Pii: S0893-6080(97)00081-6

Keywords--Model-based, Neural networks, Spectrum estimation, Entropy, Ionosphere, Radar, Propagation, Clutter. 1. I N T R O D U C T I O N Model-based neural networks utilize internal models of signals, processes, or objects in order to combine adaptive learning with a priori knowledge. One of the first model-based neural network was Widrow's Adaline (Widrow, 1959). Several neural networks based on more complicated models have been recently described for applications in classification, signal processing, tracking, and control (Perlovsky, 1987. 1994; Specht, 1990; Streit & Luginbuhl, 1990; Perlovsky & Jaskolski, 1994). In this paper, a neural network utilizing compositional, flexible model for spectrum estimation is developed and applied to characterization of ionospheric propagation effects in Doppler spectra observed by over-the-horizon (OTH) radars. The developed technique has a broad applicability for analyses of spectral data, including time-frequency spectra used in speech recognition as well as for other applications, where the model parameters are difficult to estimate, because of multiple interfering sources. Acknowledgements: This research was partially supported by the USAF under the contract number F19628-94-C-0049. Authors greatly appreciate suggestions of the reviewer, Prof. Widrow. Requests for reprints should be sent to Leonid I. Perlovsky, Nichols Research, 70 Westview St., Lexington, MA 02173, USA; Tel: 617-8629400; Fax: 617-862-9485; E-mail: [email protected] 1541 We present a new approach to model-based spectrum estimation founded on Einstein's interpretation of the spectrum as a probability distribution function (pdf) of photon frequency. This leads to a "Einsteinian" likelihood function which is different from what usually is encountered in statistical estimation. We model the spectrum as a superposition of signals from several sources using physically based models for each signal source. And, we estimate parameters of this model by using the physical principle of the maximum entropy (ME) of a photon ensemble. Section 2 discusses Einstein's interpretation of photon spectra, introduces a physical model for signal spectra compatible with Einstein's ideas, and presents the ME estimators for the model parameters. Section 3 briefly describes an architecture of Einsteinian neural network. In Section 4 the developed estimation technique is applied to radar data. The results are discussed in Section 5. 2. PHYSICAL MODEL FOR SPECTRUM ESTIMATION Einstein interpreted the electromagnetic spectrum as a probability distribution function (pdf) of the photon energy (Einstein & Hopf, 1910). A similar interpretation is valid for phonons of acoustic spectra (speech, seismic 1542 L. I. Perlovsky et al. signals, etc.) and for any signal field obeying BoseEinstein's statistics (bosons). Statistical estimation theory usually considers a pdf of the given set of data as a function of model parameters, which is called likelihood. A new spectrum modeling and estimation technique, which relates statistical estimation of a spectrum to the equilibration of a physical ensemble of photons, is introduced in this section. It is inspired by the Einsteinian interpretation of the spectrum, which considers frequency (rather than spectral values) as a random variable. We define a spectrum model in terms of the number of physical photon states as follows. A spectrum S(c0) is measured in units of energy and the energy of the individual photon is h; thus, the number of measured photons is No~ = S(o~)/ho~, (1) where w is frequency in radians per second. A pdf is proportional to an expected number of observations. For a single photon, this is proportional to the density of photon states as a function of frequency. Therefore according to Einstein's interpretation, a spectrum model F(w) is interpreted as proportional to a number of physical states for a single photon at each frequency, NF~ F(o~) = const.hw.NFco. (2) A computation of entropy of the physical ensemble, E, can be found in statistical physics textbooks, for example (Sakurai, 1985); it results in