分数阶微积分在现代信号分析与处理中应用的研究
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摘要
本文的主要研究是对分数阶微积分在现代信号分析与处理中的应用进行研究。
首先,系统地论述了连续子波变换数值实现中尺度采样间隔的确定的基本理论。按照信号的最高数字频率等于或小于π的两种情况,论证了均匀点格采样时连续子波变换数值实现中Morlet母波以及偶对称或奇对称的各阶高斯函数导数解析母波的尺度采样间隔的最佳取值,并且特别分析了著名的墨西哥帽母波的尺度采样间隔的最佳取值;另外还讨论了奇对称母波数值实现中同时所需的时间平移量;另外还研究了偶对称或奇对称的各阶高斯函数导数解析母波的相应数字滤波器的波动情况,对其波动性进行了研究;最后对连续子波变换数值实现中均匀点格采样的研究结论推广到二进点格采样和二进抽取采样两种情况。系统地论述了连续子波变换数值实现中信号时间和扫描时间之间的几何关系;论述了连续子波变换数值实现中起始扫描时间的最佳取值范围。
第二,推导并研究信号分数阶微积分的五种数值算法实现算法。首先推导并比较信号分数阶微分的幂级数数值算法、Fourier级数数值算法,并将这两种算法与经典的基于Grümwald-Letnikov定义的数值算法相比较;进而,推导具有较高精度和计算速度的基于子波变换的分数阶微积分快速数值算法;最后,以计算精度为代价进一步提高计算速度,推导基于子波变换的快速工程算法。
The dissertation mainly concerns the application of fractional calculus to modern signal analysis and processing.
In the first, it discusses the fundamental theory for ascertaining of scale sample step in implementing the numerical value of the continuous wavelet transform. And it demonstrates the optimal scale-sampling step value of even or odd symmetric analytical mother wave for all phases Gauss function’s differential coefficients and Morlet mother wave in its numerical implementing when even dot-grid sampling is adopted, under two occasions when the highest frequency is or lower thanπ. Especially, it analyzes the optical scale-sampling step of Mexico Cap mother wave. In addition, it discusses the time parallel move in implementing odd symmetric analytical mother wave, and the fluctuant of accordingly digital filter of even or odd symmetric analytical mother wave for all phases Gauss’s differential coefficient. Then it puts the results of even dot-grid sampling to binary one and binary spot-check sampling. The paper describes the geometrical relationship of signal time and scanning time in continuous wavelet transform, and its best value range for
首先,系统地论述了连续子波变换数值实现中尺度采样间隔的确定的基本理论。按照信号的最高数字频率等于或小于π的两种情况,论证了均匀点格采样时连续子波变换数值实现中Morlet母波以及偶对称或奇对称的各阶高斯函数导数解析母波的尺度采样间隔的最佳取值,并且特别分析了著名的墨西哥帽母波的尺度采样间隔的最佳取值;另外还讨论了奇对称母波数值实现中同时所需的时间平移量;另外还研究了偶对称或奇对称的各阶高斯函数导数解析母波的相应数字滤波器的波动情况,对其波动性进行了研究;最后对连续子波变换数值实现中均匀点格采样的研究结论推广到二进点格采样和二进抽取采样两种情况。系统地论述了连续子波变换数值实现中信号时间和扫描时间之间的几何关系;论述了连续子波变换数值实现中起始扫描时间的最佳取值范围。
第二,推导并研究信号分数阶微积分的五种数值算法实现算法。首先推导并比较信号分数阶微分的幂级数数值算法、Fourier级数数值算法,并将这两种算法与经典的基于Grümwald-Letnikov定义的数值算法相比较;进而,推导具有较高精度和计算速度的基于子波变换的分数阶微积分快速数值算法;最后,以计算精度为代价进一步提高计算速度,推导基于子波变换的快速工程算法。
The dissertation mainly concerns the application of fractional calculus to modern signal analysis and processing.
In the first, it discusses the fundamental theory for ascertaining of scale sample step in implementing the numerical value of the continuous wavelet transform. And it demonstrates the optimal scale-sampling step value of even or odd symmetric analytical mother wave for all phases Gauss function’s differential coefficients and Morlet mother wave in its numerical implementing when even dot-grid sampling is adopted, under two occasions when the highest frequency is or lower thanπ. Especially, it analyzes the optical scale-sampling step of Mexico Cap mother wave. In addition, it discusses the time parallel move in implementing odd symmetric analytical mother wave, and the fluctuant of accordingly digital filter of even or odd symmetric analytical mother wave for all phases Gauss’s differential coefficient. Then it puts the results of even dot-grid sampling to binary one and binary spot-check sampling. The paper describes the geometrical relationship of signal time and scanning time in continuous wavelet transform, and its best value range for
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