Signal Extraction and Processing in a Weakly Stochastic Process using Entropy Concepts with Examples

A weakly stationary Stochastic Process (S.P) can be analysed by autocorrelation function (acf) and spectral density approaches. However, the use of spectral density approach is preferred over the time domain ‘acf’ approach as the spectral representation of the phenomena may reveal useful information on hidden frequencies Significant spikes in the spectra indicate high energy concentration and these can be interpreted to mean possible periodicities in the data structure(s). For characterization of an S.P, the various techniques that are used for spectrum estimation may be classified as: 1) Fourier transformation of the auto-correlation function 2) Averaging the square of the Fourier transmation of the time series and 3) Use of a shapely tuned filters and 4) Technique based on entropy concepts (MEM). Here we focus on two techniques viz., FFT and MEM. When the data length is short, FFT or other techniques, even though based on sharply tuned filters, may not give realistic results. Therefore we look at the concepts of entropy and the maximum entropy methods. FFT and MEM methods were applied to a set of gold mineralization samples drawn from a gold mineralization in India. The sharp spikes in the spectra indicate high concentration of energy which can be interpreted to mean periodicities in the discrete mineralization.

This apart, generally speaking, if an input data (Xi) is viewed as consisting of signals (Si) and noise (Ni), it is essential to remove the noise. The resultant signal series Si = Xi-Ni. Using the techniques of statistical theory of communication, we may also employ an optimum bilateral smoothing model to remove the noise from the input. The auto-correlation function of the data may be represented by a first order Markovian model say, of the type: C exp[-A |k|], where C and A are constants which can be quantified from the model. The application of this model is illustrated in this paper with examples.