Iterative Execution of Discrete Fourier Transforms for Signal Denoising

Core Concepts

Iterative algorithms using discrete Fourier transforms are effective for signal denoising.

Abstract

The content discusses iterative algorithms based on discrete and inverse discrete Fourier transforms for signal denoising. It explores the convergence properties, generalizations, and practical utility of these algorithms. The IterativeFT method is highlighted as a powerful technique for recovering periodic spike signals in the presence of noise. Simulation studies demonstrate the superior denoising performance of IterativeFT compared to existing methods like real domain thresholding, frequency domain thresholding, Butterworth bandpass filtering, and wavelet filtering. The paper concludes by suggesting future research directions and acknowledging funding sources. Problem Statement: Family of iterative algorithms involving discrete Fourier transforms. Motivated by uncertainty principle & sparsification operation. General Convergence Properties: Key question: Combinations of h(), g(), c() functions enabling convergence. Interest in scenarios relevant to data analysis applications. Algorithm 1: Inputs: x ∈Rn, im (maximum iterations). Outputs: y ∈Rn, ic (iterations completed). Detailed definition provided. Trivial Cases: Convergence after single iteration if h() and g() are identity functions. Simplification with only one function being identity function. Generalizations: Matrix-valued input possible. Complex-valued input allowed. Alternative invertible discrete transform can replace DFT. Related Techniques: Relationship to standard DFT analysis discussed. Comparison with other iterative algorithms like ADMM, Dykstra's algorithm, EM made. Iterative Convergence under Sparsification: Subclass involves sparsification functions for h() and g(). Motivated by DFT uncertainty principle. Stable compromise between real and frequency domain sparsity achieved upon convergence. Detection of Spike Signals Using Iterative Convergence: Assessment of practical utility using simulation design. Performance evaluation relative to varying parameters like n value and sparse proportion. Comparative Evaluation of Denoising Performance: Simulation study design outlined. Comparison with existing denoising techniques - real/frequency domain thresholding, Butterworth bandpass filtering, wavelet filtering. Superior performance of IterativeFT method demonstrated through simulation results across different spike signal patterns and SNR values.

Stats

arXiv:2211.09284v3 [eess.SP] 24 Mar 2024

Quotes

Key Insights Distilled From

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

by H. Robert Fr... at arxiv.org 03-26-2024

https://arxiv.org/pdf/2211.09284.pdf

Deeper Inquiries

What are the implications of using sparsification functions in both real and frequency domains

Using sparsification functions in both real and frequency domains has significant implications for signal denoising. By applying sparsification operations iteratively, the IterativeFT method aims to strike a balance between real domain sparsity and frequency domain representation. The discrete Fourier transform uncertainty principle guides this approach, ensuring that inducing sparsity in one domain affects the other. This interplay leads to a stable compromise where the solution represents a harmonious level of sparsity in both domains. The implications are twofold: Enhanced Signal Recovery: The iterative application of sparsification functions allows for more precise recovery of periodic spike signals amidst noise interference. By iteratively refining the data through h() and g(), the method can effectively isolate and preserve signal components while suppressing noise artifacts. Convergence to Stable Patterns: Convergence is achieved when a stable pattern of sparsity emerges in both domains, indicating successful denoising without losing essential signal information. This iterative process ensures that the final output strikes an optimal balance between retaining signal features and reducing noise. In essence, using sparsification functions in both real and frequency domains enables the IterativeFT method to leverage complementary aspects of data representation, leading to robust denoising capabilities with improved convergence properties.

How does the performance of the IterativeFT method compare to other advanced denoising techniques

The performance of the IterativeFT method surpasses that of traditional denoising techniques like real domain thresholding, frequency domain thresholding, wavelet filtering, and Butterworth bandpass filtering across various scenarios: Superior Denoising Accuracy: In comparative evaluations on simulated data with different spike patterns at varying frequencies and SNR levels, IterativeFT consistently outperforms other methods by achieving near-perfect recovery of periodic spike signals even under challenging conditions. Robustness Across Signal Characteristics: Regardless of spike signal frequencies or SNR values tested, IterativeFT demonstrates remarkable denoising efficacy compared to alternative techniques such as simple thresholding or advanced filters like wavelet or Butterworth. Adaptability to Diverse Data Models: The versatility of IterativeFT extends its applicability beyond specific signal types or noise distributions due to its iterative nature involving multiple transformations on input data. Efficient Noise Suppression: Through iterative execution involving discrete Fourier transforms coupled with sparse operations in both domains, IterativeFT achieves efficient noise suppression while preserving essential signal components effectively.

How can these iterative algorithms be extended to handle more complex data structures or transformations

To extend these iterative algorithms for handling more complex data structures or transformations: Matrix-Valued Inputs: Modify Algorithm 1 to accept matrix inputs instead of vectors by incorporating two-dimensional counterparts for discrete Fourier transforms. Complex-Valued Inputs: Adapt the algorithm framework to accommodate complex-valued inputs by adjusting computations accordingly. 3 .Alternative Transformations: Explore replacing discrete Fourier transforms with other invertible discrete transforms like wavelet transform. 4 .Hypercomplex Valued Inputs - Extend support for hypercomplex-valued inputs (e.g., quaternion or octonion) by adapting transformation steps accordingly 5 .Advanced Sparsification Techniques - Integrate more sophisticated sparse operations beyond basic thresholding into h()and g()functions By expanding these algorithms' capabilities towards diverse data structures and transformations, the potential applications could be broadened across various fields requiring advanced signal processing techniques."