Sequences with Minimal Time-Frequency Uncertainty

TL;DR

This paper formulates and solves a convex optimization problem to identify sequences that are optimally compact in both time and frequency, revealing Mathieu functions as the optimal solutions and establishing a sharp uncertainty principle for sequences.

Contribution

It introduces a convex optimization approach to find maximally compact sequences and characterizes these sequences using Mathieu's harmonic cosine functions, providing a new uncertainty bound.

Findings

Maximally compact sequences are derived from Mathieu's harmonic cosine functions.

A sharp uncertainty bound for sequences is established through convex optimization.

Optimal sequences outperform traditional window functions in time-frequency compactness.

Abstract

A central problem in signal processing and communications is to design signals that are compact both in time and frequency. Heisenberg's uncertainty principle states that a given function cannot be arbitrarily compact both in time and frequency, defining an "uncertainty" lower bound. Taking the variance as a measure of localization in time and frequency, Gaussian functions reach this bound for continuous-time signals. For sequences, however, this is not true; it is known that Heisenberg's bound is generally unachievable. For a chosen frequency variance, we formulate the search for "maximally compact sequences" as an exactly and efficiently solved convex optimization problem, thus providing a sharp uncertainty principle for sequences. Interestingly, the optimization formulation also reveals that maximally compact sequences are derived from Mathieu's harmonic cosine function of order zero. We further provide rational asymptotic expansions of this sharp uncertainty bound. We use the derived bounds as a benchmark to compare the compactness of well-known window functions with that of the optimal Mathieu's functions.