XJTLU Academic Unveils New Insights on Uncertainty Principles in Time-Frequency Analysis

A recent study titled "Uncertainty principles for the short-time Fourier transform on the lattice" has been published in the Journal Mathematische Nachrichten. The research was conducted under the leadership of Dr Anirudha Poria from the School of Mathematics and Physics at Xi'an Jiaotong-Liverpool University.

Uncertainty principles are integral to various domains, including mathematics, physics, and engineering fields such as quantum theory, image processing, signal processing, optics, and more. In the realms of signal analysis and quantum mechanics, these principles are frequently discussed in the context of simultaneous time-frequency representations like the short-time Fourier transform (STFT). The STFT is a transformation that maps a function from ℝ^𝑛 to ℝ^𝑛 × ℝ^𝑛, known as the time-frequency plane or phase space. Given that the dual group of ℝ^𝑛 is identical to ℝ^𝑛, the phase space is also ℝ^𝑛 × ℝ^𝑛. For the group ℤ^𝑛, its dual group is 𝕋^𝑛, resulting in a phase space of ℤ^𝑛 × 𝕋^𝑛.

The paper introduces the STFT on ℤ^𝑛 × 𝕋^𝑛 and delves into various uncertainty principles. Specifically, it establishes the uncertainty principle for orthonormal sequences, Donoho-Stark’s uncertainty principle, Benedicks-type uncertainty principle, Heisenberg-type uncertainty principle, and a local uncertainty inequality for the STFT on ℤ^𝑛 × 𝕋^𝑛. The research culminates in deriving the Heisenberg-type uncertainty inequality through the use of 𝑘-entropy and an examination of the 𝑘-entropy localization of the STFT on ℤ^𝑛 × 𝕋^𝑛.

Dr Anirudha Poria

Looking ahead, the team intends to explore the practical applications of these uncertainty principles in signal separation issues.

“In particular, we will examine the potential for using the uncertainty principle to separate signals and investigate the sparsity-based signal separation problems, and window optimization problems using uncertainty principles.” Says Dr Anirudha Poria.

Materials provided by Anirudha Poria

Edited by Qinru Liu