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A discrete Fourier transform based on Simpson's rule
Author(s) -
Singh P.,
Singh V.
Publication year - 2012
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.1547
Subject(s) - mathematics , non uniform discrete fourier transform , discrete time fourier transform , discrete fourier transform (general) , fourier transform , discrete fourier series , fractional fourier transform , discrete frequency domain , fourier analysis , mathematical analysis , short time fourier transform , discrete sine transform , fourier inversion theorem , discrete hartley transform , discrete time signal , frequency domain , analog signal , computer science , signal transfer function , digital signal processing , computer hardware
Fourier 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. Copyright © 2012 John Wiley & Sons, Ltd.