We describe a family of iterative algorithms that involve the repeated execution of discrete and inverse discrete Fourier transforms. One interesting member of this family is motivated by the discrete Fourier transform uncertainty principle and involves the application of a sparsification operation to both the real domain and frequency domain data with convergence obtained when real domain sparsity hits a stable pattern. This sparsification variant has practical utility for signal denoising, in particular the recovery of a periodic spike signal in the presence of Gaussian noise. General convergence properties and denoising performance relative to existing methods are demonstrated using simulation studies. An R package implementing this technique and related resources can be found at https://hrfrost.host.dartmouth.edu/IterativeFT.

翻译：我们描述了一类迭代算法族，其核心在于重复执行离散与逆离散傅里叶变换。该算法族中一个值得关注的成员源于离散傅里叶变换的不确定性原理，其通过在实数域和频域数据上同时施加稀疏化操作，并在实数域稀疏性达到稳定模式时实现收敛。该稀疏化变体在信号去噪方面具有实用价值，尤其是用于在存在高斯噪声的条件下恢复周期性尖峰信号。通过仿真研究，本文展示了该方法的普遍收敛性质，以及相较于现有方法的去噪性能。实现该技术的R语言包及相关资源可在https://hrfrost.host.dartmouth.edu/IterativeFT获取。