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 B_n, 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.

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广义Born-Jordan分布和应用

Wigner分布无疑是最流行的时频表示之一。但是，它的二次性质是产生不必要的干扰或伪像。抑制这些伪像的愿望是工程师，数学家和物理学家一直在寻找相关的时频分布的原因，其中许多是科恩类的成员。其中，Born-Jordan分布最近引起了许多作者的关注，因为所谓的幻影频率得到了很大的衰减，并且总体上降低了噪声。它似乎在量子力学中也起着关键作用。中心洞察力依赖于这样一个分布的核心，该分布包含窦性心律，第一B样条B ₁的傅里叶变换。。这个想法是用样条或阶数n（用B _n表示）代替函数B ₁，当傅立叶变换时产生函数（sinc）ⁿ，其无穷大时的衰减速度随n增加。相关的Cohen内核由\（{\ Theta} ^ {n}（z_ {1}，z_ {2}）= \ text {sinc} ^ {n}（z_ {1} \ cdot z_ {2}）给出\），\（\ mathbb {N} \中的n \），相应的时频分布称为n阶广义Born-Jordan分布。我们表明，这种新的表示形式具有很大的潜力来衰减不想要的干扰效应，并且该衰减效应随n的增加而增加。。我们对这些特性的证明需要使用跨学科方法，使用来自微局部和时频分析的工具。作为副产品，研究了新的量化规则和相关的伪微积分。