Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Fourier Ratio and complexity of signals

K. Aldaleh, W. Burstein, G. Garza, G. Hart, A. Iosevich, J. Iosevich, A. Khalil, J. King, N. Kulkarni, T. Le, I. Li, A. Mayeli, B. McDonald, K. Nguyen, N. Shaffer

TL;DR

This work introduces and rigorously analyzes the Fourier ratio $FR(f)=\frac{\|\widehat{f}\|_1}{\|\widehat{f}\|_2}$ as a central, scale-invariant measure of Fourier-side structure for signals on $\mathbb{Z}_N$. It proves that signals concentrated on generic sparse spectral supports must have large $FR$, while signals with small $FR$ admit accurate low-degree trigonometrical approximants in both $L^2$ and $L^{\infty}$, and enjoy tight connections to algorithmic rate-distortion and Kolmogorov complexity. The paper establishes quantitative links between $FR$, uncertainty principles, and spectral concentration, and develops stability results under perturbations and random restrictions, alongside practical implications for missing-value imputation in time series. Complementing theory, extensive numerical experiments assess Talagrand’s and Bourgain’s constants and demonstrate real-data FR behavior near the minimal value, supporting the theory’s relevance for learnability and structure detection in signals. Collectively, the results lay a unified framework connecting spectral sparsity, approximation by low-degree polynomials, and information-theoretic aspects of learning for time-series and related signals.

Abstract

We study the Fourier ratio of a signal $f:\mathbb Z_N\to\mathbb C$, \[ \mathrm{FR}(f)\ :=\ \sqrt{N}\,\frac{\|\widehat f\|_{L^1(μ)}}{\|\widehat f\|_{L^2(μ)}} \ =\ \frac{\|\widehat f\|_1}{\|\widehat f\|_2}, \] as a simple scalar parameter governing Fourier-side complexity, structure, and learnability. Using the Bourgain--Talagrand theory of random subsets of orthonormal systems, we show that signals concentrated on generic sparse sets necessarily have large Fourier ratio, while small $\mathrm{FR}(f)$ forces $f$ to be well-approximated in both $L^2$ and $L^\infty$ by low-degree trigonometric polynomials. Quantitatively, the class $\{f:\mathrm{FR}(f)\le r\}$ admits degree $O(r^2)$ $L^2$-approximants, which we use to prove that small Fourier ratio implies small algorithmic rate--distortion, a stable refinement of Kolmogorov complexity.

The Fourier Ratio and complexity of signals

TL;DR

This work introduces and rigorously analyzes the Fourier ratio as a central, scale-invariant measure of Fourier-side structure for signals on . It proves that signals concentrated on generic sparse spectral supports must have large , while signals with small admit accurate low-degree trigonometrical approximants in both and , and enjoy tight connections to algorithmic rate-distortion and Kolmogorov complexity. The paper establishes quantitative links between , uncertainty principles, and spectral concentration, and develops stability results under perturbations and random restrictions, alongside practical implications for missing-value imputation in time series. Complementing theory, extensive numerical experiments assess Talagrand’s and Bourgain’s constants and demonstrate real-data FR behavior near the minimal value, supporting the theory’s relevance for learnability and structure detection in signals. Collectively, the results lay a unified framework connecting spectral sparsity, approximation by low-degree polynomials, and information-theoretic aspects of learning for time-series and related signals.

Abstract

We study the Fourier ratio of a signal , as a simple scalar parameter governing Fourier-side complexity, structure, and learnability. Using the Bourgain--Talagrand theory of random subsets of orthonormal systems, we show that signals concentrated on generic sparse sets necessarily have large Fourier ratio, while small forces to be well-approximated in both and by low-degree trigonometric polynomials. Quantitatively, the class admits degree -approximants, which we use to prove that small Fourier ratio implies small algorithmic rate--distortion, a stable refinement of Kolmogorov complexity.

Paper Structure

This paper contains 49 sections, 27 theorems, 333 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Structure of this paper
  3. Normalization and notation
  4. Fourier Ratio
  5. Numerical sparsity and coherence
  6. Bi-Fourier ratio
  7. Some key background results
  8. The Fourier ratio makes its first appearance
  9. Bounding the Fourier ratio from above and below.
  10. Refined lower and upper bounds on the Fourier Ratio and the classical Fourier Uncertainty Principle
  11. $L^1$ versus $L^2$ concentration
  12. Two key examples
  13. Small subgroups of ${\mathbb Z}_{pq}$
  14. Inverse Fourier transform of a random set
  15. Approximation by low degree polynomials
  16. ...and 34 more sections

Key Result

Theorem 1.1

Let $(\varphi_j)_{j=1}^n$ be an orthonormal system in $L^2(\mathbb{Z}_N)$ with $\|\varphi_j\|_{L^\infty} \leq K$ for $1 \leq j \leq n$. There exists a constant $\gamma_0 \in (0,1)$ and a subset $I \subset \{1, \dots, n\}$ with $|I| \ge \gamma_0 n$ such that for every $a = (a_i) \in \mathbb{C}^n$, where $C_T > 0$ is a universal constant.

Figures (8)

  • Figure 1: Four real-world datasets: plots show raw values (not Fourier ratios), so the $y$-axis is the observed series value.
  • Figure 2: $C(q)^{\frac{q}{q-2}}$ vs $q$
  • Figure 3: $C(q)^{\frac{q}{q-2}}$ vs $N$
  • Figure 4: $C(q)^{\frac{q}{q-2}}$ vs $q$ and $N$
  • Figure 5: $C_T$ vs $q$
  • ...and 3 more figures

Theorems & Definitions (48)

  • Theorem 1.1
  • Definition 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Lemma 1.6
  • Remark 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 38 more