Generalized transfer function estimation and informative priors for positive time-frequency distributions

In the first part of this thesis we present a new technique for estimating the generalized transfer function (GTF) of a time-varying filter from time frequency representations (TFRs) of its output. We use the fact that many of these representations can be written as blurred versions of the GTF. Our goal is to find the GTF such that when blurred, produces the TFR estimate. We use a deconvolution technique to obtain the deblurred GTF. The magnitude squared of this GTF will result in an estimate of the evolutionary spectrum (ES). Our technique is general and can be based on any TFR. The approach consists of minimizing the error between the TFR found from the data and that found by blurring the GTF. The problem as such has many solutions. We, therefore, constrain it by minimizing the distance between the resulting ES and autoterms of the Wigner distribution. Where the cross terms are suppressed using a mask function. In our work we compute the mask function using the evolutionary periodogram. Advantages of this method are: (a) it estimates the GTF without the need for the orthonormal expansion used in other estimators of the ES, (b) it does not require the semi-stationarity assumption used in the current deconvolution techniques, (c) it can be based on many TFRs, (d) the GTF obtained can be used to reconstruct the signal and to model linear time-varying systems and (e) the resulting ES estimate out performs the ES obtained by using the current estimation techniques and can be made to satisfy the time and frequency marginals while maintaining positivity.;In the second part of the thesis we develop a method for generating an informative prior when constructing a positive time-frequency distribution (TFD) by the method of the minimum' cross-entropy (MCE). This new prior uses a combination similar to the one described above of a mask function and the Wigner Distribution. It results in a more informative MCE-TFD, as quantified via entropy and mutual information measures. The procedure allows any of the bilinear distributions to be used in the prior and the TFRs obtained by this procedure are close to the ones obtained by the deconvolution procedure. Thus, if the objective is to obtain the TFR only, this procedure offers a viable alternative at reduced computational cost.