分数阶傅里叶域变尺度、多分辨分析理论与方法研究

The fractional Fourier transform has been a novel signal processing tool and attracted a lot of attention in recent ten years. As the generalization of Fourier transform, the fractional Fourier transform is adaptive for nonstationary signal processing. The short-time fractional Fourier transform analyzes the signal by means of the rotating time-frequency unit, and can achieve the time-frequency analysis with variable scale. In addition, filter bank can obtain the multi-resolution analysis in the fractional Fourier domain. Thus, this project combines filter bank in the fractional Fourier domain, adaptive filtering, and window function. And the adaptive filtering based on subband decomposition in the fractional Fourier domain is studied systematically, as well as the time-frequency analysis focused on the short-time fractional Fourier transform. The detailed content of this project includes: oversampled filter banks in the fractional Fourier domain, subband decomposition in the fractional Fourier domain, adaptive filtering based on subband decomposition in the fractional Fourier domain, short-time fractional Fourier analysis with variable scale, and etc. Through this project, the theory system of fractional Fourier transform will be enriched further, as well as the methods of adaptive filtering and time-frequency analysis. The research production has the broad application prospect in such fields as beam forming, radar target detection and recognition, transform domain communication, blind channel equalization, adaptive channel estimation, enhancing the security of digital watermarking, and so on.

分数阶傅里叶变换是近十多年来备受关注的一种新型信号处理工具，它是傅里叶变换的广义形式，适合于进行非平稳信号处理。短时分数阶傅里叶变换使用旋转的时频分辨单元分析信号，能够进行变尺度时频分析。分数阶傅里叶域滤波器组则能够实现分数阶傅里叶域多分辨率分析。因此，本项目将分数阶傅里叶域滤波器组、自适应滤波和短时加窗结合起来，系统地研究分数阶傅里叶域子带分解及其自适应滤波，以及以短时分数阶傅里叶变换为核心的时频分析理论。具体内容包括：分数阶傅里叶域过采样滤波器组、分数阶傅里叶域子带分解、基于分数阶傅里叶域子带分解的自适应滤波、短时分数阶傅里叶域变尺度分析等。通过本项目研究，可以进一步完善分数阶傅里叶变换理论体系，丰富自适应滤波和时频分析方法和手段，所得成果在波束形成、雷达目标检测与识别、变换域通信、信道盲均衡、自适应信道估计、增强数字水印系统的安全性及抗各种恶意攻击的稳健性等方面具备广阔的应用前景。

传统信号处理是以傅里叶变换为基础的，其本质问题仍是单频信号的检测和估计问题以及使用单频信号对非平稳信号进行逼近和拟合。因此，以傅里叶变换为基础的信号处理理论已经不能很好适应复杂环境下信息处理系统的高精度处理要求。分数阶傅里叶变换是傅里叶变换的广义形式，以分数阶傅里叶变换为基础的短时分数阶傅里叶变换是将信号分解到一系列加窗的线性调频基上，是使用旋转的时频分辨单元对信号进行分析，而分数阶傅里叶域滤波器组则可以进行分数阶傅里叶域多分辨率分析，因此，本项目通过研究基于分数阶傅里叶域滤波器组和短时分数阶傅里叶变换的变尺度时频分析理论，并在此基础上探索强噪声背景下非线性调频信号这类典型非平稳信号的检测和参数估计新方法，具体研究内容主要包括：1.短时分数阶傅里叶变换的优缺点、适用条件及其改进方法；2.基于Hilbert变换和横向延迟滤波的线性正则域Hilbert滤波实现；3.典型频率时变信号检测判别、多源融合辨识和SAR图像舰船目标检测及应用实践。并利用仿真和实测数据对上述三个方面工作进行了可行性和有效性的验证。本项目研究成果充实了分数域多分辨分析体系，丰富了现代时频分析理论，为通信、雷达、导航等信息系统进一步提升信号处理能力，创新处理方法提供了新思路、新途径。