Class-dependent, discrete time-frequency distributions via operator theory

Abstract

We propose a property for kernel design which results in distributions for each of two classes of signals which maximally separates their energies in the time-frequency plane. Such maximally separated distributions may result in improved classification because the signal representation is optimized to accentuate the differences in signal classes. This is not the case with other time-frequency kernels which are optimized based upon some criteria unrelated to the classification task. Using our operator theory formulation for time-frequency representations, our "maximal separation" criteria takes on a very easily solved form. Analysis of the solution in both the time-frequency and ambiguity planes is given along with an example on discrete signals.

Optimizing time-frequency kernels for classification

“…is classified via , where is an element-by-element product. Our approach to kernel design and classification is a generalization of the signal class-dependent method described in more detail before [9]- [11]. A brief overview of this previously described approach is provided as it is essential to understanding the motivations for our modifications.…”

“…This average smoothed TFR is defined as (8) is the matrix representation of the generalized TFR derived from . The class-dependent kernel is given by (9) where is the Frobenius norm of the matrix. McLaughlin and Atlas [11] have shown that the kernel that achieves this maximization can be obtained directly in the auto-ambiguity plane.…”

“…This is given by (10) is the "average" auto-ambiguity function of class , which is defined as (11) A unitary energy constraint is imposed on the kernel given by (12) The kernel that results from the maximization in (10), constrained by (12), is "1" at the single location in , where the s are most separated and "0" elsewhere. McLaughlin and Atlas [11] have shown that this corresponds to the kernel that maximizes (9), which is the class-dependent kernel.…”

“…This alternate approach allows classification of limited size data sets, with minimal performance loss compared with the linear discriminant kernel. It is important to note that using either of these approaches no longer guarantees that the resulting TFRs satisfy (9). However, the resulting TFRs are superior in terms of classification performance, and for certain signals, these kernels can be optimal for classification.…”

Individual Radio Frequency Interference Identification on VHF Radar Based on Time-Frequency and Chaotic Characteristics

“…Hippenstiel and Payal applied discrete wavelet decomposition to the research on the individual identification, but their method requires a relatively high signal to noise ratio [1]. McLaughlin proposed class time-frequency distribution for pattern classification [2]. Serinken extracted the information dimension and correlation dimension characteristics of the transient signals [3].…”