Topic
Discrete-time Fourier transform
About: Discrete-time Fourier transform is a research topic. Over the lifetime, 5072 publications have been published within this topic receiving 144643 citations. The topic is also known as: DTFT.
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TL;DR: For numerical integration the range must be made finite and any resulting Gibbs oscillations can be reduced by incorporating the sigma factor described in Part I, and at the same time this secures the additional advantage of avoiding problems associated with the convergence of the integrals involved.
Abstract: For numerical integration the range must be made finite and any resulting Gibbs oscillations can be reduced by incorporating the sigma factor described in Part I. However, the poles of J(00) may, and in many cases do, lie close to the real axis. This causes the integrand to peak over a series of small intervals lying in the neighbourhood of the poles and hence for accurate numerical integration it is necessary to take a very small step length. This limitation can be overcome by using the modified Fourier transformtti and at the same time this secures the additional advantage of avoiding problems associated with the convergence of the integrals involved. The method Lego and Sze devised to overcome this latter difficulty was to introduce an artificial forcing function they denoted by U(a, t) (see Ref. 2, p. 1032, equation 7).
45 citations
TL;DR: In this paper, the authors developed the understanding and skill necessary to recognize fractional Fourier transforms and their parameters by visually examining ray traces, and determined the differential equations governing the propagation of the order, scale, and curvature of the Fourier transform.
45 citations
TL;DR: In this paper, the Fourier transform of the circular cosine function, which can be expressed in terms of a half-order derivative of the impulse ring δ(r-f), plays a fundamental role in the spectra of radially periodic functions.
Abstract: Although the spectrum of radially periodic images is often expressed in terms of finite or infinite series of Bessel functions, such expressions do not clearly reveal the exact impulsive structure of the spectrum. An alternative Fourier decomposition of radially periodic images, in terms of circular cosine functions, is presented, and its significant advantages are shown. It is shown that the Fourier transform of the circular cosine function, which can be expressed in terms of a half-order derivative of the impulse ring δ(r-f), plays a fundamental role in the spectra of radially periodic functions. Just as any symmetric periodic function p(x) in the one-dimensional case can be represented by a sum of cosines with frequencies of f=1/T, 2/T, … [the Fourier series decomposition of p(x)], a radially periodic function in the two-dimensional case can be decomposed into a circular Fourier series, which is a sum of circular cosine functions with radial frequencies of f=1/T, 2/T, … . This result can also be formulated in terms of the spectral domain: Just as the Fourier transform of a one-dimensional periodic function consists of impulse pairs located at f=n/T (the Fourier transforms of the cosines in the sum), the Fourier spectrum of a radially periodic function in the two-dimensional case consists of half-order derivative impulse rings with radii f=n/T (which are the Fourier transforms of the circular cosines in the sum). The significance of these results is discussed, and it is briefly shown how they can be extended into dimensions other than two.
45 citations
TL;DR: In this paper, a new convolution structure for the special affine Fourier transform (SAFT) is introduced, which preserves the convolution theorem for the FT, which states that the FT of the convolutions of two functions is the product of their Fourier transforms.
45 citations
TL;DR: In this article, the authors define the quadratic algebra which is a one-parameter deformation of the Lie algebra extended by a parity operator, and investigate a model of the finite one-dimensional harmonic oscillator based upon this algebra.
Abstract: We define the quadratic algebra which is a one-parameter deformation of the Lie algebra extended by a parity operator. The odd-dimensional representations of (with representation label j, a positive integer) can be extended to representations of . We investigate a model of the finite one-dimensional harmonic oscillator based upon this algebra . It turns out that in this model the spectrum of the position and momentum operator can be computed explicitly, and that the corresponding (discrete) wavefunctions can be determined in terms of Hahn polynomials. The operation mapping position wavefunctions into momentum wavefunctions is studied, and this so-called discrete Fourier–Hahn transform is computed explicitly. The matrix of this discrete Fourier–Hahn transform has many interesting properties, similar to those of the traditional discrete Fourier transform.
45 citations