分数阶傅里叶群变换算子理论及其在信息传输中的应用

In this program, from the perspective of characteristic function,the fractional Fourier transform (FrFT) operator with single rotational angle is generalized to group fractional Fourier transform (GFrFT) operator with a angle vector, and the properties such as linearity, reversibility, order superposition, are studied. In order to improve the computational efficiency, kernel matrix is designed by directly discretizing method, and the fast alogorithm of discrete GFrFT is derived.From the point of application, FrFT has proposed a new evoluation idea for orthogonal frequency division multiplexing (OFDM) technology. However,the exponential bases of OFDM systems are directly replced by chirp bases, and it is difficult to cancell the co-channel interference, which is caused by the similar chirp bases at the same subcarrier.Fortunately, GFrFT theorem, proposed in this program, with optimal angle resource set searched by intelligent algorithm, can modulate the signals, at each subcarrier, with a group of mutually approximately orthogonal chirp bases. Therefore, in the proposed optimal GFrFT based OFDM system, the inter-stream base functions at different sub-channel are mutually orthgonal, and the intral-stream base functions at same sub-channel are mutually approximately orthogonal, which exactly provides the novel "angle-dimensional" diversity and multiplexing to information transmission.

本项目把分数阶傅立叶变换(FrFT)算子中的单一旋转角推广为角向量集，从特征函数角度给出分数阶傅立叶群变换(GFrFT)算子的定义，并研究其线性性、可逆性、阶数迭加性等；为了提高计算效率，采用直接离散化方法设计核矩阵，并获得离散GFrFT快速算法；从应用角度，虽然FrFT作为傅立叶变换的广义形式已经为正交频分复用(OFDM)技术提供了一种改进思路，但这只是把传统指数基简单替代为chirp基,同频干扰等问题未得到本质改善,这是基于单一类型基的OFDM技术本身内在缺陷的外在反应。然而本项目中提出的GFrFT理论结合由智能搜索所生成的最优角资源集,使每个子信道上利用一组近似正交chirp基进行调制,得到的基于最优GFrFT-OFDM技术具有调制数据流内部不同子信道上基函数纵向正交和数据流之间相同子信道上基函数近似正交的特性，恰好为信息传输提供新的"角度维"分集和复用增益。

针对传统正交频分复用(OFDM)信息传输系统中，以指数型特征函数作为构成能量有限信号空间的标准正交基在干扰抑制、频偏敏感性、并行传输等方面存在先天缺陷。本项目把分数阶傅立叶变换(FrFT)算子中的单一旋转角推广为角向量集，从特征函数角度给出分数阶傅立叶群变换(GFrFT)算子的定义，并研究其线性性、可逆性、阶数迭加性等；为了提高计算效率，采用直接离散化方法设计核矩阵，并获得离散GFrFT快速算法；从应用角度，虽然FrFT作为傅立叶变换的广义形式已经为正交频分复用(OFDM)技术提供了一种改进思路，但这只是把传统指数基简单替代为chirp基，同频干扰等问题未得到本质改善,这是基于单一类型基的OFDM技术本身内在缺陷的外在反应。然而本项目中提出的GFrFT理论结合由智能搜索所生成的最优角资源集，使每个子信道上利用一组近似正交chirp基进行调制，得到的基于最优GFrFT-OFDM技术具有调制数据流内部不同子信道上基函数纵向正交和数据流之间相同子信道上基函数近似正交的特性，恰好为信息传输提供新的"角度维"分集和复用增益。同时，利用该线性旋转算子在时频分析中无分量交叉相优点，实现最优旋转域的系统干扰抑制。本项目力求从内在机制角度，寻求既能继承传统OFDM 技术优势，又能对外在问题和困难予以避免的“基函数改造”的治本方式，为传统系统提供新的演进方向。