Adaptive Frequency Bin Interval in FFT via Dense Sampling Factor $α$
Haichao Xu
TL;DR
The paper addresses the rigid frequency bin spacing of FFT-based spectral analysis, which gives rise to the picket fence effect. It proposes a dense sampling factor $\alpha$ to adjust the DFT bin interval, deriving modified transform equations and highlighting that $\alpha>1$ corresponds to zero-padding while $\alpha<1$ reduces the bin count with potential information loss; it also develops an FFT-accelerated implementation for the $\alpha$-DFT and analyzes its computational complexity. The contributions include the mathematical derivation of the $\alpha$-DFT, a divide-and-conquer FFT for non-square transform shapes with complexity $T_{\alpha}(n) = \mathcal{O}(\mathcal{M} \log \mathcal{N})$, a comparison to the conventional zero-padding approach, and practical guidance on choosing $\alpha$ under data quality and resolution constraints. Overall, the approach provides flexible, efficient spectral analysis that can tailor bin spacing to signal characteristics without altering the time-domain data, potentially reducing computation for high-resolution needs.
Abstract
The Fast Fourier Transform (FFT) is a fundamental tool for signal analysis, widely used across various fields. However, traditional FFT methods encounter challenges in adjusting the frequency bin interval, which may impede accurate spectral analysis. In this study, we propose a method for adjusting the frequency bin interval in FFT by introducing a parameter $α$. We elucidate the underlying principles of the proposed method and discuss its potential applications across various contexts. Our findings suggest that the proposed method offers a promising approach to overcome the limitations of traditional FFT methods and enhance spectral analysis accuracy.