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，其核心在于重复执行离散傅里叶变换及其逆变换。该算法族中一个有趣的成员受离散傅里叶变换不确定性原理启发，其特点在于对实域和频域数据同时施加稀疏化操作，当实域稀疏性达到稳定模式时即获得收敛。此稀疏化变体在信号去噪方面具有实际效用，特别是在高斯噪声环境下恢复周期性尖峰信号。通过仿真研究，我们展示了该方法的通用收敛特性以及与现有方法相比的去噪性能。实现此技术及相关资源的 R 软件包可在 https://hrfrost.host.dartmouth.edu/IterativeFT 获取。