The Sliding Window Discrete Fourier Transform

TL;DR

This paper presents the Sliding Window Discrete Fourier Transform (SWDFT), a new method for analyzing local periodic components in time-series data, with an efficient parameter estimation procedure and a simulation study.

Contribution

The paper introduces the SWDFT for local time-series analysis, defines a 5-parameter model, and proposes a fast estimation method with initial simulation results.

Findings

SWDFT effectively captures local periodic signals.

The estimation procedure is computationally efficient.

Simulation shows promising results under noise.

Abstract

This paper introduces a new tool for time-series analysis: the Sliding Window Discrete Fourier Transform (SWDFT). The SWDFT is especially useful for time-series with local- in-time periodic components. We define a 5-parameter model for noiseless local periodic signals, then study the SWDFT of this model. Our study illustrates several key concepts crucial to analyzing time-series with the SWDFT, in particular Aliasing, Leakage, and Ringing. We also show how these ideas extend to R > 1 local periodic components, using the linearity property of the Fourier transform. Next, we propose a simple procedure for estimating the 5 parameters of our local periodic signal model using the SWDFT. Our estimation procedure speeds up computation by using a trigonometric identity that linearizes estimation of 2 of the 5 parameters. We conclude with a very small Monte Carlo simulation study of our estimation procedure under different levels of noise.