[Paper Review] Exact bounds on the amplitude and phase of the interval discrete Fourier transform in polynomial time

Marco de Angelis|arXiv (Cornell University)|May 27, 2022

Numerical Methods and Algorithms4 citations

TL;DR

This paper establishes the mathematical foundations for a polynomial-time algorithm that computes exact bounds on the amplitude and phase of the interval discrete Fourier transform (DFT). By modeling input uncertainties as intervals and using Minkowski sums of complex interval diagonals, the method fully resolves the dependency problem in interval arithmetic, ensuring rigorous and convex united sets for each harmonic, enabling verified amplitude and phase bounds in complex-valued DFT computations.

ABSTRACT

We elucidate why an interval algorithm that computes the exact bounds on the amplitude and phase of the discrete Fourier transform can run in polynomial time. We address this question from a formal perspective to provide the mathematical foundations underpinning such an algorithm. We show that the procedure set out by the algorithm fully addresses the dependency problem of interval arithmetic, making it usable in a variety of applications involving the discrete Fourier transform. For example when analysing signals with poor precision, signals with missing data, and for automatic error propagation and verified computations.

Motivation & Objective

• To provide formal mathematical foundations for an interval algorithm that computes exact bounds on the amplitude and phase of the discrete Fourier transform (DFT).
• To address the dependency problem in interval arithmetic when computing the DFT under input uncertainty.
• To demonstrate that the united set of the interval DFT is convex and can be computed via nested Minkowski additions.
• To enable rigorous and verified computations in signal processing applications involving imprecise, missing, or uncertain data.
• To support automatic error propagation and verified numerical computations in DFT-based signal analysis.

Proposed method

• Models input signals as interval vectors in ℝ^N, representing uncertainty through bounded intervals.
• Represents the DFT as a complex inner product involving interval-valued input and fixed complex Fourier coefficients.
• Expresses each term in the DFT sum as a complex interval diagonal (a convex polytope with two vertices) in the complex plane.
• Constructs the united set of the DFT output via nested Minkowski additions of these interval diagonals.
• Uses Minkowski addition properties to ensure convexity and exactness of the resulting united set in ℂ.
• Applies Proposition 4.13 to compute vertices of the united set in linear time, avoiding costly convex hull computations.

Experimental results

Research questions

• RQ1 Can the dependency problem in interval arithmetic be fully resolved when computing the DFT under input uncertainty?
• RQ2 Is it possible to compute the exact bounds on the amplitude and phase of the DFT in polynomial time?
• RQ3 What geometric structure does the united set of the interval DFT exhibit, and how can it be characterized?
• RQ4 How can the Minkowski sum of complex interval diagonals be used to represent the full range of the DFT output?
• RQ5 What conditions must be met for the phase of the DFT to be well-defined across the united set?

Key findings

• The united set of the interval DFT is a convex polytope in the complex plane, ensuring exact and rigorous bounds on amplitude and phase.
• The united set can be computed as a nested sequence of Minkowski additions of complex interval diagonals, preserving full dependency information.
• The united set has at most 2N vertices, and its vertices can be computed in linear time using Minkowski addition algorithms.
• The method fully resolves the dependency problem in interval arithmetic, enabling verified and rigorous DFT computations.
• When the united set contains the origin, the phase is undefined, which can be detected via the geometry of the convex set.
• The approach supports automatic error propagation and is applicable to signals with poor precision or missing data.