The methods of fast parallel calculation of multidimensional hypercomplex discrete Fourier transform (HDFT) are discussed. The theoretical and experimental estimation of parallelization efficiency is given. It is shown that proposed method has very high efficiency (up to 90%).

In this paper we introduce convolution theorem for the Fourier transform (FT) of two complex functions. We show that the correlation theorem for the FT can be derived using properties of convolution. We develop this idea to derive the correlation theorem for the quaternion Fourier transform (QFT) of the two quaternion functions.

We analyze the minimum achievable mean-square error in frequency-modulated continuous-wave range estimation of a single stationary target when photon-counting detectors are employed. Starting from the probability density function for the photon-arrival times in photodetectors with subunity quantum efficiency, dark counts, and dead time, we derive the Cramér-Rao bound and highlight three importa...

An analytical and concise formula is derived for the fractional Fourier transform (FRT) of partially coherent beams that is based on the tensorial propagation formula of the cross-spectral density of partially coherent twisted anisotropic Gaussian-Schell-model (GSM) beams. The corresponding tensor ABCD law performing the FRT is obtained. The connections between the FRT formula and the generaliz...

In the Wigner domain of a one-dimensional function, a certain chirp term represents a rotated line delta function. On the other hand, a fractional Fourier transform (FRT) can be associated with a rotation of the Wigner-distribution function by an angle connected with the FRT order. Thus with the FRT tool a chirp and a delta function can be transformed one into the other. Taking the chirp as add...

A method for the calculation of the fractional Fourier transform (FRT) by means of the fast Fourier transform (FFT) algorithm is presented. The process involves mainly two FFT's in cascade; thus the process has the same complexity as this algorithm. The method is valid for fractional orders varying from -1 to 1. Scaling factors for the FRT and Fresnel diffraction when calculated through the FFT...

polyethylene is a synthetic polymer which is used in our daily life for different purposes. increased use of polyethylene causes severe environmental problems. there are different methods to decrease problem caused by polyethylene for example source reduction, incineration and land filling and all of them have their own drawbacks. so, the best way to reduce the problem caused by polyethylene is...

In open-path Fourier Transform Infrared spectroscopy, the generation of a suitable background single-beam spectrum is of major concern. The Shifting Method is a derivative-like approach to correct for the background without the need to actually measure a background spectrum using the sample single-beam spectrum. A thorough study of the Shifting Method was conducted. A set of guidelines was deve...

It is shown that number theoretic transforms (NTT) can be used to compute discrete Fourier transform (DFT) very efficiently. By noting some simple properties of number theory and the DFT, the total number of real multiplications for a length-P DFT is reduced to (P — 1). This requires less than one real multiplication per point. For a proper choice of transform length and NTT, the number of shif...

A mathematical theory on the computational complexity of the Discrete Fourier Transform (DFT) is required, independently of any kind of known Fast Fourier Transform (FFT) algorithm.