【作者】 王小娟；

【导师】 齐林；

【作者基本信息】 郑州大学 ， 信号与信息处理， 2011， 硕士

【摘要】 分数阶Fourier变换(FrFT)作为传统Fourier变换的广义形式,增加了自由旋转角度参数,在保留传统Fourier的所有特性的基础上并具备新优势,更适用于处理非平稳信号。它广泛应用于科学研究和工程技术的各个方面,随着数字信号处理的迅猛发展,离散分数阶Fourier变换的研究备受关注,多种关于分数阶Fourier变换的离散化方法及相关定理相继提出。本文在基于分数阶圆周定理计算分数阶卷积的基础上对其加以改进,提出一种对长序列进行分段处理计算分数阶卷积的算法,主要内容如下：1.阐述了分数阶Fourier变换的离散算法、采样定理及离散分数阶Fourier变换特有的重要性质—chirp周期性。重点描述了分数阶圆周卷积定理,该定理是本文所提算法的重要理论基础,对基于分数阶圆周卷积定理的快速算法进行改进得到分段计算分数阶卷积的算法。2.提出分段计算分数阶卷积的新方法,该算法从分数阶圆周卷积定理出发,根据离散分数阶Fourier变换具有隐含chirp周期性这一重要性质,采用重叠相加法和重叠保留法对较长序列进行分段处理,得到一种计算离散序列分数阶线性卷积的算法。该算法能够快速计算长度相差较大的两序列的分数阶线性卷积,从而弥补了分数阶圆周卷积定理只适用于处理长度近似或相等的两序列分数阶线性卷积这一不足。并对分段序列长度的设定问题进行深入讨论。最后对本文所提算法进行仿真,结果验证了算法的正确性。3.分析线性调频信号在分数阶Fourier域的滤波问题,并阐述扫频滤波器与分数阶Fourier域滤波器之间存在的关系。在验证本文算法正确性的基础上,提出利用分数阶卷积实现LFM信号滤波的方法。将本文所提出的算法应用于滤波实现过程,对相关原理进行论述,并进行仿真实验。结果验证了滤波算法的有效性,说明算法具有更广泛的实用价值。更多还原

【Abstract】 As a generalization of the classical Fourier transform(FT), the fractional Fourier transform (FrFT) which adds a degree of freedom parameter-rotation angle and retains all the features of the classical Fourier and novel advantages moreover, is very propitious for processing of the non-stationary signals. It is widely used in scientific research and engineering in all aspects. The study of discrete fractional Fourier transform has formed a big climax, with the rapid development applications of the digital signal processing. Many scholars have derived different types of discrete fractional Fourier transform(DFrFT) and sampling theory in fractional Fourier transform domain. A novel algorithm based on fractional Fourier circular convolution theorem is proposed to deal with the fractional linear convolution. The main contributions and innovations of the dissertation are:1. The sampling theorem of the fractional Fourier transform and the hidden chirp periodicity of the discrete fractional Fourier transform is explained, the fractional circular convolution theorem which is the important foundation of the proposed algorithm. It is described in detail. The novel algorithm proposed in this paper is able to deal with the fractional linear convolution by making segment on the longer sequence.2. A novel algorithm based on fractional Fourier circular convolution theorem is proposed to deal with the fractional linear convolution, which is in accordance with the hidden periodicity of the discrete fractional Fourier transform. In the pre-processing, it applies the methods of overlap-save and overlap-add to make segment on the longer sequence. The algorithm is able to overcome the disadvantages of traditional fractional circular convolution theorem, which only can be used to calculate the convolution of two sequences with the similar length. The problem of setting the sequence length is discussed in detail. Simulation results show the effectiveness of this algorithm.3. The filtering algorithm of the linear frequency modulation (LFM) signal in the fractional Fourier domain is explored, it described relationship between the realization of the sweep filter and the filter in fractional Fourier domain. Based on the correctness of the algorithm, the application of fractional convolution algorithm on (LFM) signal filtering is proposed. The relevant principle is discussed. Analyses and simulation results also presented which show that the algorithm has a broader practical value.更多还原