一类特殊的仿射小波变换及其Poisson求和公式
Special Affine Wavelet Transforms and the Corresponding Poisson Summation Formula
摘要：
特殊的仿射傅里叶变换（SAFT）是一种分析具有更多自由度的非平稳信号的有效工具。然而，SAFT由于其全局核而无法获得非瞬态信号的局部特征，从而使SAFT在需要时频联合信息的情况下变得无能为力。为了克服这一限制，我们提出了一种高度灵活的时频变换，即特殊的仿射小波变换（SAWT），并研究了相关的常数$Q$ -属性在联合时间-频域。利用特殊的仿射傅里叶变换和算子理论，讨论了该变换的基本性质，如瑞利定理、反演公式和值域的表征。此外，还讨论了声表面波的离散对应，得到了相应的重构公式。此外，我们还驱动一个众所周知的特殊仿射维格纳分布和建议的变换之间的直接关系。接着介绍了一种新的波包变换与特殊的仿射傅里叶变换。最后，一个模拟的泊松求和公式所提出的特殊仿射小波变换。
英文摘要：
The special affine Fourier transform (SAFT) is a promising tool for analyzing non-stationary signals with more degrees of freedom. However, the SAFT fails in obtaining the local features of non-transient signals due to its global kernel and thereby make SAFT incompetent in situations demanding joint information of time and frequency. To circumvent this limitation, we propose a highly flexible time-frequency transform namely, the special affine wavelet transform (SAWT) and investigate the associated constant $Q$-property in the joint time-frequency domain. The basic properties of the proposed transform such as Rayleigh's theorem, inversion formula and characterization of the range are discussed using the machinery of special affine Fourier transforms and operator theory. Besides this, the discrete counterpart of SAWT is also discussed and the corresponding reconstruction formula is obtained. Moreover, we also drive a direct relationship between the well known special affine Wigner distribution and the proposed transform. This is followed by introducing a new kind of wave packet transform associated with the special affine Fourier transform. Towards the end, an analogue of the Poisson summation formula for the proposed special affine wavelet transform is derived.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
