttall/Uhych and Clayton, is significantly improved for short data records and low signal-to-noise ratio (SNR) by using information about the rank M of the signal correlation matrix. A source for the improvement is an implied replacement of the usual estimated correlation matrix by a least squares approximation matrix having the lower rank M. A second, related cause for the improvement is an increase in the order of the prediction filter beyond conventional limits. Computationally, the recommended signal processing is the same as for the FBLP method, except that the vector of prediction coefficients is formed from a linear combination of the M principal eigenvectors of the estimated correlation matrix. Alternatively, singular value decomposition can be used in the implementation. In one special case, which we call the Kumaresan-Prony (KP) case, the new prediction coefficients can be calculated in a very simple way. Philosophically, the improvement can be considered to result from a preliminary estimation of the explainable, predictable components of the data, rather than attempting to explain all of the observed data by linear prediction.
Estimation of Frequencies of Multiple Sinusoids: Making Linear Prediction Perform Like Max-imum Likelihood A New Analysis technique for Time series dah Modern Specfral Esrimiion: lheory & Applica-lion
  • D W Tufts
  • R Kumaresan
  • J P Burg
Tufts, D.W.. and R. Kumaresan. "Estimation of Frequencies of Multiple Sinusoids: Making Linear Prediction Perform Like Max-imum Likelihood", Proc. IEEE. vol. 70, pp. 975-989, Sept. 1982 Burg, J.P., " A New Analysis technique for Time series dah," Proc. NAlO Advanced Sfdies Inairuie on Signal Proc., Enschede, Thc Nctherlands, 1968. Cf. Kay, S.M. Modern Specfral Esrimiion: lheory & Applica-lion. Prentice Hall. New Jersey, 1987; Marple, S.L,,Jr., Digiral Specfral Analysis, Prentice Hall. New Jersey. 1987. I41 [51