Two-band fast Hartley transform

This article has been made available through the Brunel Open Access Publishing Fund.Efficient algorithms have been developed over the past 30 years for computing the forward and inverse discrete Hartley transforms (DHTs). These are similar to the fast Fourier transform (FFT) algorithms for computing the discrete Fourier transform (DFT). Most of these methods seek to minimise the complexity of computations and or the number of operations. A new approach for the computation of the radix-2 fast Hartley transform (FHT) is presented. The proposed algorithm, based on a two-band decomposition of the input data, possesses a very regular structure, avoids the input or out data shuffling, requires slightly less multiplications than the existing approaches, but increases the number of additions

Fast and direct inversion methods for the multivariate nonequispaced fast Fourier transform

The well-known discrete Fourier transform (DFT) can easily be generalized to arbitrary nodes in the spatial domain. The fast procedure for this generalization is referred to as nonequispaced fast Fourier transform (NFFT). Various applications such as MRI, solution of PDEs, etc., are interested in the inverse problem, i.e., computing Fourier coefficients from given nonequispaced data. In this paper we survey different kinds of approaches to tackle this problem. In contrast to iterative procedures, where multiple iteration steps are needed for computing a solution, we focus especially on so-called direct inversion methods. We review density compensation techniques and introduce a new scheme that leads to an exact reconstruction for trigonometric polynomials. In addition, we consider a matrix optimization approach using Frobenius norm minimization to obtain an inverse NFFT

A new Truncated Fourier Transform algorithm

Truncated Fourier Transforms (TFTs), first introduced by Van der Hoeven, refer to a family of algorithms that attempt to smooth "jumps" in complexity exhibited by FFT algorithms. We present an in-place TFT whose time complexity, measured in terms of ring operations, is comparable to existing not-in-place TFT methods. We also describe a transformation that maps between two families of TFT algorithms that use different sets of evaluation points.Comment: 8 pages, submitted to the 38th International Symposium on Symbolic and Algebraic Computation (ISSAC 2013

Type-IV DCT, DST, and MDCT algorithms with reduced numbers of arithmetic operations

We present algorithms for the type-IV discrete cosine transform (DCT-IV) and discrete sine transform (DST-IV), as well as for the modified discrete cosine transform (MDCT) and its inverse, that achieve a lower count of real multiplications and additions than previously published algorithms, without sacrificing numerical accuracy. Asymptotically, the operation count is reduced from ~2NlogN to ~(17/9)NlogN for a power-of-two transform size N, and the exact count is strictly lowered for all N > 4. These results are derived by considering the DCT to be a special case of a DFT of length 8N, with certain symmetries, and then pruning redundant operations from a recent improved fast Fourier transform algorithm (based on a recursive rescaling of the conjugate-pair split radix algorithm). The improved algorithms for DST-IV and MDCT follow immediately from the improved count for the DCT-IV.Comment: 11 page

FFT Interpolation from Nonuniform Samples Lying in a Regular Grid

This paper presents a method to interpolate a periodic band-limited signal from its samples lying at nonuniform positions in a regular grid, which is based on the FFT and has the same complexity order as this last algorithm. This kind of interpolation is usually termed "the missing samples problem" in the literature, and there exists a wide variety of iterative and direct methods for its solution. The one presented in this paper is a direct method that exploits the properties of the so-called erasure polynomial, and it provides a significant improvement on the most efficient method in the literature, which seems to be the burst error recovery (BER) technique of Marvasti's et al. The numerical stability and complexity of the method are evaluated numerically and compared with the pseudo-inverse and BER solutions.Comment: Submitted to the IEEE Transactions on Signal Processin