多窗实值离散Gabor展开与变换理论及快速算法

The traditional discrete Gabor expansion and transform with single-window suffer a limitation of the constrained time-frequency resolution governed by the Heisenberg uncertainty principle. To enhance the time-frequency resolution of the discrete Gabor expansion and transform, the multi-window real-valued discrete Gabor expansion and transform including their completeness are researched in this project. The biorthogonality condition between analysis windows and synthesis windows for the multi-window real-valued discrete Gabor expansion and transform will be derived and proved to be equivalent to the completeness condition. The fast algorithms for solving the windows will be presented based on the biorthogonality condition. Moreover, the fast series algorithms for the multi-window based real-valued discrete Gabor expansion and transform will be provided, and the fast parallel algorithms based on multirate filtering will be also presented. This project will make great contributions to the development of the theories and applications of the Gabor time-frequency analysis technology.

由于Heisenberg不确定原理的制约，传统单窗离散Gabor展开与变换的时频分辨精度受到很大限制。为了提高离散Gabor展开与变换的时频分辨精度，本申请项目将在已取得的单窗实值离散Gabor展开与变换理论基础上，研究多窗实值离散Gabor展开与变换及其完备性，推导出多窗实值离散Gabor展开与变换的窗函数双正交关系式，并证明此关系式等同于多窗实值离散Gabor展开与变换的完备性条件；研究利用此窗函数双正交关系式求解多窗函数的快速算法。考虑到实时应用，还将研究多窗实值离散Gabor展开与变换快速串行算法以及基于多抽样率滤波原理的快速并行实现算法。本项目研究丰富和完善了Gabor时频分析理论，具有重要的理论意义和应用价值。

由于Heisenberg不确定原理的制约，传统单窗离散Gabor展开与变换的时频分辨精度受到很大限制。为了提高离散Gabor展开与变换的时频分辨精度，本项目在已取得的单窗实值离散Gabor展开与变换理论基础上，研究了多窗实值离散Gabor展开与变换及其完备性，推导出多窗实值离散Gabor展开与变换的窗函数双正交关系式，并证明了此关系式等同于多窗实值离散Gabor展开与变换的完备性条件；研究了利用此窗函数双正交关系式求解多窗函数的快速算法。考虑到实时应用需要，还研究了多窗实值离散Gabor展开与变换快速串行算法以及基于多抽样率滤波原理的快速并行实现算法。本项目研究丰富和完善了Gabor时频分析理论，具有重要的理论意义和应用价值。