On the reduction of the interferences in the Born-Jordan distribution

TL;DR

This paper provides a rigorous mathematical analysis of the Born-Jordan distribution's interference reduction properties, especially addressing the directional smoothing effects and limitations using wave-front set and microlocal analysis.

Contribution

It offers the first formal microlocal analysis explaining the directional interference smoothing in the Born-Jordan distribution for general signals.

Findings

Horizontal and vertical interferences are less damped in the Born-Jordan distribution.

Wave-front set analysis clarifies the distribution's smoothing effects.

Mathematical explanation aligns with heuristic engineering observations.

Abstract

One of the most popular time-frequency representation is certainly the Wigner distribution. To reduce the interferences coming from its quadratic nature, several related distributions have been proposed, among which the so-called Born-Jordan distribution. It is well known that in the Born-Jordan distribution the ghost frequencies are in fact damped quite well, and the noise is in general reduced. However, the horizontal and vertical directions escape from this general smoothing effect, so that the interferences arranged along these directions are in general kept. Whereas these features are graphically evident on examples and heuristically well understood in the engineering community, there is not at present a mathematical explanation of these phenomena, valid for general signals in L^2 and, more in general, in the space S' of temperate distributions. In the present note we provide such a rigorous study using the notion of wave-front set of a distribution. We use techniques from Time-frequency Analysis, such as the modulation and Wiener amalgam spaces, and also results of microlocal regularity of linear partial differential operators.