Directly computing Fourier power spectra at fractional periods of real sequences can be beneficial in many digital signal processing applications. In this article, we present a fast algorithm to compute the fractional Fourier power spectra of real sequences. For a real sequence of length ofwe may deduce its congruence derivative sequence with a length of l. The discrete Fourier transform of the original sequence can be calculated by the discrete Fourier transform of the congruence derivative sequence. The relation of discrete Fourier transforms between the two sequences may derive the special features of Fourier power spectra of the integer and fractional periods for a real sequence. It has been proved mathematically that after calculating the Fourier power spectrum (FPS) at an integer period, the Fourier power spectra of the fractional periods related this integer period can be easily represented by the computational result of the FPS at the integer period for the sequence. Computational experiments using a simulated sinusoidal data and protein sequence show that the computed results are a kind of Fourier power spectra corresponding to new frequencies that cannot be obtained from the traditional discrete Fourier transform. Therefore, the algorithm would be a new realization method for discrete Fourier transform of the real sequence.

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计算分数周期傅里叶频谱的一种快速算法

在许多数字信号处理应用中，直接计算实序列的分数周期的傅立叶功率谱可能是有益的。在本文中，我们提出了一种快速算法来计算真实序列的分数傅立叶功率谱。对于长度为我们可以推导出其长度为l 的同余导数序列. 原始序列的离散傅里叶变换可以通过同余导数序列的离散傅里叶变换来计算。两个序列之间的离散傅立叶变换的关系可以推导出实序列的整数周期和分数周期的傅立叶功率谱的特殊特征。数学上已经证明，在计算整数周期的傅立叶功率谱（FPS）后，与该整数周期相关的分数周期的傅立叶功率谱可以很容易地由序列的整数周期的 FPS 的计算结果表示. 使用模拟正弦数据和蛋白质序列的计算实验表明，计算结果是一种与传统离散傅立叶变换无法获得的新频率相对应的傅立叶功率谱。因此，该算法将是实数序列离散傅立叶变换的一种新的实现方法。