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.

Ultradifferentiable functions and Fourier analysis

Abstract: Classes of non-quasianalytic functions are classically defined by imposing growth conditions on the derivatives of the functions. It was Beuding [1] (see Bjorck [2]) who pointed out that decay properties of the Fourier transform of a compactly supported function can be used for this purpose equally well. In the present article we modify Beuding's approach. More precisely, we call w: [0,00[--+ [0, oo[ a weight function if w is continuous and satisfies

Fast Convolution using fermat number transforms with applications to digital filtering

Abstract: The structure of transforms having the convolution property is developed. A particular transform is proposed that is defined on a finite ring of integers with arithmetic carried out modulo Fermat numbers. This Fermat number transform (FNT) is ideally suited to digital computation, requiring on the order of N \log N additions, subtractions and bit shifts, but no multiplications. In addition to being efficient, the Fermat number transform implementation of convolution is exact, i.e., there is no roundoff error. There is a restriction on sequence length imposed by word length but multi-dimensional techniques are discussed which overcome this limitation. Results of an implementation on the IBM 370/155 are presented and compared with the fast Fourier transform (FFT) showing a substantial improvement in efficiency and accuracy.

Discrete fractional Fourier transform based on orthogonal projections

Abstract: The continuous fractional Fourier transform (FRFT) performs a spectrum rotation of signal in the time-frequency plane, and it becomes an important tool for time-varying signal analysis. A discrete fractional Fourier transform has been developed by Santhanam and McClellan (see ibid., vol.42, p.994-98, 1996) but its results do not match those of the corresponding continuous fractional Fourier transforms. We propose a new discrete fractional Fourier transform (DFRFT). The new DFRFT has DFT Hermite eigenvectors and retains the eigenvalue-eigenfunction relation as a continous FRFT. To obtain DFT Hermite eigenvectors, two orthogonal projection methods are introduced. Thus, the new DFRFT will provide similar transform and rotational properties as those of continuous fractional Fourier transforms. Moreover, the relationship between FRFT and the proposed DFRFT has been established in the same way as the conventional DFT-to-continuous-Fourier transform.

New apodizing functions for Fourier spectrometry

Abstract: A new class of apodizing functions suitable for Fourier spectrometry (and similar applications) is introduced. From this class, three specific functions are discussed in detail, and the resulting instrumental line shapes are compared to numerous others proposed for the same purpose.

Closed-form discrete fractional and affine Fourier transforms

Abstract: The discrete fractional Fourier transform (DFRFT) is the generalization of discrete Fourier transform. Many types of DFRFT have been derived and are useful for signal processing applications. We introduce a new type of DFRFT, which are unitary, reversible, and flexible; in addition, the closed-form analytic expression can be obtained. It works in performance similar to the continuous fractional Fourier transform (FRFT) and can be efficiently calculated by the FFT. Since the continuous FRFT can be generalized into the continuous affine Fourier transform (AFT) (the so-called canonical transform), we also extend the DFRFT into the discrete affine Fourier transform (DAFT). We derive two types of the DFRFT and DAFT. Type 1 is similar to the continuous FRFT and AFT and can be used for computing the continuous FRFT and AFT. Type 2 is the improved form of type 1 and can be used for other applications of digital signal processing. Meanwhile, many important properties continuous FRFT and AFT are kept in the closed-form DFRFT and DAFT, and some applications, such as filter design and pattern recognition, are also discussed. The closed-form DFRFT we introduce has the lowest complexity among all current DFRFTs that is still similar to the continuous FRFT.