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This paper introduces the theory and hardware implementation of two new algorithms for computing a single component of the discrete Fourier transform. In terms of multiplicative complexity, both algorithms are more efficient, in general,…

The eigenstructure of the discrete Fourier transform (DFT) is examined and new systematic procedures to generate eigenvectors of the unitary DFT are proposed. DFT eigenvectors are suggested as user signatures for data communication over the…

Fast Fourier transform algorithms are an arsenal of effective tools for solving various problems of analysis and high-speed processing of signals of various natures. Almost all of these algorithms are designed to process sequences of…

The discrete Fourier transform (DFT) is of fundamental interest in photonic quantum information, yet the ability to scale it to high dimensions depends heavily on the physical encoding, with practical recipes lacking in emerging platforms…

First principles methods can provide insight into materials that is otherwise impossible to acquire. Density Functional Theory (DFT) has been the first principles method of choice for numerous applications, but it falls short of predicting…

High-dimensional datasets depict a challenge for learning tasks in data mining and machine learning. Feature selection is an effective technique in dealing with dimensionality reduction. It is often an essential data processing step prior…

Fourier transforms are ubiquitous mathematical tools in basic and applied sciences. We here report classical and quantum optical realizations of the discrete fractional Fourier transform, a generalization of the Fourier transform. In the…

The matrix expressions for every parts of a transformer are firstly described. Based on semi-tensor product (STP) of matrices the hypervectors are reconsidered and the linear transformation over hypervectors is constructed by using…

The explicit analytical expression of the Nonlinear Fourier Transform (NFT) of a finite set of data is provided. Then a simple recursion relation for the NFT is constructed as a function of the spectral parameter. These tools provide a…

We give a detailed study of the discrete Fourier transform (DFT) of $r$-even arithmetic functions, which form a subspace of the space of $r$-periodic arithmetic functions. We consider the DFT of sequences of $r$-even functions, their mean…

DFT is the numerical implementation of Fourier transform (FT), and it has many forms. Ordinary DFT (ODFT) and symmetric DFT (SDFT) are the two main forms of DFT. The most widely used DFT is ODFT, and the phase spectrum of this form is…

Exact reconstruction of an image from measurements of its Discrete Fourier Transform (DFT) typically requires all DFT coefficients to be available. However, incorporating the prior assumption that the image contains only integer values…

This paper examines the existence and region of convergence of Fourier transform of the functions of bicomplex variables with the help of projection on its idempotent components as auxiliary complex planes. Several basic properties of this…

Graph signal processing (GSP) advances spectral analysis on irregular domains. However, existing two-dimensional graph fractional Fourier transform (2D-GFRFT) employs a single fractional order for both factor graphs, thereby limiting its…

Density Functional Theory (DFT) is one of the most widely used methods for "ab initio" calculations of the structure of atoms, molecules, crystals, surfaces, and their interactions. Unfortunately, the customary introduction to DFT is often…

How could the Fourier and other transforms be naturally discovered if one didn't know how to postulate them? In the case of the Discrete Fourier Transform (DFT), we show how it arises naturally out of analysis of circulant matrices. In…

We consider the problem of finding the Discrete Fourier Transform (DFT) of $N-$ length signals with known frequency support of size $k$. When $N$ is a power of 2 and the frequency support is a spectral set, we provide an $O(k \log k)$…

There are several notions of duality between lines and points. In this note, it is shown that all these can be studied in a unified way. Most interesting properties are independent of specific choices. It is also shown that either dual…

Steady-state density functional theory, called i-DFT, is employed to compute spectral and transmission properties of general interacting nanoscale regions coupled to electronic reservoirs. Exchange-correlation functionals are constructed…

Density functional theory (DFT) is an incredible success story. The low computational cost, combined with useful (but not yet chemical) accuracy, has made DFT a standard technique in most branches of chemistry and materials science.…