Generalized Born-Jordan distributions and applications

Abstract

One of the most popular time-frequency representations is certainly the Wigner distribution. Its quadratic nature is, however, at the origin of unwanted interferences or artefacts. The desire to suppress these artefacts is the reason why engineers, mathematicians and physicists have been looking for related time-frequency distributions, many of them being members of the Cohen class. Among these, the Born-Jordan distribution has recently attracted the attention of many authors, since the so-called ghost frequencies are grandly damped, and the noise is, in general, reduced; it also seems to play a key role in quantum mechanics. The central insight relies on the kernel of such a distribution, which contains the sinus cardinalis sinc, the Fourier transform of the first B-spline B1. The idea is to replace the function B1 with the spline or order n, denoted by Bn, yielding the function (sinc)n when Fourier transformed, whose speed of decay at infinity increases with n. The related Cohen kernel is given by ${\Theta }^{n}(z_{1},z_{2})=\text {sinc}^{n}(z_{1}\cdot z_{2})$ , $n\in \mathbb {N}$ , and the corresponding time-frequency distribution is called generalized Born-Jordan distribution of ordern. We show that this new representation has a great potential to damp unwanted interference effects and this damping effect increases with n. Our proofs of these properties require an interdisciplinary approach, using tools from both microlocal and time-frequency analysis. As a by-product, a new quantization rule and a related pseudo-differential calculus are investigated.