Capus Chris, Brown Keith
Ocean Systems Laboratory, School of Engineering & Physical Sciences, Mountbatten Building, Heriot-Watt University, Riccarton, Edinburgh, United Kingdom.
J Acoust Soc Am. 2003 Jun;113(6):3253-63. doi: 10.1121/1.1570434.
The fractional Fourier transform (FrFT) provides a valuable tool for the analysis of linear chirp signals. This paper develops two short-time FrFT variants which are suited to the analysis of multicomponent and nonlinear chirp signals. Outputs have similar properties to the short-time Fourier transform (STFT) but show improved time-frequency resolution. The FrFT is a parameterized transform with parameter, a, related to chirp rate. The two short-time implementations differ in how the value of a is chosen. In the first, a global optimization procedure selects one value of a with reference to the entire signal. In the second, a values are selected independently for each windowed section. Comparative variance measures based on the Gaussian function are given and are shown to be consistent with the uncertainty principle in fractional domains. For appropriately chosen FrFT orders, the derived fractional domain uncertainty relationship is minimized for Gaussian windowed linear chirp signals. The two short-time FrFT algorithms have complementary strengths demonstrated by time-frequency representations for a multicomponent bat chirp, a highly nonlinear quadratic chirp, and an output pulse from a finite-difference sonar model with dispersive change. These representations illustrate the improvements obtained in using FrFT based algorithms compared to the STFT.
分数阶傅里叶变换（FrFT）为线性调频信号的分析提供了一种有价值的工具。本文开发了两种短时FrFT变体，适用于多分量和非线性调频信号的分析。其输出具有与短时傅里叶变换（STFT）相似的特性，但具有更高的时频分辨率。FrFT是一种参数化变换，其参数a与调频速率有关。这两种短时实现方式在a值的选择方法上有所不同。在第一种方法中，全局优化过程会参考整个信号选择一个a值。在第二种方法中，会为每个加窗段独立选择a值。给出了基于高斯函数的比较方差度量，并证明其与分数域中的不确定性原理一致。对于适当选择的FrFT阶数，所推导的分数域不确定性关系对于高斯加窗线性调频信号最小化。通过对多分量蝙蝠啁啾信号、高度非线性二次调频信号以及具有色散变化的有限差分声纳模型的输出脉冲进行时频表示，证明了这两种短时FrFT算法具有互补优势。这些表示说明了与STFT相比，使用基于FrFT的算法所获得的改进。
