A discrete Fourier transform based on Simpson's rule

Abstract

Communicated by J. BanasiakFourier analysis plays a vital role in the analysis of continuous-time signals. In many cases, we are forced to approximate the Fourier coefficients based on a sampling of the time signal. Hence, the need for a discrete transformation into the frequency domain giving rise to the classical discrete Fourier transform. In this paper, we present a transformation that arises naturally if one approximates the Fourier coefficients of a continuous-time signal numerically using the Simpson quadrature rule. This results in a decomposition of the discrete signal into two sequences of equal length. We show that the periodic discrete time signal can be reconstructed completely from its discrete spectrum using an inverse transform. We also present many properties satisfied by this transform.

Computation of a real eigenbasis for the Simpson discrete Fourier transform matrix

“…The DFT plays an important role in audio signal processing, adaptive filtering of artefacts from the human electroencephalogram and detection of fetal heartbeats from the fetal electrocardiogram 2 . We have proposed a DFT based on Simpson's numerical quadrature rule 1 and proved analogous properties to the classical DFT 3 . The eigenvalues of a matrix play an important role in the spectral resolution of functions of the matrix into their constituent components and the multiplicities give the number of linearly independent eigenvectors corresponding to each eigenvalue 4 .…”

The duality property of the Discrete Fourier Transform based on Simpson's rule

“…Recently, the Fast Fourier Transform algorithm [2] based on the classical DFT has been used in financial mathematics to evaluate option pricing [3]. In order to improve the accuracy of the computation of the Fourier coefficients of a period function we have introduced in [4] a discrete Fourier transform based on Simpson's quadrature rule [5].…”

Properties of the Simpson discrete Fourier transform

“…This transformation is the preferred tool in these areas since processing in the frequency domain is inexpensive. Having provided an alternative transformation, the SDFT [5], which may be applied to these areas of interests, it is only natural that we establish its properties.…”