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A Minimum Free-Energy Hybrid Algorithm for 2-D Spectral Estimation

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Abstract

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

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Kiernan, P., Two-Dimensional Spectral Estimation by Free Energy Minimization,Digital Signal Processing6 (1996), 160–168.We present a high-resolution 2-D minimum free energy (MFE) spectral estimation technique which is a 2-D extension of the MFE spectral estimation method. We demonstrate the performance of the technique for spectral estimation of closely spaced 2-D sinusoids in white Gaussian noise. Results from our tests on the effect of signal processing temperature illustrate that our method provides accurate low model order autoregressive spectral estimation. The method provides superior spectral estimation with similar computational burden to that achieved with the Levinson algorithm for a number of cases involving sinusoids in white noise at various signal-to-noise ratio (SNR) levels. An example is provided. The method is faster than the modified covariance method (MCV). An example is given from a number of cases involving two closely spaced sinusoids at low SNR, where MFE estimates are as well resolved as MCV estimates. MFE models may also be used for correlation extension and for field modeling and synthesis. We indicate possible extensions to the MFE method for computational efficiency improvement, a priori temperature determination, and the use of higher-order statistics.
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Chapter
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Conference Paper
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Quadratic regularization
  • A Thompson
A. Thompson, " Quadratic regularization, " in Maximum Entropy and Bayesian Methods, J. Skilling, Ed.
Applications of digital signal processing to radar Maximum entropy image restoration in astronomy Maximum entropy signal processing in practical NMR spectroscopy
  • J H Mcclellan
  • R J Purdy
  • R Narayan
  • R Nityananda
  • S Sibiski
J. H. McClellan and R. J. Purdy, " Applications of digital signal processing to radar, " in Applications of Digital Signal Processing, A. V. Oppenheim, Ed. 131 R. Narayan and R. Nityananda, " Maximum entropy image restoration in astronomy, " Annu. Rev. Astron. Astrophys., vol. 24, pp. 127-170, 1986. 141 S. Sibiski et al., " Maximum entropy signal processing in practical NMR spectroscopy, " Nature, vol. 311, pp. 446-447, Oct. 1984.
Application of a maximum entropy frequency analysis to synthetic aperture radar
  • P L Jackson
  • L S Joyce
  • G B Feldkamp
P. L. Jackson, L. S. Joyce, and G. B. Feldkamp, " Application of a maximum entropy frequency analysis to synthetic aperture radar, " in Proc. RADC Spectrum Estimation Workshop (Rome, NY), May 1978.
Line array beamfonning using linear prediction for aperture interpolation and extrapolation New results on data adaptive high resolution spectral analysis techniques
  • L S Joycesi
  • D N Swingler
  • R S Walker
L. S. Joyce, " A separable 2-D autoregressive-spectral estimation algo-rithm, " in Proc. Int. Conf. Acoust., Speech, Signal Processing (Washing-ton, DC), Apr. 1979. [SI D. N. Swingler and R. S. Walker, " Line array beamfonning using linear prediction for aperture interpolation and extrapolation, " IEEE Trans. Acoust., Speech, Signal Processing, vol. 37, no. I, pp. 16-30, Jan. 1989. 191 C. H. Chen, A. H. Costa, and C. Sue, " New results on data adaptive high resolution spectral analysis techniques, " in Proc. 1988 IEEE Int. Conf. Syst., Man, Cyhern. (Shenyang, China), Aug. 1988.
Haykin Advances in Spectral Analysis and Array Processing
  • S D Silverstein
  • J M Pimbley