Multi-dimensional real Fourier transform

Four subroutines compute one-dimensional and multi-dimensional Fourier transforms for real data, multi-dimensional complex Fourier transforms, and multi-dimensional sine, cosine and sine-cosine transforms. Subroutines use Cooley-Tukey fast Fourier transform. In all but one-dimensional case, transforms are calculated in up to six dimensions

Fast Quantum Fourier Transforms for a Class of Non-abelian Groups

An algorithm is presented allowing the construction of fast Fourier transforms for any solvable group on a classical computer. The special structure of the recursion formula being the core of this algorithm makes it a good starting point to obtain systematically fast Fourier transforms for solvable groups on a quantum computer. The inherent structure of the Hilbert space imposed by the qubit architecture suggests to consider groups of order 2^n first (where n is the number of qubits). As an example, fast quantum Fourier transforms for all 4 classes of non-abelian 2-groups with cyclic normal subgroup of index 2 are explicitly constructed in terms of quantum circuits. The (quantum) complexity of the Fourier transform for these groups of size 2^n is O(n^2) in all cases.Comment: 16 pages, LaTeX2

Fast Fourier Transforms for Finite Inverse Semigroups

We extend the theory of fast Fourier transforms on finite groups to finite inverse semigroups. We use a general method for constructing the irreducible representations of a finite inverse semigroup to reduce the problem of computing its Fourier transform to the problems of computing Fourier transforms on its maximal subgroups and a fast zeta transform on its poset structure. We then exhibit explicit fast algorithms for particular inverse semigroups of interest--specifically, for the rook monoid and its wreath products by arbitrary finite groups.Comment: ver 3: Added improved upper and lower bounds for the memory required by the fast zeta transform on the rook monoid. ver 2: Corrected typos and (naive) bounds on memory requirements. 30 pages, 0 figure

Fast Fourier Transforms for the Rook Monoid

We define the notion of the Fourier transform for the rook monoid (also called the symmetric inverse semigroup) and provide two efficient divide-and-conquer algorithms (fast Fourier transforms, or FFTs) for computing it. This paper marks the first extension of group FFTs to non-group semigroups

Efficient Quantum Transforms

Quantum mechanics requires the operation of quantum computers to be unitary, and thus makes it important to have general techniques for developing fast quantum algorithms for computing unitary transforms. A quantum routine for computing a generalized Kronecker product is given. Applications include re-development of the networks for computing the Walsh-Hadamard and the quantum Fourier transform. New networks for two wavelet transforms are given. Quantum computation of Fourier transforms for non-Abelian groups is defined. A slightly relaxed definition is shown to simplify the analysis and the networks that computes the transforms. Efficient networks for computing such transforms for a class of metacyclic groups are introduced. A novel network for computing a Fourier transform for a group used in quantum error-correction is also given.Comment: 30 pages, LaTeX2e, 7 figures include

Composite Cyclotomic Fourier Transforms with Reduced Complexities

Discrete Fourier transforms~(DFTs) over finite fields have widespread applications in digital communication and storage systems. Hence, reducing the computational complexities of DFTs is of great significance. Recently proposed cyclotomic fast Fourier transforms (CFFTs) are promising due to their low multiplicative complexities. Unfortunately, there are two issues with CFFTs: (1) they rely on efficient short cyclic convolution algorithms, which has not been investigated thoroughly yet, and (2) they have very high additive complexities when directly implemented. In this paper, we address both issues. One of the main contributions of this paper is efficient bilinear 11-point cyclic convolution algorithms, which allow us to construct CFFTs over GF$(2^{11})$. The other main contribution of this paper is that we propose composite cyclotomic Fourier transforms (CCFTs). In comparison to previously proposed fast Fourier transforms, our CCFTs achieve lower overall complexities for moderate to long lengths, and the improvement significantly increases as the length grows. Our 2047-point and 4095-point CCFTs are also first efficient DFTs of such lengths to the best of our knowledge. Finally, our CCFTs are also advantageous for hardware implementations due to their regular and modular structure.Comment: submitted to IEEE trans on Signal Processin

Fast Computation of Voigt Functions via Fourier Transforms

This work presents a method of computing Voigt functions and their derivatives, to high accuracy, on a uniform grid. It is based on an adaptation of Fourier-transform based convolution. The relative error of the result decreases as the fourth power of the computational effort. Because of its use of highly vectorizable operations for its core, it can be implemented very efficiently in scripting language environments which provide fast vector libraries. The availability of the derivatives makes it suitable as a function generator for non-linear fitting procedures.Comment: 8 pages, 1 figur