[Paper Review] A Multiscale Sub-linear Time Fourier Algorithm for Noisy Data

Andrew Christlieb, Debbie A. Lawlor+1 more|arXiv (Cornell University)|Apr 16, 2013

Sparse and Compressive Sensing Techniques18 references3 citations

TL;DR

This paper presents a novel multiscale sub-linear time Fourier algorithm for sparse signals corrupted by noise, extending a prior deterministic sparse Fourier transform. By using geometrically spaced time samples and a progressive frequency identification strategy akin to $β$-encoders, the method achieves robust error correction, enabling accurate frequency estimation at coarser sampling rates and outperforming FFTW and other sparse FFT variants across diverse noise and sparsity levels, with average runtime of $\textup{O}(k\log(k)\log(N/k))$. The algorithm is the first to demonstrate such error-correction in the sparse Fourier transform context, significantly improving performance in noisy environments without sacrificing speed.

ABSTRACT

We extend the recent sparse Fourier transform algorithm of (Lawlor, Christlieb, and Wang, 2013) to the noisy setting, in which a signal of bandwidth N is given as a superposition of k << N frequencies and additive noise. We present two such extensions, the second of which exhibits a novel form of error-correction in its frequency estimation not unlike that of the beta-encoders in analog-to-digital conversion (Daubechies et al, 2006). The algorithm runs in time O(k log(k) log(N/k)) on average, provided the noise is not overwhelming. The error-correction property allows the algorithm to outperform FFTW, a highly optimized software package for computing the full discrete Fourier transform, over a wide range of sparsity and noise values, and is to the best of our knowledge novel in the sparse Fourier transform context.

Motivation & Objective

• To address the challenge of efficiently and accurately computing the Fourier transform of signals with sparse frequency content when corrupted by additive noise.
• To extend the deterministic sparse Fourier transform of Christlieb et al. [1] to the noisy setting, where only $k \ll N$ frequencies are significant and noise is present.
• To develop a method that maintains sub-linear runtime while achieving robustness to noise through novel error-correction mechanisms.
• To outperform existing sparse Fourier transform algorithms, including FFTW and sFFT 1.0, in both accuracy and runtime under realistic noisy conditions.

Proposed method

• The algorithm uses two sets of equispaced time samples with a small temporal offset to detect and resolve aliasing, enabling direct frequency estimation for isolated components.
• A multiscale approach is introduced, using geometrically spaced time shifts to progressively refine frequency estimates, mimicking the behavior of $\beta$-encoders in analog-to-digital conversion.
• The method employs a robust aliasing test to distinguish between true and spurious frequency components, minimizing false positives even in high-noise regimes.
• Frequency estimates are refined iteratively using a progressive identification strategy that corrects errors through multi-scale sampling, enhancing accuracy without increasing computational complexity.
• The algorithm is designed to run in $\textup{O}(k\log(k)\log(N/k))$ average time, significantly faster than FFTW and prior sparse FFT methods in the noisy regime.
• A rounding-based variant is also provided, requiring $p \geq \sigma^{2/3}$ for correct frequency recovery, but this leads to higher sampling rates and slower performance.

Experimental results

Research questions

• RQ1 Can a deterministic sparse Fourier transform be extended to handle noisy signals while maintaining sub-linear runtime and high accuracy?
• RQ2 How can error correction be incorporated into sparse Fourier transform algorithms to improve robustness in the presence of noise?
• RQ3 What sampling strategy enables accurate frequency estimation at coarser sampling rates without sacrificing performance?
• RQ4 How does the proposed multiscale algorithm compare to FFTW and other sFFT variants in terms of runtime and error metrics across varying noise and sparsity levels?

Key findings

• The proposed multiscale algorithm achieves average runtime of $\textup{O}(k\log(k)\log(N/k))$, outperforming FFTW3 across all tested sparsity levels and noise values.
• The algorithm's error-correction mechanism allows it to maintain near-zero EMD($\omega$) error across a wide range of noise levels, with EMD($\omega$) values consistently below $10^{-16}$, indicating accurate frequency recovery.
• In high-noise, high-sparsity regimes, the algorithm introduces an average of only one spurious frequency per signal, demonstrating strong robustness to false positives.
• The rounding-based variant requires $p \geq \sigma^{2/3}$ for correct recovery, but this leads to slower performance due to high sampling rates, especially at moderate noise levels.
• The multiscale algorithm significantly outperforms sFFT 1.0 and AAFFT in both accuracy and speed, particularly in noisy environments where sFFT 1.0 shows non-zero EMD($\omega$) errors.
• The algorithm's performance is stable across noise levels, with no significant increase in runtime until very high noise ($\sigma = 0.512$), indicating strong resilience.