Hartley transform

About: Hartley transform is a research topic. Over the lifetime, 2709 publications have been published within this topic receiving 79944 citations.

Generalized Commuting Matrices and Their Eigenvectors for DFTs, Offset DFTs, and Other Periodic Operations

Abstract: It is well known that some matrices (such as Dickinson-Steiglitz's matrix) can commute with the discrete Fourier transform (DFT) and that one can use them to derive the complete and orthogonal DFT eigenvector set. Recently, Candan found the general form of the DFT commuting matrix. In this paper, we further extend the previous work and find the general form of the commuting matrix for any periodic, quasi-periodic, and offset quasi-periodic operations. Using the general commuting matrix, we can derive the complete and orthogonal eigenvector sets for offset DFTs, DCTs of types 1, 4, 5, and 8, DSTs of types 1, 4, 5, and 8, discrete Hartley transforms of types 1 and 4, the Walsh transform, and the projection operation (the operation that maps a whole vector space into a subspace) successfully. Moreover, several novel ways of finding DFT eigenfunctions are also proposed. Furthermore, we also extend our theories to the continuous case, i.e., if a continuous transform is periodic, quasi-periodic, or offset quasi-periodic (such as the FT and some cyclic operations in optics), we can find the general form of the commuting operation and then find the complete and orthogonal eigenfunctions set for the continuous transform.

Calculating the FHT in hardware

Abstract: A parallel, pipelined architecture for calculating the fast Hartley transform (FHT) is discussed. Hardware implementation of the FHT introduces two challenges: retrograde indexing and data scaling. A novel addressing scheme that permits the fast computation of FHT butterflies is proposed, and a hardware implementation of conditional block floating point scaling that reduces error due to data growth with little extra cost is described. Simulations reveal a processor capable of transforming a 1 K-point sequence in 170 mu s using a 15.4 MHz clock. >

An efficient unified systolic architecture for the computation of discrete trigonometric transforms

Abstract: In this paper, a novel unified systolic architecture which can efficiently implement various discrete trigonometric transforms (DXT) including the discrete Fourier transform (DFT), the discrete Hartley transform (DHT), the discrete cosine transform (DCT), and the discrete sine transform (DST) is described. Based on Clenshaw's recurrence formula, a set of efficient recurrences for computing the DXT is developed first. Then, the inherent symmetry of the trigonometric functions is further exploited to render a hardware-efficient, systolic structure. For the computation of any N-point DXT of interest, the proposed structure requires only about N/2 multipliers and N adders, thus providing substantial hardware savings compared with previous works. Furthermore, the new scheme can be easily adapted to compute any type of DXT with only minor modification. The complete I/O buffers have been addressed as well which allows for a continuous flow of successive blocks of input data and transformed results in natural order.

A fast Hough transform for the parametrisation of straight lines using fourier methods

Abstract: The Hough transform is a useful technique in the detection of straight lines and curves in an image. Due to the mathematical similarity of the Hough transform and the forward Radon transform, the Hough transform can be computed using the Radon transform which, in turn, can be evaluated using the central slice theorem. This involves a two-dimensional Fourier transform, an x-y to r-? mapping and a ID Fourier transform. This can be implemented in specialized hardware to take advantage of the computational savings of the fast Fourier transform. In this paper, we outline a fast and efficient method for the computation of the Hough transform using Fourier methods. The maxima points generated in the Radon space, corresponding to the parametrisation of straight lines, can be enhanced with a post transform convolutional filter. This can be applied as a ID filtering operation on the resampled data whilst in the Fourier space, so further speeding the computation. Additionally, any edge enhancement or smoothing operations on the input function can be combined into the filter and applied as a net filter function.

The Algorithm of the Inverse for Lie Transform

Abstract: The algorithm for Lie transforms proposed here reduce the amount of computation to be carried out in particular but typical problems of Perturbation Theory. In conjunction with the formulas for inverting and composing Lie transforms it is shown to reduce by a factor of three the number of Poisson brackets to be evaluated in a part of the implementation of the Analytical Lunar Theory.