Short-Time Fractional Fourier Transform and Its Applications

TLDR

The fractional Fourier transform is a powerful tool for chirp signal analysis but cannot locate fractional Fourier domain frequency contents needed in certain applications. This work introduces the short‑time fractional Fourier transform to overcome that limitation. The STFRFT jointly represents time and FRFD‑frequency, optimizes two‑dimensional resolution and support via a time‑FRFD‑bandwidth product, and includes an inverse transform, computational complexity, and is applied to time‑of‑arrival and pulse‑width estimation as well as STFRFD filtering. Simulations demonstrate that the proposed algorithms accurately recover time‑of‑arrival, pulse‑width, and perform effective filtering, confirming the method’s validity.

Abstract

The fractional Fourier transform (FRFT) is a potent tool to analyze the chirp signal. However, it fails in locating the fractional Fourier domain (FRFD)-frequency contents which is required in some applications. The short-time fractional Fourier transform (STFRFT) is proposed to solve this problem. It displays the time and FRFD-frequency information jointly in the short-time fractional Fourier domain (STFRFD). Two aspects of its performance are considered: the 2-D resolution and the STFRFD support. The time-FRFD-bandwidth product (TFBP) is defined to measure the resolvable area and the STFRFD support. The optimal STFRFT is obtained with the criteria that maximize the 2-D resolution and minimize the STFRFD support. Its inverse transform, properties and computational complexity are presented. Two applications are discussed: the estimations of the time-of-arrival (TOA) and pulsewidth (PW) of chirp signals, and the STFRFD filtering. Simulations verify the validity of the proposed algorithms.