XFT: A Discrete Fourier Transform
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We present in this chapter a procedure to obtain a novel discrete Fourier transform. The discrete Fourier transform obtained in this way is called XFT (from eXtended Fourier Transform) in order to distinguish it from the usual discrete Fourier transform DFT, which was studied in Chap. 2. It is shown in this chapter that the XFT appears as a quadrature of the fractional Fourier transform and that the XFT can be applied as a discrete Fourier transform to solve some simple examples. A discrete scheme for the Hermite functions, the Schrödinger equation, the Fourier cosine/sine transform, and for partial fractional differentiation is also given. The aliasing problem is discussed.
KeywordsExtended discrete Fourier transform Discrete Hermite functions Square-integrable functions Quadrature Sine/cosine transform Differentiation matrices Fast algorithms Signal processing Sampling Aliasing Partial differentiation matrices
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