Iterative execution of discrete and inverse discrete Fourier transforms with applications for signal denoising via sparsification

TL;DR

The paper addresses denoising by recovering sparse periodic spike signals using an iterative Fourier framework. It introduces IterativeFT, which repeatedly applies $h()$, $dft()$, $g()$, and $dft^{-1}()$ with convergence driven by a real-domain sparsity pattern, and demonstrates effective 1D and 2D spike recovery, often outperforming standard non-iterative thresholding and filtering methods. Key contributions include a convergence analysis under real- and frequency-domain sparsification, extensive simulation results across multiple spike models and SNRs, and an open-source R package implementing the method. The work provides a practical, robust approach to denoising periodic structures with potential extensions to other transforms and complex-valued data, offering significant impact for signal processing tasks where sparsity constraints in both domains are advantageous.

Abstract

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.

Figures (17)

• Figure 1: Mean iterations until convergence for random length $n$ vectors of $\mathcal{N}(0,1)$ random variables and $h()$ and $g()$ functions that rank order the elements according to absolute value or magnitude and set the bottom 50% to 0. Convergence is based on repeating the same pattern of sparsity after execution of $h()$ on two sequential iterations.
• Figure 2: Mean iterations until convergence for random length 500 vectors of $\mathcal{N}(0,1)$ random variables. For these results $h()$, $g()$, and $c()$ have similar definitions as detailed for Figure \ref{['fig:convergence_vs_n']}. In this case, $n$ was fixed at 500 and the sparsity proportion varied between 0.1 and 0.9.
• Figure 3: Output of the IterativeFT method on $\mathbf{x}$ vector simulated according to the model detailed in Section \ref{['sec:vector_denoising']}. The top panels show the periodic spike signal and Gaussian noise. The remaining panels show the input data and output from the $h()$ function after each iteration with the error relative to the spike signal captured as a dashed red line and quantified as mean squared error (MSE).
• Figure 4: Average MSE ratio relative to the number of cycles, b, captured in the input $\mathbf{x}$ vector. The MSE ratio is computed as $MSE_c/MSE_1$ where $MSE_c$ represents the MSE after convergence and $MSE_1$ represents the MSE after the first execution of $h()$ on the input $\mathbf{x}$.
• Figure 5: Average MSE ratio after each iteration of the algorithm.