Construction of positive time-frequency distributions

Abstract

Abstruct-A general method for constructing nonnegativedefinite, joint time-frequency distributions (TFD's) satisfying the marginals of time Is(t)l' and frequency lS(f)12 is presented. As nonnegative-definite distributions with the correct marginals, these TFD's are members of the Cohen-Posch class. Several examples illustrating properties of these TFD's are presented for both synthetic and real signals. Patrick J. Loughlin

Positive time-frequency distributions based on joint marginal constraints

“…In this correspondence, we consider an information theoretic approach for the definition of proper time-frequency distributions. It follows the pioneering work of Loughlin et al [5]- [7]. We will show that sometimes, the two classical marginal requirements used as constraints in an entropy optimization problem cannot provide sufficient information to yield a representation with improved resolution with respect to the prior.…”

“…Recently, a member of Cohen-Posh class of positive distributions was defined by Loughlin et al [5]- [7]. It was formulated as the solntion to a minimum cross-entropy (MCE) optimization problem subject to linear constraints in general and, in particular, the fulfillment of the marginal conditions.…”

“…Let us first illustrate the main limitations associated with the MCE-TFD as defined in [5]- [7] when only the marginals are used as constraints. MCE subject to the two marginal constraints in general can only improve, to a limited extend, the apriori distributions, unless signal components are oriented in "vertical" or "horizontal" lines in the time-frequency plane.…”

“…The limitations of the MCE-TFD using the marginals constraints only can be easily recognized if we analyze the classical form of the solution to the cross-entropy minimization problem. Using the method of Lagrange multipliers [7] 'All distributions for synthetic signals have been computed using 256 samples of the signal and with a 256 x 256 resolution. The Spectrogram employed a Gaussian window of variance equal to 256/3.5.…”

A class of positive isentropic time-frequency distributions

“…Cohen's indications led to research, most often in terms of variational Lagrange cross-entropy minimization procedures [6]- [8] with known constraints in order to obtain CPZ TFDs with nice properties.…”