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Ramanujan subspace pursuit for signal periodic decomposition

机译：Ramanujan子空间追求的信号周期性分解

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The period estimation and periodic decomposition of a signal represent long-standing problems in the field of signal processing and biomolecular sequence analysis. To address such problems, we introduce the Ramanujan subspace pursuit (RSP) based on the Ramanujan subspace. As a greedy iterative algorithm, the RSP can uniquely decompose any signal into a sum of exactly periodic components by selecting and removing the most dominant periodic component from the residual signal in each iteration. In the RSP, a novel periodicity metric is derived based on the energy of the exactly periodic component obtained by orthogonally projecting the residual signal into the Ramanujan subspace. The metric is then used to select the most dominant periodic component in each iteration. To reduce the computational cost of the RSP, we also propose the fast RSP (FRSP) based on the relationship between the periodic subspace and the Ramanujan subspace and based on the maximum likelihood estimation of the energy of the periodic component in the periodic subspace. The fast RSP has a lower computational cost and can decompose a signal of length N into a sum of if exactly periodic components in O(KN log N). In short, the main contributions of this paper are threefold: First, we present the RSP algorithm for decomposing a signal into its periodic components and theoretically prove the convergence of the algorithm based on the Ramanujan subspaces. Second, we present the FRSP algorithm, which is used to reduce the computational cost. Finally, we derive a periodic metric to measure the periodicity of the hidden periodic components of a signal. In addition, our results show that the RSP outperforms current algorithms for period estimation.
机译：信号的周期估计和周期分解代表了信号处理和生物分子序列分析领域的长期问题。为了解决这些问题，我们介绍了基于Ramanujan子空间的Ramanujan子空间追踪（RSP）。作为一种贪婪的迭代算法，RSP可以通过在每次迭代中从残差信号中选择并删除最主要的周期性分量，从而将任何信号唯一地分解为精确的周期性分量之和。在RSP中，基于通过将残差信号正交投影到Ramanujan子空间中而获得的精确周期性分量的能量，得出了新颖的周期性度量。然后使用该度量来选择每次迭代中最主要的周期成分。为了降低RSP的计算成本，我们还基于周期子空间与Ramanujan子空间之间的关系并基于周期子空间中周期分量能量的最大似然估计，提出了快速RSP（FRSP）。快速RSP具有较低的计算成本，并且可以将长度为N的信号分解为O（KN log N）中是否恰好是周期性分量的总和。简而言之，本文的主要贡献在于三个方面：首先，我们提出了一种RSP算法，用于将信号分解为周期分量，并在理论上证明了基于Ramanujan子空间的算法的收敛性。其次，我们提出了FRSP算法，该算法用于降低计算成本。最后，我们得出一个周期量度来测量信号隐藏周期分量的周期。此外，我们的结果表明，RSP的性能优于当前的周期估计算法。

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